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The Concept of Explanation
 9788378862048, 9788378862055

Table of contents :
Bartosz Brożek, Michael Heller, Matuesz Hohol
Introduction

Bartosz Brożek
Explanation and Understanding

Jan Woleński
Are Explanation and Prediction Symmetric?

Robert Audi
A Priori Explanation

Stanislaw Krajewski
Remarks on Mathematical Explanation

Krzysztof Wójtowicz
On the Problem of Explanation in Mathematics

Michael Heller
Limits of Causal Explanations in Physics

Andrzej Koleżyński
Pragmatism of Chemical Explanation

Dominique Lambert
“When Science Meets Historicity”
Some Questions About the Use of Mathematics in Biology

Mateusz Hohol, Michał Furman
On Explanation in Neuroscience: The Mechanistic Framework

Marcin Gorazda
The Epistemic and Cognitive Concept of Explantion in Economics.
An Attempt at Synthesis

Wojciech Załuski
The Varieties of Egoism. Some Reflections on Moral Explanation of Human Action

Willem B. Drees
Is Explaining Religion Explaining Religion Away?

Olivier Riaudel
Explanation in Christian Theology: Some Points in Common with the Human Sciences

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© Copyright by Copernicus Center Press, 2016 Editing Aeddan Shaw Cover design Mariusz Banachowicz Layout Mirosław Krzyszkowski Typesetting MELES-DESIGN Publication Supported by The John Templeton Foundation Grant "The Limits of Scientific Explanation" ISBN 978-83-7886-204-8 Kraków 2016 Publisher: Copernicus Center Press Sp. z o.o. pl. Szczepański 8, 31-011 Kraków tel. (+48) 12 430 63 00 e-mail: [email protected] www.en.ccpress.pl

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Table of Contents

Bartosz Brożek, Michael Heller, Mateusz Hohol Preface. .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   7 Bartosz Brożek Explanation and Understanding . .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   11 Jan Woleński Are Explanation and Prediction Symmetric?.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   43 Robert Audi A Priori Explanation.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   63 Stanisław Krajewski Remarks on Mathematical Explanation .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   89 Krzysztof Wójtowicz On the Problem of Explanation in Mathematics. .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   105 Michael Heller Limits of Causal Explanations in Physics. .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   123 Andrzej Koleżyński Pragmatism of Chemical Explanation .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   137

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Dominique Lambert “When Science Meets Historicity” Some Questions About the Use of Mathematics in Biology. .  .  .  .  .  .  .  .  .  .  .  .   177 Mateusz Hohol, Michał Furman On Explanation in Neuroscience: The Mechanistic Framework .  .  .  .  .  .  .  .  .   207 Marcin Gorazda The Epistemic and Cognitive Concept of Explanation in Economics. An Attempt at Synthesis.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   237 Wojciech Załuski The Varieties of Egoism. Some Reflections on Moral Explanation of Human Action. .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   267 Willem B. Drees Is Explaining Religion Explaining Religion Away?.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   283 Olivier Riaudel Explanation in Christian Theology: Some Points in Common with the Human Sciences.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   307

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Preface

Explanation is one of the most fundamental human cognitive activities. It is not surprising, since manoeuvring in the complex world we inhabit would be impossible without some comprehension of what is happening around us. In social settings, it is absolutely essential to understand the actions and beliefs of other people; otherwise, human interactions would be highly unpredictable and it would be difficult to imagine any joint undertakings. The same holds for natural phenomena. For example, once some explanation of why thunder occurs is proposed (whether in terms of the rage of gods, or through the application of modern physics), the relevant fragment of the reality is cognitively ‘domesticated’. The case of explaining thunder is instructive for two additional reasons. First, it uncovers the causal dimension of the process of explanation. We strive to identify the causes of explained phenomena, not mere statistical correlations. To know that thunder almost always occurs whenever there is lightning, may be of some practical value; however, it is much more to know that it is lightning that produces the thunder (the lightning increases the pressure and temperature in the surrounding air, which causes it to expand violently, creating a sonic wave). Second, explanation often proceeds by unification: the same kinds of causes (e.g., the same physical laws) are applied to explain a variety of different phenomena. In this way, the complexity of the world is reduced and encapsulated into a manageable set of regularities.

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Bartosz Brożek, Michael Heller, Mateusz Hohol

There is also a psychological dimension of explanation: the need to have some understanding of the occurring phenomena is deeply rooted in human nature. While there surely exist significant individual differences between people in this respect (e.g., those of us who have a high level of the need for cognitive closure are quick to accept any, even bad explanations), there is no grounds for denying that much of our conscious effort is devoted to making sense of what is going on around us. Also, the unconscious mechanisms at work in cognition are usually shaped so as to facilitate the integration of the received stimuli into one’s conceptual scheme (for example, the heuristics described by Kahneman and Tversky, although they may lead to decisions which deviate from some abstract standards of rationality, are so designed as to provide us with quick, robust explanations of the encountered phenomena). Explanation also has a more theoretical, sophisticated face – scientific explanation – which has been the focus of a fierce philosophical discussion. It owes much to the seminal paper by Hempel and Oppenheim, ‘Studies in the Logic of Explanation’. Their Deductive Nomological Model of scientific explanation provided a point of reference for the multifaceted literature, in which various aspects of the model have been modified or rejected. In recent years, however, the philosophical debate pertaining to scientific explanation is less preoccupied with the assessment of Hempel and Oppenheim’s theory, which was developed in the context of physics. Currently, the interest of philosophers has shifted towards other sciences, in particular biology, neuroscience and psychology. The pressing question is how to construct and evaluate explanations of mental phenomena – explanations, which clearly fail to fulfil the standards of explaining acceptable in physics. The present volume brings together thirteen contributions which analyse different aspects and modes of explanation. Three of them (Bartosz Brożek’s ‘Explanation and Understanding’, Jan Woleński’s ‘Are Explanation and Prediction Symmetric?’, and Robert Audi’s ‘A Priori explanation’) are devoted to conceptual problems. The fol-

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lowing six chapters inquire into the structure and limits of explanation in the sciences. Stanisław Krajewski (‘Remarks on Mathematical Explanation’) and Krzysztof Wójtowicz (‘On the Problem of Explanation in Mathematics’) consider the role of explanation in mathematics; Michael Heller (‘Limits of Causal Explanations in Physics’) analyses the problem of causality against the backdrop of the contemporary research in cosmology; Andrzej Koleżyński (‘Pragmatism of Chemical Explanation’) highlights some intriguing aspects of explanation in chemistry; Dominique Lambert (‘When Science Meets Historicity’) considers the role of mathematics in biological explanations; and Mateusz Hohol and Michał Furman (‘On Explanation in Neuroscience: The Mechanistic Framework’) inquire into the limits of mechanical explanations in the neurosciences. The final four chapters of the volume attempt to analyse the nature of explanation in social sciences and the humanities: economics (Marcin Gorazda’s ‘The Epistemic and Cognitive Concept of Explantion in Economics’), ethics (Wojciech Załuski’s ‘The Varieties of Egoism. Some Reflections on Moral Explanation of Human Action’), religious studies (Wim Drees’s ‘Is Explaining Religion Explaining Religion Away?’), and theology (Olivier Riaudel’s ‘Explanation in Christian Theology: Some Points in Common with the Human Sciences’). It is our hope that the essays collected in this volume will contribute to a better understanding of what explanation is and what are its limits. Bartosz Brożek Michael Heller Mateusz Hohol

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Bartosz Brożek Jagiellonian University Copernicus Center for Interdisciplinary Studies

Explanation and Understanding

My goal in this essay is to consider the relationship between explanation and understanding. I will defend the thesis that there is an entire spectrum of cognitive activities that deserve the name “understanding,” and that various kinds of explanation may be pivotal in some, but not in other such activities. I begin by outlining the classical Aristotelian theory of explanation and attempt to show how it relates to the Ancient Greeks’ understanding of “understanding.” I then proceed to analyze the contemporary debate pertaining to explanation, which revolves around the so-called deductive-nomological model. Although the model itself assumes no link between explanation and understanding, all of the competing accounts place emphasis on the necessity of making the link explicit. I conclude by outlining a theory of understanding which helps one to grasp why different kinds of explanation contribute to making the world more comprehensible.

1. Back to Aristotle The Aristotelian theory of explanation is often analyzed in reference to his thesis that there exist teleological explanations (cf. Leunissen, 2010). Undoubtedly, this aspect of Aristotle’s doctrine in particularly interesting, since it clearly goes against the modern and contemporary understanding of what explaining a physical phenomenon amo-

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unts to. To say that a heavy body falls because such bodies have a built-in tendency to move towards their ‘natural place’, i.e. the center of Earth, is alien to the followers of Galileo and Newton: when Aristotle thought that gravitation may be explained in terms of the goal (telos) an object strives to achieve, modern physics dispenses with teleological causes and provides an account of gravitation based on the concept of a force that acts upon a falling body. However, putting too much emphasis on the role of teleology in Aristotle’s theory of explanation, easily leads to ignoring other intriguing components of his conception, which, arguably, may help us understand what counts as explanation. I believe it is best to construe Aristotelian view as based on the following four claims: (A1) [The nature of explanandum] Explanation (in the proper sense of the word) never concerns a particular phenomenon, but always some regularity in nature, since genuine knowledge is universal, never particular. (A2) [The nature of explanans] Explanation consists in identifying the cause of the explained regularity. (A3) [The ontological scheme] There are four kinds of causes (formal, material, efficient and final), which – taken together – determine the structure of the universe. (A4) [The form of explanation] Explanation always takes the form of a deductive argument (i.e., a syllogism). It is the middle term of the syllogism that provides an explanation for the conclusion, as it identifies the cause of the explained regularity (Brożek, Brożek & Stelmach, 2013). The thesis (A1) underlines that Aristotle is not considering the kind of explanations we employ on an every-day basis (e.g., why have a person X killed his aunt or why did the lightning strike X’s house), but is rather interested in what may be called scientific explanation, i.e. explanation of regularly occurring phenomena. The insistence on the general character of knowledge is strictly connected to the most

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revolutionary insight of the Greek philosophers, from Heraclitus and Parmenides, to Plato and Aristotle. They believed that behind the ‘veil of appearances’ there exists the logos or the rational structure of the universe; although in our perception we experience constant changes, there must be universal norms according to which the changes occur. The goal of philosophy (science) is to uncover those universal laws, and not to deal with particular events. The thesis (A2) is the claim that every explanation is causal; however, it is easy to miss what Aristotle is trying to convey by saying this. It is sometimes argued that “cause” is not the best translation of the term Aristotle uses (aition) and that it should rather be rendered as ‘reason’ or ‘explanation.’ Initially, the word meant “accused” or “responsible” and was used in legal and moral contexts (LSJ The Online Liddell-Scott-Jones Greek-English Lexicon, 2011). Its semantic evolution mimics the etymological trajectory of other concepts, which came to express the idea that the universe is governed by general principles. For example, in his intriguing essay, “The Genesis of the Concept of Physical Law,” Edgar Zilsel (2003) claims that in Ancient thought the predominant way of expressing causal relations in nature was the use of the “legal metaphor”: physical phenomena were governed by the laws given by god(s). There are many traces of this mode of thinking in the Old Testament – for example in The Book of Job: “When he made a decree for the rain, and a way for the lightning of the thunder: Then did he see it, and declare it; he prepared it, yea, and searched it out” (28, 26-27). In the Hebrew original there appears the word chok, a derivative of “to engrave,” but used also for moral and ritual laws (in the Vulgate this expression is translated as “ponebat legem”).1 Interestingly, the idea of physical laws understood as God’s command is expressed also at the beginning of Corpus Iuris, where it is said that “nature has taught all animals the natural law.” Similar conceptual categories were employed by some pre-Socratic philoso  The other passages from the Old Testament where the same idea is expressed are Job 26,10 and 38,10; Psa 104, 9; Prv 8,9. 1

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phers, for instance Anaximander (“The source of coming-to-be for existing things is that into which destruction, too, happens according to necessity; for they pay penalty and retribution to each other for their injustice according to the assessment of Time”) and Heraclitus (“The Sun will not transgress his measures; otherwise the Erinyes, the bailiffs of Dike, will find him”). These remarks suffice, I believe, to substantiate the claim that for Aristotle “cause” (aition) must have been quite different from our understanding of the term. This observation finds additional confirmation when we consider the thesis (A3). Aristotle distinguishes four types of causes: material, formal, efficient and final. In his Physics, we read: It is clear then that there are causes, and that the number of them is what we have stated. The number is the same as that of the things comprehended under the question ‘why.’ The ‘why’ is referred ultimately either (1), in things which do not involve motion, e.g. in mathematics, to the ‘what’ (to the definition of ‘straight line’ or ‘commensurable’), or (2) to what initiated a motion, e.g. ‘why did they go to war? – because there had been a raid’; or (3) we are inquiring ‘for the sake of what?’ – ‘that they may rule’; or (4), in the case of things that come into being, we are looking for the matter. The causes, therefore, are these and so many in number. Now, the causes being four, it is the business of the physicist to know about them all, and if he refers his problems back to all of them, he will assign the ‘why’ in the way proper to his science-the matter, the form, the mover, ‘that for the sake of which’ (Aristotle, 2009, Book II, Part 7).

The formal cause of every existing substance is its form, or that which makes a given thing belong to a particular genus; the material cause of a thing is its material, or that out of which the thing is composed, which determines its individual, non-defining characteristics; the efficient cause is the “source of motion”; and the final cause is the end, or that which is the goal of motion (change). Two things are worth stressing in this context. First, the Aristotelian conceptual scheme em-

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braces more kinds of causes than in the contemporary philosophical and scientific discourse. It may be argued that what remained of the list of causes presented by Aristotle is only the efficient cause. However – second – what Aristotle calls the efficient cause is not necessarily what in the contemporary parlor is deemed a cause. In particular, Aristotle speaks of the efficient cause as the “source of motion,” and by motion he understand any kind of change. The examples of motion he cites are instructive, but from the contemporary viewpoint also surprising: “When the buildable, in so far as it is just that, is fully real, it is being built, and this is building. Similarly, learning, doctoring, rolling, leaping, ripening, ageing” (ibidem, Book III, Part I). When Aristotle says “cause” he is thinking of something other than when Newton or Einstein use the same word. Finally, the thesis (A4) says that the form of any genuine explanation must be deductive. When one explains some regularity in nature, one needs to take advantage of a syllogism, such as: (Major premise) Rational animals are grammatical. (Minor premise) Human beings are rational animals. (Conclusion) Human beings are grammatical. Why does this argument count as an explanation of the human ability to use language? As we have seen, Aristotle believes that to be considered an explanation an argument must identify the proper cause of the explained phenomenon. The cause in question is expressed in the middle term of the syllogism (here: “rational animals”). Human beings are grammatical, because they are rational – the cause of our ability to use language is formal, since rationality is the form of a human being. Similarly, let us consider the following example: (Major premise) What is near does not twinkle. (Minor premise) Planets are near. (Conclusion) Planets do not twinkle (quoted after Gottlieb, 2006).

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Here, the middle term is ‘being near’, and it clearly refers to the efficient cause (being near prevents planets from twinkling). Now, lets us turn to an aspect of the Aristotelian physics I have already mentioned: (Major premise) All heavy bodies move naturally towards the center of Earth. (Minor premise) Stones are heavy bodies. (Conclusion) Stones move naturally towards the center of Earth. In this argument the middle term is ‘heavy body’, and – according to Aristotle – such bodies tend to move towards their ‘natural place’, which is the center of Earth. In other words, by moving towards the center of Earth heavy bodies ‘fulfill their nature’, and so ‘heavy body’ refers here to the final cause, which provides an explanation of the free falling of stones. There is one more problem we need to address: the nature of knowledge. Aristotelian ideal knowledge (επιστημη) seems to be a huge, interconnected network of deductive syllogisms which identify relevant causes of the regularities in nature. The syllogisms demonstrate how the laws governing the universe can be derived from first principles, which are “true, primary, immediate, better known than, prior to, and explanatory of the conclusion” (Aristotle, 2009b, 71b1622). However, this picture is misleading as it suggests that for Aristotle knowledge is purely propositional, a body of sentences organized in a certain way. Myles Burnyeat forcefully argued that this is not the case – what Aristotle calls επιστημη should not be translated as ‘knowledge’, but rather as ‘understanding’ (Burnyeat, 1984). While this claim may not be fully accurate (cf. Lesher, 2001), Burnyeat has a point. Aristotle uses ‘επιστημη’ and ‘επιστασθαι’ to speak of different, but interrelated things: the intellectual capacity to understand a given phenomenon, an organized body of knowledge in a particular discipline (e.g., geometry), knowledge of an individual fact, and immediate knowledge of first principles (ibidem, pp. 47-49). In this con-

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text, James Lesher suggests that the best way to render the Aristotelian ‘επιστημη’ is to think of it as ‘expert knowledge’ or ‘disciplinary mastery’ (ibidem, p. 54). Επιστημη – at least in the most proper use of the word – is not a particular body of propositions, but rather a skill, an ability to produce the relevant syllogisms at will, and grasp interconnections between them; it is such an understanding of the given subjects which enables one to answer any question regarding it and to explain any aspect thereof. (This reading of επιστημη is compatible with the earliest uses of the word in Homer, where it means “to know how,” “to be able to”) (ibidem, p. 51). As we have seen, the theory of explanation developed by Aristotle is – at least to a certain degree – different from our contemporary way of thinking. First, he believes that the subject of explanation (explanandum) is never a concrete fact, but rather a regularity in nature. Second, to provide an explanation one needs to identify the cause of the considered regularity. Third, to do so one formulates a syllogism in which the explanandum is the conclusion, and the cause that is pivotal to the explanation is expressed by the syllogism’s middle term. Fourth, Aristotle’s concept of explanation is closely linked with his ontology. He believes that there are four kinds of causes – formal, material, efficient, and final, and any of them is fully sufficient to explain a phenomenon under consideration. Moreover, on closer inspection it transpires that the way Aristotle understands causes is fundamentally different from our way of thinking. Fifth, Aristotle’s conception of explanation, as well as his view of knowledge (επιστημη), is closely linked with understanding: one’s understanding of a regularity in nature manifests itself in the ability to produce relevant syllogisms, which explain the regularity by demonstrating how it can be derived from the first principles. Thus, the Aristotelian view of explanation may be succinctly characterized as a causal and deductive way of understanding regularities in nature within the conceptual framework of the Aristotelian ontology. It is the ontological background which ultimately determines what counts as an explanation, and what does not.

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2. The contemporary debate The contemporary debate concerning the nature of scientific explanation kicked off with the publication in 1948 of the seminal paper by Carl G. Hempel and Paul Oppenheim, “Studies in the Logic of Explanation.” Their goal was to shed some light on the function and the essential characteristics of explanation in science. They began by identifying the general structure of any explanation, in which one should distinguish explanandum, i.e. “the sentence describing the phenomenon to be explained” (ibidem, pp. 136–137), and explanans, i.e. “the class of those sentences which are adduced to account for the problem” (ibidem, p. 137). Further, they proposed the following conditions of the soundness of scientific explanation: (a) that explanandum must be a logical consequence of the explanans; (b) that the explanans must contain general laws, which must be required for the derivation of the explanandum; (c) that the explanans must have empirical contents, i.e. must be testable by experiment or observation; (d) that the sentences constituting the explanans must be true (or, at the least, highly confirmed by all the relevant evidence available) (ibidem, pp. 137–138). Thus, the general structure of scientific explanation appears roughly as follows: (1) (2)

statements of antecedent conditions general laws

(3)

description of the explained phenomenon

where (1) and (2) constitute the explanans, (3) is explanandum, and the horizontal line represents deduction. Importantly, Hempel and Oppenheim stress that the element (1) is not necessary – it is possible to imagine scientific explanation where the explanans involves no statement of individual facts, only general laws, as when one wishes to explain “the derivation of the general regularities governing the motion of double stars from the laws of celestial mechanics” (ibidem, p. 137).

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Let us contrast the model of explanation proposed by Hempel and Oppenheim (and usually referred to as the deductive-nomological, or the D-N model) with the account of explanation offered by Aristotle: (DN1) [The nature of explanandum] Unlike in the Aristotelian theory, the D-N model accounts not only for explanations of regularities in nature, but also of concrete, individual facts. Moreover, while developing a detailed formal account of scientific explanation in the technical part of their paper, Hempel and Oppenheim limit their analysis to the explanation of particular events, i.e. to those cases only, where the explanandum is a singular sentence (ibidem, p. 159). Thus, although the motivation behind the D-N model is to uncover the mechanisms of explaining both individual facts and regularities in nature, the fully developed version of the model accounts only for the explanation of particular events. (DN2) [The nature of explanans] While Aristotle believed that explanation always consists in identifying the causes of the explained phenomena, the proponents of the D-N model are somewhat ambiguous with regard to this problem. They claim that the explanans must include a general law, and go as far as saying that such a type of explanation “is often referred to as causal explanation.” However, they offer no detailed analysis of the concept of causality. Moreover, in his later writings Hempel argued that causality plays no important role in scientific explanation (Hempel, 1965, pp. 347–351). (DN3) [The ontological scheme] Similarly to the Aristotelian account of explanation, the D-N model has some ontological commitments: it is a theory of explanation developed against the background of logical positivism. It is clearly visible when one considers two facts: that, according to Hempel and Oppenheim, the explanans must be testable by experiment or observation, and that explanation must take a form of a sound deductive argument. While both those requirements seem perfectly justifiable (since we are considering scientific explanation), it is the overemphasis on the empirical and logical components of explanation, and

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some disregard for other aspects thereof, which ultimately drives the D-N model into theoretical problems. In particular, I think it is the main reason for neglecting the role of causality in the process of explanation. (DN4) [The form of explanation] The form of explanation in the D-N model is deductive. Thus, the position of Hempel and Oppenheim is the same as Aristotle’s (with the obvious caveat that the contemporary logic Hempel and Oppenheim had at their disposal is much more powerful than Aristotelian syllogistics). As we can see, the D-N model dismisses with some of the Aristotelian insights regarding scientific explanation (causality, explanation of regularities in nature rather than particular events), while retaining some others (to explain one needs to use a general law, as well as a deductive form of argument). Arguably, however, the biggest difference between the D-N model and the Aristotelian theory lies in the ontological commitments of both views of explanation. Let us recall that for Aristotle explanation was an aspects of knowledge (επιστημη), and knowledge was ultimately a kind of craft. To be able to explain something was a synonym for having a mastery and a perfect understanding of a subject. Meanwhile, the view of knowledge and science which lies behind the D-N model is de-personalized: it is, to use Karl Popper’s famous phrase, epistemology without the knowing subject. For Hempel, Oppenheim and other empirical positivists, knowledge is a set of sentences, and therefore scientific explanation can be nothing other than a certain logical relation between sentences. From such a viewpoint, the question concerning the relationship between explanation and understanding is ill-stated: understanding is a psychological process, and has nothing to do with objective logical relations obtaining between the sentences belonging to the body of scientific knowledge.2   Hempel acknowledges it explicitly when he says that “such expressions as ‘realm of understanding’ and ‘comprehensible’ do not belong to the vocabulary of logic, for they refer to the psychological or pragmatic aspects of explanation.” However, on some occasions he attempts to connect explanation and understanding, as when he says: “also,

2

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The development of the D-N model gave rise to different kinds of reactions, and to the emergence of different research traditions (cf. Salmon, 1989; Woodward, 2014). A detailed analysis of those approaches lies beyond the limits of this essay. Instead, I will concentrate on three replies to the D-N model which underline to necessity to link scientific explanation with understanding. The first may be found in Michael Friedman’s “Explanation and Scientific Understanding” of 1974. Friedman believes that while Hempel and Oppenheim are right in insisting that explanation should be deductive and objective, they fail to account for the fact that scientific explanation provides us with a better understanding of the world. Friedman further claims that the mistake hangs together with their failure to provide a theory of the explanation of scientific laws (i.e., sentences describing some regularities in nature), rather than particular events. But what does an explanation of a regularity consist in? The answer Friedman supplies is quite straightforward: to explain a regularity in nature means to reduce it (or to show its relation to) a more fundamental law. For example, the fact that heated water turns to steam may be explained by saying that “water is made of tiny molecules in a state of constant motion. Between these molecules are intermolecular forces, which, at normal temperatures, are sufficient to hold them together. If the water is heated, however, the energy, and consequently the motion, of the molecules increases. If the water is heated sufficiently the molecules acquire enough energy to overcome the intermolecular forces they fly apart and escape into the atmosphere” (ibidem, p. 5).3 But why does such a reduction of one law to the other generate understanding and comprehension? Friedman observes that the more fundamental a law we have, the more embracing it is. For example, the Boyle-Charles law, Graham’s law, Galileo’s law of free fall, and given the particular circumstances and the laws in question, the occurrence of the phenomenon has to be expected; and it is in this sense that the explanation enables us to understand why the phenomenon occurred” (Hempel, 1965, p. 413, 327). 3   It should be noted that the conception of explanation of general scientific laws Friedman offers has some problems of its own (see in particular Woodward, 2014).

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Kepler’s laws may all be reduced to the Newtonian laws of mechanics. In this way, our understanding of the world increases, since we are able to reduce the number of independently acceptable assumptions required to explain natural phenomena. Our picture of the world becomes simpler, less scattered and fragmented. Friedman stresses, however, that on such a unification view of explanation, the kind of understanding provided by science is global rather than local. Scientific explanations do not confer intelligibility on individual phenomena by showing them to be somehow natural, necessary, familiar, or inevitable. However, our over-all understanding of the world is increased; our total picture of nature is simplified via a reduction in the number of independent phenomena that we have to accept as ultimate (ibidem, p. 18).

The theory of scientific explanation developed by Friedman constitutes a modification, rather than a complete rejection of the D-N model. On the one hand, Friedman stresses that explanation mainly concerns regularities in nature rather than particular events, and that it increases our understanding of the world. On the other hand, however, he does not include causality in his account of explanation, and uses a peculiar, global notion of understanding. In other words, Friedman’s theory of scientific explanation is fully compatible with the logico-positivistic ontology of Hempel and Oppenheim, where science is considered an ordered set of sentences and there is no need to consider the role of the knowing subject. Even if Friedman speaks of understanding, it is not a psychological phenomenon, but rather a logical feature of the set of sentences we identify with science. The main insight of his analysis is, however, that given two empirically equivalent sets of scientific laws, the one in which the number of independently acceptable assumptions is smaller, is a better tool of understanding. The second important reply to Hempel and Oppenheim was supplied by Michael Scriven. In his paper of 1962, “Explanations, Predictions and Laws,” he observes that the D-N model has serious drawbacks: (a) it is

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too restrictive as it excludes much of what actually counts as explanation; (b) at the same time, it is too inclusive as it includes entirely non-explanatory schema; (c) it presupposes accounts of cause, law and probability which are basically unsound; and (d) it leaves out the notions of context, judgment and understanding, which are essential for any account of explanation (ibidem, p. 196). Much of Scriven’s criticism revolves around two basic tenets of the D-N model: that scientific explanation always involves (or should involve) a general law, and that it always is (or should be) deductive. For example, Scriven shows that the D-N model is too exclusive, partly because its insistence on providing explanation through supplying a general law. Consider the following example: As you reach for the dictionary, your knee catches the edge of the table and thus turns over the ink-bottle, the contents of which proceed to run over the table’s edge and ruin the carpet. If you are subsequently asked to explain how the carpet was damaged you have a complete explanation. You did it, by knocking over the ink. The certainty of this explanation is primeval. It has absolutely nothing to do with your knowledge of the relevant laws of physics; a cave-man could supply the same account and be quite as certain of it (Scriven, 1959, p. 456).4

To put this point more generally: Scriven believes that one often explains a particular fact by pointing toward some other particular fact. Thus, explanation does not have to include general laws, and it does not have to be a deductive argument (importantly, such nondeductive and non-nomological explanations are typically applied in science, e.g. in biology or psychology). Scriven also observes that the D-N model does not link explanation with understanding, while he believes that explanation in science is “a topically unified communication, the content of which imparts understanding of some scientific   It should be observed, however, that the explanation Scriven presents can be augmented so that in includes a general law, e.g. “Whenever knees impact tables on which an inkwell sits and further conditions K are met (where K specifies that the impact is sufficiently forceful, etc.), the inkwell will tip over.” (Cf. J. Woodward, 2003, section 2.4). 4

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phenomenon” (Scriven, 1962, p. 224). Understanding, according to Scriven, is “organized knowledge, i.e. knowledge of the relations between various facts and/or laws. These relations are of many kinds – deductive, inductive, analogical, etc. (Understanding is deeper, more thorough, the greater the span of this relational knowledge)” (ibidem, p. 225). Thus, Scriven’s account is similar to Friedman’s in that one of the goals of explanation is to increase our understanding of the world around us; but while Friedman believes it may be achieved only through unification, Scriven claims that various kinds of logical relations between sentences expressing facts and/or laws may do the job. The third critique of the D-N model that interests us here is due to Wesley C. Salmon. He claims that the problem of Hempel and Oppenheim’s approach lies in their failure to recognize the causal or mechanistic character of explanation in science (this view is implicit in Scriven’s conception, but the work of Salmon has given the idea of causal explanation a comprehensive outlook). Let us consider Salmon’s famous example: (1) All males who take birth control pills regularly fail to get pregnant. (2) John Jones is a male who has been taking birth control pills regularly. Therefore: (3) John Jones fails to get pregnant (Salmon, 1973, p. 34). Although this is a deductive argument, which utilizes a general law (1), intuitively the explanans – (1) together with (2) – does not provide an explanation for the explanandum (3). Salmon argues that the reason for the failure of this argument, and of the D-N model in general, is that explanation proceeds through identifying causes of the explained phenomena, and (1) together with (2) is not the cause of (3): John Jones fails to get pregnant because he is a male, not because he has failed to take birth control pills regularly. Salmon further attempts to present this intuitively appealing idea in a philosophically rigorous way. He claims, for example, that “an intersec-

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tion of two processes is a causal interaction if both processes are modified in the intersection in ways that persist beyond the point of intersection, even in the absence of further intersections” (Salmon, 1990, p. 7). An example is a collision of two billiard balls which modifies their motions and the modifications persist beyond the point of collision. Further, “a process is causal if it is capable (…) of entering into a causal interaction” (ibidem). Of course, the idiom Salmon develops is somewhat contrived, and his conception has some troublesome theoretical consequences (cf. Woodward, 2014). However, the important insight of his analysis is that explanation is closely linked with understanding how things work, and not necessarily with how they fit into a set of general laws. In other words: there is a (…) notion of scientific understanding that is essentially mechanical in nature. It involves achieving a knowledge of how things work. One can look at the world, and the things in it, as black boxes whose internal workings we cannot directly observe. What we want to do is open the black box and expose its inner mechanisms (Salmon, 1990, p. 18).

Salmon thinks, therefore, that scientific explanation is not necessarily deductive, and does not necessarily involve a general law. The goal of explanation is rather to uncover the mechanism behind the phenomenon at hand: as long as we do not know the mechanism, we cannot be said to understand or have explained it.

3. What is understanding? The above considerations had two goals: the minor was to outline a “map” of the contemporary debates concerning the nature of scientific explanation;5 the major was to see how the relation between explanation and understanding is construed by various theorists. As I have   The “map” is by no means comprehensive – there exist other approaches to scientific explanation. For an overview see Woodward, 2014. 5

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pointed out in the first section of this essay, for Aristotle explanation – always deductive and causal – constituted an important mechanism behind επιστημη, a kind of knowledge which requires a complete understanding of the relevant domain. When someone has επιστημη regarding some discipline, she has such a perfect grasp of the subject that she can produce an appropriate explanation for any phenomenon connected to it. Thus, for Aristotle, understanding is the ability of the human mind to capture the causal connections between regularities in nature through deductive syllogisms.6 Whoever has understanding is capable of providing explanations, whoever supplies all the explanations has a perfect understanding of the subject matter in question. The contemporary discussion on scientific explanation, which began with the famous essay by Hempel and Oppenheim, has dispensed with most of the Aristotelian assumptions; in particular, it has had little interest in the Aristotelian doctrine of the four causes. In the deductive-nomonological model championed by Hempel explanation was construed as deductive and involving a general law; however, Hempel explicitly rejects explanation through causes, as well as establishes no link between explaining (which is an objective procedure) and understanding (which he considers psychological, and hence subjective). In contrast, Friedman claimed that scientific explanation is the main tool we use in our attempts to understand the world; he believed that the explanation of differentiated phenomena through deductively deriving them from the most general laws, increases our understanding. In this, his position is similar to Aristotle’s: one should explain not only particular events, but also regularities in nature, and one should do so by establishing deductive links between the most general laws and the explained phenomena. At the same time, Friedman does not believe explanation must be causal, in which he dismisses both one of the Aristotelian criteria of explanation, as well as his ontological framework. Scriven, in turn, disagrees with Hempel and Friedman   One should remember, however, that the Aristotelian causes are quite distinct from their contemporary cousins.

6

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with regard to the deductive character of explanation: the formal links between the explanandum and the explanans do not have to be deductive, but may also be inductive, analogical, etc. Moreover, explaining does not always involve supplying a general law – it is perfectly correct to explain one particular event by citing another one. At the same time, Scriven believes that the more elaborate the network of formal links between particular facts and/or general laws one has, the better is one’s understanding of the world. Thus, it is clear that Scriven has a different conception of understanding from Friedman’s: while the latter claims that our understanding is enhanced only by knowledge which is highly unified through chains of deductive explanations leading to as few as possible general laws, the former believes that any formal connections between the nodes in our web of beliefs contribute to the quality of our understanding. Finally, in his account, Salmon is not so interested in the logical structure of explanation, but rather in a substantive aspect thereof. He claims that explaining requires the identification of the relevant causes. It is through inserting the explained phenomena into causal chains – claims Salmon – that we are able to understand them better. As we have seen, the relationship between explanation and understanding is accounted for in various ways: while Hempel sees no essential connection between the two concepts, Friedman, Scriven and Salmon believe explanation to be the key to our understanding of the world. However, they all seem to assume completely different conceptualizations of understanding: it is one thing to say that understanding is increased with the unification of knowledge, and quite another to claim that unification is just one possible way of organizing our beliefs which enhances understanding; it is still something else to say that explanation – and understanding – are achieved through identifying causes of the explained phenomena. How to assess this complicated theoretical situation? I believe that the best way to proceed is to inquire into the nature of understanding. In order to do so, I would like to begin with three different insights concerning the concept in question.

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The first insight is the idea of axiomatization in mathematics. In 1891 David Hilbert, who was the early champion of the contemporary attempts of the axiomatization of different branches of mathematics and physics, attended the lectures on the foundations of geometry delivered by Hermann Wiener. During the lectures Wiener observed that: also for geometry the return to such simplest objects and operations is important, since they may serve to construct an abstract science, one that is independent of the axioms of geometry, but whose theses go hand in hand with the theses of geometry (Wiener, 1980, p. 46).

After Wiener’s lecture Hilbert famously said that in the axioms of geometry it should be possible to replace “points, straight lines, and planes” with “tables, chairs, and beer mugs” (quoted after Corry, 2004, p. 74). I believe this statement should be read as saying that the inferential structure of the given domain – e.g., geometry – even without taking into account the substantive content of the axioms and theses involved, is the carrier of vital information pertaining to the domain. It does not need to serve as heuristic tool to uncover new truths, but it increases one’s understanding of the relevant facts: I understand under the axiomatical exploration of a mathematical truth [or theorem] an investigation which does not aim at finding new or more general theorems being connected with this truth, but to determine the position of this theorem within the system of known truths in such a way that it can be clearly said which conditions are necessary and sufficient for giving a foundation of this truth (Hilbert, 1902, p. 50, quoted after Peckhaus, 2003).

Importantly, Hilbert claims that the utilization of the axiomatic method is beneficial not only in mathematics, but also in other disciplines, such a physics:

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In this way [i.e., by means of the axiomatic treatment] geometry is turned into a pure mathematical science. In mechanics it is also the case that all physicists recognize its most basic facts. But the arrangement of the basic concepts is still subject to a change in perception (…) and therefore mechanics cannot yet be described today as a pure mathematical discipline, at least to the same extent that geometry is. We must strive that it becomes one. We must ever stretch the limits of pure mathematics wider, on behalf not only of our mathematical interest, but rather of the interest of science in general (quoted after Corry, 2006, p. 146).

Let us observe that an axiomatic system (if properly constructed, i.e. when it is consistent and complete with regard to the chosen domain such as Euclidean geometry or arithmetic), is a structured set of sentences that is unified to the greatest possible extent: the theses of the system are derived from but a few, mutually independent general laws (axioms). This seems to be the background behind Friedman’s conception of explanation and understanding – he seems to suggest that our scientific theories should resemble axiomatic systems as much as possible. What are the advantages of such a construction? Where does the better understanding of mathematics or physics resulting from axiomatization come from? Hilbert offers a number of clues. He says, for example, that – once axiomatized – the given discipline has its foundations clearly identified and fully uncovered; this, in turn, enables one to grasp all the basic concepts of the discipline and see their relations to less fundamental, derivative concepts. Hilbert believes that “the aim of every science is, first of all, to set up a network of concepts based on axioms to whose very conception we are naturally led by intuition and experience. Ideally, all the phenomena of the given domain will indeed appear as part of the network and all the theorems that can be derived from the axioms will find their expression there” (quoted after ibidem, p. 161). What must be stressed in this passage is the word “ideally.” Hilbert does not think that every discipline (of mathematics and physics) can easily be presented in an axiomatic

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form. He thinks this is an ideal one should strive to achieve. But, importantly, this is not the only, or even the main goal of scientific endeavor. It is often reasonable to proceed without axiomatization in securing a number of truths, and only then make an attempt at axiomatization in the hope of gaining more insight and understanding. As Hilbert puts it himself: The edifice of science is not raised like a dwelling, in which the foundations are first firmly laid and only then one proceeds to construct and to enlarge the rooms. Science prefers to secure as soon as possible comfortable spaces to wander around and only subsequently, when signs appear here and there that the loose foundations are not able to sustain the expansion of the rooms, it sets about supporting and fortifying them. This is not a weakness, but rather the right and healthy path of development (quoted after ibidem, p. 164).

The second insight I would like to analyze comes from a completely different philosophical tradition and has little to do with understanding in mathematics – it is Hans Georg Gadamer’s conception of understanding, which I will try to recast in more analytic terms.7 For example, when Gadamer speaks of the process of understanding as leading to “the constitution of sense (Sinn),” I believe he should be read as speaking about propositions or sentences (Gadamer, 2004, p. 164). The two key hermeneutic concepts that describe the structure of understanding are: pre-understanding or pre-judgment (Vorverstandnis, Vorurteil) and the hermeneutic circle. It is possible, or so I argue, to capture those concepts in a precise way. Of course, what I present below is only a paraphrase of the original conception, but arguably an admissible one. Gadamer nowhere defined the concept of pre-understanding. He speaks of it as a transcendental condition of understanding and criticizes the Enlightenment tradition, claiming that by rejecting pre-judg7

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  The following fragment is based on my paper (Brożek, 2014).

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ments as not based on the authority of reason, the only admissible authority, itself embraces a prejudice. However, one cannot, Gadamer continues, imagine understanding without a pre-understanding. Gadamerian pre-understanding has at least two dimensions. Firstly, everyone who engages in the interpretation (understanding) of a text is a participant in a certain culture (tradition), and so understanding and interpretation are always relative to a tradition. Secondly, pre-understanding also has an individual “flavor”: one that interprets or “poses a question to a text,” always anticipates an answer, initially ascribes some meaning to the text (cf. ibidem, pp. 277–304). These theses are far from clear and dangerously close to nonsense. What does it mean that one “poses a question to a text”? What is “the anticipation of meaning”? In what way – apart from the obvious one in which context influences interpretation – does tradition play the role of a “transcendental condition of understanding”? It is tempting to conclude that, while Gadamer may be trying to verbalize something important, the result is vague and imprecise and brings rather more confusion than insight. However, I believe that it is possible to express the intuitions that stand behind Gadamer’s obscure phrase in a more precise way. To do so, I suggest distinguishing between four kinds of pre-understanding. First, the thesis that tradition is a transcendental condition of understanding may be seen as an attempt to say that whoever interprets something must use an interpreted language. Thus, she must have at her disposal a vocabulary, syntactic rules (rules for constructing compound expressions), rules of inference and a function which maps constants to individuals belonging to the domain of language, one-place predicates to sets of such individuals, etc. Second, participation in the same tradition requires a shared set of presuppositions. Usually, it is assumed that a sentence A is a presupposition of a sentence B iff B may be ascribed truth or falsehood only if A is true. Third, two persons participate in the same tradition if they have the same or similar background knowledge, where the term usually refers to all those statements that – within the process of solving a problem

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– are assumed to be true or unproblematic. Here, I shall understand background knowledge in a similar way, as consisting of all those sentences that – at least prima facie – are taken to be true or justified. Fourth, it seems that the best way to explicate the individual dimension of pre-understanding is to treat pre-judgements as initial hypotheses, i.e. sentences capturing the sense (meaning) of the interpreted text, which one formulates at the beginning of the process of interpretation, and aims at confirming or rejecting them in due course. Given the above, if one is to interpret a text then one is in the following position: she has at her disposal an interpreted language, a set of presuppositions, background knowledge and a set of initial hypotheses. How does the process of interpretation look like? Gadamer describes it by recourse to the concept of a hermeneutic circle. He says, for instance: but the process of construal is itself already governed by an expectation of meaning that follows from the context of what has gone before. It is of course necessary for this expectation to be adjusted if the text calls for it. This means, then, that the expectation changes and that the text unifies its meaning around another expectation. Thus the movement of understanding is constantly from the whole to the part and back to the whole. Our task is to expand the unity of the understood meaning centrifugally. The harmony of all the details with the whole is the criterion of correct understanding. The failure to achieve this harmony means that understanding has failed (cf. ibidem, p. 291).

And elsewhere he adds: every revision of the foreprojection is capable of projecting before itself a new projection of meaning; rival projects can emerge side by side until it becomes clearer what the unity of meaning is; interpretation begins with fore-conceptions that are replaced by more suitable ones. This constant process of new projection constitutes the movement of understanding and interpretation (cf. ibidem, p. 263).

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Thus, Gadamer seems to believe that understanding is a constant process of putting forward, testing, and rejecting or modifying hypotheses. When one tries to understand a written text or some other cultural artifact, one formulates and interpretive hypothesis – or, more often, a number of competing hypotheses (“rival project emerging side by side”). Now, each of those hypotheses stands in some relation to one’s background knowledge. For instance, they may contradict some previously accepted propositions or a presupposition one tacitly assumes, and thus require the rejection of the hypothesis, an element of the background knowledge or a presupposition. Most importantly, Gadamer claims that “our task is to expand the unity of the understood meaning centrifugally. The harmony of all the details with the whole is the criterion of correct understanding.” I read him as saying that one should prefer those interpretive hypotheses which bring about most coherence in one’s worldview. And so, an interpretive hypothesis which contradicts one’s background knowledge should be rejected in favor of another one, which preserves consistency; on the other hand, two different interpretive hypotheses, both consistent with one’s background knowledge, should be compared in relation to the level of coherence they generate. Crucially, even if a well-constructed axiomatic system is coherent to the greatest possible degree,8 it will rarely be possible to organize one’s worldview in such a perfect way. Rather, our cognitive condition is much less ideal, and hence the inferential connections within our worldview may be of different kinds: deductive, inductive or analogical, which makes us constantly repeat our attempts at formulating better hypotheses, and achieving better grasp of the world around us. This view of understanding seems to be what Scriven has in mind when he insists that explanation – and the   I take coherence to be determined by taking into account: (a) the number of nontrivial inferential connections in our belief set and (b) the degree of its unification. There exist nontrivial inferential connections between sentences belonging to a given set if they can serve together as premises in logically valid schemes of inference. In turn, a given set of sentences is unified if it cannot be divided into two subsets without a substantial loss of information (cf. Bonjour, 1985). 8

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understanding it generates – does not have to, and rarely does, proceed through unification. The third and final insight regarding the concept of understanding comes from cognitive science. George Lakoff and his followers consider the nature of the human conceptual apparatus (Lakoff & Johnson, 1999). They claim that our cognition is largely determined by our bodily interactions with the physical environment. As a result, our fundamental concepts are concrete in character. It is only on their basis that more abstract concepts are formed, by means of metaphor. Of course, metaphor is not understood as a poetic device here; it is understood as a mechanism of “understanding and experiencing one kind of thing in terms of another” (Lakoff & Núñez, 2000, p. 5). For example, importance is understood in terms of size (“This is a big issue,” “It’s a small matter”), while difficulties are conceptualized as burdens (“I’ve got some light housework,” “He’s overburdened”) (ibidem, p. 41). “Each such conceptual metaphor has the same structure. Each is a unidirectional mapping from entities in one conceptual domain to corresponding entities in another conceptual domain. As such, conceptual metaphors are part of our system of thought. Their primary function is to allow us to reason about relatively abstract domains using the inferential structure of relatively concrete domains” (ibidem, p. 42). Mathematics provides good examples to illustrate this mechanism. According to Lakoff and Núñez, our understanding of arithmetic involves four basic, or grounding, metaphors. In all of them, the target domain is elementary arithmetic, while the source domains are: a collection of objects, object construction, using a measuring stick, and movement along a path (ibidem, p. 42; cf. Brożek & Hohol, 2014, chapter 2). For example, the first metaphor involves the following mapping of elements from the source domain (a collection of objects) to the target domain (arithmetic): a collection of objects of the same size the size of the collection

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a number



the size of the number

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bigger collection smaller collection the smallest collection putting collections together taking a smaller collection from a larger collection

→ → → → →

bigger number smaller number the unit (one) addition subtraction

One can doubt whether Lakoff’s theory is true in all its details, for example whether arithmetic is really based on the four grounding conceptual metaphors and whether they involve the same mappings that Lakoff suggests. However, the theory of embodied language need not be true in all its details in order to indicate where we should look for a viable theory of the roots of our language ability: it is undoubtedly based, even if not solely, on motor schemas, i.e. the brain circuits crucially involved in our interactions with the environment.9 This fact has a number of interesting consequences for the problems of explanation and understanding. According to Lakoff, our more basic, concrete concepts are rooted in our motor schemas and shaped by our interactions with the environment. It follows that they must encode some rudimentary understanding of causality. On the one hand, the constant observation and manipulation of objects in the world must lead to some understanding of how they interact and how to influence such interactions (e.g., when I throw a stone at the window it will probably break). On the other hand, the causal relations in question extend to the sphere of the social – we comprehend the actions of others and we learn how to influence such actions by understanding their motives, intentions, etc. Lakoff claims further that such basic, concrete concepts may be abstracted from and – via metaphorization – serve to construct more abstract concepts. What is preserved in the process is (a part of) the inferential structure of the conceptual scheme which constitutes the metaphor’s source domain. It is   Additional arguments for the role of the motor areas in the brain in our ability to use language are provided by the theory of cognitive simulation (cf. Bergen, 2012). 9

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this relation that enables us to understand abstract concepts. We know what love is, since we (partly) comprehend it in terms of the inferential scheme of the concept of journey. It is important to note that this “understanding of understanding” has nothing to do with the unification of knowledge, the use of general laws, or even establishing some kind of formal links between facts and/or general laws. To understand a phenomenon means to conceptualize it in terms of the inferential scheme of a more concrete, fundamental concept. On this account, the mathematical equations of general relativity or quantum mechanics are not the direct source of understanding, as they themselves require understanding. We comprehend them since mathematics is constructed on the basis of more concrete concepts which we are able to grasp in a more direct way. Thus, if Lakoff is right, it would explain (nomen omen!) our preference for causal explanations and the appeal of Salmon’s theory of explanation. In order to see more clearly the differences between the various kinds of understanding and explanation, let us have a look at the following example: A mother leaves her active baby in a carriage in a hall that has a smooth level floor. She carefully locks the brakes on the wheels so that the carriage will not move in her absence. When she returns she finds, however, that by pushing, pulling, rocking, bouncing, etc., the baby has succeeded in moving the carriage some little distance. Another mother, whose education includes some physics, suggests that next time the carriage brakes be left unengaged. Though skeptical, the first mother tries the experiment and finds that the carriage has moved little, if at all, during her absence. She asks the other mother to explain this lack of mobility when the brakes are off (it is taken from Salmon, 1990, p. 12).

Salmon claims that the event described in the story may be explained in two different ways. The unification-driven explanation would have to appeal to the law of conservation of linear momentum – the baby

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and the carriage constitute an essentially isolated system (with respect to horizontal motion) when the brake is off, but is linked with the earth when the brake is on. The causal explanation, on the other hand, would consist in the analysis leading to the conclusion that – when the break is off – all the forces exerted by the baby on the carriage and by the carriage on the baby cancel out (ibidem, p. 12). Thus, the former explanation requires quite abstract laws which are far from our every-day experience, while the latter are more readily comprehensible given that we are accustomed with situations, when forces cancel out, as when we are playing Tug of War. At the same time, the explanation of the event in terms of the law of conservation of linear momentum links the carriage that does not move when the brake is off with a number of different, often seemingly dissimilar situations: when a canon releases a projectile, it moves in the direction opposite the projectile; or when a particle is deflected by gravity, the gravitational field will also be modified by the particle (given some chosen reference frame). Salmon suggests that both kinds of explanation are complementary as they are connected to two different kinds of understanding. The unification view of explanation increases “global understanding,” i.e. understanding in terms of one’s worldview (Weltanschaung): “to understand the phenomena in the world requires that they be fitted into the general world-picture” (ibidem, p. 17). On the other hand, the causal-mechanistic view of explanation goes hand in hand with a different concept of “local understanding” of how a particular thing works. In light of the above considerations, Salmon’s suggestion seems tenable. However, I would like to argue that we are not dealing with two contrasting ways of “understanding understanding,” but rather with a more subtle situation. I posit that various incarnations of understanding can be depicted as a horizontal line. At one end of the line understanding is concrete – it is based on “directly embodied” concepts, i.e. those which are shaped by our interactions with the environment (both physical and social). The other end of the line represents a kind of understanding that may be called abstract: it is global understanding of a certain do-

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main of cognition achieved through a construction of an axiomatic system (or something close to it), which encapsulates all truths about the domain. There are substantial differences between the two extremes. The “concrete understanding” is pre-linguistic (i.e., it is possible to grasp the meaning of an event or of someone’s action with no use of words). Moreover, such an understanding is action-oriented – its goal is to facilitate one’s actions. It is also causal, but “cause” is not a theoretical concept here, but rather an aspect of how our brains represent events and actions. Still, this kind of understanding has a flavor of subjectivity: two individuals can understand the same event or action in quite different ways, given their varied experience, cognitive skills, goals, etc. However, since we are equipped with similar cognitive mechanisms and face similar problems, the subjective character of “concrete understanding” has substantial potential for objectivity. It is strengthened by the development of language, whose primary function is to orchestrate the conceptual maps of the members of the same linguistic community (Brożek & Hohol, 2015). But language also constitutes a key to the sphere of abstract theories. The pre-linguistic concept of “a cause” may be theorized in various ways. For example, as we have seen, Ancient Greeks construed it with the use of the term “aition,” which came from social discourse. The contemporary concept of a cause does not have this “social flavor” anymore – over the centuries, scientific practice and the reflection upon it have led to the development of a highly theoretical notion. Such conceptual shifts, together with the developments in various disciplines, especially physics, mathematics and logic, paved the way to the “abstract” concept of understanding, which lies at the other end of our line. In contrast with “concrete” understanding, the “abstract” kind is linguistic, social and objective – it is, in a sense, understanding “without the knowing subject.” To put it in different terms, “abstract” understanding is a transcendental notion – it is something we jointly strive to achieve, but with no guarantee that we will ever succeed. It puts emphasis on the formal dimension of scientific discourse and does not necessarily require making recourse to causality; unifi-

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cation is what matters. Still, unlike its “concrete” relative, “abstract” understanding is not action-oriented, but truth-oriented. Usually, it is abstract understanding that is taken as a model for scientific explanation, especially in such disciplines as theoretical physics. The reason behind this fact has to do with the universal character of scientific knowledge. A highly formalized explanation is invariant with respect to more conceptual schemes. For example, the equations of quantum physics are compatible with various ontological frameworks: they may be interpreted in the spirit of the Copenhagen school, along the lines of Bohm’s variable, or Everett’s many worlds. Thus, the equations of quantum mechanics – as an explanation of certain phenomena – are highly universal in the sense that they do not depend on the acceptance of some particular conceptual apparatus (recall Hilbert’s saying that in the axioms of geometry it should be possible to replace points, straight lines, and planes with tables, chairs, and beer mugs – the formal structure of an axiomatic system is invariant in relation to the contents or meaning of particular axioms). Of course, things get more complicated when one moves from mathematics and physics to less formalized sciences, such as biology, neuroscience or psychology. It should come as no surprise that biological or psychological explanations are different than those in the physical sciences: they are further away from the ideal of an axiomatic system and take advantage of some particular conceptual schemes. They are, therefore, less invariant with regard to ontological choices than general relativity or quantum mechanics. Finally, both ends of our line represent ideal models of understanding. Our actual attempts to understand the world lie somewhere in-between. Although our concrete, embodied conceptual scheme provides a scaffolding for the development of abstract theories expressed in language, the latter are able – through a kind of feedback loop – to influence even quite basic cognitive processes (cf. Brożek, 2013, chapter 2). For example, there is a number of studies which suggest that experts in a given domain perceive the relevant phenomena in a different way that people who are not properly trained (which nicely coincides with Aris-

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totle’s conception that explanation is a kind of understanding characteristic of those who have mastered a discipline and hence are capable of providing more abstract account of its subject-matter). Therefore, there are many faces of understanding – from the concrete and causal one, to the abstract and axiomatic, but there is no discontinuity between them; rather, they constitute a continuous spectrum. *

*

*

What is the bearing of the above considerations on the relationship between explanation and understanding? It should be stressed at the outset that there is no one correct concept of explanation – both in our every-day experience and in scientific practice various, often quite different arguments count as explanations of the phenomena at hand. This should come as no surprise, given the entire spectrum of cognitive activities that increase understanding. Some explanations may help us in concrete understanding, others in abstract, while still others lie in-between. At the same time, not every argument has explanatory power – not everything may be considered an explanation. It seems that Aristotle already identified some features of genuine explanations: their deductive character, their use of general laws, and their appeal to causal relationships. Of course, those concepts have been construed in various ways in different historical periods. Aristotle believed that general laws should be expressed in natural language, that deductive arguments always have a structure of a syllogism, and that there are four kinds of causes. The same concepts feature (in various configurations) in the contemporary debates pertaining to explanation, but they presuppose an altogether different theoretical background: the monadic logic of Aristotle is just a small part of the vast field covered by contemporary logic (classical and non-classical); general laws, at least in physics, are expressed in the language of mathematics; and our ontologies dispense with the Aristotelian four causes, presupposing a substantially different view of causality. All in all, we seem to

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be using the same type of canvas as Aristotle but utilizing different pigments, brushes and painting techniques.

References Aristotle (2009a). Physics. Translated by R. P. Hardie & R. K. Gaye, http:// classics.mit.edu/Aristotle/physics.html. Aristotle (2009b), Posterior analytics. Translated by G. R. G. Mure, http:// classics.mit.edu/Aristotle/posterior.html. Bergen, B. K. (2012). Louder than words. New York: Basic Books. Bonjour, L. (1985). The structure of empirical knowledge. Cambridge, MA: Harvard University Press. Brożek, B. (2013). Rule-following: From imitation to the normative mind. Kraków: Copernicus Center Press. Brożek, B. (2014). On nonfoundational reasoning, Zagadnienia Filozoficzne w Nauce, LVI, 5–28. Brożek, A., Brożek, B., & Stelmach, J. (2013), Fenomen normatywności. Kraków: Copernicus Center Press. Brożek, B., & Hohol, M. (2014). Umysł matematyczny. Kraków: Copernicus Center Press. Brożek, B. & Hohol, M. (2015). Language as a tool: An insight form cognitive science. Studia Humana, 4(2), 16–25 Burnyeat M. (1984), Aristotle on understanding knowledge. In. E. Berti (Ed.), Aristotle on science: the Posterior Analytics. Proceedings of the Eighth Symposium Aristotelicum (pp. 97–139). Padua: Editrice Antenore. Corry, L. (2004). David Hilbert and the axiomatization of physics. Dordrecht: Springer. Corry, L. (2006). The origin of Hilbert’s axiomatic method. In J. Renn (Ed.), The genesis of General Relativity, vol. 4. Dordrecht: Springer. Friedman, M. (1974). Explanation and scientific understanding. The Journal of Philosophy, 71(1), 5–19. Gadamer, H.-G. (2004). Truth and method. London: Continuum. Gottlieb, P. (2006). The practical syllogism. In R. Kraut (Ed.), The Blackwell guide to Aristotle’s Nicomachean Ethics. Malden & Oxford: Blackwell. Hempel, C. G. (1965). Aspects of scientific explanation and other essays. New York: The Free Press.

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Hempel, C. G., & Oppenheim, P. (1948). Studies in the logic of explanation. Philosophy of Science, 5, 135–175. Hilbert, D. (1902). Über den Satz von der Gleichheit der Basiswinkel im gleichschenkligen Dreieck. Proc. London Math. Soc., 35(50). Lakoff, G. & Johnson, M. (1999). Philosophy in the flesh. New York: Basic Books. Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books. Lesher, J. H. (2001). On Aristotelian ‘episteme’ as ‘understanding’. Ancient Philosophy, 21, 45–55. Leunissen, M. (2010). Explanation and teleology in Aristotle’s science of nature. Cambridge: Cambridge University Press. LSJ The Online Liddell-Scott-Jones Greek-English Lexicon (2011), http:// stephanus.tlg.uci.edu/lsj/. Peckhaus, V. (2003). The pragmatism of Hilbert’s programme. Synthese, 137(1–2), 141–156. Salmon, W. C. (1971). Statistical explanation. In W. C. Salmon (Ed.), Statistical explanation and statistical relevance. Pittsburgh: University of Pittsburgh Press. Salmon, W. C. (1989). Four decades of scientific explanation. In P. Kitcher & W. C. Salmon (Eds.), Scientific explanation (pp. 3–218). Minneapolis: Minnesota Studies in the Philosophy of Science. Salmon, W. C. (1990), Scientific explanation: Causation and unification. Critica, XXII(66), 3–23. Scriven, M. (1959). Truisms as the ground for historical explanations. In P. Gardiner (Ed.), Theories of history. New York: Free Press. Scriven, M. (1962). Explanations, predictions, and laws. In H. Feigl & G. Maxwell (Eds.), Scientific explanation, space and time (pp. 170– 230). Minneapolis: University of Minnesota Press. Wiener, H. (1890). Ueber Grundlagen und Aufbau der Geometrie. Jahresbericht der Deutschen Mathematiker-Vereinigung, 1, 45–48. Woodward, J. (2003). Making things happen: A theory of causal explanation. Oxford: Oxford University Press. Woodward, J. (2014). Scientific explanation. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/archives/ win2014/entries/scientific-explanation/. Zilsel, E. (2003). The genesis of the concept of physical laws. In E. Zilsel, The social origins of science (pp. 96–122). Dordrecht: Kluwer Academic Publishers.

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Jan Woleński Jagiellonian University University of Information, Technology and Management

Are Explanation and Prediction Symmetric?

The thesis that scientific explanation and prediction are mutually symmetrical (for brevity, SEP) appeared for the first time in Popper (1935, pp. 59–62) (page-reference to Popper, 1959). However, Popper rather discussed its connection with causal explanation. As it is well known, Popper’s Logik der Forschung had a highly polemic flavor. It strongly criticized the philosophy of science proposed by the Vienna Circle, especially its anti-realistic consequences. Schlick rejected the explanatory functions of science. He (and other early logical empiricists) argued that since explanation must appeal to causality, it inevitably falls into metaphysics. On the other hand, prediction is a fully legitimate function of scientific theories. This position required a special treatment of universal laws. Schlick understood them as inferential tickets, that is, schemes for formulating and expressing predictions. Consequently, universal laws are not propositions, but rather, logically speaking, propositional functions. Thereby, they do not describe anything and cannot be assessed as true or false, but rather as effective or ineffective in prediction. Hence, there is no need to invoke the concept of causality for analysis of scientific laws and theories. Popper radically opposed this approach to science. He was a realist in his philosophical heart and could not agree that scientific laws have no descriptive content. If so – he argued – we have to consider expla-

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nation as a legitimate business of science. The SEP becomes plausible under this view.1 One can observe that SEP in Popper’s hands is a by-product of a more general problem, namely the essence of scientific enterprise. It was Carl Hempel who made this thesis purely methodological and almost devoid of various metaphysical (or anti-metaphycical) ingredients (see essays collected in Hempel, 1965a, in particular Hempel, 1942; Hempel & Oppenheim, 1948; Hempel, 1958 and Hempel, 1965; all page-references are to Hempel, 1965a; Hempel alluded to the similarity of explanation and prediction in many other papers). Hempel formulated and analyzed SEP as the question of the logic of science, that is, so-called formal methodology. This approach considers science rather as a product than an activity. Typically, scientific products (theories, laws, hypotheses, etc.) are treated as linguistic entities (sentences, propositions, statements) or their ordered sets. Due to the strong use of formal logical tools in doing this kind of methodology, it is sometimes compared (see Ajdukiewicz, 1960) with metamathematics. Ajdukiewicz himself contrasted metamathematical and methodological approach to science. According to his view, the latter cannot abstract from scientists as agents of scientific investigations, their tasks, expectations and shared values related to performing research. The logic of science in the above understanding dominated the philosophy of science until the end of the 1950s and early 1960s and was mostly propagated by logical empiricists and their epigones. They assumed that the logic of science should offer a rational reconstruction of scientific enterprise. Other styles of doing the philosophy of science became popular in the 1960s and later.2 The attempts to give a rational reconstruction of science were accused of being at odds with the   Popper did not use the term “symmetry.” I will still return to terminological matters.   The dates are somehow tentative. Anticipations of the new philosophy of science appeared earlier, but the formal pattern of this field was (and still is) continued after the early 1960s. For example, Stegmüller, 1969 is a massive monograph scientific explanation and confirmation. In fact, it is a part of a voluminous series under the title Probleme und Resultate der Wissenschaftstheorie und analytischen Philosophie. 1 2

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practice of scientific research. New philosophers of science inspired by Kuhn, Feyerabend and Lakatos claimed that works in this field should be closer to the reality of actual investigations of physicists, biologists, etc. In particular, scientific activities (not scientific products) became the prevalent subject of philosophical analysis of science. This change of the style of philosophy of science has consequences for SEP, which is even not mentioned in recent textbooks of philosophy of science (see for example, Hung 2014; this is qualified as a “complete text on traditional problems and schools of thought” and has a chapter on Hempel’s conception of scientific explanations; Achinstein, 2000 is, according to my knowledge, the only recent paper in which SEP is discussed). However, I think that SEP still constitutes an interesting problem. It is my reason to recall it here.3 Hempel introduced SEP in 1942 (see Hempel, 1942, p. 234) in the following way: […] the logical structure of a scientific prediction is the same a that of scientific explanation […]. In particular, prediction no less than explanation throughout empirical science involves reference to universal empirical hypotheses. The customary distinction between explanation and prediction rests mainly on a pragmatic difference between the two: While in the case of an explanation the final event is known to have happened, and its determining conditions have to be sought, the situation is reversed in the case of a prediction: here, the initial conditions are given, and their “effect” – which, in the typical case, has not yet taken place – is to be determined. In view of the structural equality of explanation and prediction, it may be said that an explanation […] is not complete: If the final event can be derived from the initial conditions and universal hypotheses stated in the explanation, then it it might as well have been predicted, before actually happened, on the basis of a knowledge of the initial conditions and the general laws.   This paper uses some material from my article (Woleński, 1979).

3

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The label “structural equality of explanation and prediction” has appeared in Hempel (1942) for the first time. The term “symmetry” in this context appeared in Hempel (1965, p. 367); the name “the explanation/prediction symmetry thesis” occurs in Salmon (1989, p. 24). The analysis of explanation and prediction is continued in Hempel and Oppenheim (1948). In particular this important paper presents the deductive–nomological model of scientific explanation. This model has the following scheme: (*)

C1, C2, …, Ck L1, L2, …, Lr _____________ E

The symbols in (*) mean: C1, C2, …, Ck – initial conditions, L1, L2, …, Lr – universal laws (or hypotheses), E – a sentence about the explained fact. C1, C2, …, Ck plus L1, L2, …, Lr form the explanans (means to explain something), while E – the explanandum (what is to be explained). The vocabulary used in the theory of definition (definiens, definiendum) motivates the description of (*). Yet there is an ambiguity. Customarily we say about explaining facts by universal natural regularities.4 However, the analysis of (*) given by Hempel and Oppenheim assumes (they are clear about that) that C1, C2, …, Ck, L1, L2, …, Lr and E are linguistic entities (I will speak about sentences). The same is suggested by the mentioned analogy with the theory of definitions. Note that the conditions of adequacy for the nomological explanation stated by Hempel and Oppenheim (see below) are meaningless under the ontological reading of (*). 4   I neglect explanation of laws (or regularities). Explanations of facts via non-universal items will be mentioned in later parts of this paper. Also I do not enter, except making very general remarks, into the concept of universality of laws.

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Hempel and Oppenheim propose four necessary conditions of adequacy (that is, correctness) for deductive-nomological (D–N) explanation: (1) E must deductively derivable from the conjunction of sentences forming explanans. Symbolically: (C1  C2  …  Ck)  (L1  L2  …  Lr) ├ E; (2) The explanans must include general laws and initial conditions; the presence of both is indispensable for deriving E; (3) The explanans must possess empirical content; (4) The explanans must be true. The conditions (1) – (3) are termed as logical, and the condition (4) – as empirical. Here we have an interesting example of how the logic of science was understood in the Vienna Circle. Strictly speaking, (1) is the only condition deserved to be called logical, because the relation of derivability comes from logic. Being a general (universal) law is a methodological property, eventually having a definite logical form (general laws should be closed by the universal generalization). However, this formal requirement is essentially supplemented by such additions like, fo instance, that universal laws are independent of spatial and temporal coordinates. The concept of empirical content, fundamental for logical empiricism, was considered by representatives of this movement as directly derivable from “the logic of scientific discourse.” I used the commas because the idea of empirical content, independently of its various explications (verifiability, falsifiability, testability, definability in directly empirical language, etc.) employ several extralogical concepts. Simply speaking, the principle of empirical content (or the empirical meaningfulness) can be considered as logical only under a very wide understanding of logic. Hempel and Oppenheim continue (p. 249): Let us note here that the same formal analysis, including the four necessary conditions, applies to scientific pre- diction as well to explana-

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tion. The difference between the two is of pragmatic character. If E is given, i.e. if we know that the phenomenon described by E occurred and a suitable set of statements C1, C2, …, Ck, L1, L2, …, Lr is provided

afterwards, we speak of an explanation of the phenomenon in question. If the latter statements are given and E is derived prior to the oc-

currence phenomenon it describes, we speak on of a prediction. It may be said, therefore, tha an explanation of a particular event si not fully adequate unless its explanans, if taken account of in time, could have served as a basis for predicting the event in question. Consequently, whatever will be said in this article concerning the logical characteristics of explanation or prediction will be applicable to either, even if only one of them should be mentioned.

This quotation practically contains the same thoughts which occur in Hempel (1942) (see above), including some ambiguity of the concept of prediction as far as the issue concerns its temporal and epistemic aspects (I did not point out this problem on the occasion of quoting Hempel (1942) – I will later discuss this issue). Hempel and Oppenheim did not use a special name for, so to speak, logical equivalence of explanation and prediction. The phrase “the thesis of the structural identity (or of symmetry) of explanation and prediction” appeared in Hempel (1965, p. 367). Since Hempel employed it in the index in Hempel (1965a) as the reference guide for all places where (*) is applied to explanation and prediction, one can consider this vocabulary as final, at least in Hempel. The name “structural identity of ….” is very strong. As applied to the relation holding between explanation and prediction, it indicates that concrete explanations and predictions satisfy (*) as their logical scheme. In other words, both operations are logically indistinguishable. Independently whether (2), (3) and (4) can be conceived as purely logical (I guess that they could not) , they apply to explanation as well as to prediction. Hence, the validity of SEP is not threatened by extralogical ingredients in the conditions of adequacy stated by Hempel and Oppenheim.

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Hempel (1965) is a summary of his previous works on explanation. We find in this study some novelties. Hempel says: […] the thesis under discussion should be understood, of course, to refer to explanatory and predictive arguments. Thus construed, the thesis of structural identity amounts to the conjunction of two sub-teses, namely (i) that every adequate explanation is potentially a prediction in the sense indicated above; (ii) that conversely every adequate prediction is potentially an explanation.

Invoking the qualification of potentiality was caused by some criticisms of Hempel pointing out that predictions are expressed by single sentences. In order to meet this objection, Hempel stressed that we should distinguish predictive arguments and predictive sentences. For instance, I can say consistently with ordinary language that the sentence “Tomorrow will be raining” expresses a prediction. Hempel insists that we should consider predictive argumentations, in the given case, an inference of the statement about tomorrow raining from some premises. Interesting enough, nobody confuse E (a description of a given phenomenon to be explained) with explanatory argumentation. Thus, comparing explanation and prediction requires that both operations must be conceived as arguments (or modes of reasoning). Perhaps a more important objection motivating introducing potentiality concerns (4) (see Scheffler, 1959, p. 44ff and Stegmüller, 1969, pp. 158–161). The condition of truth looks as too strong. For instance, it is enough, particularly in the case of prediction, that the explanans is sufficiently justified. A weaker form of (4) says that the explanans is justified (confirmed, tested, etc.) or considered as true. However, the interpretation predictions as arguments does not liquidate all doubts. Consider Hempel’s quoted elucidations. He says (in Hempel, 1942) that “in the case of an explanation the final event is known to have happened, and its determining conditions have to be sought” but “the situation is reversed in the case of a prediction: here, the initial conditions are given, and their ‘effect’ – which, in the typi-

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cal case, has not yet taken place – is to be determined.” A similar point concerning the temporal difference between explanation and prediction occurs in Hempel & Oppenheim (1948). The explanans is formulated “afterwards” in the case of explanation, but if “E is derived prior to the occurrence of the phenomenon it describes, we speak on of a prediction.” In Hempel (1965, pp. 366–367) we read: […] scientific explanation (of the deductive-nomological kind) differs from scientific prediction not in logical structure, but in certain pragmatic aspects. In one case, the event described in the conclusion is known to have occurred, and suitable statement of general law and particular fact are sought to account for it; in the other, the latter statement are given and the statement about the event in question is derived from them before the time of its presumptive occurrence.

Two things are to be observed. Firstly, the comment concerning explanation employs epistemic language (“is known”) supplemented by an ontological vocabulary (“to have occurred,” that is to have existed), but the talk about prediction has no reference to knowledge but only to “before the time of its presumptive occurrence.” One can claim this jargon is affected by the ordinary concept of prediction as referring to future events. However, and it is the second point, scientists uses the term “prediction” also as referring to already existing phenomena. For example, it is customary to say that the general theory of relativity predicts that (a) gravity bends light rays, and (b) the frequency radiation varies with the gravitational potential (it follows from the equivalence of gravitational and inertial mass). Of course, we cannot say that both predicted phenomena did not occur before formulating the related E. This example shows that the temporal aspect of prediction is secondary. It happens that predictive arguments concern future events, but also they can forecast already existing facts. Hence, pragmatic circumstances are rather related to the scope of knowledge as an epistemic state than to occurence of explained or predicted phenomena. By the way, this observation sup-

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ports SEP, to some extent, because it shows that pragmatic circumstances of prediction and explanation are secondary.5 Several authors tried to formulate general as well as concrete counterexamples against SEP. They can be naturally divided into arguments against the subthesis (i) every adequate (that is satisfied all conditions of adequacy) explanation, and arguments against the subthesis (ii) every adequate prediction is a potential explanation; Hempel was entirely conscious of various difficulties and he himself divided counterexamples into those directed against (i) and those directed against (ii). General arguments typically point out that the D–N model has limited applications to some important kinds of scientific explanation, namely (the names should not be considered as mutually exclusive) statistical, inductive, functional, teleogical, appealing to emergency or genetic (not in the sense of biology, but genesis of something). For example, if the D–N model is inherently connected with so-called covering (that is, universal) laws, serious difficulties appear, when we try to define strictly universal genetic, functional, historical, emergent or teleogical laws. For example, Popper observed at the beginning of his scientific career that Darwin’s evolutionary theory and psychoanalysis are asymmetrical with respect to explanation and prediction, because they explain facts, even deductively, but do not lead to predictions. A more complete discussion of dubious cases requires an elaboration of many difficult and complex ideas, like, for example, statistical laws, inductive logic, emergent phenomena, genetic chains, historical regularities, functions (not necessarily in the mathematical sense) or purposive behavior; this collection can   Due to a wide (perhaps even too wide) understanding of the logic of science in the Hempel program, one could propose to limit SEP to the application of (*) without additional adequacy constraints. This view, although closer to the contemporary standard account of formal logic (mathematical) than the view of logical empiricism (shared by Hempel as well), is in my opinion too restrictive at least as the starting point of a discussion about the symmetry thesis. Now I can motivate my preference to speak rather about the symmetry thesis than the structural identity. By fiat, we can, of course, establish that both labels are equivalent, but the latter is much more challenging from the logical point of view. 5

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be easily increased by adding determinism, indeterminism, chance or causality. All these problems are surrounded by deep philosophical environments. Take, for instance, the problem of historical regularities. Their existence is a condition sine qua non for historical laws.6 Yet the status of history as a science (“science” is here understood as a counterpart of Wissenschaft in German) and the historical reality was hotly discussed in German philosophy in the second half of the 19th century (and later). The distinctions of Kulturwissenschaften (the humanities) and Naturwissenschaften (science in its customary English understanding) as well as idiographische Wissenschaften (consisting of descriptions of individual facts or sometimes historical generalizations) and nomothetische Wissenschathen (establishing universal, that is, nomological laws) became fundamental for the mentioned debates. If the (classical) humanities are only idiographic, they explain not by covering laws, but otherwise, for example, by tracking genetic chains. Under this view, history is not productive in forecasting the future or its possibilities are very limited in this respect. Most historians were satisfied with this understanding of history. On the other hand, sociologists and economists contested the idiographic model as inadequate for theoretical social sciences and leading to their degradation in rank. This example from the history of the philosophy of science shows that the scope of SEP has several links with the deepest problems of the philosophy of science. Let me illustrate the question of how the issues of explanation/ prediction are rooted in the environment of the philosophy of science by one example. I take so-called statistical explanation (SE) as the case, which is fairly instructive. Hempel did much work about SE and its problems. He introduced the concept of SE (see Hempel,   On the occasion, note that “regularity” belong to ontological vocabulary, but the term “scientific law” to the methodology of science. Hence, historical laws, if any, are linguistic items, but historical regularities not. Consequently, we should not speak about laws of nature, but about laws of science. Of course, these problems can be easily clarified or solved by adopting suitable linguistic conventions, but they should not be entirely neglected, because it happens that important intuitions are behind regulative definitions. 6

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1962; Hempel, 1965). SE falls under the following scheme (Hempel, 1965, p. 390): (**)

p(G/F) = r Fi ═════════ (r) Gi

where the symbol p(G/F) = r means that the statistical probability of F relative to G is equal to r, Fi refers to the case that i is F, Gi – to the case that i is G. Finally, the double indexed stroke ═════════ (r) symbolizes the inductive entailment having the degree r. The explanans in (**) consists of a statistical law ascribing the probabilistic measure to the frequency of items possessing a property F in a set G. Fi describes the initial condition, but Gi is the explanadum. Generally speaking, an explanation is statistical provided that in its explanans occurs at least one statistical law and this law is relevant for derivation of Gi. Clearly, (**) is a structurally similar replica of (*), although modified by introducing statistical concepts. Here is an example (see Hempel 1965, p. 388). I quote: Let the experiment D (more exactly, an experiment of kind D) consists in drawing, with subsequent replacement a ball from an urn containing 999 white balls and one black, all the same size and material. We might then accept the statistical hypothesis that with respect to outcomes “white ball” and “black ball,” D is a random experiment in which the probability of obtaining a white ball is p(W, D) = .999.

Consequently, we concretize (**) by (5) p(W, D) = .999 Dd ═════════ (.999) Wd

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Hempel concludes that (5) illustrates a good statistical explanation, because r is high (close to 1 in this case). Note on the occasion that we also have deductive statistical explanations (see Hempel, 1965, pp. 380–381). For instance, the argument that flipping of a fair coin yields the head-side with probability ½, provided that outcomes are statistically independent is deductive by all methodological standards of deduction. This example, as well as its general background, raises several doubts (they do not concern statistical deductive explanations,). First of all, it is unclear whether r in p(G/F) = r and r in ═════════ (r) mean the same. The former expresses statistical probability, but the latter – logical probability. Carnap in his inductive logic assumed that both kinds of probability, statistical and logical, satisfy the same abstract principles, in particular, Kolmogorov’s axiomatization of the probability (mathematical) theory. Hempel follows the Carnapian pattern of thinking about the logical probability. Leaving details aside, this option has very weak points, for instance, related to the logical probability of general hypotheses (see also below; let me remind that Popper rejected inductive logic just for its inability to copy with this problem) . Moreover, if Fi can be reasonably considered as the initial condition and an individual event, Gi does not admit such an interpretation, at least if statistical probability matters. In fact, the correct statistical reasoning about white and black balls concerns samples of balls, not individual balls. We can say that if we draw 10 balls at random, 10 of them will be white in almost all cases. In another example, one can additionally illuminate the problem. Assume that, according to medical research, x patients in the sample of y suffering from a sickness z recovers after taking the medicine t and that the related frequency is expressed by r. Consider the situation of a doctor d asked by a patient x’ “Is the probability of my recovering after taking t equal to r?” The correct answer should be “Not, the general law says that not about individual patients, but samples of them. Moreover, if a sample is more numerous, the percentage of recovering persons is closer to that expressed by r.” Furthermore, it is also unclear why cases with a

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small r are not explanatory. If r is small but quantitatively established, p(G/F) = r and Fi explain why the frequency of F’s among G’s is not high. The plausibility of (**) as a general scheme of successful explanations essentially depends on solving many questions pertained to inductive logic and extended use of statistical methods. Although some hopes are associated with the Bayes theorem and the subjective interpretation of probability as applied to individual cases, the present state of arts motivates a very skeptical position concerning the plausibility of the scheme (**). As far as the issue concerns concrete counterexamples, it is easy to conceive that they intend to show that there exist legitimate explanatory cases, which do not suffice for predictions (counterexamples with respect to the subthesis (i)) or that we can point out legitimate predictive cases, which have no explanatory power or their predictive strength appears as very small (counterexamples with respect to the subthesis (ii)). Three counterexamples against (i) were formulated in Scriven, 1959, 1959a and 1972 (see also Grünbaum, 1961, Chapter 9 for further examples, and Achinstein, 2000 for a discussion of the problem). Firstly consider the scheme of the individual (or singular) causation A is the only cause of B. Scriven illustrates (6) by Syphilis is the only cause of paresis. According to Scriven, if someone suffers from paresis, it can be well explained that this person previously got syphilis. On the other hand, syphilis is very rarely correlated with paresis, that we cannot soundly predict that a syphilitic person will suffer from paresis. Hence, because-sentences related to (5) (individual causal sentences) are effective tools for explanation, but their role in predictions is minimal. Scriven’s second example concerns the situation of killing not faith-

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ful wife by a very jealous husband. Although we know that the exceptional jealousy was the only cause of murdering and can explain the latter event by the state of psyche of the killer, it would be not rational to expect (to predict) that jealousy causes killing in most cases. The third example concerns the falling of a bridge. We can explain this event by point out that a fragment of a given bridge was damaged or by observing that too many heavy trucks simultaneously used the bridge or discovering the fatiguing of the metal. Every explanation (or its sum) is OK, but its predictive power is not very great, because we can know about the cause of the falling of the bridge just after of its catastrophe. Thus, Scriven concludes that not every explanation, even fairly successful, can be transformed into an effective predictions and, hence, not every explanation forms a potential prediction.. Hempel (see Hempel, 1965) commented Scriven’s counterexamples. In general, he argued that all three examples represent incomplete explanations. Explaining paresis via syphilis is incomplete because it does not explain why the correlation of both is so rare. Other two examples are, according to Hempel, incomplete because our psychological or technological knowledge is essentially limited and thereby causes difficulties in related predictions. Hempel also touches the evolution theory and observes that its combination with molecular genetics can increase the predictive strength. However, it is difficult to consider Hempel’s rejoinder of Scriven as something more than hopes. Clearly, we can improve our predictions in dubious cases. For instance, if we know that bridge is fragmentary damaged, that many heavy trucks approaching the bridge or knowing that metal was fatigued. The sum of this circumstances, if known, certainly, makes our prediction of falling the bridge more reasonable than every of separate basis. Similarly, if we know that A is jealous and impulsive, the prediction that he can kill his unfaithful wife is better grounded than in the case, when we know that he is jealous without information about his temperament. Although new information can be relevant for the quality of predictions, all counterexamples offered by Scriven justify the conclusion that explanation and prediction are not symmetric

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in this sense that the strength of the former is greater than the latter. Hempel also commented Popper’s mentioned objection concerning the theory of evolution. He pointed that supplementing Darwin’s classical theory by data coming from molecular genetics (it is proposed in the synthetic evolutionary theory), could lead to a better understanding of mechanisms of biological development and to increase predictive instruments. However, it is still a project, not a scientific reality. Hempel regards objections against the subthesis (ii) as more serious than against the first component of SEP. In particular, he comments the objection stating that genuine predictions, but based on a finite collection of data, has no predictive power. According to Hempel, this objection can be resolved by appealing to Carnap’s idea of inductive inference, which does not depend on universal laws. Indeed, Carnap in order to avoid the problem of logical probability of strictly universal hypotheses (r for them is equal to 0 or close to this value and no further empirical investigation can change this situation) construed inductive reasoning in such a way that universal sentences do not play any role in probabilistic inferences. Hempel observes that since prediction based on a finite set of data has as its outcome a statement about other sample, unknown at the moment of performing the argument in question, are counterparts of definite logical (inductive) principles, they can be qualified as general laws, however, not of the strictly universal character. Hempel seems to be conscious that invoking general laws without the mark of universality constitutes a proviso solution. In fact, it requires a weakening of (**). Anyway, Hempel agrees that the validity of (ii) is still an open question. He is right but his proviso is essentially at odds with the initial intuitions concerning SEP. In spite of many objections against SEP I see a possibility of defending it not only in the case o D–N explanations. SE looks as the crucial case. One possibility consists in conceiving statistical explanations as incomplete D–N explanations. This view is related to so-called hidden determinism as a key to interpretation of statistical physics. It became difficult to defend it after the rise of quantum mechanics, but some contemporary philosophers and physicists are its

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advocates. Most philosophers of science probably share the following evaluation (Nagel, 1961, p. 23): It is sometimes maintained that that probabilistic explanations are only temporary halfway stations on the road to the deductive ideal and do not therefore constitute a distinct type. All that need be done, so it has been suggested, it to replace the statistical assumption in the probabilistic explanations by a strictly universal statement […]. But, though the suggestion is not necessarily without merit and may be good for further inquiry, in point of fact, it is extremely difficult in many subject matters to assert with even moderate plausibility strictly universal laws that are not trivial and hence otiose. Often the best that can be established with some warrant is a statistical regularity. Accordingly, probabilistic explanations cannot be ignored, on pain of excluding from the discussion of the logic of explanation important areas of investigation.

If we agree that SE are not reducible to D–N explanations and hence indispensable in science, the problem whether that fall under SEP becomes crucial. The following path seems to the only possibility to argue that SEP holds also from SE. The scheme (**) should be replaced by erasing (r) from ═════════ (r). Consequently, the double stroke will refer to the ordinary (non-deductive, and thereby fallible) statistical inference (induction) and the entire (**) would be interpreted via statistical probability. Thus, p(G/F) = r symbolizes an established statistical regularity with respect to a relation between Gs and Fs (the problem whether it universal or not loses its importance), Fi is to be replaced by G/Fi as referring to a investigated sample and Fi, replaced by F/Gj as referring to a sample to be explained or predicted. Formally, it is covered by (***)

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The inference according to (***) allows to extend SEP to SE without additional problems. And we can say after Adolf Grünbaum (Grünbaum, 1961, p. 311) that “with respect to the symmetry thesis [at least as valid for theoretical systems, statistically-nomological or universally–nomological – J. W.] Hempel ab omni naevo vindicates.” Although SEP has under this proposal a smaller scope than in the case of Hempel’s original proposal, it seems to code an important methodological fact.7 It also contributes to a solution of an epistemic ambiguity of statistical reasoning. What about the situation that r = ½? It leads to the conclusion that p(G/non-F) = r as well. However, there is no paradox that frequencies of F and non-F are equal. Explanations and predictions of F as well as and its complement are equally admissible.

References Achinstein, P. (2000). The symmetry thesis. In J. A. Fetzer (Ed.), Science, explanation, and rationality. The philosophy of Carl G. Hempel (pp. 167–186). Oxford: Oxford University Press. Ajdukiewicz, K. (1960). Axiomatic systems from the methodological point of view. Studia Logica, IX, 205–218. Repr. in K. Ajdukiewicz (1978), The scientific world-perspective and other essays, 1931–1963 (pp. 282– 294). Dordrecht: Reidel.   It means that genetic, functional, teleological, etc. explanations do not fall under SEP. I do not deny that they can also serve as predictive bases, but their explanatory strength is greater than their predictive function. For example, we can genetically explain why changes in Poland and other European communist countries occurred around 1989, but their prediction was practically unrealistic in June of this year. One American political expert was asked at the end of June 1989 for his prediction of the further political development in Poland. It is important that this story happened after Polish election (June 4) in which the communists lost, but they still had a majority in the Polish parliament. The answer was this “Now you must be patient, because a long period of coexistence with communists must be expected.” One month later the first non-communist Prime-Minister (Tadeusz Mazowiecki) was appointed. Zbigniew Brzeziński was the mentioned American expert. Although he possessed all the accessible information relevant to diagnose the future situation in Poland, his prediction was dramatically incorrect. I tell this story not to blame Brzeziński, but to demonstrate how difficult political forecasts are. 7

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Ajdukiewicz, K. (1978). The scientific world-perspective and other essays, 1931–1963. Dordrecht: Reidel. Feigl, H., & Maxwell, G. (Eds.). (1962). Minnesota studies in the philosophy of science, vol. III. Minneapolis: University of Minnesota Press. Fetzer, J. A. (Ed.). (2000). Science, explanation, and rationality. The philosophy of Carl G. Hempel. Oxford: Oxford University Press. Gardiner, P. (Ed.). (1959). Theories of history. New York: Free Press. Grünbaum, A. (1973). Philosophical problems of space and time. Dordrecht: Reidel. Hempel, C. G. (1942). The function of general laws in history. The Journal of Philosophy, 35–48. Repr. in C. G. Hempel (1965a), Aspects of scientific explanation and other essays in the philosophy of science (pp. 231–245). New York: The Free Press. Hempel, C. G. (1958). The theoretician’s dilemma. A study in the logic of theory construction. In Feigl, Scriven, Maxwell, 1958, pp. 37–98. Repr. in Hempel, 1965a, pp. 173–226. Hempel, C. G. (1962). Deductive nomological vs. statistical explanation. In H. Feigl, & G. Maxwell (Eds.), Minnesota Studies in the Philosophy of Science, vol. III (pp. 98–169). Minneapolis: University of Minnesota Press. Repr. in C.G. Hempel (2000), The philosophy of Carl G. Hempel. Studies in science, explanation, and rationality (pp. 87–145). Oxford: Oxford University Press. Hempel, C. G. (1965). Aspects of scientific explanation. In C. G. Hempel, Aspects of scientific explanation and other essays in the philosophy of science (pp. 331–496). New York: The Free Press. Hempel, C. G. (1965a). Aspects of scientific explanation and other essays in the philosophy of science. New York: The Free Press. Hempel, C. G. (2000). The philosophy of Carl G. Hempel. Studies in science, explanation, and rationality. Oxford: Oxford University Press. Hempel, C. G., & Oppenhein, P. (1948). Studies in the logic of scientific explanation. Philosophy of Science, 15, 135–175. Repr. in C. G. Hempel (1965a), Aspects of scientific explanation and other essays in the philosophy of science (pp. 245–295) (with Postscript). New York: The Free Press. Hung, E. (2014). Philosophy of science. Complete. A text on traditional problems and schools of thought. Boston: Wadsworth. Nagel, E. (1961). The structure of science. Problems in the logic of scientific explanation. New York: Harcourt, Brace & World.

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Popper, K. (1935). Logik der Forschung. Wien: Springer. Eng. tr., Popper, 1959. Popper, K. (1959). The logic of scientific discovery. London: Hutchinson. Salmon, W. (1989). Four decades of scientific explanation. Minneapolis: University of Minnesota Press. Scheffler, I. (1969). The anatomy of inquiry. Philosophical studies in the theory of science. New York: Knopf. Scriven, M. (1959). Truisms as the grounds for histroical explanation. In P. Gardiner (Ed.), Theories of history (pp. 443–475). New York: Free Press. Scriven, M. (1959a). Explanation and prediction in evolutionary theory. Science, 139, 477–482. Scriven, M. (1962). Explanations, predixtions and laws. In H. Feigl & G. Maxwell (Eds.), Minnesota studies in the philosophy of science, vol. III, (pp. 170–230). Minneapolis: University of Minnesota Press. Stegmüller, W. (1969), Wissenschafliche Erklärung und Begründung. Berlin: Springer. Woleński, J. (1979). Wyjaśnianie i przewidywanie (Explanation and prediction). Prace Naukoznawcze i Prognostyczne, 3(24), 5–19. Repr. in J. Woleński (1996), W stronę logiki (pp. 251–265). Kraków: Aureus. Woleński, J. (1996). W stronę logiki (On the side of logic). Kraków: Aureus.

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Robert Audi University of Notre Dame

A Priori Explanation

Most of what we ordinarily want explained is empirical. It is often a singular phenomenon, say the collapse of a bridge. Sometimes it is general, as with the regular ebb and flow of the tides. But in logic, in pure mathematics, and in at least some of philosop­hy, there are a priori propositions in need of explanation, say that it is a mistake to believe that knowledge is simply justified true belief. For empirical cases, there is some reason to say that science is sovereign: not in the sense that its application is required for every explanation of an empirical phenomenon, but in the sense that by and large explanations of empirical phenomena are testable, or at least defeasible, by sufficient scientific information. This holds, at any rate, if a scientific explanation appropriately takes account of everyday propositions knowable on the basis of common-sense observations, such as observations of color and shape, mass and text­ure, and sounds and smells. Without the basic epistemic authority of perception – an authority that is central for empirical justification and empirical knowledge – scientific knowledge of the high-level kind so widely admired is not possible. Is science, however, similarly authoritative in the realm of a priori explanations? Are a priori explanations scientific at all? Here philosophical naturalists and other thinkers are divided. On one understanding of philosophical naturalism, truth by its very nature is ultimately amenable to scientific confirmation or disconfirmation, even if some truths do not belong to any particular science or even have any significance for scientific inquiry. Thus, genuine explanations must

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be considered broadly scientific. For those who reject philosophical natur­alism, there may be truths, and indeed explanations, that, even if they should be naturalistically analyzable, are not empirical and not appraisable using scientific method.

1. Framing the issue: Preliminary considerations My focus here will be on explaining why.1 Even with this restriction – which is appropriate to the main philosophical issues at stake in the theory of explanation – there is great diversity. The most evident is perhaps in mode of formulation: explanations can be given propositionally, say in writing; they may also be given orally or perhaps even gesturally, as where one points to a fallen tree to answer the question why someone was injured. There may be no semantic or epistemological differences among such explanatory appeals to propositions; but there are pragmatic differences. These will not concern this chapter, and my purposes will be adequately served by concentrating on explanations as a kind of propositional structure. In focusing on explaining why, I have in effect indicated that the targets of explanation I am concerned with are true propositions or facts conceived as expressible by true propositions (I do not claim an identity between facts and propositions). The propositions may be a priori or empirical, necessary or contingent, modal or non-modal, say of the forms of ‘necessarily, p’, or ‘It could be the case that p’, or ‘probably, p.’ I hasten to add that presupposing factivity in the target, hence that only what is the case can be explained, does not entail factivity in the explanans, the proposition(s) or fact(s) providing the explanation. Even if only what is true can be explained it does not follow that only what is true can explain. This is why we can say such things as that her explanation is better than his, even where the two expla  My position on many aspects of explanation in general is provided in (Audi, 2014), which also contains pertinent references to other work on this topic.

1

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nations are incompatible, so that at most one of them avoids a falsehood in at least one of its constituents. Indeed, where we speak of inference to the best explanation, we must allow for the possibility of falsehood in an explanans: an inference to the “best” explanation can yield falsehood in the explanatory statement(s) inferred. Even apart from this, if some explanations are better than others, it is to be expected that at least some that are not the best may have a false constituent. Such abductive contexts will not be considered in this chapter, though its results can be applied to them. If, for instance, there can be a priori explanations, then the search for the “best” one in a given case can be understood not to presuppose that only factive a priori explanations are genuine. If one prefers to use the term ‘presumptive explanations’ for explanatory statements which, however plausible they may be, have an explanans entailing a falsehood, my results will accommodate that terminology. One further question is whether a genuine explanation must be such as to explain the phenomenon in question to a comprehending inquirer – one who understands both the target and the discourse containing the explanans. I doubt this, on the ground that this semantic understanding does not guarantee actually understanding why p, where p is the target, even if, under appropriate conditions, it does entail sufficiency for understanding why p is the case. What can be said here is that if an explanation, E, of why p is the case, is an explanation for S (a particular person), then if it is sound the person understands why p. But not every sound explanation offered to us is an explanation for us. This will become clearer in the discussion of understanding below.

2. A priori explanation, proof, and theoretical method The expression ‘a priori explanation’ is not a standard term, though any plausible understanding of it would represent it as entailing an explanation having some a priori element. Clearly, an explanation

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can have more than one a priori element. I will call it wholly a priori provided both the explanans and the explanandum are a priori. Thus, in logic we might explain why p is a theorem or, in pure math, why the even numbers are (in a certain sense) as numerous as the integers. (They correspond to the integers one to one, even though the set of the former may seem intuitively more “dense” than the set of the latter when judged by the number of unitary versus even elements in any finite sequence of integers.)2 We can also explain, however, why the latter seems false, at least to some people; and that target of explanation is empirical, as will be at least one proposition in the explanation, say that people confuse one-one correspondence with density in the sets of numbers in question, though the entire explanation will also have a statement of the a priori mathematical truth that appears false, for instance that equi-numerous sequences can be represented by differentially dense subsets. Is there any brief way to characterize the a priori for our purposes? I believe it will suffice to conceive a priori propositions as, in the basic case, self-evident and, in non-basic cases, either selfevidently entailed by a self-evident proposition or provable in the strong sense of being derivable from a self-evident proposition by a series of self-evident steps. The conception of self-evidence I have in mind here is that of a truth such that (1) in virtue of adequately understanding it one is justified in believing it (whether one does or not) and (2), if one believes it on the basis of an adequate understanding of it, then one knows it.3 This conception of the a priori does not automatically yield an account of (prima facie) justification or a priori knowledge, but it can do this if we supplement it with plausible assumptions.4 For our pur  As wholly a priori explanations are described here, they may be inductive as well as deductive. It seems appropriate to leave this open, if only because it could in some cases be a priori that the explanandum is probable given the explanans. 3   For explanation and defense of this view, with various references to relevant literature, see (Audi, 1999). 4   This view is defended in (Audi, 2010, chs, 5–6) 2

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poses, however, the notion of explanation and a priori cases of it is central. I am not here attempting to explicate the concept of explanational knowledge: knowing an explanation of something in a sense that entails knowing why the explanandum holds. The present concern, which is with what constitutes a priori explanation, is apparently prior. Even if it is not prior, my aim (in part) is to clarify know­ ledge why by appeal to the notion of understanding why rather than the other way around. Explanation and proof Logically as opposed to pragmatically, explanations have been taken to be arguments. A major reason for this view is that appealing to the explanans in a sound explanation is conceptualizable as a way of show­ing the truth of the explanandum. A paradigm of showing is proving, and the argumental conception of explanation suggests that a priori explanations, at least, might be seen as proofs. Let us explore this idea. Arguably, every genuine proof in logic or pure mathematics is wholly a priori, though I do not see how to prove that higher-order thesis and will not provide reasons beyond those implicit in the overall discussion to follow. Are there any a priori explanations in philosophy, and might any count as proofs? (I take it as clear and probably uncontroversial that not every sound proof is an explanation.) Is it a priori, for instance, that knowledge is not simply justified true belief? And if so, is this both explained and proved by noting that someone could (1) have a justified true belief that it is noon, based on correctly reading a stopped clock that the person justifiedly takes to be reliable, yet (2) not know that it is noon, though (as the clock says) it happens to be exactly that time? I see no reason to answer negatively. The explanation seems both to answer the question why knowledge is not simply justified true belief and to prove that from a priori propositions (I take the statement of the counterexample to be a priori and true).

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To be sure, proof, at least in philosophy, is an epistemic notion, in the sense that to prove p by appeal to q requires that q be both true and knowable, indeed knowable in a certain way.5 The premise(s) need not in fact be known – at least if we allow that someone could prove that p by citing q when simply hypothesizing q, where q is known to someone else or at any rate knowable in the right way. To deny this would imply that one could not in this way unknowingly prove something; but such possibilities are surely to be allowed (and likely historically instantiated). Some would say that q must even be “certain” to serve as a premise in a genuine proof. Insofar as one finds this certainty requirement plausible, one is likely to doubt that genuine proof is possible at all in philosophy. I see no reason to endorse this requirement – though we might plausibly do so for the subclass of proofs naturally called “rigorous” (and usually formal). There is also a question whether we should relativize the notion of knowability, and correspondingly the notion of proof, so that q is a proof of p only for someone who can know that q (and come to know that p on the basis of q. These complexities go beyond our concerns here (the notion of proof can easily occupy an entire paper). With or without relativization, only a certain kind of strong skeptic would deny that there can be some kind of proof in philosophy. Strong naturalists, to be sure, will deny that there are either a priori explanations or a priori proofs, but I am taking our examples (together with previous work, such as some cited in my Epistemology) to make it reasonable to believe that there are such explanations and proofs. If the notions of proof and, correspondingly, provability are taken in a technical sense in which any theorem of a (non-defective) formal system is provable, then the common (and arguably standard) epi­ stemic sense of ‘proof’ is inapplicable. Moreover, trivially, any (formally) logically necessary proposition is provable, in the sense that it   For a recent informative, non-technical discussion of mathematical proof, see (Murawski, 2014). His distinction between formal and informal proof is particularly relevant to the notions of proof common in philosophy.

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is formally derivable from any set of premises, yet it does not follow that it is knowable on the basis of the premises of the proof.6 They need not count as any kind of evidence for it or may have nothing to do with it even in content. Indeed, the validity of the proof may be based on the (formal) necessity of its conclusion, as opposed to the possible knowledge of the conclusion’s being potentially based on knowledge of entailing premises. Thus, even if theoremhood entails provability, it does not entail knowability on the basis of a proof.7 There is also a point here concerning explanation: a proof in the formal sense in question need not have explanatory power. Where the premises have no contentual connection to a logically true conclusion, they provide neither evidence for it nor any understanding of why it is the case. If, e.g., the explanandum is a formal truth, it is provable from any set of premises regardless of contentual relations. This kind of “proof,” even if it happens to have true premises, contrasts with the proofs in mathematics and logic that do have some explanatory power regarding their conclusion. Explanation, theory, and scientific method Can a priori explanations of any kind be considered scientific? Certain­ ­ly they are not as such unscientific, in the sense that they show ignorance of or disregard for, any method or result of empirical science. Indeed, scientific inquiry presupposes such mathematical and logical propositions as admit of a priori explanation, or proof (or both), along   That the system be a formal one is crucial here. A formally necessary proposition, say that p v not-p, can easily be proved by deriving a contradiction from its negation. But in a non-formal system one might not be able to prove that nothing is round and square (I assume that this is non-formally necessary); proving this would apparently depend on what non-formal premises are allowable. 7   This assumes that in a formal sense every formally logically necessary proposition is a theorem of some formal system (or at least assuming there is one whose language enables its formulation). Note that I am not taking every metaphysically necessary proposition to be provable in the formal sense. If it is synthetic a priori that nothing is round and square, we can see that this is strictly implied by any proposition, but that would not imply our being able to provide a formal proof of it. 6

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the lines I have described. We might also ask what is the status of the following principles, plainly presupposed in any kind of intellectual inquiry, scientific or other: ‘Apparent counterexamples disconfirm the relevant generalizations’ or, to take a less problematic counterpart, the proposition that if there is something that is both F and not G, then it is false that all Fs are Gs?8 This is surely a priori, even self-evident. Could such propositions be empirical in a sense requiring an ultimate basis in non-intellective experience such as sense-perception? Imag­ine trying to disconfirm 7 + 5 = 12 by finding counterinstances in physical addition processes, such as counting five nails, next seven more, then combining the two groups and counting the whole group. One apparent counterinstance would not suffice; to frame a good negative case, we would at least need to rely on counting the “counter­ instances,” thus presupposing that, say 1 + 1 + 1 = 3 (and also the reliabil­ity of perception and, likely, of memory). In this context it should be useful to explore why some may doubt whether there is genuinely a priori explanation. In my view, philosophy in at least the English-speaking world has for some decades been dominated to a significant degree by two (sometimes tacit) beliefs: first, a belief in scientific method as the paradigm of an objective, rational method of seeking truth; secondly, by an associated belief, or presupposition, that philoso­phy must, in methodology as well as doctrine, either be viewed as a branch of science or at least take account of the prog­ress of science. This is not to say that the (or a) method of science, or some interpretation of scientific method, has become the dominant philosophical method, though what is now called “experimental philosophy” is perhaps evolving in that direction.9 In any case, there is a widely held assumption – which I’ll call the assumption of the philosophical primacy of scientific method – ­that scientific method is the best model of the rational pursuit of truth, in a sense implying   The first principle is problematic because ‘apparent’ must have an elusive epistemic reading of a certain kind or there will be exceptions. 9   For a general explanation of and case for the value of experimental philosophy, see (Knobe & Nichols, 2008). 8

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both that one’s philosophical method, if not itself scientific, should bear an appropriate resemblance to scientific method, and that one’s philosophical results are probably mistaken if they are at odds with, or unable to account for the possibility of, well-established scienti­fic findings. Let me elaborate on this primacy assumption by suggesting some of its apparent implications in the major areas I am focusing on: epistemology, philosophy of action, metaphysics, and methodology.10 My main point here is that we must carefully distinguish scientific method from something of which it is an immensely impressive special case: theoretical method. The former is empirical and, broadly speaking, experimental. The latter is the more general method of building and rebuilding theories in rela­tion to data: roughly, raising questions, hypothesizing, comparing and evaluating hypotheses in relation to data, revising theories in the light of the com­parisons and evaluations, and adopting theories through assessing com­peting accounts of the same or similar problems. This distinction has not always been recognized or fully appreciated. For one thing, if, like most philosophers and scientists, we have empiricist leanings, they would have us see scientific method as the only kind of theoretical method, at least outside logic and mathematics. But the­oretical method is not the property of empiricism; rationalists can also use it, and so can both non-philosophers and philosophers uncommitted with respect to, say, empiricism, rationalism, and pragmatism. The theoretical method is very old in philosophy, as ancient as systematic philosophy itself, as witnessed by, say, Plato’s dialogues. It is, or is a major part of, the best general method in philosophy; and it is likely to remain with us as long as theoretical and, especially,   In the next few paragraphs I draw on (Audi, 1987). What I call theoretical method is quite similar to the method of reflective equilibrium as helpfully described by (Scanlon, 2014). There is dispute about whether the latter method is coherentist rather than, as I would argue, moderately foundationalist. The most important points about each method are neutral on that matter. But notice this. The phrase ‘considered judgments’ apparently implies something like initial credibility; and a coherentism that countenances that has a  significant foundationalist element. Initial credibility allows for intuitive non-inferential judgments. 10

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philosophical, reflection continues. On the other hand, there are signs that the assumption of the primacy of scientific method, and with it the often tacit view that scientific meth­od is the only rational theoretical method outside logic and mathematics, are being reassessed and may be weakening. If this chapter on the right track, we must not take the prominence of scientific methods and explanations as paradigms to give the impression that they are the only instances of theoretical methods and of genuine explanations.

3. A priori elements in empirical explanation The term ‘empirical explanation’ may be no more common than ‘a priori explanation.’ I take empirical explanations to be those whose target – the apparent fact to be explained – is empirical. The question here is not what principles govern empirical explanations. Consider what has been called a “salience principle”: If F is a member of a homogeneous partition of G and G was bound to obtain, then F does not call for explanation.”11 If, for example, autumn leaves falling from a tree form a light cover on the grass below, that phenomenon does not call for explanation, whereas their falling into a pattern that represents two poetic lines would cry out for explanation. The salience principle is apparently a priori; the leaves’ falling into a poetic expression would call for empirical explanation. Could an empirical explanation have a priori premises? It could have one – thereby having an a priori element – but the term naturally applies only where the entire explanans is empirical. Take an example. One might be puzzled about why I think a friend is justified (to some degree) in believing that there are two tree stumps some 100 meters from us, when you and I are sure we see only one in the same place. A satisfactory answer might be this. Though he does not have 11   See White, 2005, p. 3. This paper contains much discussion of both kinds of explanation and principles governing explanatory practice.

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any inkling of it, he has recently been beset by a vi­sual defect that, under these conditions, causes him to “see double”; so he has as vivid an impression of two stumps as we do of one, and (with no counterconsiderations affecting us) we are all justified in beliefs held on the basis of clear visual experience. This explanation appeals both to an epistemic principle that seems a priori and to empirical propositions indicating that a believer is subsumable under the principle. A similar if more controversial case concerns mentalistic explanations of action. Consider asking why someone sold a stock a mere week after buying it. An answer might be that he wanted to make a quick financial gain and believed that selling it then was necessary for doing that. Here both explanans and explanandum are empirical; but the psychological principle connecting them is arguably not: it is roughly that when a normal agent, S, wants something, G, (on balance), has the ability and opportunity to A, and believes that A-ing is required to realize G, S tends to A. This principle may be argued to be a priori, given an appropriate specification of agential normality, ability and opportunity, and perhaps other variables. It may be that the most one could show is that even with such specifications, only a tendency to A follows, but that might still allow one to regard the proposition as a priori. One would then have to consider the explanation inductive, since S’s A-ing is of course not entailed by S’s tending to A.12 Here is a different kind of reason to think that there is an a priori element in certain explanations. There are surely deductive explanations – those in which the connection between explanans and explanandum is, in the broad sense, logically necessary. Simple if idealized cases would be subsumptive ones, such as ‘The piston rose because the gas locked in the chamber housing the piston was heated, and whenever that happens, the gas expands and raises the piston.’   I have defended these and other points about the psychological generalization in question in some detail; first in (Audi, 1973) and later in various papers, some also reprinted in (Audi, 1993). One challenge here is to explain how to square the apparent testability of the generalization with its status as partly constitutive of the concept of wanting and, on that count, as apparently in some sense a priori. 12

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Here the entailment is formal and is plainly a priori. But must an a priori entailment connecting explanans and explanandum be formal? I doubt that. Consider someone who has no experience with mechanics but does know some elementary geometry. The person asks, of a round pipe, ‘Why won’t it fill the square hole of the same width that we need to plug?’ Here is an explanation (assuming rigidity and other physical properties). Let the diameter of the pipe be D, say 2 inches, and the side of the square of the same width also be D. we can explain why the pipe cannot smoothly enter the square, i.e. completely fill its area. The square’s area is 4 (D squared). Where D is 2, we can see that the area of a cross-section of the pipe is pi times the radius (which is 1) squared, far less than 4 and self-evidently smaller than 2 times 2. It is also self-evident that a figure contained within another cannot wholly fill the space of the latter if its area is smaller than that of the latter. Here we have self-evident propositions that do not seem to be provable from logical premises alone and are not formal truths.

4. Explanation and understanding So far, my concern has been to show how an explanation can be a priori in whole or in part: directly a priori where an element in the explanans or explanandum is a priori; indirectly a priori where the explanatory relation between them is a priori (as might be required to meet the criteria for some kinds of explanation). Another aspect of the theory of explanation is left open by the points so far stressed: its connection with understanding. If this connection is a priori, i.e., if it is an a priori truth that a genuine explanation must provide a basis for understanding its explanandum, then we might say that one of the constraints – indeed, one of the partly constitutive conditions – on successful explanations is a priori. It does appear that a genuine explanation must produce understanding or at least be a potential basis of it, in the sense that a comprehending acceptance of the explanation by an appropriate recipi-

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ent would yield understanding. Is meeting this condition empirical? I believe it is not. Imagine someone’s saying such things as: I understand your explanation why p and I accept it as sound, but I still have no understanding of why p. If this means ‘I don’t feel I understand why p’, we can allow it. But the question of what yields a sense of understanding, as opposed to actual understanding, is not at issue here. On the matter of actual understanding, note that the person explaining why p could, in telling the other why p, give a kind of correct account of why p. Here, if I believe q, the explanans, and see that it constitutes a sound explanation of why p, I then know why p if I believe p on this basis, even if I still feel puzzled about p. To be sure, if there are mistaken explanations that are nonetheless genuine explanations or if there are genuine explanations that are not “best” among the alternatives, then we must grant that some explanations provide more (or a better basis for) understanding than others. But granting this is qualifying rather rejecting the thesis that providing understanding is an a priori condition for genuine explanation. Can one, however, know why p is the case and not understand why it is so? To be sure, there is perhaps a kind of understanding of p itself the lack of which is compatible with understanding why it is the case. Perhaps someone could understand why one person, say, a mother, loves another, say, her child, yet not understand love at all well, hence not well understand the proposition that she loves the child. But if this holds, it is not really an objection: we can simply say that, first, a sound explanation of a proposition p by another proposition q must provide, in an appropriate recipient, a certain minimal level of understanding of its explandum, and, second, the minimal level may be low relative to the richness of the content of the explanandum, but must be such that the recipient understands the explanatory statement p because q. The distinction just made between actual understanding and feeling one understands is an indication that understanding is not entirely

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a phenomenal element nor subjective.13 One point here is surely that understanding is connected essentially with inference. Not just any inferences are relevant: to infer that p or q from p is possible without understanding p (as opposed to presupposing that this disjunction is truth-valued). But to infer or be disposed to infer propositions that indicate constitutive elements in p does tend to reveal understanding of p. If the proposition is that Natasza is highly rational, understanding it should lead to a disposition to infer that she does not hold obviously inconsistent sets of beliefs, that she can readily see certain self-evident consequences of what she believes, that she cognitively responds to deliverances of her senses, and much more. Understanding also manifests itself in rejecting certain propositions: Natasza will tend not to accept sweeping assertions that are obviously incompatible, or even just probabilistically incompatible, with what she believes; she will detect poor arguments; she will tend to form beliefs in response to evidences presented to her, and more. These points about understanding are merely suggestive, but they help to show why not every (sound) proof provides understanding of what is proved, relative to any given level of understanding – the level appropriate to perfect omniscience is not in question here. If we assume that formal proof is closed under substitution of logical equivalents of any of its premises, it could employ unintuitive equivalents of intuitive premises and, if the explanandum is only minimally understood to begin with, the explanans may do little or nothing to enhance the kinds of inferential and other tendencies that partly constitute understanding. The different explanans may indeed reduce understanding. Two points will add clarity here. First, if we do not keep the level of understanding fixed in making such comparisons, then the substitution of one or more equivalents might not eliminate understanding; for even if a premise is not intuitively equivalent to one it supplants in 13  The theory of understanding is a large topic and only a few points can be made here. For general discussion that takes account of much of the relevant literature, see (Grimm, 2011).

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a proof, a person of great enough understanding might see the equivalence and might, as with the original proof, still gain an understanding of why the conclusion holds – even if aware that perspicuity would be better served by avoiding the substitution. Second, regardless of the understanding, proof requires retaining the exact explanandum in question: if that is changed we do not have an equivalent proof but at best a proof of an equivalent.14 Granted, when proof is understood epistemically, no such closure holds. Knowledge and other propositional attitudes are not closed under substitution of logical equivalents. Hence, there is no good reason to expect proof in the epistemic sense to survive substitution of just any logical equivalent of a premise. We would not have an equivalent proof, in the epistemic sense of ‘proof.’ There may of course be many proofs epistemically adequate to show that p; but they will not be epistemically equivalent unless their premises are interchangeable salva veritate in cognitive contexts such as those of beliefs of propositions. It should also be noted that there are different kinds of explanational capacities that go with understanding. If I understand why p and realize that p holds because q does, I know why p and have an explicit understanding of why p is the case. But suppose I know that p and see what explains it or, perhaps not quite equivalent to this, see why what explains it holds. I might, for instance, observe someone looking unhappy on being introduced as a speaker and see why this is so, in the sense that I am aware of the person’s accomplishments being subtly minimized. Here I need not believe that this fact explains the unhappiness; indeed, I need not even believe that the introducer is minimizing the accomplishments. Perceiving a phenomenon does not entail believing that it obtains, much less believing particular propositions about it. Still, if asked why the speaker was unhappy I will   I  am assuming that propositions are individuated finely enough so that logically equivalent propositions need not be identical. I leave open here whether facts are individuated this finely; but facts under a description presumably are; and these might be argued to be what is explained where the explanandum is a fact. For those, closure under logical equivalence would not hold. 14

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come to believe, and to know, that the accomplishments were minimized and that this explains the unhappiness. Call the understanding I have before considering the matter structural understanding. This occurs where one’s cognitive system contains information that, provided one’s conceptual capacities suffice to understand an explanation citing that information, yields – given an appropriate stimulus – both explanatory knowledge and explicit understanding of the phenomenon explained by that information. The distinction between structural and explicit understanding is neutral with respect to kind of explanation but is most commonly applicable in simple cases in which a state of affairs is explainable in terms of causative factors. Among the clear examples of these explanations are some that subsume a phenomenon under a universal generalization, as with explaining why the volume of gas whose size determines a piston’s vertical movement expands by saying that the gas was heated and gases expand when heated. This explanation makes use of apparent causal connections but it also proceeds by representing gasses as a subcategory of things that expand when heated. Such categoriality makes subsumptive reasoning possible. Does categoriality play a special role in understanding? It does in certain cases. One is that in which the subcategory is a priori included in the containing one. This holds with the category of those who avoid belief of obviously contradictory propositions being within that of those who are rational. Another case of categorical subsumption is being a causal consequence; the category of gases, for instance, is included that of things that expand when heated. Categoriality, whether a priori or empirical, makes subsumptive reasoning natural. Subsumptive reasoning, in turn, is highly intuitive: the corresponding conditional of the cited explanation of expansion of a gas is luminously self-evident, where the corresponding conditional of an argument is simply the conditional whose antecedent contains the conjunction of the premises (in the order of their occurrence in the argument) and whose consequent is the conclusion. The conditional represents an AAA syllogism in the first figure – BARBARA – the

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form of the beloved ‘Socrates is mortal’ example that has been cited in the teaching of logic perhaps since ancient times.15 A great deal might be said about the connection between categoriality and understanding, but here I must be content with just one further point. The idea bears both on subsumptive explanations and on other, less simple kinds. It concerns a property that is relative to a level of mental capacity but, I believe, important at any level: thinkability, the possibility of being focally held before the mind at a single time. If the corresponding conditional of a syllogistically valid explanation is thinkable then, other things equal, we tend to understand the explanation better. Perhaps we only feel that we do, but I would resist that minimal interpretation of the data. Note that the explanatory conditional will be not only self-evident, but immediately so, meaning roughly that we can see its truth at a mental glance: we need not reflect, as we do with some self-evident propositions, say that first cousins share a pair of grandparents; nor need we depend on memory, which, as skeptics will remind us, is fallible. Moreover, if a subsumptive explanation has explanatory premises so lengthy or complex that we must depend on memory to see its meaning (or that of its corresponding conditional), we are at least not likely to have the sense of understanding. This may be a subjective matter, but it may bear on our inferential tendencies in a significant way. The connection of those with understanding is not a subjective matter.16 Both categoriality and thinkability bear on simplicity. I would speculate that both count toward simplicity, though they are of course not the only factors that do. If a preference for simpler explanations, other things equal, is justified, that might be a matter of the accessibility of thinkable explanations and the intuitive character of subsump  The categorical syllogism with three A propositions in the first figure (in Barbara) – as illustrated by ‘All humans are mortal; Socrates is human; hence Socrates is mortal’ – is so intuitive and so commonly instantiated in philosophy that an outside observer might think that, no matter whom they are married to, philosophers are in love with Barbara. 16   Thinkability may be usefully compared with what Thomas Tymoczko calls surveyability. See his (Tymoczko, 1979). Surveyability and its relation to proof are also discussed by Murawski (2014, pp. 47–50). 15

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tive ones as manifested in the self-evidence of their corresponding conditional. Similar considerations apply to preferences for simpler proofs, which confirms the view that, as I have stressed, explanation, proof, and apriority are importantly connected.17

5. Explanation of normative propositions I shall suppose with intuitionist moral philosophers that there are a priori moral propositions. One is the principle that lying is prima facie wrong, which we may here take as equivalent to the proposition that there is a moral reason to avoid lying. Since reasons may be overridden by counter-reasons, the second formulation, which is framed in terms of reasons, takes appropriate account of the prima facie qualification. Many demand an explanation of the truth, and certainly the apriority, of these propositions. I have addressed these matters else­where. Here I am concerned with the possibility that such a priori principles are explainable. A natural thought is that the categorical imperative will suffice – Kant himself treats some of the moral propositions in question as derivable from it.18 To be sure, one might need to do some regimentation to achieve a valid deduction, for instance to suppose that intentionally breaking promises to people tends to fail to treat them as ends and that all (intentional) failures to treat people as ends are prima facie wrong. These premises together entail that intentional promisebreaking is prima facie wrong. A further point is that the categorical imperative is not clearly a priori (if it is a priori) and, even if it is, not self-evident. Does this matter?   Simplicity has been a major topic in philosophy of science and indeed in much of philosophy at least since Aquinas, and, in other ways, at least since Descartes and Leibniz. Richard Swinburne is among the contemporary philosophers who make much of its epistemic significance (see, e.g., Swinburne, 2004; 2011, esp. 6–11). 18   This is implicit in his treatment of his four famous examples in sec. 2 the Groundwork. For confirmation of this deductivist reading (see, e.g. Wood, 2007). 17

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Can an a priori proposition be explained only by one that is self-evident or at least as readily known? Consider the square peg case. Are we not here explaining something by appeal to something else less readily known than the proposition to be explained? Perhaps the point is not obvious here, but there is at least some reason to regard as self-evident the no-fit proposition about square pegs and round holes with their diameters of equal length with the sides of the squares, and no good reason to view as self-evident the geometrical theorems needed for proving the proposition. In any case, let us proceed to a normative case. Consider a different subsumptive explanation of why lying is wrong (for simplicity I assume here that ‘wrong’ means ‘prima facie wrong’): lying ill-befits the communicative relation; there is reason to avoid whatever does that; hence, there is reason to avoid lying. Note that the major premise here, if self-evident at all, is not clearly “better known” (strictly, more readily knowable) than what is to be explained by subsumptive appeal to it. This case is useful in addition because it illustrates how a high degree of vagueness, as with ‘ill-befits’, does not necessarily divest a proposition of explanatory power. Given what has emerged in this section, then, we have in effect answered two important questions. Must sound explanations of self-evident propositions have self-evident explanantia? and Is the self-evident not explainable? To the first, we have seen that the answer may be negat­ ive, though even if it is not, something close to the self-evidence requirement may hold: it may be that the explanans must be a priori (as perhaps with certain fittingness claims), hence (on my view) at least provable by appeal to what is self-evident. On the second question, we can be more confident: self-evidence does not preclude explainability or indeed explanation having a self-evident corresponding conditional.19   This section is intended to go partway toward allaying Philip Stratton-Lake’s pessimism: “I cannot offer a positive characterisation of what sort of explanation recommenders [such as the fact that one promised to A] provide of deontic facts (though I take some comfort from the fact that no one else has any idea either (Lake, 2011)). My “Skepticism about the a priori” (Audi, 2008) sketches the ontology of the kinds of a priori propositions that figure in the kinds a priori explanations illustrated above. For extensive further discussion of the epistemology and ontology of such propositions, see (Chudnoff, 2013). 19

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6. Explanatory ultimacy and the grounding of explanatory facts Does everything true admit of explanation? (I mean, explanation why, not what or any other kind.) There is some reason to think not, as with propositions such as that everything is identical to itself and that no proposition is both true and false. It seems reasonable to think that p admits of explanation only if it makes sense to wonder why p. Making sense here is a matter of conceptual appropriateness, not mere semantic intelligibility, as applies to any linguistic expression admitting of correct and incorrect translation. If it does not make sense to wonder why p, an explanation of why p would apparently not make sense either – it could not qualify as an answer to a genuine why-question.20 This rationale would go with the idea that an explanation should be a basis for yielding understanding. There are propositions that are explanationally self-sufficient. This is not to say they are self-explanatory, though it may be one interpretation of that phrase; it is to say that understanding them indicates, in itself, why they are so. Perhaps explanational self-sufficiency is based on another property: a proposition’s being comprehensionally compelling – such that one cannot understand it without seeing that and, in some sense, how it is true. Perhaps such a proposition would also not admit of explanation. The point is not that one cannot wonder why p if one needs an explanation of what p says – though that is true. The point is that there are some propositions, such as those just cited, such that seeing what they say suffices for seeing enough regarding their content to rule out genuinely wondering why they are so. Could a person wonder why a particular thing is not a different thing unless the person did not really see what self-identity is?   The point here is not the same as the point that if we, or some particular person, cannot intelligibly wonder why p, then we (or that person) cannot understand an explanation why p – I take that to be person-relative and a pragmatic point. 20

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It may seem that the point is this: that what does not admit of explanation is thereby self-explanatory. To be sure, even if some propositions are self-explanatory, it would not follow that they do not admit of explanation in terms of something else. This would be a kind of explanational overdetermination. But one might think that what does not admit of explanation is simply the self-explanatory. I doubt this, in part but only in part because I find the notion of the self-explanatory unclear. In my view, ‘self-explanatory’ is a misleading term in suggesting that something explains itself. Taking ‘selfexplanatory’ literally, applying it would commit one to the thesis that self-explanatory propositions are themselves what explains why they are true, where this thesis would entail countenancing as explanations some statements of the form of p because p. The inadmissibility of such claims as explanations should indicate why explanation is properly considered irreflexive. I believe that what we might loosely want to call self-explanatory is either (a) both self-evident and readily understood or (b) otherwise not in need of explanation for anyone who has the concepts in question or (c) where something is said to be self-explanatory in a context, not in need of explanation, in that context, for anyone having the concepts in question. But this alone does not entail not admitting of explanation. It may be, however, that what does not admit of explanation is always both self-evident and readily understood. Suppose something is self-explanatory. What needs no explanation may also be able to receive it. It would not follow that it is explanatorily ultimate. One might think, however, that explanational self-sufficiency is a case of explanatory ultimacy, where this is a status belonging to propositions that can explain something else but cannot be explained by appeal to anything further. Suppose something could be explanationally self-sufficient yet unable to explain anything else. Then it might be in a benign sense unexplainable, but not explanatory. I doubt, however, that any propositions of the kind in question cannot explain anything whatever. Even instances of the

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principle of non-contradiction can explain why those who deny them are mistaken.21 Suppose it is true that any explanationally self-sufficient proposition can explain something. That leaves open whether everything explainable must have an explanational foundation, in the sense of being explanatorily traceable to something itself explanationally either not in need of explanation or not admitting of it. Schematically, if p is explainable, then there must be some proposition q such that either q explains why p or there is a chain of such explanatory relations beginning with q and terminating with p. Consider grounding explanations, as illustrated by certain normative principles, such as the promissory one stated earlier, whose antecedent specifies a grounding condition and whose consequent ascribes a property grounded in that condition. I have taken certain of these principles, which embody such a conditional, as holding a priori. Presumably that is because of an a priori grounding relation. But even if grounding is a necessary relation, in the sense that propositions ascribing it are necessary, neither this relation nor that between any further ground and the one it specifies need be ascertainable a priori. Is promising, for example, which grounds a reason to do the promised deed, itself grounded? Presumably it is: one promises to A in virtue of communicating in a certain way that one will A, and, as so used, ‘in virtue of’ expresses the grounding relation. Need the chain beginning with these two linked items and tracing back to more and more basic grounds ever end? I assume the grounding relation is explanatory, so if it can figure in a non-terminating chain, then there are explaining factors (grounds) that can explain without ever being ultimately explained or even explainable. I leave open whether there are such chains and also whether the fact that one element grounds another may itself be in need of explanation (I assume that some such facts admit of explanation).22   Perhaps the main explanatory work is done here by the higher-order principle that anything having the form of ‘p and not-p’ is false and any belief with the relevant content is mistaken. If so, then the proposition that p and not-p still plays an explanatory role. 22   For a general account of grounding, see (Audi, 2012). 21

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I do not think, then, that it is a priori true that grounding chains must end with an ultimate ground. Nothing in this chapter precludes leaving this hypothesis open (as I am). Note, however, that the possibility of further explanation does not entail a need for it. I doubt there are infinite explanatory chains – in the propositional sense – I am not speaking of psychologically realized chains, as with speech acts expressing explanations.23 But even if there must be an end to each, it of course does not follow that each ends in some one place. The appeal of explanational foundationalism may help to explain some of the appeal of the ontological argument. Proponents take it that the crucial premise that God is the greatest conceivable being is self-evident and neither needs nor even admits of explanation. Perhaps some also think this of the premise that a being just like God except for being non-existent would be less great. Now the argument is so economical that one can occurrently consider its corresponding conditional. If it is plausible to maintain that if each premise is explanatorily foundational and that its corresponding conditional is think­able in this way, then we might have a fully a priori explanatory argument for God’s existence. We might not only get by an a priori route to an affirmation of the existence of God, but also proceed from premises so solid that we cannot even comprehendingly wonder why they are true. Whatever we say about the ontological argument, it appears that there are some a priori propositions that do not admit of explanation. Can any empirical propositions meet this condition? What about the proposition that I am conscious now? If it means now-as-opposed-toearlier, when (I am told) I was on a drug, that is one thing; we may certainly explain that fact in terms of how long the drug is effective. But suppose I want to know simply why I am conscious now, rather than why I was not unconscious under the drug for a longer time? Well, I am open-eyed and aware of my environment. But such awareness presupposes consciousness. Perhaps the (Russellian) proposition   Various kinds of infinitism has been defended in recent years. For detailed cases see (Klein, 2011, pp. 245–256; Aiken, 2010). 23

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that I am conscious now is comprehensionally compelling and thus such that to consider it comprehendingly entails knowing why it is true. But do I know why I am conscious now, as opposed to being unable to doubt it? Granted, I can know sufficient conditions for it, such as neurophysiological facts about myself. But even if they explain the fact that I am conscious now, I do not normally know that fact on any such inferential basis. This is apparently one of the special cases in which one can explain in an indirect way what one knows in a direct way but (at least occurrently) cannot rationally doubt. The comprehensional compellingness view would not be that the comprehensionally compelling must be optimally understood. Understanding comes in degrees, but explainability presupposes that it cannot be altogether lacking. It still seems that explaining why I am conscious now – if it does not come to explaining, for example why I am conscious given that I have been struck on the head with a baseball bat – is not clearly possible. This is a good place to reiterate that even if there are propositions that are unexplainable in the order of determination (roughly, that of why-questions), it does not follow that these are unexplaniable in the order of explication (roughly, that of what-questions). Thus, if it should be impossible to explain why God is the greatest conceivable being, it may be quite possible to explain what this means and, more broadly, implies. Explication is here parallel to proof: as we can prove much from an axiom even if it is unprovable and we thereby exhibit its fruits, we can explain much by appeal to what is unexplainable and thereby exhibit what it in some sense “says.”

7. Conclusion We have seen, through exploring a number of examples, that science is not the only explanatory realm. But it would be a mistake to think – as a dogmatic naturalist might – that the only important contrast between kinds of explanation is between the scientific and the unscientific. A priori explanations are non-scientific; but they need in no way

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be such that philosophers or others who are scientifically oriented should reject them. Logic and pure mathematics, after all, are essential for science – and for scientific explanations – but contain a priori explanations. We have also considered the normative realm. Here too, not only in the moral domain but also in the epistemic and psychological domains, there are explanations that seem to be a priori and nonscientific, but in no way unscientific. Through the use of theoretical method, philosophy, as well as science, can offer explanations. Some of these exhibit the power of a priori reason, but I have not maintained that doing so is a condition for the success of philosophical explanations. I have argued for a related point: the philosophical realm, particularly as including the normative domain, is apparently one in which there can be proof, even if not formal proof, as in logic and pure mathematics. We have seen, to be sure, that not every proof corresponds to or provides an explanation, just as not every explanation (as probabilistic cases illustrate) corresponds to a proof. But explanation of any kind is essentially connected in some way with understanding. This holds for scientific explanation as well as for other kinds. Scien­ tific understanding is an important and indispensable kind, but it is not the only kind, and its scope is not unlimited. Its possibility rests on the possibility of explanations of other kinds. A limitation, of course, is not a defect; and this point applies as much to the philosophical and the normative outside science as to science itself.24

References Aiken, S. (2010). Epistemology and the regress problem. London: Routledge. Audi, P. (2012). Grounding: toward a theory of the in-virtue-of relation. Journal of Philosophy, 109(12), 685–711.   This paper has benefited from discussion with many people at the Kraków conference in which it was presented in an earlier version. I would particularly mention discussion with Richard Swinburne, as well as Bartosz Brożek and Sonja Rinofner-Kreidl. 24

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Audi, R. (1973). The concept of wanting. Philosophical Studies, 24, 1–21. Audi, R. (1987). Realism, rationality, and philosophical method. Proceedings and Addresses of the American Philosophical Association, 61, 65– 74. Audi, R. (1993). Action, intention, and reason. Ithaca: Cornell University Press. Audi, R. (1999). Self-evidence. Philosophical Perspectives, 13, 214–228. Audi, R. (2008). Skepticism about the a priori: Self-evidence, defeasibility, and cogito propositions. In J. Greco (Ed.), The Oxford Handbook of Skepticism (pp. 149–175). Oxford: Oxford University Press. Audi, R. (2010). Epistemology: A contemporary introduction. London and New York: Routledge. Audi, R. (2014). Naturalism, normativity and explanation. Kraków: Copernicus Center Press. Chudnoff, E. (2013). Intuition. Oxford: Oxford University Press. Grimm, S. (2011). Understanding. In S. Bernecker, & D. Pritchard (Eds.), The Routledge companion to epistemology (pp. 84–95). New York: Routledge. Klein, P. D. (2011). Infinitism. In S. Bernecker, & D. Pritchard (Eds.), Routledge Companion to Epistemology (pp. 245–256). London: Routledge. Knobe, J., & Nichols, S. (2008). Experimental philosophy. Oxford: Oxford University Press. Murawski, R. (2014). On proof in mathematics. In M. Heller, B. Brożek, & Ł. Kurek (Eds.), Between philosophy and science (pp. 41–59). Kraków: Copernicus Center Press. Scanlon, T. M. (2014). Being realistic about reasons. Oxford: Oxford University Press. Stratton Lake, P. (2011). Recalcitrant pluralism. Ratio, XXIV(4), 364–383. Swinburne, R. (2004). The existence of God. Oxford: Oxford University Press. Swinburne, R. (2011). God as the simplest explanation of the universe. Philosophy, suplement, 68, 3–24. Tymoczko, T. (1979). The four-color problem and its philosophical significance. Journal of Philosophy, 67, 57–83. White, R. (2005). Explanation as a guide to induction. Philosopher’s Imprints, 5(2), 1–29. Wood, A. W. (2007). Kantian ethics. Cambridge: Cambridge University Press.

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Stanisław Krajewski University of Warsaw

Remarks on Mathematical Explanation

The term “mathematical explanation” has two distinct meanings. First, it is about the explanation of mathematics, an account of how the mathematical sciences illuminate their own subject matter. The second meaning is explanation by mathematics, the use of mathematical techniques to illuminate other subject matters. While in literature the latter meaning, the one dealing with the applications of mathematics, used to be discussed much more than the former, explanation within mathematics has also recently been studied. Some of these studies, by Paolo Mancosu and others, are briefly surveyed below. In Section 1 a principal problem of explanation in mathematics is discussed, namely how difficult it is to characterize explanatory proofs as opposed to those that demonstrate but do not explain. In the present paper, the stress is on the unavoidability of a psychological aspect, or rather foundation, of the process of explanation. In Section 2, some examples of explanation in the applications of mathematics are briefly considered, including applications of mathematical theories to other mathematical theories. No survey of relevant literature is attempted. Here again the subjective, psychological, or human, basis of explanation is stressed. By subjective or psychological foundations I understand the need to take into account as a fundamental dimension of any explanation the human agent who wants to understand. This means that explanation is meant to provide an understanding for some human beings or perhaps human beings in general. The human dimension is essential.

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In logic this approach is commonly called logical pragmatics. According to this view, the perspective of those who use logical tools must be taken into account. This means that syntactic and semantic levels are not enough, meanings are not just meanings in themselves but are meanings for someone. In mathematical explanation in both of the senses just mentioned, this approach can be called subjectivism (or perhaps even psychologism). One could notice that there exist differences between the human factor, the psychological aspect, and the subjective approach. Whilst this is true, the differences are not pondered here since they point in broadly the same direction. In the present paper the thesis on the inevitability of subjectivism is defended. It states that the needs, peculiarities and limitations of human beings are an indispensable aspect of mathematical explanations. This thesis is simple, and possibly uncontroversial, but still worth emphasizing. While it does not contradict the remark contained in the competent survey by Mancosu (2011), namely, that it can be the case that “mathematical explanations are heterogeneous and that no single theory will encompass them all,” it does propose the existence of a universal aspect that is claimed to be present in all explanations. I am not suggesting that only psychological, subjective aspects of explanations exist; rather, the thesis is that the objective and subjective aspects coexist, and the latter are never absent, and, indeed, are central to any attempt to explain explanation.

1. Proofs which explain In recent decades “philosophers of mathematics have turned their attention more and more from the justificatory to the explanatory role of proof,” “diversity of notions of proof and explanation.” (Hanna et al., 2010, p. 2) A serious challenge to subjectivism is suggested by Mark Steiner’s paper (1978), probably the first modern paper devoted to explanation within mathematics. The author analyzed the difference be-

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tween mathematical proofs that “merely demonstrate” and those that, in addition, offer an explanation as to why the thesis is true. He tries first to define the difference by invoking, after Georg Kreisel and Solomon Feferman, generality and abstractness. This attempt fails, argues Steiner: while in some cases more general proofs are more explanatory ones, there are examples where this is not the case. He gives two proofs of a Euler equation as an argument. The same example shows that “discoverability” is not the right criterion that characterizes the explanatory proofs. A further possibility that comes to mind, the ability to visualize proofs, is dismissed by Steiner in a peculiar way: “Aside from possible counter-examples, however, this criterion is too subjective to excite.” (Steiner, 1978, p. 143) My feeling is that this remark reveals the motivation behind Steiner’s work. It is an attempt to define explanation in objective terms, without reference to the psychological or any other human dimensions. So it seems to be directed against the thesis of the present paper. Let us see how he proceeds. Steiner’s proposal is as follows. To explain the behavior of an entity one should refer to “the essence or the nature of the entity.” To avoid the problems raised by the idea of an essence Steiner proposes the concept of “characterizing properties,” which are “unique to a given entity or structure within a family or domain of such entities or structures.” (The notion of family is left undefined.) Now “an explanatory proof” is defined as one making “reference to a characterizing property of an entity or structure mentioned in the theorem, such that from the proof it is evident that the result depends on the property. It must be evident, that is, that if we substitute in the proof a different object of the same domain, the theorem collapses; … In effect, then, explanation is not simply a relation between a proof and a theorem; rather, a relation between an array of proofs and an array of theorems…” (Ibidem). According to Steiner, in addition to being dependent on a characterizing property the proof that explains is “generalizable”: “an explanatory proof depends on a characterizing property of something mentioned in the theorem: if we ‘deform’ the proof,

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substituting the characterizing property of a related entity, we get a related theorem.” (Ibidem, p. 147) In addition to presenting the above construction, Steiner suggests that explanations other than explanation by a proof can be described using the notion of characterizing property. The other varieties of explanation, for example by a theory, that is, a “global” explanation, are not discussed further in that paper, so I will not pursue it here. Incidentally, the approach like that in Balacheff (2010) is also ignored here. According to it, explanation is a general category, and proofs, including mathematical proofs, form a subclass, a subcategory of explanations. To additionally justify his proposal, Steiner indicates that the initial criteria first coming to mind when the issue of explanation is raised can be seen as plausible using his approach. Thus, for instance, “generality is often necessary for capturing the essence of a particular and the same goes for abstraction” (1978, p. 146). The helpful role of characterizing properties in the process of discovery of proofs results from the fact that “it is often the case that a characterizing proof is intuitive enough to serve as an instrument of discovery.” Coming back to the issue of visualization, Steiner remarks that a “characterizing property is likely to be visualizable (as is certainly the case with a geometrical property)” (1978, p. 146). In my view this is a very inadequate recognition of the role of visualization. It is not just in geometry where figures are essential – in practice, if not in theory (but their theoretical significance can be reclaimed – cf. Manders, 2008a and 2008b). I think that it is important to acknowledge the appearance of more general figures – visual arrangements, mental pictures representing proofs or rather some key points or moves in mathematical proofs. This seems to be a common practice among mathematicians. The pictures are imprecise, hazy, unstable, messy, often in the process of moving, difficult to describe in detail and to transmit to others, but they are important, presented to colleagues, serve to focus on the mechanics of the proof. They seem unavoidable for human mathematicians.

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One could say that the importance of those mental pictures is only psychological. What would, however, be meant by the term “only”? It could mean that they are usually not included in formal papers or in textbook presentations. True, but this does not mean that they are not part of real-life mathematics! It can also mean that in mathematics itself we study conceptual relations and the way humans approach them is inessential to those relations, the thing itself. However, there are myriad relationships among mathematical entities. We select and study those that make sense to us human mathematicians. It makes no sense to take an arbitrary equation of order 1000, of length 100 000, with 1000 unknowns and to say that solving it is a good mathematical problem. It could be for a superhuman mind, as quadratic equations are for us, but for us the huge equation would usually remain meaningless. Mathematics is ultimately human. The subjective, psychological aspect cannot be overcome. The issue of pictures providing proofs, or pictorial proofs has been noticed by philosophers of mathematics, for instance by George Polya. Among relatively recent examples is the collection of papers Mancosu et al. (2005), papers by Giaquinto (2008a, 2008b), as well as the more popular book of Brown (2008), where examples are given of visualizing more than just geometrical facts. Of course pictures can be misleading. Still, I believe that Brown is right in saying, “pictureproofs are obviously too effective to be dismissed and they are potentially too powerful to be ignored” (Brown, 2008, p. 47). The more general issue, introduced above, of the mental picturing of proof ideas, of that hazy pictures present in our imagination when doing proofs has not been properly analyzed so far, even though it is so common and described clearly already in the classic book by Hadamard (1952). There are other more general considerations that analyze the concept of pictorial proof. Thus Nordmann (2010) takes seriously Wittgenstein’s suggestion that one can regard proof as a picture and as an experiment. “A proof, one could say, must originally have been a kind of experiment – but is then simply taken as a picture” (Wittgen-

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stein, 1978, p. 110; III, 23). A picture must be surveyable, like a calculation, and an experiment, like a proof by reductio ad absurdum, changes the domain of the imaginable. Nordmann claims that “a proof is a picture and also that it is an experiment,” and that this “complementarity is necessary” (2010, p. 194). “When we see a proof, we see a picture” (2010, p. 202). More specifically, “the opposition between picture and experiment expresses well what is only clumsily hinted at by opposing static versus dynamic, synchronic versus diachronic, justificatory versus exploratory aspects of proof” (2010, p. 203). At the same time, Nordmann is sure that “to be a proof a proof needs to be convincing, of course” (2010, p. 193). The use of the psychological term “convince,” as a well as the psychological term “surveyable” and the human term “experiment,” not to forget the term “picture,” is a clear indication that the subjective understanding of explanation is unavoidable under this approach. * It is not only proofs which can be explanatory. I would guess that all of the types of mathematical activities mentioned by Giaquinto (2005, p. 85) can help explain something mathematical. These activities include Discovery, Formulation, Application, Justification, Proving theorems, Motivation of definitions/axioms, Representation, and of course Explanation itself. So far nobody has tried to develop a systematic account of the explanatory aspects of these activities. One can find scattered remarks, for example about definitions that clearly can be more or less explanatory: “natural,” revealing, grasping the crux of the matter, etc. When such accounts appear I suspect they will have to be based on psychological foundations. According to Giaquinto, Explanation can be divided into Subjective and Objective, with the latter being “stronger.” A pictorial proof of Pythagoras’ theorem and a counting dots argument showing the commutativity of addition “instantiate only a rather weak, subjective kind of explanation. This is explaining in the sense of making a fact

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more intuitively compelling to a person than it was before.” At the same time, says Giaquinto, “Mancosu and others suggest that there is a stronger kind of explanation in mathematics, one that has an objective basis.” (Giaquinto, 2005, p. 79). I agree that objective aspects can exist, but, as already mentioned, I do not agree with the implied suggestion that that explanation can be fully objective, with no psychological dimension. Philip Kitcher has attempted to define the explanatory power of mathematical theories rather than individual proofs. Significant generalizations of theories, as distinct from trivial ones, can serve as an example. In his own words, “The account I have offered distinguishes three types of explanation in mathematics. The discussion of generalization in Section IV indicates how we can sometimes explain mathematical theorems by recognizing ways in which analogous results would be generated if we modified our language... My notion of rigorization introduces a second type of explanation: explanation by removal of previous inability to recognize the fine structure of connections. Finally, Section VI is explicitly concerned with explanation by unification” (Kitcher, 1984, p. 227). Kitcher’s later idea of the universally applicable “explanatory unification,” that is “deriving descriptions of many phenomena using the same pattern of derivation again and again, and, in demonstrating this, it teaches how to reduce the number of types of facts that we have to accept as ultimate (or brute)” (Kitcher, 1989, p. 432). This concept is discussed by Hafner and Mancosu (2008). It seems that all of Kitcher’s proposals are based on the assumptions that explanation is expressed in human language, which means it is for us, humans, and does not live in an objective suprahuman realm. The psychological side of explanation is accepted. * To conclude this section, let us come back to the challenge implicitly posed by Steiner’s approach – I say implicitly since possibly

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this was not his actual view; in his book (1998) he defends the opinion that mathematics is anthropocentric. The approach visible in his former paper suggests that properly explained mathematical explanation avoids subjectivism. Now, it is not too hard to tell why this is misleading. The vision of the objectivity of mathematical explanation seems to be based on the vision of a formal proof; formal proof is an objective entity which needs no human or other subject to remain a proof. Such proofs are, however, not only impractical but also the very opposite of explanatory proofs. Every real proof is more or less explanatory because it appeals to human mathematicians. It is more explanatory if it reveals more to humans, in terms accessible to the people of the given period. It does reveal objective relationships, and the concept of characterizing properties can be helpful to characterize explanatory aspects of some proofs – though not necessarily all proofs (cf. Hafner & Mancosu, 2005). Yet each time we objectivize some feature there remains a subjective element behind it. Formal proofs can be called “proofs” only because there is a certain relationship between them and our human age-old proof procedures. I believe that the phenomenon of a subjective basis, a psychological remnant behind the objective and even formal concepts is very important. It makes all science, including mathematics, a human endeavor, and mathematical explanation is a part of that endeavor.

2. Some other attempts to explain Mancosu (2008a) reviews discussions about the ways mathematics explains natural phenomena. It is the relatively well known “indispensability argument,” introduced by Quine and Putnam (see Colyvan, 2011 for references), that provides the point of departure for contemporary discussions. According to that argument, mathematics is indispensable in every description of physical realities, and probably also of more involved biological and other realities; this means we need to presuppose mathematical realities in order to do science. In other

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words, since an explanation can be only successful if the explanans is true, that is, refers to a reality, mathematical entities are justified by their unavoidability. On a superficial level, mathematical formulations of physical and other realities can be understood as the discovery of objective regularities. Equations provide an account of the internal relationships of phenomena. The approach assumed in the indispensability argument is, however, not devoid of a subjective dimension since it is based on an analysis of descriptions. And descriptions are always descriptions for someone; we produce them, or rather choose those that seem the best. The mathematical formulation is never unique. For example, the calculus as we know it could have been developed with full precision in another style, for instance in an intuitive version of Abraham Robinson’s non-standard analysis. Thus the human dimension appears naturally. One can still ask: Is the fact that mathematical descriptions are meant for human scientists really essential? I think so, and to dramatize this issue let me propose an example. In robotics it is very difficult to construct a bipedal walking robot that would walk more or less like a human being. The way to achieve this task seems to be via a successful description of a human walking, a description that can be used by robot builders. Descriptions are mathematically advanced and it is probably fair to say that solutions of complex systems of equations are necessary to formulate the task that engineers must implement. Even a simpler, so called “intuitive,” method requires the builder to combine sinusoids (cf. Han, 2012). Sophisticated mathematical training is needed to provide an explanation of the walking movements that would be sufficient to reproduce walking in a robot. At the same time, even the most mathematically illiterate person can walk. Would it make sense to state that every walking person unconsciously solves equations? The affirmative answer seems rather ridiculous. At most, one could say that the mathematical problem posed by the description of walking is solved by ignoring it. In real life we do not describe walking, we just do it. It is when we, human scientists, want to recreate walking that we need appropriate descriptions, which

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presumably must involve rather advanced mathematics. Descriptions are not internal to the phenomena we want to explain but result from our human approach to the phenomena. * Some problems about explanation arising in applications of mathematics can be observed within mathematics, when one mathematical theory is applied to another. Even the indispensability argument seems to have a parallel. It could seem that a method similar to the indispensability argument would work to ground mathematical Platonism or realism with respect to various mathematical entities. Namely, one could justify the existence of more complex entities by reference to simpler, better grounded ones, assuming the more complex are necessary to analyze the simpler ones. Mancosu observed that to explain scattered results about familiar objects (for example, natural numbers) one uses more abstract entities (for example, analytic functions), and this gives us a good reason to believe in the existence of those abstract objects. He is, however, cautious: giving good reason is not a proof. “I am not endorsing the indispensability argument for mathematics,” he says, it is, however, “of interest” (Mancosu, 2008a, p. 140). He also says something that has clearly a “psychological” sense; he mentions “the intended audience” for the argument, viz. those who are realists about some mathematical entities, and, in addition, are not committed to a foundational position forbidding the entities postulated by the explanation. The subjective aspect appears naturally. Interesting problems arise when explanation of mathematical issues is attempted by taking advantage of mathematical logic. The famous Matiyasevich-Robinson-Davis-Putnam theorem (MRDP theorem) of 1970 provided, among other things, polynomial (more precisely, Diophantine) definitions of recursive (and even recursively enumerable) sets of natural numbers. This result can be applied to one of the most important and interesting sets among those that are easily definable but could never be defined by a traditional arithmeti-

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cal formula, namely the set of prime numbers. Incidentally, the concept of being a prime number is so simple that I claim it can be understood by very small children: the number of marbles or beads is prime if they cannot be arranged into a regular rectangular pattern. The distribution of prime numbers is very wild. Mathematicians have found asymptotic regularities, but no formula producing prime numbers. Now, due to the MRDP theorem, using the Diophantine definition of prime numbers: n is prime if and only if there exists a solution for p(n, x1, ..., xk) = 0, where p(x0, x1, ..., xk) is a fixed polynomial, we can use a beautiful trick proposed by Hilary Putnam and get a polynomial q(x0,x1,...,xk) that, for positive arguments, produces as its positive values precisely the prime numbers. The definition of q(x0, x1, ..., xk) is as follows: x0(1– p(x0,x1,...,xk)2). A word of caution: the negative values of q form a strange set having nothing to do with prime numbers; and actually no polynomial can have all prime numbers and only prime numbers as its values. Thus we get a formula producing all prime numbers and only them, if the negative outcomes are ignored. Does it mean that we have the formula people have been looking for? Not really. To us this polynomial q is too complicated to be useful. It exists but only theoretically. The proof of its existence is a great achievement but it does not fit our human capabilities. Perhaps it could satisfy a superhuman mind that can deal with such huge polynomials as we do with short, simple polynomials, but not us. For the hypothetical superhuman mind the prime numbers would receive an explanation that it is not available to us. The human aspect of the situation cannot be overcome. Another famous mathematical example is of a distinctly metamathematical character. In consequence of the work of Kurt Gödel and Paul J. Cohen the status of the continuum hypothesis (CH) has been settled: CH is independent of the accepted set-theoretical axioms, say ZF, that is, it is neither provable nor disprovable on their

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basis. The natural question in our context reads: Does the proof of independence give an explanation of the status of CH? The answer is not unambiguous. Undoubtedly the independence results provide very important information. For some mathematicians it is enough, for them explanation is complete, they are ready to treat the question whether CH is true as unanswerable. In result the question about truth becomes meaningless. Yet to others the naïve question about the truth of CH is meaningful. Some of the best experts maintain that we need to find the truth about CH. Paul Cohen said as much in his lecture in Moscow where, having received the Fields medal, he presented the independence of CH. It is well known that Gödel invested a lot of effort into attempts at disproving CH. At present Hugh Woodin explores varieties of the so-called generic multiverse, and believes the status of CH can be settled in this way. The paper reviewing the work done largely by himself, ends with the following statement (Woodin, 2011, p. 117): Finally, the extension of the Inner Model Program to the level of one supercompact cardinal will yield examples (where none are currently known) of a single formal axiom that is compatible with all the known large cardinal axioms and that provides an axiomatic foundation for set theory that is immune to independence by Cohen’s method. This axiom will not be unique, but there is the very real possibility that among these axioms there is an optimal one (from structural and philosophical considerations), in which case we will have returned, against all odds or reasonable expectation, to the view of truth for set theory that was present at the time when the investigation of set theory began.

To the above-mentioned scholars the explanation of CH still eludes us. It is clear that the difference in attitude between the two groups of mathematicians is not due to any difference in factual knowledge. It is due to different attitudes or philosophical approaches. The human factor is essential, the objective interrelations of formal concepts are insufficient.

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* In physics no explanation can be made without mathematics. Yet even the simplest examples from classical physics reveal the presence of a human aspect. The presence of equations, even very successful ones, is not by itself sufficient to offer explanation, at any rate not a full explanation. Is a fundamental equation like F = ma or F = gm1m2r-2 sufficient to explain the entities it involves? There at least two major problems here. First, while certain relations among the entities involved are shown, the nature of mass, force or gravitation is not disclosed; they are explained neither by the equations nor by the classical theory around them. This has been a well-known property of physics for the past three centuries. True, in twentieth century physics the gravitational force has been explained as a curvature of space, but also in the present-day physics there are basic components whose nature remains unknown to us. The success of experimental predictions is sufficient to justify theories. This state of affairs has been well summarized by the well-known dictum “Shut up and calculate!” It is attributed to Richard Feynman or Paul Dirac. Most probably also in the future we will have to live with this lack of explanation combined with empirical success. The second major problem concerns the form of the equations. Why is it so simple? There are other equations that presumably would give exactly the same experimental results, for example, F= (1+ 2-1000000)ma or F=gm1m2r-2.0000001. (Possibly the number of zeroes needs to be adapted to be sure the agreement with experimental results is not affected, but the point remains.) What is the reason for our belief in the simple form of equations? Is it purely aesthetic? Then the subjective aspect would be primary. Or is it due to the fact that we can handle only simple formulae well? For a superhuman mind, the equations of a very high order, for example of several millions, could perhaps be accessible, not for us. The limitation shows that the human factor is crucial. Of course, one can hope to provide a cogent explanation of the simple form of the equations. Sometimes

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one can show how they originate from some imaginary experiments, indeed – to use the term used in Section 1 – moving mental pictures. Thus the factor r2 in Newton’s formula for gravity can be explained by a visualization of the mechanism that gives rise to the equation: if, for example, gravitational force is imagined as tentacles going in all directions then the density of these tentacles is obviously decreasing proportionally to the square of the distance from the point imagined as their source. This process of visualization is again a human way to deal with the issue. It includes an objective mathematical structure but it is used to illuminate our human understanding of the situation. The resulting explanation can be seen as satisfactory if it provides a humanly graspable picture. Inevitably a psychological level is reached. The above selection of examples is by no means comprehensive or even representative. I believe, however, that other examples would also reveal subjective aspects, the unavoidability of a human factor, the presence of a psychological level behind each explanation.

References Balacheff, N. (2010). Bridging knowing and proving in mathematics: A didactical perspective. In G. Hanna, H. N. Jahnke, H. Pulte. Explanation and proof in mathematics. Philosophical and educational perspectives (pp. 115–136). New York: Springer. Brown, J. R. (2008). Philosophy of mathematics, a contemporary introduction to the world of proofs and pictures (2nd ed.; 1st ed. 1999). New York & London: Routledge. Colyvan, M. (2011). Indispensability arguments in the philosophy of mathematics. Stanford Encyclopedia of Philosophy. Retrieved from http:// plato.stanford.edu/entries/mathphil-indis/#1> (accessed Dec 29, 2013). Giaquinto, M. (2005). Mathematical activity. In P. Mancosu (Eds.). Visualization, explanation and reasoning styles in mathematics (pp. 75–87). Dordrecht: Springer. Giaquinto, M. (2008a). Visualizing in mathematics. In P. Mancosu, K. F. Jørgensen, S. A. Pedersen (Ed.). The philosophy of mathematical practice (pp. 22–42). Oxford: Oxford University Press.

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Giaquinto, M. (2008b). Cognition of structure. In P. Mancosu (Ed.). The philosophy of mathematical practice (pp. 43–64). Oxford: Oxford University Press. Hadamard, J. (1952). The psychology of mathematical invention in the mathematical field. New York: Dover Publications. Hafner, J., & Mancosu, P. (2005). In P. Mancosu, K. F. Jørgensen, S. A. Pedersen (Eds.). Visualization, explanation and reasoning styles in mathematics (pp. 215–250). Dordrecht: Springer. Hafner, J., & Mancosu, P. (2008). Beyond unification. In P. Mancosu (Ed.). The philosophy of mathematical practice (pp. 151–176). Oxford: Oxford University Press. Han, J. (2012). Bipedal walking for a full-sized humanoid robot utilizing sinusoidal feet trajectories and its energy consumption, dissertation. Virginia Polytechnic Institute. Retrieved from http://scholar. lib.vt.edu/theses/available/etd-05042012-155706/unrestricted/Han_ Jeakweon_D_2012.pdf > (accessed Dec. 25, 2013). Hanna, G., Jahnke, H. N. & Pulte, H. (2010). Explanation and proof in mathematics. philosophical and educational perspectives. New York: Springer. Heller, M., & Woodin, W. H. (Eds.) (2011). Infinity. New research frontiers. Cambridge: Cambridge University Press. Kitcher, Ph. (1984). The nature of mathematical knowledge. Oxford: Oxford University Press. Mancosu, P. (Ed.) (2008). The philosophy of mathematical practice. Oxford: Oxford University Press. Mancosu, P. (2008a). Mathematical explanation: Why it matters. In P. Mancosu (Ed.). The philosophy of mathematical practice (pp. 134–150). Oxford: Oxford University Press. Mancosu, P. (2011). Explanation in mathematics. Stanford Encyclopedia of Philosophy, Retrieved from http://plato.stanford.edu/entries/mathematics-explanation/. Mancosu, P., Jorgensen, K. F., & Pedersen, S. A. (Eds.) (2005). Visualization, explanation and reasoning styles in mathematics. Dordrecht: Springer. Manders, K. (2008a). Diagram-based geometric practice. In P. Mancosu (Ed.). The philosophy of mathematical practice (pp. 65–790). Oxford: Oxford University Press. Manders, K. (2008b). The Euclidean diagram. In P. Mancosu (Ed.). The philosophy of mathematical practice (pp. 80–133). Oxford: Oxford University Press.

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Nordmann, A. (2010). Proof as experiment in Wittgenstein. In G. Hanna, H. N. Jahnke, H. Pulte. Explanation and proof in mathematics. Philosophical and educational perspectives (pp. 191–204). New York: Springer. Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34, 135–151. Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Cambridge, MA: Harvard University Press. Wittgenstein, L. (1978). Remarks on the foundations of mathematics. 3rd ed. Oxford: Basil Blackwell. Woodin, W. H. (2011). The realm of the infinite. In M. Heller, & W. H. Woodin (Eds.). Infinity. New research frontiers (pp. 89–118). Cambridge: Cambridge University Press.

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Krzysztof Wójtowicz University of Warsaw

On the Problem of Explanation in Mathematics

1. Introductory remarks The problem of explanation is one of the main topics in the philosophy and methodology of science. The situation is quite different in the case of mathematics. According to a quite widespread view, in mathematics, “to explain” means simply “to give a (rigorous) proof.” In other words: a mathematical fact is explained, when a proof is presented – and the notion of explanation (exceeding the mere existence of the proof) becomes extraneous. In my opinion, this is a vast oversimplification. Providing a proof of a fact does not always seem to explain it and to exhaust the mathematical problem in its depth. Mathematicians are often confronted with the problem of the “true nature” of mathematical phenomena – i.e. with conceptual problems which often cannot be solved just by presenting a proof. In informal discussions, questions like: “but why does this mathematical fact really obtain?;” “what is the true meaning of this theorem?;” “I see that this proof works – but why?” or “what is the true reason – behind the formal machinery?” etc. occur frequently.1 They are of course not formal, mathematical questions, but are concerned rather with the more general conceptual order within mathematics. And the answer: “The   Other examples are: Why should this concept be defined in such a way? Why is this definition more natural? Why is this notion so fertile? Etc. 1

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ultimate reason is the existence of a formal proof within the theory” can be considered satisfactory by mathematicians only in an act of despair, where there are no other possibilities in sight.2 They know, that there are proofs – but they would also like to know, what do they really express, what the deep reasons behind them are. They expect the proofs to reveal the interplay and dynamics of mathematical ideas. If e.g. an algebraic and a topological criterion are equivalent, the mere existence of any proof (of this equivalence) will not be satisfactory – mathematicians will rather seek for a proof, which reveals the interplay of ideas and the deeper reasons. The questions concerning mathematical proofs mentioned above are not mathematical questions, and we cannot expect to solve them with mathematical means. They are rather conceptual problems of a much more general, methodological and philosophical character. Admittedly, notions like “the deep reason for...,” “the true nature of...” etc. seem to be rather vague, and we still cannot provide a satisfactory characterisation of them. They defy precise definitions, not to mention formalization. Perhaps this is one of the reasons why philosophers of mathematics do not discuss them very often.3 Many philosophers of mathematics prefer to concentrate on topics where the discussion can be made more precise, and possibly clarified by means of formal methods (or even formalized). Nevertheless, I think that the problem of explanation in mathematics deserves attention, and the aim of this paper is to convince the reader that it is so.

  Of course, the situation might be different when we are interested in purely prooftheoretic phenomena – in such cases the existence of a proof can be considered to be “the ultimate answer.” But in general it is not the case. 3   In (Rota, 1997) the author discusses in his article the importance of such notions in the philosophical analysis of mathematics. In particular, he emphatically claims, that “The notion of understanding, that is used in informal discussion but quashed in formal presentation, will have to be given a place in the sun.” (Rota, 1997, p. 195). 2

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2. Mathematical explanations in science One of the first problems that perhaps comes to our mind when discussing the problem of explanation in mathematics is the problem of the mathematical explanation of non-mathematical, i.e. physical, chemical, biological (or perhaps even social and psychological) facts. Examples of such situations are common, and the existence of such explanations is perhaps the essence of any advanced science. Statements like “this phenomenon obtains because a certain integral equation has a solution” or “this is so because of the convergence of a certain stochastic process” etc. are considered to provide explanations of physical facts.4 They are carried out by introducing mathematical concepts and appealing to mathematical facts.5 When analyzing such situations, we are interested rather in the philosophical problem of the applicability of mathematics, and the problem of explanation becomes a special case of the general philosophical problem concerning the relation between mathematics and the physical world. The problem is indeed quite general, it even has some implications for the ontological discussion: the explanatory power of mathematics within science is important for the indispensability argument(s). They are formulated roughly in the following way: the postulation of entities of a certain kind results in the increase in explanatory power (and other theoretical virtues of scientific theories) – so we can use this fact as an argument for the existence of these entities. Applying this scheme to mathematical objects (which seem to be indispensable in science), we arrive at the thesis of mathematical realism. So the notion of explanation is also important for the ontological discussions.6   “...The trajectory of this physical process is such-and-such because a certain functional attains its minimum / because a smooth transformation of a n-dimensional manifold has a fixpoint / because there is a singularity /...” There are plenty of examples of this kind. 5   Of course, this always happens in a certain theoretical setting: we assume, that a physical process is modelled by an integral equation or a stochastic process etc. 6  The locus classicus for the contemporary discussion of the indispensability arguments in philosophy of mathematics are the writings of Quine (and Putnam). There is a huge discussion concerning the indispensability argument, where (Field, 1980) gave an impetus to this phase of the discussion, there are countless monographs and articles (by e.g. Hellman, Maddy, Shapiro, Colyvan, Balaguer, Burgess, Rosen – to mention only a few). 4

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The problem of applicability of mathematics, and of the role of mathematical models in explaining and modelling physical (biological, chemical, social or psychological etc) phenomena is one of the central topics in the philosophy of mathematics, but will not be discussed in this article.7 Here I will concentrate on the problem of explanation within mathematics, when mathematics is considered to be an intellectual enterprise an sich, not only as a part of the conceptual machinery of science. The question becomes therefore: how to explain certain mathematical (rather than scientific) phenomena by appealing to other mathematical phenomena – and what does it mean.8

3. A psychological problem? There seems to be a natural temptation to classify the problem of explanation in mathematics as merely psychological. According to this approach, solving the problem of mathematical explanation amounts to providing a satisfactory analysis of the psychological phenomena in mathematicians’ minds. So in order to understand what explanation is, we will look for the criteria of epistemic comfort of the mathematicians, of their “grasping” a concept, or of their understanding a theorem, a proof, a theory. This seems to be a very natural – and safe – solution of the problem. On the one hand it does not dismiss the category of explanation as meaningless (it is an undisputable fact, that mathematicians use it, so this fact deserves attention), but one the other hand is saves us from never-ending, non-conclusive discussions   Cf. e.g. (Steiner, 1998; 2005) for the discussion of the general problem of applicability of mathematics. In (Baker, 2005) an example of mathematical explanations within biology is discussed. 8   In (Mancosu, 2001) five questions are formulated: “1. Are there explanations in mathematics? 2. What form do they take? 3. Is mathematical explanation a novelty in philosophy of mathematics? 4. What are the philosophical accounts of mathematical explanation? 5. What is the relationship between mathematical explanation and theories of scientific explanation?” (Mancosu, 2001, p. 98). These questions will not be analysed here in a systematic way, but they might be helpful as a general guide in the discussion. 7

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involving ill-defined notions. If we define “to explain a mathematical phenomenon” as “to provide epistemic comfort to a certain group of people called mathematicians,” that means, that we have simply translated our problem of explanation into psychological terms. Then it might be considered to be just a special case of a more general phenomena in cognitive psychology, or it might be considered a non-reducible experience, or maybe there are other possibilities – but in all these cases it is the psychologist or the cognitive scientist who is responsible for providing an answer. The (philosophical) problem of explanation is explained away, so to speak. The problem of cognitive processes going on in the mathematician’s mind is a fascinating subject (and still a terra incognita), but I think, that the notion of explanation is not a purely psychological category, and should be analyzed in philosophical (or methodological) terms. And certainly the problem of explanation is not a pedagogical problem concerning inventing “user-friendly” presentation of mathematical principles (e.g. by drawing pictures or telling stories). It is important in itself, but of course it is not a philosophical problem.

4. Questions concerning explanation in mathematics Consider some examples of questions involving explanation in mathematics. All they have the form “Explain, why a”: Explain, why... • ...there are infinitely many prime numbers? • ...a given equation has no solution? • ... a certain fact obtains in dimension 4 only? • ...the continuum hypothesis is independent from set theory? • ...a sentence is not provable in T, but is provable in T*? • ...Fermat’s theorem is true? • ...the sequence of distributions of random variables converges to the normal distribution? ...etc.

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This group of questions is by no means homogenous and providing even a rough classification can be considered to be a first step in our analysis. Depending on the level of generality and the context, these questions might be interpreted as concerning: the identification of the motivations for accepting certain assumptions; finding a suitable interpretation of the given situation as a special case of a more general situation; justifying the choice of a certain set of primitive notions; the choice of the proper conceptual setting of a mathematical discipline etc. It is not possible to give a detailed analysis of the problem in its full generality in a short paper. In what follows I will concentrate on two problems: 1. The explanatory versus the non-explanatory character of mathematical proofs (which might be labelled as the local problem of explanation). 2. The problem of an explanatory conceptual recasting of a mathematical discipline (which has a more global character).

5. Explanatory character of mathematical proofs “What is mathematics primarily made of? There are, roughly speaking, two schools. The first school holds that mathematics consists primarily of facts (...) The second school maintains instead that the theorems of mathematics are to be viewed as stepping stones, as more or less arbitrary stoppers that serve to separate one proof from the next. Proofs are what mathematics is primarily made of, and providing such proofs is the business of the mathematician” (Rota, 1997, pp. 187–188). So, is mathematics primarily about proofs or about theorems? The question might be formulated as: what contributes more to mathematical knowledge: studying proofs or studying theorems?9 Of course, there   Is my increase in mathematical knowledge and understanding greater after studying a three pages long proof of one theorem, or after studying a (three pages long) list of theorems without proofs?

9

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are no theorems without proofs, and no proofs without a result, so we cannot really separate these two kinds of entities. But still, even if the claims of Rota are somewhat metaphorical, the problem of the role of proof in mathematics is well posed. It is discussed in a detailed way in the insightful article (Rav, 1999), where the author formulates the thesis, that “theorems are in a sense just tags, labels for proofs, summaries of information, headlines of news, editorial devices” (Rav, 1999, p. 20) – and metaphorically describes theorems as headlines, whereas the inside story is provided by proofs. So proofs are the essence of mathematics. And their role consists not only in assuring us, that a certain mathematical fact obtains, but also (and perhaps even primarily) in providing understanding. This fact can be seen even more clearly when we take into account the fact, that mathematicians also frequently re-prove theorems. According to Dawson, “different proofs of theorems bolster confidence not only in the particular results that are proved, but in the overall structure and coherence of mathematics itself” (Dawson, 2006, p. 281). A new proof often makes it possible to provide a better understanding of the whole conceptual machinery and the role of the particular theorem within a network of ideas, concepts and techniques. It is therefore rather the interplay of ideas which is important, and not the particular result itself. So, in order to understand the role of proof in mathematics, we certainly have to discuss the problem of their explanatory content. The distinction between explanatory and non-explanatory arguments has a long history, dating at least to Aristotle, and in the mathematical folklore it is used widely. Mathematicians often claim that some proofs leave them with an uneasiness about the true reason for some fact: they are formally correct, but do not provide any insights into the “nature of things.”10   Mordell expresses it in the following way: “Even when a proof has been mastered, there may be a feeling of dissatisfaction with it, though it may be strictly logical and convincing: such as, for example, the proof of a proposition in Euclid. The reader may feel that something is missing. The argument may have been presented in such a way as to throw no light on the why and wherefore of the procedure or on the origin of the proof or why is succeeds” (Mordell, 1959, p. 11) (quotation from (Mancosu, 2008, p. 142)). 10

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The problem of the explanatory value of mathematical proofs can be given a clear formulation in the case of computer-assisted proofs. There are many computer-assisted proofs and the most famous one is probably the proof of the four-color theorem, given by Appel, Haken and Koch.11 To make the problems even more clear, consider a thought experiment, in which we ignore the physical limitations and imagine that we have a super-computer at our disposal (which is e.g. 10100 times faster than our computers). We can use this supercomputer to generate sequences of formal expression (in particular formal proofs) within the formal system in question, e.g. within PA (Peano Arithmetic) or ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) – or other formal systems. Let us assume, for the sake of the argument, that the computer simply checks, by brute-force computation, all the possible computational paths, looking for the (hypothetical) proof of (the formalized version of) an open mathematical problem, like Riemann’s hypothesis.12 Even if one of these searches succeeds, and the computer prints out the proof: 152,734 pages of ZFC-formulas, it is not clear, whether this would provide any epistemic profit, any insight into the conceptual structure of mathematics.13 The problem of the epistemic status of computer proofs has been discussed in the philosophical literature mainly in the context of the computer-assisted proof of the four-color theorem.14 There is still   Their results were presented in (Appel & Haken, 1977) and (Appel, Haken & Koch, 1977), refined by Allaire (1977). In the original version it required ca. 1200 hours of the computer’s work. There are many other computer-assisted proofs, and probably the most widely known among them is Hales proof of Kelper’s conjecture, concerning optimal packing of balls in a box (Hales, 2000, 2005). 12  Our supercomputer could simply check by brute force computation all possible proofs one after another. Provides it is quick enough, sooner or later it will find the proof or disproof of Riemann’s hypothesis (of course provided it is not independent of ZFC). In fact, the supercomputer will after some time reconstruct the entire mathematical knowledge of mankind A.D. 2015 and start producing new results. 13   Of course, this fact would motivate our search: if we knew, that a formal proof of Riemann’s hypothesis exists within ZFC, we would search for the proof rather than trying to disprove it. But it seems rather clear, that the epistemic advantage of the mathematicians would be rather very limited. 14   The first philosophical reactions to this problem are: (Tymoczko, 1979; Swart, 1980; Teller, 1980; Detlefsen & Luker, 1980; Krakowski, 1980; Lewin, 1981). 11

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some feeling of uneasiness – and some mathematicians claim, that we still do not know the reasons, why the four-color theorem is true.15 And even if we agree, that the computer provides only a kind of computational assistance, providing the “computational missing link,” which is of no real importance for the conceptual analysis (which is perhaps the pragmatic attitude of many mathematicians) the situation would be quite different in the case of the (hypothetical) fully-formalized computer proof of some new theorem. Would such a proof explain anything? Boolos gives an interesting example of a proof of a simple statement whose length in first-order logic exceeds all reasonable bounds, but has a very simple proof within second-order logic. And we would rather claim, that we know of the result and understand, why it is true, because of the knowledge of the simple second-order logic proof, not because we know, that in certain formal system a proof exists (Boolos, 1987). The source of our understanding is not the formal first-order proof, but rather the idea behind it – which can be grasped in a very simple way within the second-order system, but not in the system, where the formal proof is formulated. But in the case of this example of Boolos we understand the situation, and the formal first-order proof is secondary. But what if we didn’t know the conceptual formulation and were presented with a formal proof of a certain theorem (provided by a computer)? Under what conditions can we have a genuine epistemic profit from studying such a proof? A necessary condition seems to be the possibility of translating the proof into the “mathematese,” i.e. the language of high-level mathematical concepts, like continuity, Borel measure, differentiable manifold, martingale, complex differentiability etc. If the automatic proof-system generated a translation into our mathematical language, we could perhaps gain some understanding of the situation in question.16 Of course, it could happen that   “Mathematicians are on the lookout for an argument that will make all computer programs obsolete, an argument that will uncover the still hidden reasons for the truth of the conjecture” (Rota, 1997, p. 186). 16   We could think of an user-friendly interface, which translates the formal proof from the language of ZFC into the language of e.g. complex analysis. 15

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we considered the proof artificial, brute-force, tricky, ugly etc., but at least we could understand it.17 And understanding requires certainly more than just accepting every single step of the formal proof – we need also to grasp the whole conceptual structure.18 Apart from the problem of the length and surveyability of proofs, there is another important question, concerning status of formal theories (like ZFC or PA or ZFC+V=L) within mathematics. It is commonly accepted, that ZFC provides a foundation for mathematics – in the sense, that virtually all mathematical concepts can be reconstructed within ZFC.19 ZFC can serve as a tool for formalizing mathematics, but for everyday mathematics, ZFC seems to be only a tool for formalizing mathematics (if someone wishes to formalize), but not a source of new ideas and concepts. Its role is quite special (or even peculiar): purely set theoretic concepts (of a large cardinal number, an inner model, iterated forcing etc.) almost never appear in other mathematical disciplines like differential equations or probability theory. This is in a stark contrast with the fact that usually there is an intense “transfer of ideas” between mathematical disciplines. So ZFC can serve some methodological purposes, like providing (formal) foundations for mathematics, but real proofs are never formulated within the formal framework of ZFC. ZFC can be used for logical reconstruction of (fragments of) mathematics, but a mathematical can be perfectly happy within his realm of differential equations, not worrying (and not even knowing) about logicians performing formal reconstructions of this mathematical discipline. A number theorist need not worry about non-standard models for PA.20 Mathematical ideas   A very distant analogy is that of chess-playing programs: their moves are sometimes judged as “unnatural” – even if they make (objectively) best moves. In the case of computer proofs, this feeling of unnaturalness would be probably magnified to a great extent. 18   The issue of surveyability of proofs is discussed in detail in (Bassler, 2006). 19   E.g. the notion of function is reconstructed as a set of ordered pair, the continuum is reconstructed via equivalence classes of... etc. 20   If we understand number theory in the standard way, i.e. as a theory of what is going on in the standard natural numbers, than it might happen, that knowledge about metamathematical properties of formalized first-order versions of number theory is not essential at least for some of the investigations. 17

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important in topology, analysis, algebra are rarely of purely set-theoretic origin, and the formal reconstruction of e.g. a probabilistic notion within ZFC does not really explain the motivations, does not explain, what that notion really is. So understanding a mathematical concept has almost nothing to do with reconstructing it within ZFC, and the process of translation does not have much to do with the process of explaining in mathematics. Real mathematical proofs are not formal – they have a semantic content, their essence lies within the conceptual, rather than the formal realm.21 A mathematician looking for the reasons, why a certain mathematical fact obtains, will certainly not accept the answer: “This theorem is true, because it can be formally proved within ZFC – and this is the ultimate reason.” Mathematicians will rather try to identify the conceptual background, the essential steps within the proof – or perhaps the common threads in diverse known proofs and similar phenomena within other areas of mathematics (to identify “the deep reasons”). They will try to isolate key concepts, key definitions, will try to “immerse” the particular result into the familiar conceptual background, to identify its localisation on their charts of the mathematical world. And from this point of view, the mere presence of the formal proof of Riemann’s hypothesis (remember it is 152,734 pages long) does not increase their mathematical knowledge. They get probably some motivation, but the knowledge itself is of a non-constructive kind: we know that a proof exists, but we are not able to identify the proof in a simple way. We would have to “extract” the real proof from the formal proof, which – taking its length into account – would not be easy. So the problem of the status of formal theories providing a conceptual framework for a discipline is far from simple.

  Rav discusses the notion of the “Hilbert bridge” between the conceptual proofs of real mathematics, and their formal counterparts in some formal theory (like PA or ZFC). He stresses the fact, that even if such a bridge exists, i.e. even if mathematical proofs can be formalized, the essence of mathematics consists in the semantic content of the proofs, not in the formalization itself (Rav, 1999). 21

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6. Global explanation The problem of the explanatory role of proofs has a local character. But in mathematics we frequently encounter problems which rather have a global character: the problem of isolating some core notions in a discipline, the problem of finding a satisfactory conceptual (re)formulation of a whole field, the problem of identifying the proper axioms for a theory etc. There are many historical examples, e.g. the problem of finding the proper conceptual presentation of calculus (which was developing for a long time before any rigorization took place) or of isolating the proper axioms of probability theory. We sometimes claim that a certain notion seems to be the right one or that a certain definition is the most natural one.22 We might therefore ask, whether the presentation of a discipline is explanatory or nonexplanatory. This problem is difficult to grasp in a precise way: what does it mean, that a theory is presented in an explanatory or a non-explanatory way? What are the criteria of being an explanatory presentation? Nevertheless I think it is a sound one. Basic principles, concepts, axioms, definitions etc. can usually be chosen in different ways. An interesting example is presented in (Mancosu 2001), where Pringsheim’s presentation of complex analysis is discussed.23 According to Mancosu, “The original approach to complex analysis defended by Pringsheim is based on the claim that only according to his method it is possible to “explain” a great number of results, which in previous approaches, in particular Cauchy’s, remain mysterious and unexplained” (Mancosu, 2001, p. 108) (my emphasis). There are three standard approaches (or perhaps: presentations or definitions) to complex analysis: (1) via the concept of complex dif  We can usually generate alternative definitions, equivalent to the original one. Quite often auxiliary lemmas, which provide alternative, equivalent characterisations of the notion in question are proved. But even if there are lots of such characterisations, we usually chose “the definition” not just by accident, but because it seems to be most natural, it sheds light on the entire field etc. 23   Pringsheim’s original papers are (Pringsheim, 1920; 1925). 22

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ferentiability; (2) via Cauchy-Riemann differential equations; (3) via the notion of (complex) power series. All these three approaches lead to the same class of functions (so all the notions of analyticity are coextensional), and the equivalence of definitions is one of the basic theorems taught in the standard complex analysis course. Pringsheim proposes a still different approach, where the notion of the mean value of taken as primitive.24 The theory of analytic functions receives a different formulation, as the notion of analyticity is defined in a different way, and the development of the theory proceeds along different lines. The difference does not of course consist in proving different theorems, but the notion of analyticity is viewed from a different angle, so to speak. According to Pringsheim, “we gain an advantage not to be underestimated over the common approach (...) – namely that basic insights which appear in Cauchy’s theory as sensational results of a mysterious mechanism performing miracles as it were, receive within our approach their natural explanation” (Pringsheim, 1925, p.v), (quotation from [Mancosu, 2001, p. 109] emphasis by Mancosu).25 Pringsheim’s approach didn’t get much popularity, and in the textbooks we find only the definition(s) of analytic functions in terms of complex differentiability, or Cauchy-Riemann equations, or (less often) in terms of power series. Nevertheless, the methodological principles and motivations of Pringsheim’s approach are very interesting from the point of view of the problem of explanation. The aim is to give such a reformulation of a theory that the known results are presented as natural – and to provide an explanation of the whole discipline. So the project is very ambitious: the aim is not just to provide an explanatory proof of   The precise definition of mean value is not important. What is important, is the fact, that it is one of notions present in the theory of analytic functions, and in the standard presentation the facts involving mean value are theorems, which are proved starting from the standard definitions. But here they are taken as basic, rather than derived truths. 25   And he also claims elsewhere “here it becomes apparent again that the application of more elementary methods gives a clearer insight into the working of the fundamental results (...)” (Pringsheim, 1920, p. 152) (quotation from Mancosu, 2001, p. 110)). 24

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a certain theorem, but to present an explanatory conceptual recasting of a theory. This is done in order to gain conceptual clarity, to identify important connections and – in this broad sense – to provide an explanation of what the core concepts and ideas of the theory are.26 The problem of explanation – both in the “local” and in the “global” case – is connected with the problem of the purity of methods. It is a common fact, that we combine methods from various mathematical disciplines in order to solve open questions. We use complex analysis in the calculus, probability theory in differential equations, algebraic geometry in number theory etc. But in some cases it might be claimed, that introducing notions from the outside (of a given mathematical discipline) somehow obscures the situation.27 Consider any combinatorial identity concerning e.g. Newton binomial coefficients.28 Such results can be usually proven by induction (the proof can be quite tedious and boring, as we have to perform long calculations), or using combinatorial interpretation (which can be quite ingenious and make the proof tricky), or perhaps using probabilistic notions (e.g. the notion of random variable, independence, limit theorems etc). Probabilistic proofs can sometimes be much simpler than the purely combinatorial ones, which can be very tedious. If we can prove a theorem by appealing to probabilistic notions and facts (say: the variance of the sum of independent random variables equals the sum of their variances), we arrive at the desired result. But from the purely combinatorial point of view, the notions of variance and independence of random variable are extraneous. Such proofs introduce   This is a general phenomenon when looking for axiomatic versions of mathematical theories. An axiomatic version of a theory is never developed from scratch, it is possible only after we have collected some experience and obtained some insights into the general structure. So looking for an acceptable axiomatic formulation of a theory can also be viewed as a kind of explanatory procedure. The data to explain are the already known results, and the explanans is provided by the axiomatic, “tidied-up” version of the theory – where the primitive concepts and the fundamental truths have been identified. 27  Arana provides an analysis of the notion of extraneousness of notion in proof in (Arana, 2009). 28   Newton binomial coefficients have a straightforward combinatorial interpretation, as the number of combinations of a given set. 26

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notions, which are not inherent to the combinatorial content of the theorem. And even if these proofs are shorter, it might be claimed that the primary combinatorial situation gets somehow obscured. These remarks apply to other areas of mathematics as well. It might be claimed, that the purity of methods enhances explanations, and at least can be one of the important factors.29 It is neither a necessary nor a sufficient condition, but nevertheless the problem of purity seems to be relevant to the discussion.30

7. Final remarks The problem of explanation in mathematics can be analysed on various levels. On one level we meet the problem of the explanatory character of proofs. Proofs play a central role in mathematics, but their role consists not only in producing derivations, but (perhaps even more importantly) in displaying the underlying conceptual structure. In particular, purely formal proofs are (at least in many cases) not satisfactory in this respect. It can be seen very easily if we consider computer-assisted proofs (especially if we consider a thought experiment where a super-computer produces a proof of an inaccessible length). An important problem in this context is also the problem of the relationship between the “ordinary mathematical language” proofs (as known from lectures, seminars and textbooks) and their formal reconstructions within e.g. ZFC. The same problem emerges when we con  Finding an elementary proof of an already known theorem can be important and illuminating. In fact, in 1950 the Fields medal was awarded to Selberg for developing methods allowing the elementary proof of the Prime Number Theorem. And of course, from the point of view of a logician, finding new proofs within weaker formal systems is an important area of study. E.g. in reverse mathematics we try to isolate weakest settheoretic axioms necessary to prove ordinary mathematical theorem ((Simpson, 1999) is the classical monograph). 30   But there is also another side of the coin: in many cases, the application of abstract methods clarifies the conceptual situation, as we can view a certain “local” fact as the expression of a general principle (e.g. fix-point principle or self-reference or a topological phenomenon etc.). 29

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sider the “global” problem of explanation, i.e. the problem of providing a satisfactory, explanatory conceptual recasting of a mathematical discipline. A formal reconstruction need not be an explanatory reconstruction (in fact – it usually isn’t). Finally, the problem of the purity of proof methods should also be taken into account in this context. The notion of explanation is important for the philosophy of mathematics – but it is still neglected in the mainstream discussions. I hope that this article has helped to convince the reader that it deserves a thorough discussion.

References Appel, K., & Haken, W. (1977). Every planar map is four colorable, part I: discharging. Illinois Journal of Mathematics, 21, 429–490. Appel, K., Haken, W., & Koch, J. (1977). Every planar map is four colorable, part II: reducibility. Illinois Journal of Mathematics, 21, 491–567. Allaire, F. (1977). Another proof of the four–color theorem. Part I. In D. McCarthy, H. C. Williams (Eds.). Proceedings of the Seventh Manitoba Conference on Numerical Mathematics and Computing (pp. 3–72). Winnipeg: Uthilitas Mathematica Pub. Arana, A. (2009). On formally measuring and eliminating extraneous notions in proofs. Philosophia Mathematica, 17, 189–207. Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114(454), 223–238. Bassler O.B. (2006). The surveyability of mathematical proof: a historical perspective. Synthese, 148, 99–133. Boolos, G. (1987). A curious inference. Journal of Philosophical Logic, 16, 1–12. Dawson, J.W., Jr. (2006). Why do mathematicians re-prove theorems. Philosophia Mathematica, III, (14), 269–286 Detlefsen, M., & Luker, M. (1980). The four color-problem and mathematical proof. Journal of Philosophy, 77, 803–820. Hales, T.C. (2000). Cannonballs and honeycombs. Notices of the American Mathematical Society, 47(4), 440–449. Hales, T.C. (2005). A proof of the Kepler conjecture. Annals of Mathematics. Second Series, 162(3), 1065–1185.

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Krakowski, I. (1980). The four-color problem reconsidered. Philosophical Studies, 38, 91–96. Levin, M. A. (1981). On Tymoczko’s argument for mathematical empiricism. Philosophical Studies, 39, 79–86. Mancosu, P. (2001). Mathematical explanation: problems and prospects. Topoi, 20, 97–117. Mancosu, P. (2008). Mathematical explanation: Why it matters. In P. Mancosu (Ed.), Philosophy of mathematical practice. Oxford: Oxford University Press. Mordell, L. (1959). Reflections of a mathematician. Montreal: Canadian Mathematical Congress. Pringsheim, A. (1920). Elementare Funktionenlehre und komplexe Integration. Sitzungsberichte der mathematisch–physikalischen Classe der k. b. Akademie der Wissenschaften zu München, 145–182. Pringsheim, A. (1925). Vorlesungen über Zahlen– und Funktionenlehre, Zweiter Band, Erste Abteilung: Grundlagen der Theorie der analytischen Funktionen einer komplexen Veränderlichen. Leipzig & Berlin: B.G. Teubner. Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7, 1999, 5–41. Rota, G.-C. (1997). The phenomenology of mathematical proof. Synthese, 111, 183–196. Simpson, S. (1999). Subsystems of second order arithmetic. New York: Springer Verlag. Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Cambridge, MA: Harvard University Press. Steiner, M. (2005). Mathematics – application and applicability. In S. Shapiro (Ed). The Oxford handbook of philosophy of mathematics and logic (pp. 625–650), Oxford: Oxford University Press. Swart, E. R. (1980). The philosophical implications of the four-color problem. American Mathematical Monthly, 87, 697–707. Teller, P. (1980). Computer proof. Journal of Philosophy, 77, 797–803. Tymoczko, T. (1979). The four-color problem and its philosophical significance. The Journal of Philosophy, 76(2), 57–83.

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Michael Heller Copernicus Center for Interdisciplinary Studies

Limits of Causal Explanations in Physics

1. Introduction There are many misunderstandings and false opinions regarding the problem of causality in physics. The prevailing view among philosophers of science is that there are no causal explanations in the sciences, and in physics in particular. What we usually term “causality” reduces, in fact, to a repeatable time sequences of events. Practicing physicists, on the other hand, often use causal language when they speak about what they are doing in their laboratories and in their theoretical research. The aim of this paper is to deal with the causality problem in physics by taking both sides of the dispute seriously. Philosophical views on causality strongly depend on Hume’s analysis of this problem and it is hard to find a philosophical paper dealing with causality without reference to Hume. This is why I feel obliged to conduct my own analysis with an eye on Hume’s philosophy. It would be too simplistic to draw far-reaching conclusions from the way physicists speak about their work; however, to totally dismiss the usage of the language would be equally imprudent, especially if we do not only listen to what physicists speak, but also look at what they are doing, i.e., if we look at the method they are using in their work. It is sometimes claimed that the greatest achievement of the mathematized empirical sciences is the method they have elaborated. This method tells us something about the world which it so successfully explores. My strategy in the present paper is to confront

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Hume’s views on causality with an analysis of this problem based on an insight into the mathematical-empirical method of modern physics. In sections 2 – 5, I focus on Hume; section 6 is a brief intermezzo on the Humean echo in contemporary philosophy; a longer section 7 confronts Hume-like philosophy with the method of modern physics.

2. Impressions and ideas Hume’s philosophy is well known (at least to specialists), but in what follows I will make an attempt to look at it from a rather specific point of view. I shall be interested in those aspects of his doctrine that later on contributed to shaping people’s regards on philosophy of science, especially on the problem of causality and causal explanations. I shall limit my analysis of Hume mainly to his An Enquiry Concerning Human Understanding (Hume, 1910) (only once referring to his A Treatise of Human Nature) since not only did he himself confess that this work superseded his earlier ideas (see Archie, 2005), but also it is exactly this work that has most often been quoted. Hume was a radical empiricist, but many of his views did not have much in common with what is usually called Humean empiricism. He believed that our cognition is based on perceptions, however he understood them broadly. They comprise everything that is somehow registered by our consciousness, such as sensual perceptions, but also our conscious thoughts. Hume divides all perceptions, with respect to “their different degrees of force and vivacity,” into two classes: impressions and ideas: “...our more lively perceptions, when we hear, or see, or feel, or love, or hate, or desire, or will are called impressions” (Hume, 1910, Sec. 2); all others are ideas. This classification is not very sharp, but it seems sufficient for Hume’s purposes. Our body is subject to the laws of inertia and to other laws of nature, but our thinking does not seem restrained by any limitations of matter: “the thought can in an instant transport us into the most dis-

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tant regions of the universe; or even beyond the universe, into the unbounded chaos, where nature is supposed to lie in total confusion” (Ibidem).1 But it is only an illusion, in fact our mind “is really confined within very narrow limits,” and all its creative power “amounts to no more than the faculty of compounding, transposing, augmenting, or diminishing the materials afforded us by the senses and experience” (Ibidem). In other words, all our ideas are but more or less crippled copies of our impressions. The conclusion is that we should beware ideas that could not be reduced to impressions. By the way, should we also be wary of Hume’s doctrine since it evidently cannot be reduced to impressions? To cope with this objection we should make recourse to the distinction between language and metalanguage, but Hume himself did not have this tool at his disposal. In his criticism of various philosophies based on ideas, Hume undertook an attempt to formulate the laws of association of ideas. From our present point of view this attempt is nothing more than a faraway presentiment of current achievements in the field of psychology and the neurocognitive sciences. Among the laws distinguished by Hume, such as the laws of resemblance and contiguity in time or place, there is a law associating ideas that we call cause with those we call effect. As we can see, the principle of causality, in Hume’s view, is not based on impressions but has its origin in an association of ideas. We are here at the root of Humean criticism of causality.

3. Matters of fact In Hume’s view, we form our convictions either by inquiring into relations between ideas or by ascertaining “matters of fact.” To the first kind belong “the sciences of Geometry, Algebra, and Arithmetic”   It is an interesting remark regarding Hume’s cosmological views: our universe is a domain of order in an otherwise infinite space of chaos. This view was not uncommon at that time. 1

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(Ibidem, Sec. 4, Part 1). We arrive at their statements through the operations of thought (“either intuitively or demonstratively”), independently of what occurs, or does not occur, in the world. We acknowledge “matter of facts” in a completely different way. “The contrary of every matter of fact is still possible; because it can never imply a contradiction” (Ibidem). Therefore, we cannot accept “matters of fact” by pure thought; and not all “matters of fact” are accepted by “the present testimony of our senses, or the records of our memory” (Ibidem). Hume thinks that philosophers have neglected this problem, and claims that “all reasonings concerning matter of fact seem to be founded on the relation of Cause and Effect” (Ibidem). Only owing to the principle of causality are we able to go beyond the testimony of our senses and memory. At this point Hume quotes an example which, later on, many thinkers shall repeat for various purposes: “A man finding a watch or any other machine in a desert island, would conclude that there had once been men in that island,” and adds the comment: “And here it is constantly supposed that there is a connexion between the present fact and that which is inferred from it. Were there nothing to bind them together, the inference would be entirely precarious” (Ibidem). The occurrence of two facts which we interpret as a cause and effect can be ascertained by senses or memory, but the causal nexus itself between these two facts cannot be ascertained in this way. Hume pronounce this view with a great conviction; he quotes several “proofs” on its behalf, but they do not go beyond an eloquent presentation of some examples and should be regarded as an intelligent rhetoric. Moreover, they are convincing only if one remains on a purely “extrinsic” understanding of classical physics. Even today, armed with a deeper understanding of modern physics, we should agree with Hume’s conclusion that the principle of causality is not a priori, but we have to look at his “proofs” in a completely different light. Here is an example of Hume’s rhetoric: “When I see, for instance, a billiard-ball moving in a straight line towards another; even suppose motion in the second ball should by accident be suggested to me, as

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the result of their contact or impulse; may I not conceive, that a hundred different events might as well follow from that cause? May not both these balls remain at absolute rest? May not the first ball return in a straight line, or leap off from the second in any line or direction? All these suppositions are consistent and conceivable. Why then should we give the preference to one, which is no more consistent or conceivable than the rest? All our reasonings a priori will never be able to show us any foundation for this preference” (Ibidem). If we remain only at the level of sensual perception, nothing could be objected against Hume’s wonder, expressed in his rhetoric questions, but physics, even at his time, was not limited to sensual perception.

4. Laws of nature Hume poses the question: Is not the principle of causality somehow rooted in the laws of nature? Yes, but “all the laws of nature, and all the operations of bodies without exception, are known only by experience,” and it is the principle of causality that organizes our experience. It is impossible to break this vicious circle. According to Hume, the principle of causality indeed plays a vital role in our arriving at the formulation of laws of nature. The main obligation of the human mind in our effort to understand the world “is to reduce the principles, productive of natural phenomena, to a greater simplicity, and to resolve the many particular effects into a few general causes… .” It is, therefore, a search for a sort of an ultimate theory. However, in Hume’s view, this search is doomed to fail. “These ultimate springs and principles are totally shut up from human curiosity and enquiry. Elasticity, gravity, cohesion of parts, communication of motion by impulse; these are probably the ultimate causes and principles which we shall ever discover in nature; and we may esteem ourselves sufficiently happy, if, by accurate enquiry and reasoning, we can trace up the particular phenomena to, or near to, these general principles.” The general conclusion cannot be but pessimistic: “as to

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the causes of these general causes, we should in vain attempt their discovery” (Ibidem). Even the use of mathematics cannot change this state of affairs. For Hume it was geometry that was the best developed part of mathematics, but even geometry is powerless in this respect. Laws of nature are discovered solely by experience, “and abstract reasonings are employed, either to assist experience in the discovery of these laws, or to determine their influence in particular instances, where it depends upon any precise degree of distance and quantity.” The inner machinery of the world is inaccessible for us. Neither mathematics, nor any other “abstract reasonings in the world could never lead us one step towards the knowledge of it” (Ibidem). The well known positivistic adage savoir pour prevoir (ascribed to Compte) is also present with Hume, but in his thinking it is entangled in speculations about causality. All our reasonings concerning the past are based on the relationship “cause-effect,” but anything we know about this relationship we know from experience, and this sort of knowledge presupposes that the future will be similar to the past. “In reality, all arguments from experience are founded on the similarity which we discover among natural objects, and by which we are induced to expect effects similar to those which we have found to follow from such objects” (Ibidem, Sec. 4, Part 2). Hume argues that the basis of this reasoning is “custom or habit.” “Perhaps we can push our enquiries no farther, or pretend to give the cause of this cause; but must rest contented with it as the ultimate principle, which we can assign, of all our conclusions from experience” (Ibidem, Sec. 5, Part 1). And to avoid any misunderstanding: “All inferences from experience, therefore, are effects of custom, not of reasoning” (Ibidem). And further: “Custom, then, is the great guide of human life. It is that principle alone which renders our experience useful to us, and makes us expect, for the future, a similar train of events with those which have appeared in the past. Without the influence of custom, we should be entirely ignorant of every matter of fact beyond what is immediately present to the memory and senses” (Ibidem).

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5. Causality and temporal sequence It is a commonplace to ascribe to Hume a view that we are unable to grasp the causal nexus between events but only their temporal sequence. All our knowledge comes from experience, and experience cannot tell us that B propter A, but only that B post A. Post hoc ergo propter hoc (“after this, therefore because of this”) is a common fallacy. These views can be found in Hume’s earlier work A Treatise of Human Nature.2 Later on Hume looked on this work with reserve, but the conviction that causality is in fact reduced to the temporal sequence is consonant with his views expressed in the Enquiry. What does experience tell us? We observe a collision of two bodies: “Motion in one body is regarded upon impulse as the cause of motion in another. When we consider these objects with utmost attention, we find only that the one body approaches the other; and that the motion of it precedes that of the other, but without any, sensible interval” (Hume, 2000, Part 3, Sec. 2). And that is all we can infer from the testimony of senses. In what we usually call causality, one may identify two elements: first, a contiguity of what we call cause and its effect and, second, the temporal precedence of what we call cause with respect to what we call its effect. Hume insists that it must be temporal succession rather than contemporaneity: “The consequence of this [of admitting contemoraneity] would be no less than the destruction of that succession of causes, which we observe in the world; and indeed, the utter annihilation of time. For if one cause were co-temporary with its effect, and this effect with its effect, and so on, it is plain there would be no such thing as succession, and all objects must be co-existent” (Ibidem). We can see here a seminal idea of the causal theory of time: without the succession of causes there would be no time. Usually this idea is ascribed to Leibniz and its newer version to Reichenbach, and Hume is not mentioned in this context.   The first anonymous edition in 1739; a contemporary edition (Hume, 2000).

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Are contiguity and succession enough to speak about causality? It seems that there should also exist a certain necessary nexus between cause and effect since an effect must be, in a sense, produced by its cause. If we approach a fire, our senses register a feeling of warmth, but our senses are not able to register any nexus between fire and our feeling, they do not perceive that fire produces or creates the feeling of warmness. The only thing we can assert based on sensual perceptions and memory is the fact that any time I approach my hand to a fire I experience the feeling of warmness. Hume speaks about “constant conjunction” between fire and this feeling. To sum up, causality can be reduced to three elements: contiguity, temporal succession and a constant conjunction between what we call cause and effect. This is what senses and memory can tell us about causality. “From the mere repetition of any past impression, even to infinity, there never will arise any new original idea, such as that of a necessary connexion; and the number of impressions has in this case no more effect than if we confined ourselves to one only” (Ibidem, Part 3, Sec. 6).

6. Humean supervenience In some influential circles of contemporary philosophy it is fashionable to be Humean. Tim Maudin characterizes this trend in the following way: “There is, in some contemporary metaphysics, an explicit preference, or desire, or in some cases demand, for ‘Humean theories.’” Humean, or ‘empiricist,’ theories of law and of chance are sought; theories that posit irreducible nomic or modal or dispositional or causal facts are dismissed as unHumean” (Maudin, 2010, p. 50). Maudin quotes David Lewis who presents “Humean supervenience” as the core idea of the Humean worldview. “It is the doctrine that all there is to the world is a vast mosaic of local matters of fact, just one little thing and then another. (…) We have geometry: a system of external relations of spatiotemporal distance between points. Maybe points of spacetime itself,

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maybe point-sized bits of matter or aether fields, maybe both. And at those points we have local qualities: perfectly natural intrinsic properties which need nothing bigger than a point at which to be instantiated. For short: we have an arrangement of qualities. And that is all. All else supervenes on that” (quotation after Ibidem, pp. 50–51). “Humean supervenience” implies the doctrine of separability which asserts that “all fundamental properties are local properties and that spatio-temporal relations are the only fundamental external physical relations” (Ibidem, p. 51). Or in a more picturesque way: “As the whole picture is dominated by nothing more than the values of individual pixels plus their spatial dispositions relative to one another, so the world as a whole is supposed to be decomposable into small bits laid out in space and time” (Ibidem). In the rest of his essay Maudin argues that such a doctrine cannot be upheld in the view of recent advancements in quantum physics. As it is well known, quantum theory strongly suggests that the world is, in fact, non-separable. “If quantum theory is even remotely on the right track, then the best physical theories will continue, as they do now, to posit fundamental non-Separable states of affairs” (Ibidem, p. 53). If so, then Maudin’s rhetoric question “Why be Humean?” acquires its force. I would go even further: “Humean supervenience” is not only spectacularly foreign to the results of quantum theory, but it is hard to be reconciled with the very method of modern physics.

7. Humean supervenience and modern physics The fashion of “being Humean” is an interesting example of the “custom or habits” remaining in drastic contrast with what happens in real science. Empiricism of the Humean type (let us not hesitate to call it naïve empiricism) cannot be reconciled with the method employed by modern physics. In principle, even Hume could see this if he looked more closely upon the method of Newtonian phys-

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ics. However, we do not reproach him for this neglect since even the most eminent physicists of those times were too busy with obtaining new results to spare time for a more detailed scrutiny of the method they were employing. This is why they so often pronounced simplified declarations. We should probably agree with Hume that in its everyday activity our brain can do nothing more than “connect, shuffle, augment or diminish” what we know from senses and memory. But in physics we have learned how to overcome this constraint. The miracle of the mathematical-empirical method consists in the fact that the mathematical model in question gives us more on its output than we gave it on its input. A good example is provided by quantum mechanics. What in this context could mean “reducing the theory to sensual perception?” Of course, the observable quantities (socalled observables) play in the crucial role in quantum mechanics, but they are represented by purely theoretical entities, the hermitian operators in a Hilbert space. Moreover, relatively few of these operators correspond to quantities that could really be measured. All the rest are “virtual observables” to which something measurable could, in principle, correspond, but we have no idea what this could be. Nevertheless, these “virtual observables” are essential elements of the formal structure of the theory; without them the theory could not work. By dealing with this abstract structure we obtain very concrete and practical results. Modern particle accelerators, the electronics industry, and the host of gadgets without which our daily life could hardly be imagined, testify to the prolific creativity of abstract thinking. If we restrict our visual horizon to sensual perceptions, then indeed it is difficult to see something more than a contiguity of cause and effect in causality, their temporal sequences, and repeatability of this sequences.3 However, this strategy is very distant from   However, we should not forget introspection that suggests that we ourselves are often the causes of various effects.

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what is going on in physics. If there is a correspondence between the mathematical structures of physical theories and the structure of the world (as the successes of physical sciences strongly suggest) then at least some inferences within this mathematical structures correspond to some temporal sequences of events in the world. Such inferences, and the temporal sequences of events corresponding to them, could rightly be termed causal dependencies. And they cannot be reduced to the contiguity of certain phenomena, their temporal succession and repeatability. Mathematical inferences within the formal structures of physical theories (at least some of them) model something that escapes sensual perception, and which is essential for the causal nexus. Contiguity and temporal successions are, as it were, external appearances of this nexus. Moreover, they need not always accompany causal dependencies. Causality, as modeled by mathematical structures, can differ from our everyday imaginations. For instance, contiguity can be replaced by distant correlations (entanglement phenomena), and temporal succession can be put into question (time delayed experiments). In the philosophy of mathematics there is a big dispute concerning the existence of mathematical objects. The view, usually referred to as mathematical Platonism, asserts that mathematical objects really exist, and even more so than material objects. Frege, one of the great supporters of this view, described mathematical objects as those which are “lacking causal powers.” Many contemporary philosophers accept this description but, unlike Frege, treat it as an argument against mathematical Platonism. If mathematical objects are “lacking causal powers,” they could not be responsible for any effects in the material world, and consequently they are superfluous in explaining the world. Michael Dummett describes this view in the following way: “It is a common complaint about abstract objects that, since they have no causal powers, they cannot explain anything, and the world would appear just the same to us if they did not exist: we can therefore have no ground to believe in their existence” (Dummett, 2002,

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p. 22).4 In light of the method employed by modern mathematical physics, both Frege’s description of mathematical objects and the argument of contemporary philosophers are totally misplaced. The detailed analysis of the mathematical-empirical method of physics shows something exactly opposite, namely that the causal powers that matter seems to possess come entirely from mathematical objects (or structures5). Let us consider an example taken from nuclear physics. In nuclear physics, a reaction is known, called beta minus, in which neutron decays into proton emitting electron and antineutrino n → p + e−− + ν e  . The theory of weak interactions tells us that that this happens because the down quark, being a part of neutron, transforms into the up quark, emitting an intermediating boson W- which, in turn, decays into an electron and antineutrino. The entire process had been read out of the mathematical structure of the physical theory of weak nuclear interactions (with a suitable interpretation of this structure) and then verified experimentally. Does this structure only describe the beta minus decay process? That is to say, material particles behave in agreement with their “nature” and it just happens that there exists a mathematical structure that correctly (but only up to a certain approximation) describes what particles are doing? Such an interpretation, although logically possible, is entirely at odds with the method of physics and its implementation in doing real science. First of all,   After presenting this view, Dummett adds the following comment: “For Frege such a complaint would reveal a crude misunderstanding. He gave as an example of an object that is abstract but perfectly objective, the equator. If you tried to explain to someone who had never heard of it what the equator was, you would certainly have to convey to him that it cannot be seen, that you cannot trip over it, and that you feel nothing when you cross it. If he then objected that everything would be exactly the same if there were no such thing as the equator, and that therefore we can have no reason for supposing it to exist, it would be clear that he had still not understood what sort of object we take the equator to be.” 5   I do not want to enter into the dispute here regarding what is ontologically prior: mathematical objects or mathematical structures. 4

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there is nothing like a “material particle.” Neutron, proton, W- boson, down and up quarks are only “places” in a mathematical structure and outside this structure are lacking any meaning. The physical “nature” of these particles is determined entirely by the mathematical structure in question (suitably interpreted). Therefore, the beta minus decay is not described by this mathematical structure, but it is prescribed by it. Even the saying that particles execute the program contained in the mathematical structure would be inadequate. Rather particles themselves are, as it were, elements of the program. What does this mean “particles themselves?” A mathematical structure gives us no insight into the “intrinsic nature” of its elements. Any element of the structure is entirely determined by the network of relations constituting the structure. If such an element is interpreted as an elementary particle, its “physical nature” is given by the entirety of relations, and only by them, in which this element enters with all other elements of the structure. Therefore, all properties of elementary particles, together with their “causal powers,” come from a given mathematical structure interpreted as the structure of a given physical theory. Mathematical structures have causal powers virtually. Causal powers become real if a given mathematical structure is interpreted as a structure of a certain part or an aspect of the world. However, this interpretation cannot be reduced to our more or less arbitrary decree; it is determined by the method physics has elaborated in the course of its history and seeks to perfect in all current research. In this sense, causal explanations do not essentially differ from the mathematicalempirical method itself. The limits of causal explanations coincide with the limits of this method.

References Hume, D. (1910). An enquiry concerning human understanding. Harvard Classics, Vol. 37. New York: Collier & Son.

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Archie, L. (2005). Hume's considered view on causality. Doi philsci-archive.pitt.edu/archive/00002247 Hume, D. (2000). A treatise of human nature. Oxford: Oxford University Press. Maudin, T. (2010). The metaphysics within physics. Oxford & New York: Oxford University Press. Dummett, M. (2002). What is mathematics about? In D. Jacquette (Ed.), Philosophy of mathematics. An anthology. Oxford: Blackwell.

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Andrzej Koleżyński AGH University of Science and Technology Copernicus Center for Interdisciplinary Studies

Pragmatism of Chemical Explanation

1. Introduction Chemistry is a science devoted to the synthesis, transformation and characterization of pure and synthesized compounds. This endeavor makes chemistry a unique and indispensable part of modern science, the fruits of which are commonly used by the whole of society. Today, the number of substances registered in the Chemical Abstracts Service Registry exceeded 83 million1 and about 15,000 new ones are added every day! This number is remarkable by itself but, what is even more incredible, the number of new substances discovered or synthesized is growing exponentially, doubling roughly every 13 years over the period spanning the last two hundred years (Schummer, 1999) and chemistry is now not only the biggest discipline, but also bigger than all other natural and social sciences and technology combined and in a single year (2000) over 900,000 chemical papers were published (Schummer, 2006).

  CAS Registry includes unique organic and inorganic chemical substances (inter alia pure substances, salts, minerals, mixtures, coordination compounds, polymers and alloys), reported in literature back to early1800s. 1

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Despite the obvious importance and a long history of chemistry as a science, philosophy of chemistry is a relatively new discipline within modern philosophy of science (the first electronic journal devoted solely to philosophy of chemistry i.e. HYLE started to be published in 1995, while the first printed journal Foundations of Philosophy in 1999). It still lacks, obviously, the depth characterizing twentieth-century philosophy of physics, but is maturing fast and becoming an autonomous and important part of the contemporary philosophy of science. It is not, however, as if the chemical concepts and theories were absent throughout the history of philosophy, since they are present in many places in the works of philosophers, either as examples proving someone’s point or as standalone topics of discussion and chemists themselves, have often engaged in discussions concerned with theoretical and methodological issues, which we easily recognize today as a part of the philosophy of chemistry. A lot of the attention of philosophy of chemistry, much like the typical philosophical study of any other particular sciences, is devoted to the analysis of central concepts and methods. The most central concept in modern chemistry – perhaps right after or at the same level of importance as a chemical substance – is chemical structure, itself inseparably linked to the idea of chemical bond. The latter is used in accounting for a wide variety of chemical phenomena ranging from basic physical and chemical properties of substances to whether a particular chemical reaction under study will occur under given conditions (e.g. temperature, pressure, substrates number and concentration etc.), what reaction pathway will be followed and which - if any - intermediate states will appear along the reaction pathway chosen. And yet, the chemical bond has always been such an elusive idea that it demands itself an explanation within the discipline and from the advent of quantum mechanics one of primary motivation for the quantum treatments of molecules (and to some extent also solids) has always been the wish for an ultimate explanation of chemical bonding. The problem of explanation and its role in science has troubled philosophers of science for

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centuries2 but serious philosophical discussions really began in the middle of the 20th century. The first important contribution to this topic is due to Carl Hempel, who popularized the so called covering law or deductive-nomological (DN) model of scientific explanation (Hempel, 1965) in which explanations are just deductive arguments linking explanandum (thing to be explained) to one or more laws. Thus a theory able to provide quantitative results in agreement with an observed phenomenon on the basis of existing laws would be equivalent to the explanation of this phenomenon. From the beginning, philosophers of science had a lot of serious objections to its validity and provided plenty of arguments and examples supporting their doubts. For statistical laws which by definition are excluded from DN model (explanation as deduction from deterministic laws), Hempel formulated inductive-statistical (IS) model where the relation between explanans and explanandum is inductive, i.e. any IS explanation will be considered good as long as its explanans results in high probability of its explanandum. The other approach, the highly influential Wesley Salmon’s statistical relevance (SR) model (Salmon, 1971), takes into account the intuitive observation that not all properties are important for explanation: in the case of every single phenomenon or process there are some statistically relevant and irrelevant properties (or relationships) and only the former are explanatory (and are named properties making a difference for an explanandum). A few years later Salmon developed a new account, the causal mechanical (CM) model of explanation, where mechanisms are composed of processes and interactions identified in terms of statistical relevance relations and transmitted marks (Salmon, 1984) and exchanges of conserved quantities (Salmon 1998), which inspired other philosophers of science to explore the notion of mechanism and mechanistic explanation in philosophy of biology and philosophy of neuroscience (Machamer et. al., 2000) as well as in the philosophy of chemistry (Goodwin, 2007, 2012). According to the third important approach to   See, e.g. (Boyd et al., 1991).

2

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scientific explanation, van Fraassen’s pragmatic model of explanation (van Fraassen, 1977), explanations are simply answers to why-questions and what really counts as relevant to a why-question depends on its pragmatic context – as I will try to show in this essay - this appears to be best fitted for the actual state of affairs in chemistry. In the following sections I will shortly describe how the crucial concepts of chemical structure and chemical bond have been developed in classical chemistry, show their importance – together with the notion of chemical reaction and reaction mechanism – for modern chemistry and then, after discussing the foundations of quantum chemical approximations necessary for the practical application of Quantum Mechanics to real systems, I will make an attempt to answer the question of if, and how, the above classical models of explanation can describe an actual explanation process in various subbranches of chemistry (since – in my opinion – the other important models of scientific explanation, i.e. unificationist approach and functional model, are not especially suitable for chemistry, they will not be discussed here).

2. Chemical structure and bonding The concept of chemical structure lies deep at the heart of modern chemistry. Originally this concept was introduced in organic chemistry, the purely experimental branch of chemistry, which emerged from practical studies of natural substances extracted from plants and animals and today focuses mostly on the chemistry of carbon compounds – mostly, since in recent decades a growing number of organic chemists have been attracted by organosilicon compounds – organic compounds containing carbon-silicon bonds, first discovered in the second half of the nineteenth century. Before the concept of molecular structure was introduced in organic chemistry, the usual way of representing newly discovered organic compounds was by means of Berzelian formulas, serving as “productive tools on paper or “paper tools”

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for creating order in the jungle of organic chemistry” (Klein, 2003, cited by Hendry, 2012a). Such formulas allowed chemists not only to describe and organize organic compounds, but also to discover an important feature of some compounds, namely isomerism. The existence of isomers, different compounds with an identical chemical formula (elemental composition) led chemists to the conclusion that the properties of a particular compound depends not only on its chemical composition (described by chemical formula) but also, in some way, on the spatial arrangement of constituting atoms, the particular way they are connected and thus interact with each other, making up a molecule or compound, i.e. they depend also on the internal structure of this compound. Such a structure, as a fixed entity with a clearly defined spatial arrangement led to another very important chemical concept, the concept of chemical bond. Today, both these concepts are universally used by chemists for the description, analysis and prediction of chemical (and to some extent physical) properties of not only molecules (in both organic and inorganic chemistry) but also of crystalline solids and glasses. But this begs the question: what is chemical structure? Why and how do atoms form particular chemical structure? The structural theory in organic chemistry began to emerge slowly in the first half of the 19th century, but its crystallization and first successful applications are due to August Kekulé (1857), who applied the idea of ‘valency’, formulated originally by Edward Frankland (1852) (the concept that atoms of any given element have definite “saturation capacity,” i.e. they can only link to limited number of atoms of the same or other element, in order to make up a molecule – the idea directly related to Dalton’s Law of Multiple Proportions [Dalton, 1808]), to “tetratomic (tetrabasic)” carbon, which – he postulated – is able to link to four other atoms, including another carbon (carbon catenation). This simple hypothesis allowed him to reduce the number of independent saturated hydrocarbons to one, homologous series. A few years later, he used these concepts together with the idea of double and triple bonds (which he introduced together with Emil Erlenmayer, in order to explain the structure of unsatu-

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rated hydrocarbons, alkenes, alkynes and their derivatives (Brock, 1992, ch. 7)) to aromatic compounds, postulating the famous hexagonal structure for benzene (Kekulé, 1865/66). Kekulé’s structural theory (mainly the concept of carbon chains) was a big step forward in the understanding, unification and taxonomy of organic compounds, but it lacked graphical representations (except the occasionally used “sausage formulas”) (Brock, ibidem). Such diagrammatic representations were first introduced in 1858 by Archibald Scott Couper and Alexander Butlerov (the latter one popularized the notion chemical structure), but only after their modification and extension done by Alexander Crum Brown in his M.S. thesis and published three years later (Brown, 1864), these graphical formulas – popularized by Edward Frankland in his textbook (Frankland, 1866a) – became “Frankland’s notation,” which later became modern structural formulas (Ritter, 2001), an indispensable part of structural theory’s language. Such two dimensional, graphical formulas have, however, serious limitations: they are unable to display the details of the spatial atomic positions within a molecule (molecular structure). Why this can be a problem was shown first by Johannes Wislicenus in 1873. His work on the isomeric lactic acids led him to the discovery of substances with identical formula and structure, but different physical properties. In the case of compounds showing such geometrical isomerism (as he called this property) structural formulas are unable to explain the observed difference in physical properties and Wislicenus suggested that this difference could be explained by the different spatial arrangement of constituent atoms in these isomers (Brock, ibidem). Wisclicenus’ idea inspired Jacobus H. van’t Hoff and led him to the concept of three dimensional structure of tetrahedrally coordinated carbon atoms accounted for optical activity and isomers (van’t Hoff, 1874/75) (he shared credit for this discovery with Joseph Le Bel, who independently developed a theory describing the relationship between optical activity and molecular structure (Le Bel, 1874)), which laid the foundations for stereochemistry. After that, the chemical structure theory started to be increasingly popular among chemists and today the con-

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cept of chemical structure (molecular as well as periodic crystal one, possessing additionally translational symmetry), a semi-rigid collection of atoms, distributed spatially in a special way and held together by a set of chemical bonds, undoubtedly one of the most far-reaching concepts ever developed in science3, is ubiquitous in chemistry. The chemical structure theory turned out to be very fruitful from the beginning of its application in organic chemistry, but at the same time, chemists began to struggle with the idea of chemical bond: there was no account for it, neither a precise description nor an explanation of its properties. The idea of a chemical compound being the result of electrical forces at play has quite a long history in chemistry and dates back to the beginning of the 19th century and Jöns Jacob Berzelius’ electrochemical theory formulated on the basis of his experiments with voltaic pile, newly invented by Alessandro Volta, which led him to the discovery that many compounds can be decomposed by electric current into a pairs of electrically opposite, electronegative acidic and electropositive basic constituents (i.e. ions). His theory was ultimately abandoned in favor of two competing theories: the theory of radicals and the theory of types, since the rise of organic chemistry provided arguments against the hypothesis that chemical composition is due to the electrical forces (Coulomb forces are non-directional and couldn’t explain the make-up of chemical structure by ionic constituents). The problem of what is responsible for the formation and stability of chemical compounds remained, however, unresolved. In 1866, the newly introduced concept of chemical structure and its diagrammatic representation led Frankland to the idea of bonds linking atoms in molecules (Frankland, 1866): By the term bond, I intend merely to give a more concrete expression to what has received various names from different chemists, such as   No generalization of science, even if we include those capable of exact mathematical statement, has ever achieved a greater success in assembling in a simple way a multitude of heterogeneous observations than this group of ideas which we call structural theory. G.N. Lewis (1923) 3

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an atomicity, an atomic power, and an equivalence. A monad is represented as an element having one bond, a dyad as an element having two bonds, etc.. It is scarcely necessary to remark by this term I do not intend to convey the idea of any material connection between the elements of a compound, the bonds actually holding the atoms of a chemical compound being, as regards their nature much more like those which connect the members of our solar system.

But as we can see, he thought of bonds as merely a useful notion and not really an existent entity of any kind. Only after J.J. Thomson’s discovery of the electron in 1897 (with his “plum pudding model” of the atom) followed by Rutherford’s experiments from 1909 and his model of an atom composed of “central charge N·e surrounded by compensating charge of N electrons” (Rutherford, 1911) and Bohr’s planetary model of the atom (Bohr, 1913), the idea of electric forces being responsible for bonding in chemical compounds was revived, paving the way for the first modern theory of a chemical bond, formulated in 1916 by G.N. Lewis (Lewis, 1916). In this seminal paper, Lewis presented the concept of cubical atom (an atom built up of a concentric series of cubes with electrons at each corner), formulated an idea of covalent bond, non-polar (in homogeneous molecules) and polar (in heterogeneous ones, possessing dipole moment) consisting of a shared pair of electrons and introduced the so called Lewis dot structures, the diagrams utilizing the octet rule defined by Richard Abegg in 1904 (Brock, 1992) and depicting valence shell electrons as dots and bonds created by these electrons as well as lone pairs of electrons that may exist in a molecule as pairs of dots. He also described double bonds as being formed by two cubical atoms sharing a face. In the case of triple bonds, which couldn’t be accounted for by cubical atom, Lewis – inspired by tetrahedral directional bonds formed by carbon – proposed the modification of cubical atom in which electron pairs where located in a middle of four cube edges building up tetrahedron. This allowed him to explain not only multiple bonds in organic compounds: single, double and triple bonds built up by two

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carbon atoms sharing corner, edge and face of tetrahedron, respectively, but also “the phenomenon of free mobility about a single bond which must always be assumed in stereochemistry” (Lewis, 1916, p. 780). A few years later these ideas were presented in a broader, historical context in his famous monograph Valence and the Structure of Atoms and Molecules, where he summarized his concept of bond in a following way4 (Lewis, 1923, p. 79): Two atoms may conform to the rule of eight, or the octet rule, not only by the transfer of electrons from one atom to another, but also by sharing one or more pairs of electrons...Two electrons thus coupled together, when lying between two atomic centers, and held jointly in the shells of the two atoms, I have considered to be the chemical bond.

Despite its simplicity and the obvious discordance with the picture of electrons in atoms and molecules provided by quantum mechanics, Lewis’ concepts of chemical bond as shared electron pair and dot structures survived – in practically an intact form – the test of time and are still commonly accepted and used by chemists in chemistry classes and laboratories, as a useful tool for the description, analysis and prediction of molecular structure (the latter especially in organic chemistry). The next natural step in classical theory of chemical bond was made in 1957 by two chemists, Ronald Gillespie and Ronald S. Nyholm, who developed a theory called VSEPR (Valence Shell Electron Pair Repulsion), devoted especially to stereochemistry of inorganic molecules, relating the number of electron pairs surrounding one or more central atoms bonded to two or more other atoms with molecular geometry, which allowed the prediction of a molecular shape assumed by given molecule (Gillespie-Nyholm, 1957).

  His monograph contains also two important concepts, namely his electron-pair theory of acid-base reactions with the definition of Lewis acid and Lewis base and the concept of hydrogen bond, formulated by his student, Maurice L. Huggins (Brock, 1992) 4

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The ideas of chemical bond presented above can be applied almost intact to supramolecular chemistry, polymer science and periodic molecular solids, but one can find there yet another type of bonding, which is due to the complexity of these compounds and the interactions between otherwise neutral molecules, i.e. weak intermolecular bonds with two types of interactions: ion-dipole (permanent and induced) and dipole-dipole interactions, generally called van der Waals interactions or forces (permanent dipole – permanent dipole, permanent dipole – induced dipole and induced dipole – induced dipole, called London dispersion forces). In the case of non-molecular periodic solids one can observe yet another type of bonding, namely ionic and metallic bonds. One could say that ionic bonds were covered by Lewis’ model as polar bond, but only in the case of crystalline solids, the old idea of ionic bonds as purely electrostatic interactions between ions with opposite charge found a hospitable environment: due to the spherical symmetry of Coulomb potential, each ion is surrounded by ions with an opposite charge, forming a periodic structure of the type depending on two main factors: relative charges of ions and their relative sizes. This direct relation between ionic bonds and a structure determined uniquely by charges and sizes of constituent ions can be described by the following five simple rules published by Linus Pauling (Pauling, 1929): 1. The Nature of Coordinated Polyhedra: A coordinated polyhedron of anions is formed about each cation, the cation-anion distance determined by the radius sum and the coordination number of the cation by the radius ratio.5 2. The Number of Polyhedra with a Common Corner. The Electrostatic Valence Principle: In a stable coordination structure the electric charge of each anion tend to compensate the strength of the electrostatic valence bonds reaching to it from the cations at   Thus for coordination number (c.n.) equal 4 we have tetrahedron, for c.n.=6 octahedron, c.n. = 8 cube, and so on ….

5

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the centers of polyhedral of which it forms a corner; that is, for each anion6:

= ζ

zi

∑= ∑s ν i

i

i i

si – strength of the electrostatic valence bond. 3. The Sharing of Edges and Faces: The presence of shared edges, and particularly of shared faces, in a coordinated structure decreases its stability; this effect is large for cations with large valence and small coordination number, and is especially large in case the radius ratio approaches the lower limit of stability of the polyhedron. 4. The Nature of Contiguous Polyhedra: In a crystal containing different cations those with large valence and small coordination number tend not to share polyhedron elements with each other. 5. The Rule of Parsimony: The number of essentially different kinds of constituents in a crystal tends to be small. These simple rules turned out to be very useful in e.g. silicate chemistry and are still in use today in teaching as well as in laboratories. The last type of bonding, i.e. metallic, can be found only in some crystals. The main property of such bonding is taking electron delocalization to its extreme – valence electrons are completely delocalized and the electronic structure of metals show no band gap between valence and conduction bands with the latter ones partially filled and appreciable density of states at Fermi energy. One can depict metallic crystal as a structure of positively charged atomic cores embedded in a sea of highly mobile valence electrons, responsible for very good   This principle was a starting point for the Bond Valence Model proposed by I.D. Brown and used successfully for various classes of materials to examine the accuracy of experimentally refined crystal structure, determine the oxidation states and identify structure and bonding instabilities, to name just the few applications – for details see (Brown, 2009) and references within. 6

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electric and thermal conductivity (this picture of the periodic distribution of positive ions stabilized by a sea of free valence electrons is quite realistic for cesium crystal, but in the case of other metals it is much more approximate (e.g in case of transition metals, where d electrons are much more localized, behaving as unpaired electrons occupying atomic orbitals and showing interesting magnetic properties).

3. Chemical reaction and the reaction mechanism I started this assay by describing chemistry as a science devoted almost completely to the synthesis, transformation and characterization of pure and synthesized compounds. It is not surprising then, that parallel to the concepts of element, structure and bond, the notion of chemical reaction emerged and the idea of chemical mechanism started to develop. In the second half of the 19th century it started to become more and more clear to organic chemists that the sole information about chemical composition and connectivity is not sufficient to uniquely characterize (at least some of) chemical compounds and one has to also take into account the particular spatial arrangement of all constituent atoms i.e. chemical structure. Analogously, in the case of many organic reactions, in order to describe them properly, the chemical structure of substrates and products has to be taken into account and chemists started to write down organic reactions between compounds using diagrammatic representations of their chemical structures. It was quite a big conceptual step, but the account for chemical reaction was not available until the last century. The first successful attempt to formulate a systematic representation of organic reactions is due to C.I. Ingold who defined reaction as electrical transaction, which takes place by virtue of some predominating constitutional affinity either for atomic nuclei or electrons on the part of a reagent; or perhaps for both, i.e. the atoms themselves (Ingold, 1934). He distinguished two categories of reactions (homolytic and heterolytic) depending on the source of a pair of bonding

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electrons (both substrates supply one electron in the former or both electrons are supplied by one of substrates in the latter) and this allowed him to define three different types of chemical reaction: addition, substitution and elimination. Additionally, he divided all reactions into three groups: unimolecular, bimolecular and intramolecular, where rate determining step involved only one species, two species or two centers reacting with each other (separated in space, but belonging to the same molecule), respectively. This classification became very popular and is still in use in organic chemistry but in order to describe any chemical reaction completely, one has to provide the answer to three questions: if the reaction occur (spontaneously or not; which involves thermodynamic analysis and energy difference between substrates and products), how (this is accounted for by the reaction mechanism) and how fast (the domain of kinetics). For any reaction to occur, the substrates involved must be converted to products, which means that existing bonds (or at least some of them) have to be broken in order to form new ones. The details of this process and structural changes taking place during such transformation are at the heart of chemistry and the knowledge about the way molecules change structurally during a given reaction is crucial for understanding these details. At first, the typical approach in organic chemistry involved geometrical considerations based on structural properties of substrates and products, usually carried out for diagrammatically written equation of reaction equilibrium with use of curved arrows with one or two barbs to indicate bond breaking and electron transfer and straight arrow to show reaction direction. Today chemists use quantum chemistry methods to determine structural changes as a path on potential energy hypersurface, calculated usually only for some chosen, possible configuration of the atoms assumed on their way from substrates to products. Because most reactions involve bond breaking and forming as consecutive or parallel processes, all intermediate configurations have higher energies. One can connect these consecutive energy points on energy hypersurface to form reaction pathway which, when reaction progresses, ascends to the con-

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figuration with maximum energy (transition state) and then descends to energy minimum related to product configuration. The difference between the initial energy and transition state energy is called activation energy (energy barrier) and is the energy which is necessary to be provided to the system for a reaction to occur. In some cases the reaction can pass along some local minima on energy hypersurface – these minima correspond to reaction intermediates. The energy differences between all energy minima (for substrates, products and intermediates) allow one to predict approximate rate of reaction (the answer to third question, how fast) and to determine which reaction step is rate determining – from all transition states, this is the one with the highest energy level on reaction profile (two dimensional graph of configuration energy as a function of reaction progress). The presented approach can be applied equally well in the case of – sometimes very complicated – chemical reactions in solids, the vast subject studied by solid state chemistry, with specific to solid state reaction types (e.g. synthesis, thermal decomposition, spinodal decomposition, topotactic reaction, sintering – with or without liquid and/or gaseous phases involved) (Smart & Moore, 2012) and kinetics (nucleation, geometric contraction, diffusion limited, reaction order etc.) (Galwey and Brown, 1998). Today, the notions of chemical structure, chemical bond, chemical reaction and reaction mechanism are used on a daily basis in virtually every subfield of chemistry as tools for the characterization of various compounds, pure and complex, natural and synthetic, in order to better understand and explain their properties and when some process involving matter transformation occurs, to describe it, recognize the reaction mechanism and kinetics and sometimes even predict the outcomes of similar processes. But unlike in the 19th and at the beginning of the 20th century, today – due to exponentially increasing computer power – chemists rely more often on the formalism of quantum mechanics applied to molecular and crystalline species. The problem is, however, that it comes with a price of necessary crucial approxi-

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mations to QM formalism to make it applicable in practice and the following section is devoted to this problem.

4. From quantum mechanics to quantum chemistry As we could see in the above (very brief) description, the notions of chemical structure and chemical bond are ubiquitous in chemistry and are used commonly by chemists, materials scientists and physicists (at least the concept of structure, since the notion of chemical bond is a constant source of controversy) in everyday scientific activity. The problem is, however, that these classical chemical concepts are mostly qualitative, which – in the eyes of some scientists – disqualifies them to some extent (or makes them suspicious and of limited usefulness). So the question arises – can quantum mechanics, mature and extremely well tested theory of micro-world be applied to chemical compounds, providing a justification for above (irreplaceable) concepts and turning them into quantitative ones, allowing application of the laws of quantum physics to scientific explanation for all chemistry endeavors, focused on synthesis and characterization of new substances? A prima facie answer – instantaneous and provided without much thinking – should be “yes.” At least, if we take seriously the following famous and – in my opinion extremely bold and evidently false – statement of Dirac (Dirac, 1929): The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.

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Today it is well known that both sentences raise serious doubts about their validity– the first one postulating – incorrectly in my opinion – the possibility of (both ontological and epistemological) reduction of chemistry to physics (conf. Hendry, 2012b) and the second one, expressing an unfulfilled hope for scientific explanation of complex system properties by means of simple approximations within quantum mechanics. Before answering the question I posed a few lines earlier, let me start first with a short description of some of the fundamental properties of Quantum Mechanics. As we know, at the heart of this theory (in its wave mechanics’ formulation) lies the time dependent Schrödinger equation of the general form: ∂Ψ ˆ i = H Ψ (1) ∂t

ˆ r, t )=T( ˆ r )+V( ˆ r, t ) is Hamiltonian of the system (this is genwhere H( eral expression and variable dependence is explicitly shown here) and Ψ – wave function (wave vector, state vector) – is continuous, square-integrable, single-valued function of the coordinates of all the particles and of time, and from which all possible predictions about the physical properties of the system can be obtained. Thus the main goal of QM consists in finding the wave function associated with a particle or a system – once we know it, we gain the access to “complete” knowledge about the system. In the case of molecules and crystals, we are usually interested in properties of the system under consideration being in its ground state (equilibrium with energy minimum) or one of its excited states. Assuming that in such case potential (and thus Hamiltonian) does not change with time i.e. V (r,t) = V (r), we can treat time and position coordinates as independent variables and express the wave function of the system as a product of two functions, first dependent only on time and second only on ψ ( r )T (t ) . As a result, we can seppositions of all particles Ψ ( r, t ) = arate variables and solve two equations independently, one for ψ ( r ) and one for T (t ) , which leads us to the famous linear homogeneous

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partial differential equation called time independent Schrödinger equation (TISE) of the form:

ˆ ψ ( r ) = Eψ ( r ) (2) H with the following solution of the full time dependent Schrödinger equation:7 Ψ( r= , t ) ψ ( r )T= (t ) ψ ( r )e − iEt /  (3) In order to solve eq. (2) we have to define first the Hamiltonian of the system:

2 2 ˆ ˆ ˆ ˆ = Hζ ( r= ∇ + Vζ ( r ) H ) T(r) + Vζ (r)= − 2m

(4)

where Tˆ is kinetic energy operator, depending only on a system (molecule, crystal) composition, i.e. number and type of constituent atoms (and has exactly the same form for every system consisting of same ˆ is potential energy operator, denumber and type of atoms) and V pending on a system composition and external conditions (external fields) i.e. environment.8 Thus general form of the Hamiltonian for the isolated system consisting of M atoms with N electrons is: ˆ = Tˆ + V ˆ = H =

2

Zα Z β e 2 2 1 2 2

-

∑ 2 α m

α

∇α -

2me

∑∇ + ∑ ∑ α α β i

i

>

rαβ

− ∑∑ α

i

Zα e 2 e 2 (5) + ∑∑ riα j i > j rij

  One can easily see that despite the fact that wave function Ψ of the system is continuously oscillating with frequency dependent on the particular eigenvalue E corresponding to the eigenfunction ψ describing given state (ground or excited), the probability density P(r,t) is constant in time (that’s why we call such solution stationary state): 2 2 P(r , t ) = Ψ (r , t ) = ψ *(r )e + iEt / ψ (r )e − iEt /  = ψ *(r ) ψ (r ) = ψ (r ) . 7

 The ζ subscript by total and potential energy operator symbolizes such environmental dependence and in usual case of isolated system is simply omitted. 8

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with the wave function of this system:

ψ ( r ) = ψ ( r1 , r2 , . . . ., rN , Rα , Rβ , . . ., RM ) (6) where α = 1,2,…..M and i = 1,2,…,N. First two terms in eq. (5) define the kinetic energy operators for atomic nuclei and electrons, respectively, while the remaining three define potential energy operators for nucleus-nucleus, nucleus-electron and electron-electron interactions respectively. Eqs (2), (5) and (6) define the general form of TISE for isolated system consisting of M atoms with N electrons. As we can see, this equation is far too complicated to be solved exactly (in fact, due to electron-electron interactions in atoms and additionally to nucleus-nucleus interaction for multi-atomic systems, the only exact, analytical solutions exist for hydrogen atom or hydrogen-like ion). This is, what probably Dirac had in mind, talking about the underlying laws of chemistry and equations being too complicated to be solved in practice. In order to be able to get any solution and make QM usable in practice, one has to follow Dirac’s suggestion and make some simplifying approximations. Historically, the first and most important

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one and with grave consequences for the ability to explain molecular or crystal structure was the so called Born-Oppenheimer Approximation (BOA) utilizing the simple observation that since nuclei are much heavier than electrons (approximately 103-105 times, depending on the number of protons and neutrons making up particular nucleus) they move much slower than electrons and thus – from electrons’ point of view– they seem to be fixed (clamped). So, the idea goes, we can treat the motions of electrons and nuclei as independent, separate variables describing their position and write the system wave function (6) as a product of two functions, the one dependent only on nuclei positions (nuclei wave function ψ N ( R ) ) and the other one on positions of electrons (electronic wave function) ψ el ( r; R ) 9 ψ ( R, r ) = ψ el (r ; R) ψ N ( R) . This allows us to solve separately two following TISE’s:

ˆ ψ ( r; R ) = E ψ ( r; R ) H el el el el (7a)



ˆ ψ ( R) = E ψ ( R) H N N N N (7b)

with the electronic and nuclear Hamiltonians defined as: 2

2

2

 Z e e ˆ = ∇i2 − ∑∑ α + ∑∑ H ∑ el (8a) 2me i riα i j i > j rij α 2  1 2 ˆ = ∇α + U ( R ) H - ∑ N (8b) 2 α mα We have to solve the eq. (7a) first, and its solutions leads us to the very important concept of a Potential Energy Hypersurface U(R): Zα Z β e 2 U R E V E ( ) = + = + ∑ ∑ r el NN el (9) α α >β αβ

  But notice: the electronic wave function depends also on the particular fixed spatial arrangement of all nuclei! 9

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Once we have the Potential Energy Surface (PES) we can solve the nuclear Schrödinger equation (7b). The solution of the nuclear TISE allows us to determine a large variety of molecular properties (e.g. vibrational energy levels, phonon dispersion in crystals, etc.). But this is just the first step, since by employing BOA we’ve simplified our Hamiltonian a little, but we still have to solve electronic TISE for our multielectron system, which is impossible to be done exactly10 and we need to make further approximations. Since we cannot calculate very complicated electron-electron interactions exactly, let us assume that electrons are independent from each other and every single electron is moving in an effective single particle potential due to all of the fixed nuclei and the average electronic density distribution of all remaining electrons. This simple idea, independent electrons approximation (IEA), was first formulated by R.D. Hartree (Hartree, 1928) and led to the development of famous Hartree-Fock SCF method, consisting of a set of self-consistent11 single-particle equations with multi-electron wave function defined as a product of one-electron wave functions (Hartree product). Since such a simple product does not fulfill the requirements for the electronic wave function of the system to be anti-symmetric (due to Pauli’s Exclusion Principle, which electrons – as indistinguish  The reason being the last term in Hamiltonian – eq. (8a), namely multicenter electron-electron interactions 11   Self consistent means that we have to do our calculation by means of self consistent field method (SCF): since every electron is regarded as moving independently in 10

n( r ')



−∑ α + dr ' and thus obeys the an effective single-particle potential VH: VH ( r ) = r ∫ r − r'  1  single particle Schrödinger equation of the form − 2 ∇ + V (r ) ϕ (r ) =ε ϕ (r ) , its energy depends on the potential VH which in turn is defined by constant potential due to all nuclei and the electronic potential due to averaged electron density of remaining electrons, we have here a sort of circular dependence and we have to solve our system of N single particle SE in a loop. We start with the definition of one electron orbitals, calculate electron density n(r) and thus the effective potential VH and then solve the eigenproblem for every single particle TISE. The new eigenvectors obtained as a solutions are used to calculate new electron density and effective potential and the entire procedure is repeated until the change in sum of eigenvalues (total energy) or other quantities of interest, like e.g. total electron density is negligible (or less than precision required for our calculations). In this way we get self consistency between electron density distribution due to effective potential and the effective potential determined by the same electron density. 2

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able fermions – have to obey) while the linear combination of different products does, the original Hartree product was later replaced by antisymmetric-by-definition Slater determinant (with (spin)orbitals as its elements, occupied by up to 2 electrons each). By applying these two approximations (BOA and IEA) to initial TISE, we’ve made a big step toward the practical applicability of QM to real systems. But one thing remains unresolved – we’ve assumed electrons to be independent particles, but we still to have define the functions describing (spin)orbitals in Slater determinant – the state vectors of these electrons. Since we do not know the exact form of such functions, we are forced to make some assumptions about their shape and usually define these functions as a linear combination of some analytical functions12 and search for the best form (the best ex-

  In principle, one can choose any functions, but in practice, due to the numerical efficiency, only limited set of various analytical functions is used – depending on the system of interest, the functions which “mimic” the electron distribution of real system best are chosen, since this can greatly decrease the number of calculation cycles and thus the time necessary to carry out the calculations. For a molecular system, the usual choice is to use atomic orbitals (AO) – one can safely assume that when atoms approach each other and start to form the molecule, the resulting electron density distribution will change, but will resemble – even in case of valence electrons – the original atomic distribution and thus the linear combination of such atomic orbitals, defining the shape of molecular orbital should be the most efficient one in molecular system. But again – since the exact form of atomic orbitals is not known, one can approximate AO by means of linear combination of some simple functions – originally Slater type orbitals were used, but later, due to their numerical inefficiency, they were substituted in most quantum chemical codes by Gaussian type orbitals, functions poorly describing the density distribution (so we need more of them in linear combination), but definitely more efficient numerically, which more than enough compensates their inefficient shape. Such Gaussian type orbitals can be single Gaussian functions (primitives) or their linear combination (contracted) ones. Currently, many predefined basis sets (depending on a definition of atomic orbitals by means of Gaussian functions) is available and the user is responsible for a choice of the basis set (theory level) best suited for the system being studied. In case of periodic solids, where one can find both extreme cases (metals and molecular crystals) and a whole lot of intermediate ones, electrons can be described by plane waves (best suited for metals) or atomic orbitals (best for molecular crystal). One can use either (having in mind that their efficiency will be quite different depending on the crystal studied) or some kind of hybrid functions like (L)APW, ASW, (L)ASO, (L) MTO, to name the few – see (Nemoskalenko, 1999) for more details. 12

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pansion coefficients for a chosen set of analytical functions) employing the variational principle. Since the original HF method does not (by definition) take into account electron correlation effects, in order to get the results in better agreement with experiments, a few extensions (Configuration Interaction, Møller-Plesset Perturbation Theory, Coupled Cluster Theory) have been proposed in following years. Such extensions improved considerably the quality of calculations, but at the expense of a significantly higher demand for computer resources and calculation time. HF Method with post HF extensions is still commonly used (especially by chemists interested in molecular compounds), but in recent years the Density Functional Theory (DFT) approach, started with the seminal paper of Hohenberg and Kohn came to dominate computational physics and a great part of computational chemistry. It follows from eq. (8a) that within the regime of BOA, the Hamiltonian of N-electron system is completely defined by N (extreme terms of eq. 8a) and external potential Vext (middle term). According to Hohenberg and Kohn (Hohenberg and Kohn, 1964), there is a one to one correspondence between total electron density of N-electron system, 2 r ) N ∫ Ψ( r, r2 , , rN dr2  drN , and external potential Vext (i.e. defined as n(= there is not only a dependence of n(r) on Vext, which is obvious, but also Vext on n(r) – not so obvious!). What is more, the total energy of the system of N interacting electrons under the influence of external potential Vext is a functional of electron density n(r)

E= [n( r )] F [n( r )] + ∫ n( r )Vext ( r )dr (10)

and reaches ITS minimum for N-electron ground state electron density in this particular, external potential Vext(r)13. An incredible result provided by DFT is a proof of the existence of an universal functional   The ground state energy can be determined by means of variational principle, since for every possible electron density n’(r), which is not an eigenvalue of external potential Vext(r), Ev [n '( r )] > Ev [n( r )] . 13

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of electron density F(n(r)) (eq. (10)), completely independent of external potential. Due to this fact, instead of dealing with N-electron wave function, we can work with a 3 dimensional function of electron density. Thus, the complexity of the problem scales linearly with system size (for HF method it scales as N2-N3, and for post HF extensions even as N7). Unfortunately, the exact form of this functional is unknown and shortly after their first paper, Kohn and Sham (KohnSham, 1965) reformulated the above problem, substituting the system of N interacting electrons with a reference system of N non-interacting electrons, but having the same total electron density distribution, which in the latter case is defined as a sum of one-electron orbitals (Kohn-Sham orbitals): N /2



n( r ) = 2∑ ψ i ( r ) (11) 2

i =1

ψi(r) are the solutions of a system of one-electron Schrödinger (KohnSham) equations14

 2 2  ∇ + VKS  ψ i ( r ) =ε i ψ i ( r ) (12) −  2m 

* obeying the orthonormality condition ∫ ψ i ( r ) ψ j ( r )dr = δ ij . In this way the problem of finding the exact form of universal functional F(n(r)) was substituted by a problem of finding the Kohn-Sham functional VKS, defined as F [n( r )] =Ts [n( r )] + EH [n( r )] + E XC [n( r )] , where consecutive terms correspond to the kinetic energy of the system of N non-interacting electrons, classical Hartree electrostatic energy and exchange-correlation energy15. Such partitioning of the universal functional F(n(r)) allows us to separate the first two terms 14   The existence of unambiguous Kohn-Sham potential VKS with the ground state electron density n(r) is a direct consequence of the Hohenberg-Kohn theorem. Analogously to HF equations, due to the circular dependence of one-electron eigenvectors ψ i ( r ) on KS potential, which, in turn, depends on n(r), depending – via eq. (11) – on all eigenvectors ψ i ( r ) , this set of equations (KS equations) have to be solved self-consistently. 15   Consisting of non-classical electron-electron interaction term and a difference in kinetic energy of N-electron interacting and non-interacting systems.

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which can be calculated exactly (and account for the lion’s share of the total energy) and the third one, covering all of the complex effects due to multi-electronic interactions. Our entire ignorance of the system is contained in this last term and – since we do not know its exact form – we have to use various approximate functional in actual calculations.16 It follows from the above short description that the concept of chemical structure is clearly present at the heart of both classical Quantum Mechanics with multi-electron wave function describing the state of the system and Density Functional Theory with 3-D total electron density, but neither quantum mechanics and DFT nor their approximations (HF and KS methods) do not account for chemical structure: they do not have it as an outcome, a result of calculation. It is, in fact, the other way around – the chemical structure is necessary input data in both theories. Without a pre-assumed molecular (or crystal) structure these theories simply cannot provide us with any useful results. But there is also a difference between these two approaches. Quantum mechanics assumes nothing about the structure and treats molecule or crystal as set of particles interacting via Coulomb forces and Born-Oppenheimer approximation becomes absolutely necessary in description of any real system, while DFT starts, in fact, from Born-Oppenheimer approximation, which lies at the heart of Hohenberg-Kohn theorems (one to one correspondence between total electron density and external potential, which in case of isolated system is simply the Coulomb potential generated by spatially distributed atomic cores). Their failure in the case of chemical bonds is even more severe – both approaches, HF and KS tell us nothing about chemical bonds, they simply do not use it, define it, account for it – chemical bond is   A lot of scientists search for the best approximation to this functional. Unfortunately, the rule one-fits-for-all doesn’t apply here and today many different approximations, suitable for different purposes, are available. This may be seen as a serious drawback, but in practice only a limited number of such approximate functionals are used in most cases and their advantages and drawback are well known and accepted. 16

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non-existent for them. Both can provide us with some information about bonding (e.g. cohesion in crystal) but this is just global, overall information, based on a difference between the total energy of a system and the sum of total energies of its all constituents treated as isolated atoms. This is – of course – very important and useful information, but it cannot account for local properties, i.e. properties of particular chemical bonds. From the advent of Quantum Mechanics physicists and chemists tried to apply its formalism to a simple molecular system in order to explain chemical bonding, which led to the development of two – opposite at first sight – approximations:17 Valence Bond Theory and Molecular Orbital Theory. In the first one, proposed by Linus Pauling (Pauling, 1928) inspired by Lewis’ idea of covalent bond and Heitler-London’s treatment of H2 molecule (Heitler and London, 1927), bonding in molecules is realized by overlapping atomic orbitals occupied by single electrons (which resemble Lewis’ idea of shared electron pairs). In order to achieve better atomic orbital overlapping in molecules (the better overlapping, the lower total energy) Pauling introduced the concept of hybridization, a mixing of the atomic orbitals into new hybrid orbitals with different shape, more suitable for electron pairing in 3D molecular systems. This concept, together with the idea of resonance structures, allowed him to define resulting molecular wave function as a superposition of pure states corresponding to classical canonical forms (Lewis’s resonance structures). The resulting wave function, the resonance hybrid, gives the results which are in better agreement with the experiment, but the concept of local, well defined bond is completely lost on the way. In the Molecular Orbital approach developed by Hund and Mulliken, instead of overlapping atomic orbitals, electrons are completely delocalized and occupy molecular orbitals, spanning the entire molecule. The first and still very popular approximation of molecular orbitals used in practical applications was the so called linear combina17   Detailed description of both approximations can be found in every quantum chemistry textbook.

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tion of atomic orbitals aproximation (LCAO-MO), but today the other functions (e.g. plane waves or hybrids in solids) are used as well. Molecular Orbital Theory turned out to be very successful and nowadays – in one version or the other – almost completely dominates quantum chemistry and physics. But again – the classical idea of chemical bond as localized directional entity, an irreplaceable tool for understanding chemical structure, symmetry and reactivity in organic molecules since a middle of ninety century, is virtually non-existent in this method (and similarly in DFT with Kohn-Sham orbitals, analogous to molecular orbitals).

5. Are classical models of explanation suitable for chemistry? The concept of chemical structure and bonding is omnipresent in chemists’ attempts to explain the properties of matter and one can wonder, if the way chemical explanation is carried out, can be described by one of standard models of scientific explanation developed by philosophers in second half of 20th century. Quantum mechanics is a theory which proved to be extremely successful in description of micro-world and thus DN model with the laws of quantum mechanics as explanans immediately comes to mind as the most suitable one. The problem is, however, that – as we have seen in previous section – strict formalism of quantum mechanics is not used in chemistry at all. In order to make QM useful in practice, some crucial simplifications have to be done, resulting in the development of Quantum Chemistry (QCh), which differs significantly from QM: quantum chemistry, as a model describing matter in the atomic scale, is based on assumptions which – to some extent – are even contradictory to those underpinning QM (treating molecular or crystal system as isolated one, electrons and nuclei as moving independently and electrons as independent to each other, the latter resulting in substituting N-electron wave function with a product of N one-electron ones and instead of solv-

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ing one TISE, one has to solve a system of N TISE equations). Since QM postulates that all particles are correlated, the model describing a molecule or crystal in QCh is drastically impoverished and in consequence is valid only on from-case-to-case basis i.e. in combination with some conditions deduced not from QM but from empirical results18 and such conditions cannot be considered as “laws” in classical sense and thus QCh fits neither DN nor causal model of explanation (Hunger, 2006). But the problem is even more grave – quantum mechanics poses a challenge to the very notion of molecular shape, which doesn’t seem to arise unless it is put in by hand via BO approximation. Therefore, one of the crucial concepts of modern chemistry is not an outcome of the theory, but an initial value put into the QCh framework by hand (Woolley, 1985) and resulting chemical structure corresponds to global minimum on the BO potential energy hypersurface.19 But – of course – this does not mean that “the central idea of classical chemistry – namely molecular structure – is somehow ‘wrong’. Nor does it mean that quantum mechanics is ‘wrong’. Both assertions are plainly ludicrous. What it does mean is that the eigenvalues and eigenfunctions of the Coulomb Hamiltonian for a collection of electrons and nuclei – the notional starting point of quantum  “The underlying theory, the N-particle Schrödinger equation, is believed because of the success of wave mechanics in explaining the properties of simple systems, the hydrogen atom, the hydrogen molecule and so on. For more complicated systems we use approximations – comparison with experiment is then regarded as a test of the approximation, not of the theory. We are doing here a remarkable sort of empirical mathematics, testing approximations not by examining the situation mathematically but by doing experiments! Of course the fact that the Schrödinger equation gives a good account of simple systems does not as a matter of pure logic entitle us to infer its accuracy in treating complicated systems, so there is an extra inductive step here leading us from simple to complicated systems” (emphasis added) (Redhead, 1980). 19  “(W)e have to specify an appropriate starting conformation to make sure that the geometry optimization will not get stuck in a local minimum on the Born–Oppenheimer hypersurface. More empirical knowledge, in form of the chemist’s intuition and/ or some experimental results about typical bond lengths and angles, enters the whole enterprise. In a proper DN-explanation this knowledge should stand at the end of a deduction and not at its beginning. It is an ironic consequence of this that the empirical validation of ab initio models has not been seen as a validation of the underlying theory but rather as a validation of the introduced approximations” (Hunger, 2006) 18

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chemistry – without the Born-Oppenheimer approximation, (or equivalently, a conscious decision to disregard the indistinguishability of identical nuclei) do not provide a basis for a quantum theory of chemistry” (Sutcliffe and Wooley, 2006). While for chemists the problem of quantum mechanics’ inability to predict chemical structure creates an unsolvable conundrum, for the physicist, not especially interested in the precise chemical nature of a matter (local properties, bonds, local structure, local order, reactivity etc.) and occupied mostly with global properties of condensed matter (electrical, magnetic, optical: e.g. electrical conductivity, superconductivity, thermal conductivity, phonons, stress tensor, EOS – all more or less ignoring identity of atomic constituent), studied by diffraction experiments and various spectroscopic methods, this is a much less important problem. But again – molecular structure and bonding are irreplaceable ideas in chemistry and the fact is, that DN and causal models of scientific explanation cannot be applied directly to chemical explanation. Another commonly used and indispensable notion in chemistry, i.e. chemical reaction and the explanation of reaction mechanism and thus elucidation of its kinetics seem to fit best the causal mechanical (CM) model of scientific explanation. The notion of mechanism is commonly used in both, biology and chemistry, but in a quite different way. In biology, mechanisms play central role in a study of living organisms, biologists attempt to explain various types of phenomena by discovering mechanisms which “(…) are sought to explain how a phenomenon comes about or how some significant process works. (…) Mechanisms are entities and activities organized such that they are productive of regular changes from start or set-up to finish or termination conditions” (Machamer et. al., 2000). Once such mechanism is proposed, it can be tested and when it holds up, it may be used for controlling and/or designing certain related biological entities (Goodwin, 2012), but at its heart, the role of mechanism in biology is to “(…) describe at some appropriate level the evolutionarily contingent process by which some important biological operation or function is realized” (Ibidem). In chemistry, on the other hand, one should

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think about a mechanism and its role as a tool of inquiry instead of its goal. Obviously, chemists are very interested in finding how and under what conditions a given reaction occurs, but since it is usually impossible to grasp the whole picture, i.e. a complete set of interdependencies of interactions among substrates, products, reaction intermediates (working sometimes as catalysts or inhibitors), reaction conditions (temperature, concentration, pressure) and kinetics, they focus their attention mostly on those aspects of a chemical reaction which are crucial for “understanding” the transformation process and can be useful in solving given synthesis problems and – in some cases – allow them to use it for inventing novel compounds. In case of organic chemistry, Goodwin distinguished two different types of reaction mechanisms, i.e. thick and thin mechanism (Ibidem). The former deals with a continuous reaction pathway i.e. line connecting consecutive points on potential energy hypersurface related to intermediate atomic configuration and describing their changes during given chemical reaction (and fits quite well with Salmon’s account of causal processes as continuous transfers of conserved quantities [Salmon, 1984]), while the latter focuses only on discrete set of points on energy surface and the reaction is decomposed into a set of reaction intermediates, semi-stable molecular species (corresponding to some local extrema on energy hypersurface) ultimately yielding reaction products.20 Thick mechanism seems to describe the actual reaction pathway, but since its experimental confirmation is practically impossible and such explanation is not always necessary, organic chemists very often restrict themselves to thin mechanism and focus on characterization of only a few specific structures (transition states and stable reaction intermediates) as this suffices to produce a satisfactory chemical explanation of a given reaction and its mechanism, identify the reaction rate determining step as well as allow predictions in the   Which fits much better with a notion of mechanisms as being “composed of both entities (with their properties) and activities. Activities are the producers of change. Entities are the things that engage in activities. Activities usually require that entities have specific types of properties” (Machamer, 2000) 20

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case of similar reactions. According to Goodwin, it is necessary to use both conceptions of mechanism in organic chemistry, since the “thick conception of mechanism allows for the application of the theoretical models of transformations to chemical reactions, while the thin conception facilitates a structural analysis of the energetics of the reaction” (Goodwin, 2012) and abandoning either of them would impoverish the explanation of chemical reactions significantly. In solid state chemistry, the explanation of the reaction and determination of its mechanism is definitely more complicated than in organic chemistry (or inorganic, for that matter), since due to the number of factors influencing the reaction (number of phases, various mass transport mechanisms, 1D and 2D defects, external conditions – temperature, pressure, etc.), it is quite often very difficult to even determine the main reaction mechanism (sometimes there are more than one), but one can find a clear analogy with Goodwin’s idea of thin and thick mechanisms. Solid state chemists use various experimental (XRD and neutron diffraction, thermal analysis and spectroscopic methods – IR, Raman, NMR, etc.) and theoretical (pure geometrical considerations useful in case of topotactic reactions, e.g. thermal decomposition of alkali metal oxalates, ab initio calculations, molecular dynamics, diffusion reaction simulations, semi-empirical continuous models of reaction mechanism, etc.) methods to determine a reaction pathway, dominating mechanism and kinetics, to characterize transition states (activation barrier) and structural properties of intermediate and final species. So basically, just like in organic chemistry, due to the fact that it is simply impossible to determine full continuous pathway along which chemical reaction in solid proceeds, solid state chemists are forced to constrain themselves to characterize only discrete points on energy hypersurface, corresponding to most important intermediate species and final products (and usually only the latter, since it is very difficult, or even impossible in many cases of heterogeneous reactions, to determine such reaction intermediates). So, returning to the question posed in the title of this section: taking into account the above considerations the answer for this question

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is simply no – all but one (the pragmatic model of explanation – discussed in next section) of the classical models of scientific explanation, seem to be inapplicable (at least not directly applicable, without modifications) in a description of the actual way chemists explain the properties of compounds and chemical reactions. But how about the pragmatic approach?

6. A pragmatic approach to explanation in chemistry According to van Fraassen, explanations are answers to why-questions and, of relevance for the answer to why-question, depend on the context in which it was asked and on the particular interest of the person asking and thus there is no objective basis for one, generally applicable selection (van Fraassen, 1970). His stance was even more extreme, as he challenged the notion that scientific explanation is even the primary aim of scientific inquiry. Most practicing chemists will agree without hesitation with the statement that the explanation they seek is of the form of context dependent why-question, but definitely not that it is as subjective as van Fraassen claimed. When chemists face the practical problem, i.e. why-question related to the actual chemical situation they start with the definition of the goals they want or have to achieve, followed by a well-thought-out strategy to achieve them. This includes an analysis and determination of what they need to do, what is the best method to obtain the given material, how the process of synthesis has to be carried out, which additional steps have to be undertaken, what kind of experimental measurements will be necessary to characterize the obtained products, if theoretical calculations can provide useful information which will allow them to achieve the goal more efficiently, etc. Chemists don’t look for an ultimate explanation of some reaction or material properties nor do they use experiments to prove some model or theory – it is the other way around: they need some tools to achieve their goals and thus use experiments, measurement techniques and theory as tools.

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And as every good craftsman uses appropriate tools (and only those tools which are necessary and the most efficient) to manufacture a required product, chemists will use various synthesis and experimental methods, models, representations, theories,21 computers with particular software, etc., to obtain a desired product, and will use only those which are absolutely necessary – they will not use computationally very demanding ab initio calculations if simpler, semi-empirical or empirical models are sufficient to get the right answers concerning reaction mechanism details or materials properties.22 The same applies to experimental methods used for material characterization – only those methods will be used which can provide answers to partial whyquestions concerning the properties of a final product (e.g. in the case of solid oxide fuel cells materials – electrical /ionic and electronic/ conductivity and for thermoelectric materials – electrical and thermal conductivity, carrier concentration and Seebeck coefficient; the properties studied, in both cases, in relation to electronic structure and its dependence on chemical composition, defects type, admixture location and thus usually, X-Ray diffraction, IR, Raman and impedance spectroscopy will be, among others, employed) or revealing the information about possible reaction mechanism, reaction intermediates, phase transitions, mass transport mechanisms. Let me emphasize this once more – chemists use theory and experiment to get the answers   There are also, however, some chemists, mainly computational chemists – aiming at developing chemical theory based on systematic refinement and decreasing number of approximations – the approach called Galilean idealization (Weisberg, 2012). Weisberg quotes the opposite stance expressed by one of the most prominent theoretical chemist, Roald Hoffman: “I took a different turn, moving from being a calculator . . . to being an explainer, the builder of simple molecular orbital models ... (and) I feel that actually there is a deeper need than before for (this) kind of work ... analyzing a phenomenon within its own discipline and seeing its relationship to other concepts of equal complexity.” 22  “Why then do not we just talk about high-level theoretical calculations and ignore the simple theory? We must choose the model that is sufficiently accurate for our computational purposes, yet still simple enough that we have some understanding of what the model describes. Otherwise, the model is a black box, and we have no understanding of what it does, perhaps even no idea whether the answers it produces are physically reasonable” (Carroll, quoted by Weisberg, 2012, p. 359). 21

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required in order to realize their goals (synthesis and characterization of functional materials possessing specific, precisely defined properties) in most efficient way. And if simpler tools will turn out to be sufficient, they will not try to use more sophisticated ones to get more precise values, since this will be costly and completely unnecessary from the point of view of the aim of their work.23 Let me show a few practical examples: physicists, when doing ab initio calculations, usually optimize the geometry of the system, since this is required by approximations they use. Chemists, on the other hand, sometimes need to use experimental geometry (e.g. from XRD data) in order to analyze the local properties of the system (bonding, geometry, strains induced by deviation of the structure from ideal one), which is important for the determination of places in the structure most prone to initiate chemical reaction. But when they want to analyze other properties, e.g. IR and Raman spectra, in order to assign bands in experimental spectra to particular normal modes of vibration and visualize such modes, they do – of course – carry out these simulations for optimized geometry. Moreover – when they use HF method, the resulting spectrum is usually compressed slightly a posteriori (how much it is scaled in practice, depends on the level of theory, i.e. basis set used in calculations and default values for various basis sets are available) in   In case of quantum chemistry, Coulson describe this attitude in a following way: “(t)he role of quantum chemistry is to understand these concepts and show what are the essential features in chemical behavior. These people are anxious to be told why, when one vinyl group is added to a conjugated chain, the UV absorption usually shifts to the red; they are not concerned with calculating this shift to the nearest angstrom; all they want is that it should be possible to calculate the shift sufficiently accurately that they can be sure that they really do possess the fundamental reason for the shift. Or, to take another example, they want to know why the HF bond is so strong, when the FF bond is so weak. They are content to let spectroscopists or physical chemists make the measurements; they expect from the quantum mechanician that he will explain why the difference exists. But any explanation why must be given in terms which are regarded as adequate or suitable. So the explanation must not be that the computer shows that D(H–F) >> D(F–F), since this is not an explanation at all, but merely a confirmation of experiment. Any acceptable explanation must be in terms of repulsions between nonbonding electrons, dispersion forces between the atomic cores, hybridization and ionic character” (Coulson, 1960), emphasis added. 23

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order to get better agreement with the experiment.24 For many years, in order to calculate force constants and frequencies of normal modes of vibrations (thus IR and Raman spectra), chemists used Wilson’s GF method defining various idealized force fields suitable for different systems or their sub-parts. When quantum chemical packages allowing the simulation of vibrational spectra started to be commonly available, this method became obsolete, but even today, if one wants to simulate vibrational spectrum for a system, which is too big even for today’s supercomputers, the usual way to get some insight into their properties, used in the past with success when employing Wilson GF method, is to cut out a smaller part of the system, a cluster or subsystem and carry out respective calculations for such a simplified system with the hope that the results will still provide some important information (which is indeed, quite often, the case). To put this last point in a numerical perspective: in case of zeolites, microporous, aluminosilicate minerals, even for simplest one, Linde Type A zeolite, a unit cell contains over eight hundred atoms and the calculation of vibrational spectra would require tens of GB of computer memory and would take many months of continuous calculations,25 which is – at least for now – completely impractical and one can try to carry out such calculations for simplified structural elements, i.e. sodalite cage, aluminosilicate sublattice etc. assuming that resulting important and characteristic vibrations (their frequencies and intensities) will be only slightly modified providing – despite the high level of idealization – useful information, which will allow a better understanding of experimental spectra of natural or synthesized counterparts. And the last example: I’ve mentioned earlier that in order to get practical knowledge about the reaction and its mechanism for a given chemi  This procedure cannot be used within DFT formalism, since unlike in HF method where deviations of calculated Hessian eigenvalues (frequencies of normal modes of vibrations) have the same sign and are proportional to absolute value of these eigenvalues, in DFT these deviations are random with opposite signs. 25   Rough estimates done by myself for Crystal09 and CASTEP software packages, running on supercomputer located in Wroclaw Supercomputer Center, a part of PLGRID infrastructure. 24

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cal reaction, chemists do not need to possess the information about the entire reaction pathway on energy hypersurface, since only a few relevant configurations really matters. In practice, the analysis of such a reaction and reaction mechanism and identification of transition states, reaction intermediates and the final product(s) can be done by means of geometrical (structural) considerations, by quantum chemical calculations carried out for relevant configurations, by the CarParinello method, by molecular mechanics methods, by Monte Carlo molecular modeling, by employing ELF topological analysis combined with Rene Thom’s catastrophe theory,26 or any other method, which chemists will find effective for this particular system – it is up to them to choose the appropriate method, usually as a compromise between required accuracy, resources and effectiveness in providing required explanation.27

7. Conclusions Classical deductive-nomological and causal-mechanistic models of explanation proposed by Hempel and Salmon seem to be not especially appropriate for chemical explanation. While the former fails completely, the latter can be used – to some extent – to describe chemical reactions and their explanation by means of reaction mechanism, in the form of a thick and thin mechanism corresponding to Salmon’s CM and Machamer’s models of scientific explanation. The actual form of chemical explanation fits best van Fraassen’s pragmatic model of explanation with one stipulation that such an explanation is not as personal and devoid of value for science, as he claims. Chemists use explanation in order to better understand chemical reactions and the properties of their product and to be able to easily carry out   See for example (Krokidis et al., 1997)   The most sophisticated and complicated methods are not always the best ones as a source of insight and explanation due to epistemic opacity of simulations, conf. (Humphreys, 2004) 26 27

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similar ones and more efficiently synthesize new substances and modify existing ones to gain new, desired properties. They often choose simplified models and idealizations not because they are too lazy to learn respective mathematical formalism, but because normally, the vast majority of systems they are dealing with are much too complicated and depend on too many interrelated variables and factors in order to be able to be treated without simplifications and approximations. Moreover, the outcomes of a given reaction or the properties of some products depend strongly on the way the reaction proceeds, small variations of temperature, concentration, pressure, on small changes in initial substrate composition, mixing method and efficiency (in case of reactions in solid state also on grain sizes, initial grinding and mixing methods etc.) which are virtually impossible to be taken into account by mathematical formalism. On the other hand, due to “structural stability” and insensitivity to (some of the) idealizations and abstractions (used by the way all the time by physicists, as well), meaning that some of these variables and factors can be safely ignored, chemists – in order to better understand the properties of the system under study, i.e. chemical compound or chemical reactions – can focus on most important (and relevant to reaction mechanism) factors only. Thus the pragmatic approach to explanation shown by chemists seems to be fully justified – it is absolutely pointless to waste time and energy to achieve an impossible level of explanation (or possible partially or to very small extent), if the outcome in case of success will not provide any substantially new insight and understanding of the properties of the system of interest.

References Bohr, N. (1913). On the constitution of atoms and molecules, Part I. Philosophical Magazine, 26(151). 1–24. Brock, W. H. (1992). The Fontana history of chemistry. London: HarperCollins Publishers Ltd.

Concept_AF.indd 172

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173

Brown, A. C. (1864). On the theory of isomeric compounds. Transactions of the Royal Society of Edinburgh, 23, 707–720. Brown, I. D. (2009). Recent developments in the methods and applications of the bond valence model. Chem Rev., 109(12), 6858–6919. Coulson, C. A. (1960). Present state of molecular structure calculations. Rev. Mod. Phys., 32, 170–177. Dirac, P. A. M. (1929). Quantum mechanics of many-electron systems. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 123(792), 714–733. Dalton, J. (1808). New system of chemical philosophy. Manchester. Frankland, E. (1852). On a new series of organic bodies containing metals. Philosophical Transactions of the Royal Society of London, 142, 417–444. Frankland, E, (1866a). Lecture notes for chemical students: Embracing mineral and organic chemistry. London: John Van Voorst. Frankland, E. (1866b). Contributions to the notation of organic and inorganic compounds. J. Chem Soc., 19, 377–395. Galwey, A. K., & Brown, M. E. (1998). Kinetic background to thermal analysis and calorimetry, Chapter 3. In M. E. Brown (Ed.), Handbook of thermal analysis and calorimetry, Vol. 1, Principles and practice. Amsterdam: Elsevier Science B.V. Gillespie-Nyholm, R. J., & Nyholm, R. S. (1957). Inorganic stereochemistry. Quart. Rev., 11, 339–380. Goodwin, W. M. (2007). Structural formulas and explanation in organic chemistry. Found. Chem., 10, 117–127. Goodwin, W. M. (2012). Mechanisms and chemical reaction. In A. I. Woody, R. F. Hendry, & P. Needham (Eds.), Handbook of the philosophy of science, Vol 6: Philosophy of chemistry (pp. 309–327). Amsterdam: North Holland. Hartree, D. R. (1928). The wave mechanics of an atom with a non-Coulomb central field. Math. Proc. Cambridge Phil. Soc., 24, 89–110. Hendry, R. F. (2012a). The chemical bond. In A. I. Woody, R. F. Hendry, & P. Needham (Eds.), Handbook of the philosophy of science, Vol 6: Philosophy of Chemistry (pp. 293–307). Amsterdam: North Holland. Hendry, R. F. (2012b). Reduction, Emergence and Physicalism. In A. I. Woody, R. F. Hendry, & P. Needham (Eds.), Handbook of the philosophy of science, Vol 6: Philosophy of chemistry (pp. 367–386). Amsterdam: North Holland.

Concept_AF.indd 173

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174

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Heitler, W., & London, F. (1927). Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik. Zeitschrift für Physik, 44(6–7), 455–472. Hempel, C. (1965). Aspects of scientific explanation and other essays in the philosophy of science. New York: Free Press. Hohenberg, P., & Kohn, W. (1964). Inhomogeneous electron gas. Phys. Rev. B, 136(3), 864–871. Humphreys, P. (2004). Extending ourselves. New York: Oxford University Press. Hund, F., & Mulliken, R. S. (1928). Zeitschrift für Physik, 51 (1928), 759; 63 (1930), 719; R.S. Mulliken, Phys. Rev., 32 (1928) 186, 761; 33 (1929), 730; J. Chem. Phys., 1 (1933) 492; 3 (1935), 375. Hunger, J. (2006). How classical models of explanation fail to cope with chemistry – the case of molecular modeling. In D. Baird et al. (Eds.), Boston studies in the philosophy and history of science, Vol. 242, Philosophy of chemistry. Synthesis of a new discipline (pp. 129–156). Dordrecht: Springer. Ingold, C. K. (1934). Principles of an electronic theory of organic reactions. Chem. Rev., 15, 238–274. Kekulé, F. A. (1857). Über die s. g. gepaarten Verbindungen und die Theorie der mehratomigen Radicale. Liebigs Annalen der Chemie und Pharmazie, 104(2), 129–150. Kekulé, F. A. (1865). Sur la constitution des substances aromatiques. Bulletin de la Societe Chimique de Paris, 3, 98–110. Kekulé, F. A. (1866). Untersuchungen über aromatische Verbindungen. Liebigs Annalen der Chemie und Pharmazie, 137(2), 129–197. Klein, U. (2003). Experiments, models and paper tools: Cultures of organic chemistry in the nineteenth century. Stanford, CA: Stanford University Press. Kohn, W., & Sham, L. J. (1965). Self-consistent equations including exchange and correlation effects. Phys. Rev. A, 140(4), 1133–1138. Krokidis, X., Noury, S., & Silvi, B. (1997). Characterization of elementary chemical processes by catastrophe theory. J. Phys. Chem. A, 101, 7277–7282. Le Bel, J. A., (1874). Sur les relations qui existent entre les formules atomiques des corps organiques, et le pouvoir rotatoire de leurs dissolutions. Bull. Soc. Chim. Fr., 22, 337–347. Lewis, G. N. (1916). The atom and the molecule. J. Am. Chem. Soc., 38(4), 762–785.

Concept_AF.indd 174

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Pragmatism of Chemical Explanation

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Lewis, G. N. (1923). Valence and the structure of atoms and molecules. New York: Chemical Catalog Co. Machamer, P., Darden, L., & Craver, C. F. (2000). Thinking about mechanisms. Phil. Sci., 67(1), 1–25. Nemoshkalenko, V. V., & Antonov, N. V. (1999). Computational methods in solid state physics. Boca Raton, FL: CRC Press. Pauling, L. (1928). The nature of the chemical bond and the structure of molecules and crystals: An introduction to modern structural chemistry. Ithaca: Cornell University Press; 3 edition. Pauling, L. (1929). The principles determining the structure of complex ionic crystals. J. Am. Chem. Soc., 51(4), 1010–1026. Redhead, M. (1980). Models in physics. British J. Phil. Sci., 31, 145–163. Ritter, C. (2001). An early history of Alexander Crum Brown’s graphical formulas. In Boston studies in the philosophy and history of science, Vol. 222, Tools and modes of representation in the laboratory sciences, Kluwer Academic Publishers. Rutherford, E. (1911). The scattering of α and β particles by matter and the structure of the atom. Philosophical Magazine, Series 6, Vol. 21, 669– 688. Salmon, W. (1971). Statistical explanation. In W. Salmon (Ed.), Statistical explanation and statistical relevance (pp. 29–87). Pittsburgh: University of Pittsburgh Press. Salmon, W. (1984). Scientific explanation and the causal structure of the world. Princeton: Princeton University Press. Salmon, W. (1998). Causality and explanation. New York: Oxford University Press. Schummer, J. (1999). Coping with the growth of chemical knowledge: Challenges for chemistry documentation, education, and working chemists. Educación Quimica, 10, 92–101. Schummer, J. (2006). The philosophy of chemistry. From infancy toward maturity. In D. Baird et al. (Eds.). Boston studies in the philosophy and history of science, 242, Philosophy of chemistry. Synthesis of a new discipline. Dordrecht: Springer. Smart, L. E., & Moore, E. A. (2012). Solid state chemistry: An introduction. Fourth Edition. CRC Press, Taylor & Francis Group. Sutcliffe, B. T., & Wooley, R. G. (2012). Atoms and molecules in classical chemistry and quantum mechanics. In A. I. Woody, R. F. Hendry, P. Needham (Eds.), Handbook of the philosophy of science, Vol 6, Philosophy of chemistry (pp. 387–426). Amsterdam: North Holland.

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van Fraassen, B. C. (1977). The pragmatics of explanation. Am. Phil. Q.,14, 143–150. van’t Hoff, J. H. (1874). Sur les formules de structure dans l’espace. Arch. Neerl. Sci. Exactes Nat., 9, 445–454. van’t Hoff, J. H., (1875). La chimie dans l’espace. Rotterdam: P. M. Bazendijk. Woolley, R.G. (1985). The molecular structure conundrum. J. Chem. Educ., 62(12), 1082–1984.

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Dominique Lambert University of Namur

“When Science Meets Historicity”

Some Questions About the Use of Mathematics in Biology

1. From physics to biology The aim of this paper is not to give a survey of all mathematical theories used in biology today, but to think a little bit about the possibility to explain in depth the nature of living phenomena using mathematical concepts. We know that every explanation in physics today is provided by mathematical theories. It is not possible to really explain what weak force is and why it appears without using the mathematical concept of (local gauge) symmetry and this, in turn, means using group theory. When physicists speak about electromagnetic fields, they are immediately referring to an antisymmetric tensor of order 2. And when they are discovering such a tensor in their mathematical theory, they straightforwardly associate it with a physical field. It is really difficult to make a clear-cut distinction between mathematics and physics because mathematics is deeply embodied inside physical explanation. A natural question arises here: can we think about such an identification between biological concepts and mathematical notions? Are the fundamental notions of biology reducible to mathematical concepts (perhaps not the ones used by physicists!)? On can object, at this stage, that there exists something that remains not completely fixed by mathematics in physics. We can think

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about initial conditions for example. But ultimately the physicists’ dream would be to find a way to derive these conditions from a more fundamental mathematical framework. The question arises then if it could be completely possible to build a mathematical explanation without introducing such purely contingent conditions (i.e. not deduced from some mathematical necessities, laws)? In cosmology it is perhaps possible, at least in principle, to think about a theory that goes without initial conditions or that derives them logically (many physicists are searching for just such a thing!). But in engineering sciences (continuum mechanics for example), it is more difficult to describe the ageing of a building using only laws and not initial conditions (which are contingent, not explained by the formalism we are using). The example of physics shows that the usual situation is to distinguish between mathematical equations (that model a phenomenon) and initial or boundary conditions (that model some constraints that control or initiate the phenomenon). Generally, we have no mathematical theory deriving the latter conditions, even if the dream would be to discover it (if there is no logical obstruction to discover such “final theory,” but that’s another problem). In biology, an explanation of what is a living being as such resorts necessarily to evolution, it means a history. But history is built by and from particular conditions in time (initial conditions) and in space (environments: boundary conditions). We cannot restrict ourselves to the description of what is a living being now! We cannot understand it without reference to its evolution and to all these conditions that have constrained its history. The question then becomes: does there exist a mathematical modeling of these particular evolutions? The question can be addressed not only for the long evolution of species, but also for our individual short-time evolution that conditioned all our decisions. What we are is not explained totally by general laws, but also by very particular events that build our history and determine a part of our personality. Can we think about a mathematical theory which describes our own behavior, our own decision process for example? It is very difficult to believe that such a formalized theory exist, because it would mean

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that a kind of universal and formal theory of human history also exists. But let us try to take the question seriously by following as far as possible the lines of mathematical explanation.

2. What is naturally explained in biology using only a priori mathematical theories? The questions we are facing here are connected to a more general one: in which situations and epistemic fields can we speak about the “unreasonable effectiveness” of mathematics? We know that Eugen Wigner (1960, pp. 1–14) wonderfully emphasized the enigma of the unreasonable effectiveness of mathematics in the natural sciences. The latter has in fact not solved this enigma but he has shown clear illustrations of this effectiveness. The solution is not so easy because it depends on the philosophy of mathematics. For example, if mathematics is considered as pure and arbitrary constructions made by humans, the solution of the enigma will be very difficult. How can we explain that human intellectual games have, very often, such effectiveness in the process of describing the physical realities existing outside human brains? If we are considering that the essence of physical reality “is” mathematical, then we can understand that mathematics are able to describe our universe adequately. But in this case you have to explain what does the expression “the essence of reality is mathematical” mean? We do not have enough space or competence here to begin a philosophical discussion about this famous enigma. We will restrict ourselves only to some modest remarks. Mathematical models are efficient if they are able to describe, and in some way to reproduce, some features of “elements of reality.” But we will see that abstract (structural) properties of mathematics enable us to represent what the intrinsic and specific features of any elements of reality are. First of all, we know that Mathematics offers the possibility to represent objects and the relations between them. This is shown for

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example by category theory (see Awodey, 2010) that allows us to describe adequately many parts of mathematics by collections of objects and collections of arrows (transformations, relations, etc.) linking these objects. Therefore, we see that mathematics has this ability to represent and reproduce the “coherence” of some domain of objects: this coherence being translated formally by the existence of this net of arrows. We can say, using other words, that mathematics is here able to represent a “form” in a philosophical sense. The notion of structure, which is very important in formal science, is an example of this notion of coherence. This notion has two facets: one is “synchronic” and the other one is “diachronic.” The words are not completely adequate, but we use them for commodity. “Synchronic coherence” means a “static” point of view, describing the net of links between objects that constitutes a specific mathematical domain given as a whole. “Diachronic coherence” means a “dynamical” point of view, describing the evolution (“trans-formation”: modification of form!) of a domain or the action on this domain. Let us give one example and consider a category of structures of a given nature. The arrows are the morphisms that preserve the structure’s nature. These determine the diachronic coherence (the form, the essence) of the category. But we can imagine acting on this category to transform it “coherently” into another. This can be done using a “functor” between both categories. This functor can be viewed as an arrow in a category of functors (whose arrows are “natural transformations”) that represent the way to transform (or translate) categories. It is interesting to note that mathematics is also able to represent invariants. A mathematical field is very often characterized giving quantities, structures, etc., which remain the same when performing particular transformations. We know, after the famous “Erlangen program” of Felix Klein, that large classes of geometries can be described using transformation groups and their characteristic invariants. Very often what we call “deep mathematics” is a conceptual domain that exhibits many invariants. In category theory, we meet plenty of “commutative diagrams” expressing that something is pre-

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served, that something remains invariant, when passing from one context to another. Category theory is a kind of mathematical theory (view!) of mathematics. It allows us to exhibit general and abstract properties characteristic of mathematics as such. The fact that this theory emphasizes very clearly that mathematics is able to formally represent, coherence (form), invariance and action (transformation), is of great importance for us (Lambert, 1997, pp. 161–178). The reason is the following. You cannot say having identified an element of reality if you have nothing that remains invariant (with respect to at least one point of view). If the “thing” has no stability at all, it cannot be identified as such and such element of reality. Furthermore, you cannot say that having grasped “this thing” if you cannot express what you are identifying as “this thing.” In other words, grasping an element of reality is at the same time what grounds this unity that is called “this thing.” This ground is the form (the principle of the coherence defining the given thing) or the essence (what characterizes “this” thing). Finally, usually, we are persuaded that an element of reality exists if it can interact with us or if we can interact on it. This shows us that the abstract structure of mathematics (as disclosed by category theory for example) is such that it is connaturally adapted to descriptions of elements of reality. What is a priori formally needed to describe and represent elements of reality is in fact the formal characteristics of mathematics as such. Of course, this alone does not solve Wigner’s enigma. In fact we have to explain why such particular structures, such specific invariants or such transformations are able to fit so adequately such a set of experimental or observational data? We have not the pretention to explain this. Let us simply note that it is not possible to explain this effectiveness without taking into account the historical dimension. One hypothesis to shed light on the enigma is to think that there exists a two-sided relation between mathematics and natural sciences. Mathematics, due to its formal characteristics, is pre-adapted (a priori) to represent an

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element of reality as we have said above. But, at the same time, natural sciences shape mathematics and feed it with “empirical information.” Mathematics is thus progressively co-adapted to empirical content. And some mathematical theories (but not all mathematics!) are step-by-step transformed and also deformed to fit new elements of reality it has able to represent. Efficient mathematical models are thus progressively and historically selected and adapted to some part of empirical contents. The crucial point for our paper here is that any element of reality can be efficiently mathematized if we are able to identify from this element something that remains invariant, and something that exhibits a form of synchronic or diachronic coherence (spatial or temporal relations between objects that constitute this reality). If we think about the sciences which are different from physics: economics, psychology, history, etc., we can check that it is not at all obvious to discover some invariants (that could be identified and measured!), characteristic of the phenomena these disciplines are studying. We can for example identify some global invariants characterizing the global behavior of an assembly of interacting people (this can be done using game theory). But it is very difficult to discover the invariant characterizing the (economical, psychological, etc.) behaviors of a particular person. Concerning biology, we have to take into account the fact that many very efficient mathematical models have entered this field (Knight & McDowell, 2002, pp. 244–246; Modelling..., 2002; Hoppensteadt & Peskin, 1991). As a matter of illustration we can give now some examples. At the molecular level, the shape of the macromolecules can be described by topology. For example, knot (and ribbon) theory and its topological invariants are very efficient to understand the action of enzymes (topoisomerases) on the DNA (see for example: Murasugi, 1996, pp. 171–196, 266–283; Pohl, 1980, pp. 20–27; Weber, 2002, pp. 328–347). The importance of topology comes from the fact that the shapes of the molecules give them the ability to perform some func-

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tions. Furthermore, topology describes situations that remain invariant under continuous changes and this is crucial to describe plasticity that is one of the important properties of living beings. Plasticity is indeed the ability of a system to be deformed without loosing some kind of coherence (Lambert, 2009, pp. 27–43; Lambert & Rezsohazy, 2005). This resistance against continuous deformations is what we call the robustness, and it is described by some invariants. Topology describes and classifies precisely these kind of invariants. Gene networks (that is responsible for the genes regulation mechanisms) can be described adequately by graph theory. The topological properties of gene networks (random graphs, “little world” graph (see the wonderful book Watts, 1999), scale free graph, etc.) can explain their robustness or their fragility when mutations occur (see for example Ravasz et al., 2002, pp. 1551–1555). Graph theory is very powerful but this theory is not immediately concerned with biology! The choice of the adequate entities to be linked (the vertices of the graph) and the choice of the kind of relation to be taken into account are crucial. If you don’t do that correctly, you can get beautiful mathematical results but that perhaps have little or no biological pertinence. On such graphs we can introduce “dynamical” information describing the way genes are interacting. Then, the coupling between graph theory and dynamical systems theory provides us with some very interesting results. Some time ago, for example, the Belgian biologist René Thomas (Thomas, 1981, pp. 180–193; Thomas & DʼAri, 1990; Thomas, 1998, pp. 479–485; Thomas & Kaufan, 2001, pp. 170– 179) conjectured that the existence, in such networks, of a positive retroaction circuit is a necessary condition to get “mutistationnarity” (to have several stationary states, corresponding to different behaviors of the living system) (Laurent & Kellershohn, 1999, pp. 418–423). This conjecture of René Thomas was proven by the great mathematician Christophe Soulé (2003; 2006, pp. 13–20, Kaufman, Soulé & Thomas, 2007, pp. 675–685), providing deep roots for an explanation of some biological fundamental phenomena (cell differentiation for example). These mathematical tools can be applied with success to

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many biological networks linking proteins or sophisticated molecular engines (see for example: Vidal, 2001, pp. 333–339). Mathematical models are also very interesting to study the global shape of living beings or of their components. An example is given by the classification of all possible shapes of the liposomes using optimization and differential geometry techniques (Willmore functional) (Peliti, 1997, pp. 171–188; Sackman, 1997, pp. 189–212; Michalet et al., 1994, pp. 1012–1018). The determination of the shape of the liposome’ surface (formed by a double layer of phospholipids) is based on the search of the optimization of a quantity representing the curvature energy of the liposome’s surface under the following constraints: constant volume and constant surface’s mean curvature (which is related to the difference between the number of phospholipids constituting the surface of the outer face and of the inner face). It is interesting to note that all the shapes discovered as solutions of this optimization problem are observed in nature or in experimental situations. Amongst the nicest (and exotic) shapes we find the “Dupin cyclides,” that are deformations of a torus. But we have also solutions resembling to the red blood cell. The dynamics of cell membranes (oscillations, etc.) that are so important for cell interactions can be described adequately using random surfaces theory in statistical mechanics (Wheater, 1994, pp. 3323–3353). We also know the classical example of geometry and optimization to classify the shapes of the diatomae, beautifully described and depicted by Ernst Haeckel. We find also today many wonderful mathematical works on the shapes of the capsids that are protein shells of viruses (see for example: Mannige & Brooks, 2009, pp. 8531–8536). All those works about biological shapes have their roots in D’Arcy Thompson’s (1917) work and in particular in his famous book: On Growth and Form. But it is interesting that, at that time, this book has nearly no influence on biologists. It is related to the fact that it was in fact a pure geometrical or physical description of living systems but without any connection with the biological processes leading to the forms and growths of plants and animals. Now the

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situation has changed because mathematical descriptions of biological shapes and growths can be linked, via what we call “Evo-Devo” (the science linking evolution and development), to genetics (master genes for example) or to some known biochemical processes. The efficiency of such mathematical models is connected not only to the fact that they describe biological structures and dynamics, but also to the fact that their notions and concepts can be related to real and observed biological mechanisms. Mathematical modeling is not an artificial and extrinsic vestment superimposed to living entities, but is deeply rooted into known biological realities. Today, a nice illustration of an adequate modeling of biological shapes (and of their development) is given by the Turing model (Turing was inspired by D’Arcy Thompson!) based on reaction and diffusion equations (see the onderful book: Murray, 1989). Using this model, we get a classification of the shapes occurring on the butterflies’ wings, or on the coats of mammals, etc., and we can also understand deeply the origin of some important spatial domains that structure the embryo development. It is interesting to be careful here. Before using the Turing model, we have to check if it is relevant! We can indeed get empirically observed results (i.e. we reproduce the shapes we are observing), without any connection with the biological reality and processes (to apply the Turing model you need to have, for example, a domain- a syncytium for example- in which chemical substances can diffuse and react). Sometimes, the analysis of biological shapes asks for a deep understanding of the biological tissues that bear these shapes and of the particular physical processes that are occurring in the tissues. It is the case for the explanation of the shape of the scales that exist on the crocodile jaw. We need here information coming from histology (describing the nature of the jaw tissue) combined with a model describing how cracks (that generate the polygonal lines separating the scales) appear during the crocodile’s development. We cannot be satisfied using only random graph theory, in order to describe the set of polygons built by the scales! This would be a description but not a satisfactory explanation. To explain means to disclose some deep mechanism, structure or

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process responsible for the phenomena. When a connection is established between some relevant biological mechanism and some mathematical notions, then we can apply the latter to get some useful and meaningful (with respect to the biologist!) results (see: Milinkovitch et al., 2012, pp. 78–81). Of course, during the first stage of the research, when we are not able to find a deep biological explanation of the emergence of some structure, shapes, etc., it can be interesting to have an “outer” description, namely purely geometrical, topological or based on physical constraints. Because, this description can pave the way to some suggestions, leading afterwards to a biological explanation. Shapes, for example, are in fact revealing something of the way the animal adapts to the environment. A nice example of an interesting geometric and physical description which is also useful for biologists is the case of the Indian Star Tortoise (Geochelone elegans)! The form of the shell is a very good approximation of the so-called “mono-monostatic body,” the existence of which was conjectured by the famous mathematician Arnold and which was proven to exist by Hungarian mathematicians (Várkonyi & Domokos, 2006a, pp. 34–38; Várkonyi & Domokos, 2006b, pp. 255–281). This convex body, also called “Gömböc,” has only one stable point and one unstable point, allowing the tortoise to turn easily when unfortunately placed on its shell. This nice description is not in fact a biological explanation of the emergence of the shell. Genetics and “Evo-Devo” (Carroll, Carroll & Klaiss, 2005) mechanisms are still needed, but, certainly, the extraordinary feature of the shells could be an evolutionary adaptation to some environmental constraint, physics and geometry revealing not the biological substrate but the traces of the environment and its specific constraints. At a physiological level, we know many important and efficient mathematical models very well. We think, for example, of the models (of the theory of control) describing retroaction loops responsible for the homeostasis (Brockett, 2001, pp. 189–219). But we can also think of those fluid mechanical models giving the description of the

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flow of the blood (see for example: Pedley, 2000, pp. 105–158). Here we are facing a phenomenon that is purely physical and that is not characteristic of biological reality as such (if this term has some intrinsic pertinence?) We could refer here also to the modeling of some complex systems like the brain (using formal automata or neuron networks) or the immunological system. All these systems can be tackled using the theory of complex systems,1 giving some very interesting results and hints. An organism is very difficult to model (even the cell itself is already a very complex system!), it comes from the fact that it involves many systems located at many different levels and scales. But we can still use mathematical frameworks coming from multiscale physics and statistical mechanics. Here, a global description, neglecting the fine biological details, is perfectly correct. This is due to the fact (a situation that is very frequent in multilevel systems) that some details present at a micro-scale are not relevant at a macro-scale. And the system behaves as if the micro-variables were not present at all. It is very important to have a global description of a biological system. Some important properties are emerging only at the level of the whole (for a biologist phenotype is as important as genotype!). Mathematical models have to be used also in what we call now “whole-istic biology”2 integrated many partial and local descriptions. We cannot speak about mathematical models in biology without alluding to evolution theory (Lambert, 2011, pp. 347–357). In this context, we know that the very interesting models coming from dynamical systems theory (the description of ecosystems by the Volterra-Lotka models for example), from evolutionary game theory (Maynard Smith, 1982; Thomas, 2005; Akin, 1987, pp. 1–93) and   We can think here as a matter of illustration to the very interesting early attempts made by (Kauffman, 1993). The use of Boolean automata networks constitutes an example of very interesting attempts to model complex systems build of many simple interaction “engines.” 2   (Whole-istic..., 2002). See also (Gierer, 2002, pp. 25-44), for a philosophical analysis of this new perspective. 1

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from the theory of landscapes in optimization3 (which gives good explanation of some phenomena of canalization in evolution). Combinatorial methods are also required in order to model phylogenic trees and to reproduce the classification of the species. We will come back to this problem later. But we can address here the problem of whether the quoted mathematical theories are sufficient to convey an a priori modeling of the evolution of a tree, or bush if you prefer? We see here that all these methods produce some general features of evolution, describe some universal regularities at work during all evolutionary process. But if evolution is in fact governed and controlled by such regularities, it is not at all completely fixed by the latter. It depends on some random events and stochastic environments. At each level, the use of stochastic processes is thus a necessity for understanding the role and the scope of random noises (see for example: McAdams & Arkin, 1999, pp. 65–69). There are many examples of very efficient uses of probabilistic methods in genetics.4 Of course the choice here for a probability measure or for the type of stochastic process is not simply a mathematical problem. It is related to the knowledge coming from observation or experimentation. A similar problem occurs in finance. If you want to compute the financial risk you are taking when you buy such option, you have to model the time evolution of the stock market price. In order to achieve this goal you have to choose a specific stochastic process modeling prices fluctuations. There is no a priori theory to make this choice. You have to rest on some past observations and data. But the consequences of this choice on the risk computation can be very different if you choose a “Brownian motion” rather than a “Levy flight” (Schlesinger, Klafter   (Stadler, 1995, pp. 78–163). The application of lanscape theory in biology has its roots in Waddington’s work (Waddington, 1957, p. 29 [cf. figures 4 and 5]). 4  The backward Kolmogorov equation, used by Motoo Kimura for example, allows us to predict the probability of fixation, after a certain number of generations, of a mutation that was initially present in the population at a given frequency. The structure of this equation derives in fact from a stochastic differential equation, which reveals something on the random way such mutation is diffusing inside the population (Gardiner, 1997, p. 55; Watterson, 1996, pp. 154–188; Kimura, 1983). 3

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& Zumofen, 1999, pp. 1253–1259). After a financial crash, you can adapt your model and correct it but here we have to accept that we have no theory grounding our choice of a specific stochastic process, because we have no theory of the complete behavior of the human economic agent allowing mastering all “local” fluctuations. We have to make a remark at this point. If we are analyzing all the mathematical models of biology we quoted above, we may note in fact that they are based on some systems described by physics (or by chemical physics). We thus reach here a kind of tautology: what is reduced to physical systems can be described and explained by physics (and then by mathematics, which is the language of physicists). But then we are losing the track of our biological subject! The question is: in principle can we reduce descriptions and explanations of the specific features of biological systems completely to mathematical models? Another question could be also addressed: if this reduction is possible, do there exist models that are specific to biology (in comparison with physics)?5 This question could be formulated as follows: does there exist new mathematics coming exclusively from biological motivations? Another question is to know if there are some intrinsic (epistemological) limits preventing a complete mathematical explanation in biology? We will tackle this question now.

3. What is difficult to explain in biology using only a priori mathematical theories? Here we should make some distinctions. It is possible to describe many things mathematically but we have to distinguish, according to René Thom, description and explanation. We also should not confuse predictions and explanations. We can describe without explain  The French physicist Jean-Marc Levy-Leblond thinks that the relation between mathematics and physics is unique, and that we cannot have a similar situation with other fields like biology or social sciences (See Levy-Leblond, 1982, pp. 195–210). 5

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ing anything (statistical laws which described many situations in human sciences are not at all explanations). But we can predict without really explaining (let us think about the dimensional analysis in fluid mechanics). To describe a system, we need to identify the central concepts and relate them to some mathematical frameworks. But to explain the phenomena arising in a system, we need to produce a deduction from some principles or laws (which have already received some meaning: this is a fundamental point on which the explanatory power of a theory rests) and furthermore, we have to connect the results of the deductions to some measures (in order to test the theory and to link it to reality). But practically, we need more: we have to have some additional information concerning initial conditions and also boundary conditions (the environment!) These conditions are very often not deduced at all from a deep theoretical framework, but are accepted as some “contingent” data needed to feed the “deduction machinery”! As it is well known, the distinction between general laws and initial conditions was emphasized by Eugen Wigner in his Nobel Lecture (Wigner, 1963): The regularities in the phenomena which physical science endeavors to uncover are called the laws of nature. The name is actually very appropriate. Just as legal laws regulate actions and behavior under certain conditions, but do not try to regulate all actions and behavior, the laws of physics also determine the behavior of its objects of interest only under certain well defined conditions, but leave much freedom otherwise. The elements of the behavior which are not specified by the laws of nature are called initial conditions. These, then, together with the laws of nature, specify the behavior as far as it can be specified at all: if a further specification were possible, this specification would be considered as an added initial condition.

There are some situations where we can explain the presence of such and such conditions (initial or boundary) using other sciences.

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Astronomy or geology can be used to explain why some environments exist where such kind of life has or could have evolved. Here a part of the explanation escapes biology because it is conveyed by other sciences. In fact, the complexity of biological systems necessarily requires explanations coming from many different epistemic fields. We have here to take seriously into account what Evelyn Fox Keller said about “explanatory pluralism” in her book: Making sense of life. Explaining biological development with models, metaphors and machines (Fox, 2003, p. 300): The central concern of this book has been with the de facto multiplicity of explanatory styles in scientific practice, reflecting the manifest diversity of epistemological goals which researchers bring their task. But I also want to argue that the investigation of process as inherently complex as biological development may in fact require such diversity. Explanatory pluralism, I suggest, is now not simply a reflection of differences in epistemological cultures but a positive virtue in itself, representing our best chance of coming to terms with the world around us.

But there are also some situations where it is difficult, even outside biology, to identify the reason why initial and boundary conditions (for example firing some mutational process) are arising. Let us note that knowing that such conditions produce such phenomenon is not the same thing as knowing why such conditions were present to trigger the processes leading to the phenomenon. Here, part of the explanation escapes from biology because some conditions cannot be tracked by science (maybe at this precise moment of its history?) Darwinian theory in its contemporary form (synthetic evolution theory) gives an adequate explanation of the emergence of species and of all the biological structures we find in living beings. Here, biological explanations needs information coming from outside. Astronomy and geology can explain adequately why some environments arose, allowing life to emerge and some particular species to be selected (the

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position and the stability of the Earth orbit explained by celestial mechanics; isolations of islands explained by plate tectonics; the fall of an asteroid explained by combination of physical phenomena, etc.). But even in these sciences, we are confronting with some initials and boundary conditions that are not totally produced by the theoretical explanatory framework. We fall each time to some “contingency.” The problem we are facing here is not at all only an abstract one. We want to give an example in which we can apprehend the difficulty of a complete logical modeling in human biology: the formal description of the human (moral) decision process (Kowalski, 2011, pp. 155–165). This problem arises in the context of what are called the “moral machines” (Wallach, 2009) (or artificial moral agents) in very recent robotics. In our societies we have introduced many robots and progressively we have given them many abilities to “take some decisions” without any human mediation. But in fields where these decisions imply important consequences with respect to the security of human beings and to the stability of economy or of States, it is crucial that all robotized decisions processes be regulated by ethical assessments and norms. Some engineers thought therefore to implement ethical programs in the robots software which was able to mimic moral human judgment and decision. How is it possible? (Lambert, 2012, pp. 28–34) One part of such programs is built on procedures verifying that laws are satisfied. But of course legality does not exhaust morality. Then, we can think about procedures checking the adequacy with some ethical principle (respect of autonomy, …). The decision process can be modeled, for example, by deontic logic (Kalinowski, 1972), using logical operators of permission or obligation. We could think here of a neurophysiologic (bottom-up) approach tempting to grasp something of the brain processes committed in the decision processes and to translate this inside a program (using neuron networks theory for example). We will restrict ourselves here to assessing the (top-down) approaches which are trying to model the global decision process using formal logics, leaving aside the problem of the mathematical description of the brain mechanisms.

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The question is now whether it is really possible to adequately model the human decision process using only mathematical formalisms implemented inside a computer program endowed with some artificial intelligence capabilities. Sometimes, it is important to resort to such programs, because the situations oblige us to decide in a very short time on the grounds of a great amount of very complex informations and data. These situations are well known in medicine and the computer can provide us very efficient assistance here. But we know that there exist some situations where we are confronted to ethical dilemmas related for example to conflicts of goods. How is it possible to model the moral decision in these cases? The answer could be given using game theory or optimization theory (computing the ratio between the “good” and the “evil” inside an utilitarianist approach). Many problems can be addressed at that stage. One of them is the following: it is not always possible to find a solution of a complex optimization problem and generally there does not exist a general algorithm to solve such problems. Another one is the fact that human beings make judgments on the basis of some values or very long learning experiences which are difficult to translate inside a program. Even concerning law, it is really difficult to model the judge’s decision process in a tribunal using only deontic logic. In situations of doubt or dilemmas, the judge must build his decision using some kind of creative process which trespasses on the limits of an a priori algorithm. In these situations, it often happens that the decision is neither a pure random choice (the computer could model this without any problem!) nor a pure product of automatic deductions, starting from some information (stored in some data bases) and respecting some deduction rules given a priori. Aristotelian philosophers could have referred here to “phronêsis,” a kind of judgment animated by prudence (Aubenque, 1963). The latter allows human beings to apply rules but in some very complex situations, where this application is not at all straightforward or automatic.6 In this situation,  “prudentia appliquat universalem cognitionem ad particularia” (Thomas Aquinas, Summa Theologiae, Ia, IIae, q. 49, a. 1., Ad Primum). 6

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there are some rules, but there are no rules concerning the application of the rules in some contingent context! As in the situations considered above, we face here something that escapes the generality of a priori rules or laws, but that is absolutely crucial. Let us note that the decisions in such situations are not random choices, because it depends ultimately on the history of the man who decides: for example the details of his learning, of the value system he has accepted, of the encounters he has made, etc. It is probably impossible to model adequately the decision process because we cannot identify some invariant or law that characterizes a history which is mainly contingent! It is thus time now to face this contingency problem.

4. Mathematical models and contingency: a scientific or philosophical problem? We are facing here a situation where some contingent information is needed to explain phenomena. The first thing to say is that some initial or boundary conditions seem contingent because they cannot be explained by biology alone. Such conditions are, in fact, perfectly explained by other sciences: geology (for selections of some environments), astronomy (for selections of sites compatible with life emergence and evolution), quantum physics (for an explanation of some mutations), etc. But, in those sciences too, we note that we are facing the status of boundary and initial conditions. Are they completely explained (by a deep theoretical frame) or do we accept them as irreducible data (which are necessary to trigger the deductions but are not deduced from the theory)? Here we have also some options. We can say that, in principle, we will find (in the future) a deep formalism enabling us to explain the conditions (and then to eliminate the contingency). But we can also pretend that there are some objections to suppress the distinction between laws and (initial or boundary) conditions. These two options in fact correspond to two famous theses: on the one hand “all

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is necessary” and on the other hand “there are necessities but there are also irreducible contingencies.” Epistemologically, we encounter here a difficulty because it is impossible to prove that an event is purely contingent. We can say that it does not follow a known law, but it is really difficult to pretend that it is not a product of some necessity at all. We can say that proper biological phenomena cannot be completely explained by biology alone because such phenomena (the emergence of life or the evolutions of species for example) cannot be properly understood without referring to cosmology, astronomy, geology, … We can also say that all which is reducible to physics can be explained by mathematics apart from that which comes from some initial and boundary conditions (which are necessary for the theory but are not explained by it). What is properly biological is in fact dependent, to a high degree, on initial and boundary conditions. This fact is a feature of “historical sciences.” In those fields, we can explain, a posteriori, many phenomena using general laws and assuming some conditions (discovered empirically and not deduced theoretically). The precise shape of the evolutionary tree or bush is due to general mechanisms, but also to some precise conditions related to the details of the history of the tree and its particular environment. It is possible to predict all possible shapes, but it is nearly impossible to predict a priori the precise shape of the evolution of a tree or bush from now. We face a similar situation in human history or economics. A posteriori we can introduce some meaning into the evolution of societies, but it is very difficult to give an a priori model of human history. This is because it depends very sensibly on conditions (on fluctuations…) that we are unable to model. Some fluctuations and conditions follow some statistical laws. These regularities can be adequately described by (classical or quantum) stochastic processes or by game theory. If there exists no way to go beyond probabilistic descriptions, we reach the following conclusion: we could have a complete mathematical description or even

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explanation, but which could not be a deterministic one. Let us then suppose that there exists a fundamental purely stochastic level. Then, a complete mathematical explanation (in physics, biology, etc.) could not go beyond probabilistic considerations. In this case we would accept contingencies as a pure fact given by nature. A difference between fundamental physics and biology is the fact that biological phenomena are more intrinsically dependent on (initial, historical and boundary) conditions (that cannot be explained completely inside biology). This situation is also specific to all of the “historical sciences,” namely the sciences whose explanations are obliged to take into account the past of the systems. In continuum mechanics we meet such situations where we have to take into account some memory of the past story of the systems in order to adequately describe the ageing of a structure, etc. The integration of this memory is not a problem for mathematics (you can use integro-differential equations for example), but what is more difficult is to produce a model explaining why such a micro-event (crack, fracture, defect, etc.) appeared here rather than there, at this time rather than earlier or later? It is in fact impossible and we introduce probabilities based on some experiments or observations. Thinking a little bit about this situation, we can check that biology is not a particular case at all and that the problem is in fact the same in physics. The crucial point is a classical epistemological one. Is the distinction between fundamental laws and initial or boundary conditions a fundamental one or not? If yes, we are in a situation where we philosophically accept a universe where we have necessities but also contingencies (not absorbed into necessities). If not, we are expecting, still philosophically, that we are in a universe controlled by necessity (in this case contingency is only a product of our ignorance). In fact it seems impossible to prove that something is the product of a pure necessity or of a pure contingency. We can only prove that some events follow such laws (inside the limits of observations) if we introduce such and such conditions (not derived from

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the law). Of course, the problem is deeper because we have to distinguish between a scientific (epistemological) necessity which is in fact a regularity (the fact that events are emerging due to scientific laws) and a metaphysical (or ontological) necessity. The same distinction is true for contingency. We ought to analyze the problem in depth here in order to know if we can really think (from a pure logical point of view) of a complete description using only laws and principles (containing even a derivation of all initial or boundary conditions). It is not the place here to enter this problem, but in physics this problem is not new and was addressed by Einstein and Eddington. The latter thought, in his “fundamental theory,” about a pure algebraic derivation of all parameters and constants of physics (which can be considered as some initial information necessary to explain the phenomena by the way of fundamental equations). The completion and logical pertinence of such a program ought to be considered. Analyzing the problem of the efficiency of mathematics in biology, we are discovering in fact that it could be important, at least with respect to epistemology, to make a distinction, inside each natural science, between what we can call a historical part and a not historical part. In the latter, explanations can be tackled using laws and general principles (the explanatory power flows here from deductions starting from the principles; the latter being considered as the source of some self-sufficient meaning). In the former, the historical part, explanation cannot be brought by the ab initio principles but rather on the contrary by an a posteriori hermeneutics. This tries to give some meaning to what remains purely contingent at the level of general and universal laws. Some philosophy of science can choose in principle to deny every contingent character, but it can make another choice, namely to accept that we need two types of explanation. The first type would be based on mathematical deductions reflecting the presence of ontological necessities. The second type would be based on information reflecting the irreducible presence of ontological contingencies.

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Of course we have to avoid a danger related to the temptation to refuse to push the mathematical explanation as far as it could deciding that such an such condition is contingent. For example, at some time the forms on mammals’ coats or of the geometrical shapes occurring on the wings of the butterflies have been considered as purely contingent structures. Yet we know now that these forms and shapes can be explained and classified using the Turing model. It is thus important not to stop the quest for a mathematical explanation using the pretext of contingency. If we avoid this danger, we can think about two kinds of explanations. One would be based on a priori principle and laws (as in fundamental physics) and the other one would be connected to an a posteriori reasoning, justifying the present existence of phenomena by some choice of some past conditions. The latter is based on a logical argument of coherence. For example: we have no fundamental law explaining why the details of an evolutionary tree or bush are as they are, but, using evolution theory and some contingent conditions (changes of environments, etc.), we can convey a coherent history of life. The relative importance of a priori or a posteriori explanation is not the same in each scientific field. For example, in social sciences you have an important part of a posteriori explanations. On the contrary, in fundamental physics, a priori explanation based on principles and universal laws is more important. In between, in engineering sciences, we can face situations (for example description of ageing of structures) where we cannot give a complete a priori explanation of the existence of crucial fluctuations that determine the evolution of the system in depth. Biology involves both types of explanations and depending on the subsystems under consideration, we can find theories which look like fundamental physical ones (for example, explanation of the shape of such a cell or animal, using variational principles and symmetries) or which look more like historical ones (for example, when we try to explain the main bifurcations in evolution or the decision processes of a human being). We can also say that biological systems are multiscale systems (Schnell, Grima & Maini, 2007, p. 141) with each scale having its

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relative autonomy and requiring its own explanatory field. Therefore the epistemological problem of explanation in biology is related not only to the historical nature of living systems (with all their contingent initial and boundary conditions) but also to the fact that biological realities are in fact complex multiscale systems. The characteristic levels of the latter, being relatively autonomous from one another, ask for different and independent explanatory schemes, coming from very different sciences.

5. The mathematics faces historicity in biology and… in physics The multiscale nature of complex living beings explains why the search for an original unique mathematical scheme attempting to grasp the nature of the biological systems as such is very difficult. Perhaps the only way to go in this direction is to be inspired by statistical mechanics and the “renormalization group philosophy” coming from field theory, by which physicists understand how to build effective theory and how to link descriptions made at different scales. The historical character of biological system, with all the contingent conditions that constrained them, explains why a complete mathematical theory of biological systems is probably not reachable. We believe that, in the future, biology will be more and more, invaded by mathematical models. But it will be difficult to derive inside these models initial and boundary conditions. Yet these conditions are what give to biological systems their own nature (biology is evolution!). Then even if we can have efficient descriptions of biological systems based on sets of equations, it will lack a mathematical explanation of what is essential to biology, to history (and history is built on very particular conditions that escape to descriptions given by universal laws, etc.) Ultimately, a solution of the problem concerning the possibility of a total mathematization of biology depends sensibly on metaphysical

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positions. Accepting an ontology which assumes that all phenomena are determined by metaphysical necessities, we can dream of a “final mathematical theory” of biology. Nevertheless, this dream seems up to now beyond our scope. But if we start from a more modest ontology saying that in nature we have necessity, but, at the same time, we have also an irreducible contingency, we are led to another conclusion. Namely: we can have explanations based on very efficient mathematical models (equations, structures, etc.) but it will be impossible, in principle, to get a complete mathematical explanation of all the essential conditions that determine general and particular evolutions, histories. It is not our subject here to enter into a defense or in a refutation of such or such ontology. It will lead us far beyond the limit of this little contribution. It is sufficient for our purpose to say that the epistemological approach of the question of the possibility of a complete mathematization of biology very often rests on the two main ontological options described above. If we take this into account, we can try to avoid two pitfalls. The first one could be related to the adoption of an ontology defending complete necessity in the universe. This could lead to searching for a general abstract mathematical theory, neglecting to take into account the particularities of biological phenomena as such. This was sometimes the case in theoretical biology. In this case the link with the phenomenal roots are lost and we face a situation where we superimpose, from the outside, a mathematical vestment on some data. It was sometimes the case when one tried to apply the famous “catastrophe theory” of René Thom (1977) to ethology or to linguistics for example. In this case, the general and very deep theory was often not able to make some bridge to measurable relevant quantities. We encounter here a situation where we can explain without being able to predict adequately! But of course we cannot accept an explanation that is not empirically adequate. The second pitfall is symmetrically opposed to the first one. This is the tendency in biology to focus on a set of particular data or levels

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refusing the use of mathematics as far as it can go. An extreme attention to phenomenal data can block the search for very deep and fruitful mathematical explaining models. The conclusion of this is that we have to know that a complete solution of the problem we are tackling depends very sensibly on a metaphysical horizon (option?), but we have to be very careful not to be trapped by our own ontological presuppositions. In fact, this problem of the mathematization of biology could be a nice area for “experimental philosophy” to test ontological assumptions – for example on the basis of how they can help us to enlighten a deep understanding of new empirical phenomena. An analysis of mathematics in biology reveals that the difference between the latter and physics is perhaps not so important. What is fundamental for a deep biological explanation is in fact evolution first of all and we have seen that the latter is shaped by (partly stochastic) environmental constraints. What is often lacking in biology is an understanding of (the reason for the existence of) these conditions. In physics the dream of a complete absorption of conditions in a “final theory” seems more attainable. But in fact it is not obvious at all. And one of the problems in physics today is to understand the value of the (special or not) initial conditions in cosmology and the meaning, the nature and also the origin of the quantum fluctuations. Physics, if we think a little bit, is also a historical science! If the distinction between laws and conditions stressed by Wigner were fundamental (and linked with an ontology based on necessity and contingency), then physics would be not so far from biology. And correlatively the problem of its mathematization, and that of all other sciences, would be essentially the same. Our question would become then: how is it possible to produce a rational approach (a science!) shedding some light on these historical (initial and boundary) conditions in the case where you are not able (in principle) to derive them from a unified theory? The answer is not so easy because it would lead, alongside a priori deductive schemes, to a kind of a posteriori hermeneutical method as it is the case in history. But could we accept this in the natural sciences?

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It does not correspond to the “dream for a final theory” but not all dreams correspond to reality…

References Akin, E. (1987). The differential geometry of population genetics and evolutionnary games. In S. Lessard (Ed.), Mathematical and statistical developments of evolutionary theory (pp. 1–93). Dordrecht: Kluwer. Aubenque, P. (1963). La prudence chez Aristote. Paris: P.U.F. Awodey, S. (2010). Category theory. Oxford: Oxford University Press. Brockett, R. (2001). New issues in the mathematics of control. In B. Engquist, & W. Schmid (Eds.), Mathematics unlimited-2001 and Beyond (pp. 189–219). Berlin: Springer. Carroll, S. B., & Carroll, J. W. (Illustrator), Klaiss, J. P. (Illustrator) (2005). Endless forms most beautiful: The new science of evo devo and the making of the animal kingdom. New York: Norton. Fox E. (2003). Making sense of life. Explaining biological development with models, metaphors and machines. Cambridge: Harvard University Press. Gardiner, C. W. (1997). Handbook of stochastic methods for physics, chemistry and the natural sciences. Berlin: Springer (second edition). Gierer, A. (2002). Holistic biology. Back on stage? Philosophia Naturalis, 39, 25–44. Hoppensteadt, F. C., & Peskin, C. S. (1991). Mathematics in medicine and the life sciences. Berlin: Springer-Verlag. Kalinowski, G. (1972). La logique des normes. Paris: P.U.F. Kauffman, S. (1993). The origin of order. Self-organization and selection in evolution. New York: Oxford University Press. Kaufman, M., Soulé, Ch., & Thomas, R. (2007). A new necessary condition on interaction graphs for multistationarity. Journal of Theoretical Biology, 248, 675–685. Kimura, M. (1983). The neutral theory of molecular evolution. New York: Cambridge University Press. Knight, J., & McDowell, N. (2002). Bridging the culture gap. Nature, 419, 244–246. Kowalski, R. (2011). Computational logic and human thinking. How to be Artificially Intelligent. Cambridge: Cambridge University Press.

Concept_AF.indd 202

2016-02-19 09:21:41

“When Science Meets Historicity”

203

Lambert, D. (1997). L’efficacité des mathématiques est-elle déraisonnable. Comptes rendus de l’Académie royale de Belgique, VIII (6ème série) 1-6, 161–178. Lambert, D. (2009). Readings of biological plasticity. In M. Slomka (Ed.), Nauka – wiara. Rola filozofii, filozofia przyrody i nauk przyrodniczych 1 (pp. 27–43). Lublin: Wydawnictwo KUL. Lambert, D. (2011). Some epistemological questions concerning the application of Mathematics in Evolution Theory. In G. Auletta, M. Leclerc, & R. A. Martinez (Eds.), Proceedings of international conference “Biological Evolution. Facts and theories. A critical appraisal 150 years after “The Origin of Species”, Roma, 3/7-03-09 (pp. 347–357). Roma: Gregorian and Biblical Press (Analecta Gregoriana, 312). Lambert, D. (2012). Is there any formal argument against moral machines? In A. Bagood, (Ed.), Human fragility and christian anthropology: Faith and culture today (pp. 28–34). Dudweiler: Pirrot-Verlag. Lambert, D. & Rezsohazy, R., (2005). Comment les pattes viennent au serpent? Essai sur l’étonnante plasticité du vivant, Paris: Flammarion, Nouvelle Bibliothèque Scientifique (new edition: [2007]. Paris: Flammarion, Collection “Champs”, n°750). Laurent, M., Kellershohn, N. (1999). Multistability : a major means of differentiation and evolution in biological systems. TIBS, november, 418– 423. Levy-Leblond, J.-M. (1982). Physique et mathématiques. In Penser les mathématiques (pp. 195–210). Paris: Seuil. Mannige, R. V., & Brooks, Ch. L. (2009). Geometric considerations in virus capsid size specificity, auxiliary requirements, and buckling. PNAS, May 26, 106 (21), 8531– 8536. Maynard Smith, J. (1982). Evolution and the theory of games. Cambridge: Cambridge University Press. McAdams, H. H., & Arkin, A. (1999). It’s a noisy business! Genetic regulation at the nanomolar scale. TIG, February, 65–69. Michalet, X., Jülicher, F., Fourcade, B., Seifert, U., & Bensimon, D. (1994). La physique des liposomes. La Recherche, 269, octobre, 1012–1018. Milinkovitch, M. C., Manukyan, L., Debry, A., Di-Poï, N., Tzika, A., Martin, S., Lambert, D., Singh, D., & Zwicker, M. (2012). The spatial organisation of crocodile jaw scales is generated by physical cracking. Science, 339(6115), 78–81. Modelling complex biological systems. (2002). BioEssays, 24. Murasugi, K. (1996). Knot theory and its applications. Basel: Birkhäuser.

Concept_AF.indd 203

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Murray, J. D. (1989). Mathematical biology. Berlin: Springer. Pedley, T. J. (2000). Blood flow in artries and veins. In G.K. Batchelor, H.L. Moffat, & M.G. Worster (Eds.), Perspectives in fluid dynamics. A collective introduction to current research (pp. 105–158). Cambridge: Cambridge University Press. Peliti, L. (2003). Shapes and fluctuations in membranes. In H. Flyvbjerg, J. Hertz, M. H. Jensen, O. G. Mouritsen, & K. Sneppen (Eds.), Physics of biological systems. From molecules to species (pp. 171–188). Berlin: Springer. Pohl, W. (1980). DNA and differential geometry. Math. Intelligencer, 3, 20–27. Ravasz, E., Somera, A. L., Mongru, D. A., Oltvai, Z. N., & Barabasi, A.-L. (2002). Hierarchical organization of modularity in metabolic networks. Science, 297, 1551–1555. Sackman, E. (1997). Bending energy concept of vesicle and cell shapes and cell transitions. In H. Flyvbjerg, J. Hertz, M. H. Jensen, O. G. Mouritsen, & K. Sneppen (Eds.), Physics of biological systems. From molecules to species (pp. 189–212). Berlin: Springer. Schnell, S. Grima, R., & Maini, Ph. (2007). Multiscale modeling in biology. American Scientist, 95, 141. Shlesinger, M. F., Klafter, J., & Zumofen, G. (1999). Above, below and beyond Brownian motion. Am. J. Phys., 67(12), 1253–1259. Soulé, Ch. (2003). Graphic requirements for multistationnarity. Seminar given at the Université Libre de Bruxelles (4-2-2003). Soulé, Ch. (2006). Mathematical approaches to differentiation and gene regulation. C.R. Paris Biologie, 329, 13–20. Stadler, P. F. (1995). Towards a theory of landscapes. In Complex systems and binary networks (Guanajuato Lectures, Mexico 1995) (pp. 78– 163). Berlin: Springer. Thom, R. (1977). Stabilité structurelle et morphogenèse. Essai d’une théorie générale des modèles. Paris: InterEditions. Thomas, R. (1981). On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. Springer Ser. Synergetics, 9, 180–193. Thomas, R. (1998). Laws for the dynamics of regulatory networks. Int. J. Dev. Biol., 42, 479–485. Thomas, R., & D’Ari, R. (1990). Biological feedback. Boca: CRC Press. Thomas, R., & Kaufman, M. (2001). Multistationnarity, the basis of cell differentiation and memory. I. Structural conditions of multistationnarity and other nontrivial behavior. Chaos, 11, 170–179.

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Thomas, V. (2005). Evolutionary game theory, natural selection, and Darwinian dynamics. Cambridge: Cambridge University Press. Thompson, D’A. W. (1917). On growth and form. Cambridge: Cambridge University Press (second edition, 1942). Várkonyi, P. L., & Domokos, G. (2006a). Mono-monostatic bodies: the answer to Arnold’s question. Mathematical Intelligencer, 28(4), 34–38. Várkonyi, P. L., & Domokos, G. (2006b). Static equilibria of rigid bodies: dice, pebbles and the Poincare-Hopf Theorem. J. Nonlinear Sci., 16, 255–281. Vidal, M. (2001). A biological atlas of functional maps. Cell, 104, 333–339. Waddington, C. H. (1957). The strategy of the genes. London: Allen & Unwin. Wallach, W., & Allen, C. (2009). Moral machines. Teaching robots right from wrong. New York: Oxford University Press. Watterson, G. A. (1996). Motoo Kimura’s Use of Diffusion Theory in Population Genetics. Theoretical Population Biology, 49, 154–188. Watts, D. J. (1999). Small worlds. The dynamics of networks between order and randomness. Princeton: Princeton University Press. Weber, C. (2002). Questions de topologie en biologie moléculaire (commentaires de A. Stasiak et J. Dubochet). In Où en sont les mathématiques?, pp. 328–347. Paris: Vuibert/S.M.F. Wheater, J. F. (1994). Random surfaces: from polymer membranes to strings. J. Phys. A: Math. Gen., 27, 3323–3353. Wholeistic biology. Science, 295 (2002). Wigner, E. (1960). The Unreasonable Effectiveness of Mathematics in the Natural sciences. Communications in Pure and Applied Mathematics, 13, 1–14. Wigner, E. P. (1963). Events, laws of nature, and invariance principles. Nobel Lecture, December 12 (“The Nobel Prize in Physics 1963”. Nobelprize.org. Nobel Media AB 2014. Web. 18 Aug 2014. http://www. nobelprize.org/nobel_prizes/physics/laureates/1963/).

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Mateusz Hohol Institute of Philosophy and Sociology, Polish Academy of Sciences Copernicus Center for Interdisciplinary Studies

Michał Furman Pontifical University of John Paul II Copernicus Center for Interdisciplinary Studies

On Explanation in Neuroscience: The Mechanistic Framework

It is a truism to say that some concepts are extremely popular in science and philosophy. “Mechanism” is undoubtedly one of them and the roots of its popularity can be associated with the rise of modern science and rejection of the Aristotelian conceptual framework in the explanation of nature (Kuhn, 1957). Although Rene Descartes and Isaac Newton presented different conceptions of cosmology, in the field of philosophy of science they were both supporters of mechanicism (Heller, 2011; Heller & Życiński, 2014). “The universe is a mechanical clock” metaphor was tempting, because it promised that the laws of mechanics could be applied both to the universe as a whole and to its elements, including the human body. This may be exemplified by William Harvey’s model of blood circulation, in which the heart is understood as a mechanical pump (Craver & Darden, 2013). The seventeenth century scholars agreed in principle that the physiology of living organisms can be explained mechanistically. Descartes put it this way: I do not recognize any difference between artifacts and natural bodies except that the operations of artifacts are for the most part performed

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by mechanisms which are large enough to be easily perceivable by the senses – as indeed must be the case if they are to be capable of being manufactured by human beings. The effects produced by nature, by contrast, almost always depend on structures which are so minute that they completely elude our senses (Descartes 1983/1644, IV, § 203).

However, early modern mechanicists differed in their approach to the human mind. According to Descartes, mind was the mental substance (res cogitans) and – in contrast to the physical body (res extensa) – remained beyond the reach of mechanicism. In turn, another eminent mechanicist Julien Offray de La Mettrie argued that Descartes “had counted one substance too many” (Draaisma, 2000, p. 71). Thus, La Mettrie decided that nothing stood in the way of exploring the mind with mechanistic methods. Contemporary mechanicism in the biological sciences has sided with La Mettrie, extending its reach to brain functions and human cognition. William Bechtel, one of the most important theorists of mechanicism in neuroscience, puts it very aptly: Here modern science has taken a different path, outdoing Descartes at his own endeavor by finding mechanistic explanations for mental as well as bodily phenomena. Cognitive scientists, and their predecessors and colleagues in such fields as psychology and neuroscience, assume that the mind is a complex of mechanisms that produce those phenomena we call “mental” or “psychological” (Bechtel, 2008, p. 2).

Thanks to Thomas Kuhn we know that scientific revolutions are associated with conceptual incommensurabilities – after the revolution, a given concept changes its meaning in relation to the state before the revolution (Kuhn, 1962). This relates not only to physical concepts, such as “simultaneity of events,” but also to metascientific concepts – those which are the subject of research in the philosophy of science. Even contemporary philosophy of science itself seems to be a child of the metascientific revolution, an event perpetrated on the one hand by philosophers (primarily members of the Vienna Circle and their main

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critic, Karl Popper), and on the other by scientists, especially physicists (Życiński, 1988). As a result of the metascientific revolution, the concept of mechanicism changed its modern meaning. However, a contemporary “new mechanicism,” typical primarily for the philosophy of biological sciences, differs from the seventeenth century “clockwork universe.” Our investigations will focus on the latest installment of mechanicism and the role of the mechanisms in neuroscientific explanation (cf. e.g. Bechtel, 2008; Craver 2007, Craver & Darden, 2013). In this article, we set ourselves the following objectives. In the first part of the work we characterize the paradigm of mechanistic explanation in contemporary life sciences, narrowing the investigation to neuroscience. We indicate the general characteristics of the mechanistic framework and differences that occur in the various versions of mechanism. We present examples of scientific explanations which have successfully managed to apply the mechanistic framework. In the second part, we present norms of mechanistic explanation which allow us to distinguish explanatory models from non-explanatory models, which serve as heuristics. In the third part, we present a multilevel model of spatial navigation in accordance with the norms of mechanistic explanation – a model emerging from the latest discoveries in neuroscience. In the last, fourth part of the paper we discuss some philosophical and methodological issues of mechanicism.

1. Towards a mechanistic explanation Explanation of phenomena is one of the main tasks of science. However, there is no consent among methodologists of science regarding what scientific explanation is. The starting point for discussion is usually a neo-positivist conception of explanation, referred to as the deductive-nomological model or covering law model, formulated by Carl Hempel and Paul Oppenheim (1948; cf. Hempel, 1966). In this sense, explanation of the phenomenon involves performing reasoning having the following features:

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(1) explanans are the premises of reasoning (2) explanandum is a conclusion (3) there is at least one law in the set of premises (explanans) (4) the presmises have empirical content. If all the above conditions ((1) – (4)) are met, we are facing a full-blooded explanation of the phenomenon. Explanations that do not meet the condition (4) are not incorrect, but are referred to as potential. In practice, they are often valuable because of their heuristic function. Explanatory reasoning in deductive-nomological model takes the following form: L1, L2,…, Lr C1, C2,..., Ck Explanans sentences _____________ E Explanandum sentence L1, L2, ..., Lr represent laws; proper explanation requires obligatory reference to at least one law. C1, C2, ..., Ck, represent sentences related to the initial conditions. Sentence E refers to the phenomenon being explained. Sentences referred to as the laws must relate to general regularities, and thus they take the following logical form: (∀x) [F(x) → G(x)], which should be read: for all x, when x is subjected to F-conditions, x acts in a G-way. Hempel writes: The laws required for deductive-nomological explanations share a basic characteristic: they are, as we shall say, statements of universal form. Broadly speaking, a statement of this kind asserts a uniform connection between different empirical phenomena or between different aspects of an empirical phenomenon. It is a statement to the effect that whenever and wherever conditions of a specified kind F occur, then so will, always and without exception, certain conditions of another kind, G. (Hempel, 1966, p. 54).

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However, this scheme works well only in case of nomothetic sciences – scientific disciplines formulating laws. The prototype of nomothetic science is physics although in biological sciences, especially in neuroscience and medicine (Marcum, 2008; Juś 2014) one can find laws, as a rule they are not nomothetic. Universal, deterministic and mathematicised laws rarely occur in these sciences. Neuroscientists sometimes refer to the general laws of evolution and genetics in their articles (e.g. Hardy-Weinberg principle); however, they cannot be regarded as prima facie neuroscientific. This means that the deductivenomological model of scientific explanation can only be used occasionally in the case of these disciplines. Admittedly, “weaker,” statistical variations of this approach have been developed, according to which explanation involves deductive (deductive-statistical model) or inductive reasoning (inductive-statistical model), but they also have not found widespread use within the framework of biological sciences, particularly neuroscience (Craver, 2007; Bechtel & Richardson, 2010). Moreover, it appears that these models of explanation lead to paradoxes (cf. Bromberger, 1966). Problems with the nomological-deductive model of scientific explanation and its weaker variations have led to new insights on the nature of explanation (Salmon, 2006). Supporters of the deductive-nomological model have also been accused of the fact that they ignore the role of causality in scientific explanation – a role pointed out by Aristotle. Michael Scriven (1962) also noted that the reference to the law in explaining the phenomena may be helpful, but not mandatory. For example, to explain the phenomenon of a blue stain on the carpet, it is sufficient to note that it was caused by spilled ink. In this case, the identification of the cause equals the explanation of the phenomenon. In practice, scientific explanations generally relate to more complex phenomena in which the identification of the cause is not as easy. In addition, at least since David Hume (1975/1748) “causality” is one of the most problematic concepts in philosophy. Although intuitively causality is related to contact in space and time in the sense that causes occur prior to their effects, it is difficult to distinguish causality from correlation.

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The theory of causality, which is often used in contemporary analyses of scientific explanation (cf. Craver, 2006; Craver & Darden, 2013; Juś, 2014; Miłkowski, 2013a), was formulated by James Woodward (2003). The starting point of this theory is the requirement of connection of explanations, which are to be provided by science, with experiments. According to Woodward: “We are in a position to explain when we have information that is relevant to manipulating, controlling, or changing nature (…). We have at least the beginnings of an explanation when we have identified factors or conditions such that manipulations or changes in those factors or conditions will produce changes in the outcome being explained” (ibidem, pp. 9–10). Woodward argues that mere observation does not say anything about causality. However, knowledge of causality is possible and can be acquired through experimental manipulations (cf. Miłkowski, 2013b; Pearl, 2000). Causal explanations demonstrate how modification of the value of one variable modifies the value of another variable or a set of variables: the claim that X causes Y means that for at least some individuals, there is a possible manipulation of some value of X that they possess, which, given other appropriate conditions (perhaps including manipulations that fix other variables distinct from X at certain values), will change the value of Y or the probability distribution of Y for those individuals. In this sense, changing the value of X is a means to or strategy for changing the value of Y (Woodward, 2003, p. 40).

Woodward’s theory of causal explanation is applicable in case of basic neuroscience and cognitive neuroscience, as these are par excellence experimental disciplines in which, depending on the level of complexity, various research methods are used: single-units recording, bioelectric brain-activity measurements (electroencephalography, EEG) or magnetic brain-activity measurements (magnetoencephalography, MEG), indirect neuroimaging (e.g. functional magnetic resonance imaging, fMRI), or temporary deactivation of brain structures (transcranial magnetic stimulation, TMS). The conception of causal explanation

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can also be applied to computational neuroscience and neurorobotics. In computational neuroscience, models are created and tested in computer simulations, in which the manipulated variables influence other variables. In the other one, physical models are constructed and then tested in the environment (Miłkowski 2013a, 2013b). What is more, the theory of causal explanation can easily be extended to include contextual aspects (for contextual account of scientific explanation, see Van Fraassen, 1980). This is because neuroscience discoveries often find their application in neurology, neuropsychology and clinical psychiatry. It must also be noted that the above-outlined approach to causality was clarified mathematically, and its practical application is facilitated by a special programming language (cf. Pearl, 2000). William Wimsatt (1972, p. 67) states that “at least in biology, most scientists see their work as explaining types of phenomena by discovering mechanisms.” This is indicated by frequent appearance of the term “mechanism” in the titles and abstracts of scientific publications in the field of biological sciences – especially neuroscience. Furthermore, a specific form of causal explanation called the “mechanicism” – or to distinguish it from proposals of early modern philosophers (such as Descartes), the “new mechanicism” – is adopted in many monographs (cf. e.g. Bechtel, 2008; Craver 2007; Craver & Darden, 2013; Miłkowski, 2013a) devoted to explanation in neuroscience and cognitive science. Therefore mechanicism is not an artificial framework developed by methodologists of science, but rather a result of the coevolution of philosophical inquiries and practice of scientists. To date, thanks to the mechanical framework, scientists have managed to formulate explanations of biological phenomena such as: cellular respiration and Krebs cycle (Bechtel, 2005, Bechtel & Abrahamsen, 2007), circadian rhythms (Bechtel, 2010), fertilization (Craver & Darden, 2013), genetics and heredity (Darden, 1991; Darden & Maull, 1977), protein synthesis (Machamer, Darden & Craver, 2000). As for neuroscience, inter alia the following mechanistic explanations have been formulated: action potential (Craver, 2006, 2008), chemical transmission at synapses (Machamer, Darden

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& Craver, 2000), memory and learning (Bechtel, 2009, Craver, 2002, 2003, Craver & Darden, 2001). According to the mechanistic strategy, explanation of the phenomenon does not depend on its logical inference from the initial conditions and laws of nature, but on the identification of the causal structure generating it. Generally speaking, the mechanism is a system composed of multiple parts (Bechtel & Richardson, 2010), which as a whole manifests a pattern of action (disposition), e.g.: The mechanism of chemical neurotransmission, a pre-synaptic neuron transmits a signal to a post-synaptic neuron by releasing neurotransmitter molecules that diffuse across the synaptic cleft, bind to receptors, and so depolarize the post-synaptic cell (…). Descriptions of mechanisms show how the termination conditions are produced by the set-up conditions and intermediate stages. To give a description of a mechanism for a phenomenon is to explain that phenomenon, i.e., to explain how it was produced (Machamer, Darden & Craver, 2000, p. 3).

According to supporters of mechanism, the disposition of a complex system should be explained as the result of the interaction of the individual parts of the mechanism or the structure of the processes taking place in it (Miłkowski, 2013b). Although the majority of supporters of mechanism would agree with this general characteristic, individual theorists propose solutions which differ in the details. Let us examine three alternative interpretations of the mechanism formulated by Stuart S. Glennan (G), William Bechtel (B) and Peter Machamer, Lindley Darden and Carl F. Craver (MDC): G: A mechanism underlying a behavior is a complex system which produces that behavior by the interaction of a number of parts according to direct causal laws (Glennan, 1966, p. 52). B: A mechanism is a structure performing a function in virtue of its component parts, component operations, and their organization. The

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orchestrated functioning of the mechanism is responsible for one or more phenomena (Bechtel, 2008, p. 13). MDC: Mechanisms are entities and activities organized such that they are productive of regular changes from start or set-up to finish or termination conditions (Machamer, Darden & Craver, 2000, p. 3).

Although at first glance these definitions appear similar, they emphasize different elements. The first two – G and B – emphasize that the mechanisms are identified on the basis of phenomena/behavior for which they are responsible. However, there are differences between them. G emphasizes the primary role of the interaction between the parts of the mechanism. It also puts particular emphasis on the role of direct causal laws. Glennan’s approach is of a reductionist nature – laws functioning at a given level are explained by reducing them to lower level laws, until the fundamental laws of physics are reached. On the other hand, the B approach – as emphasized by Bechtel himself – “focused on the ‘functions’ (operations) that parts perform” (p. 13, footnote 4). With this approach, the need for reference to direct causal laws disappears – causality occurs at the level of interaction between the parts. Last approach, MDC, seems to take into account the claims of both G and B. MDC emphasizes the role of activities in the mechanisms, while these activities are not necessarily scientific laws in the strict sense. We operationally choose the definition of mechanism proposed in MDC approach. It should be emphasized that in this approach mechanisms should not to be understood in reference to mechanical systems based on the principle of action and reaction. Machamer, Darden and Craver argue that the mechanisms are complex structures constituted by both entities and activities. According to MDC approach it is assumed that entities and activities exist in reality and can be identified by empirical research. What’s more, it is a pluralistic form of scientific realism: you cannot reduce activities to entities and vice versa. MDC is therefore inconsistent with both substantialism (e.g. Aristotle) and processual-

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ism (e.g. Whitehead), because each of these approaches proclaims the fundamental nature of only one of these categories: entities or activities. For this reason mechanicism according to MDC approach is defined as dualistic. Machamer et al. (2000, p. 3) characterizes their approach more precisely as follows: Activities are the producers of change. Entities are the things that engage in activities. Activities usually require that entities have specific types of properties. The neurotransmitter and receptor, two entities, bind, an activity, by virtue of their structural properties and charge distributions (…). The organization of these entities and activities determines the ways in which they produce the phenomenon. Entities often must be appropriately located, structured, and oriented, and the activities in which they engage must have a temporal order, rate, and duration. For example, two neurons must be spatially proximate for diffusion of the neurotransmitter. Mechanisms are regular in that they work always or for the most part in the same way under the same conditions. The regularity is exhibited in the typical way that the mechanism runs from beginning to end; what makes it regular is the productive continuity between stages. Complete descriptions of mechanisms exhibit productive continuity without gaps from the set up to termination conditions. Productive continuities are what make the connections be- tween stages intelligible. If a mechanism is represented schematically by A → B → C, then the continuity lies in the arrows and their explication is in terms of the activities that the arrows represent. A missing arrow, namely, the inability to specify an activity, leaves an explanatory gap in the productive continuity of the mechanism.

As can be seen in the above passage, in the MDC approach the concept of the laws of science is replaced by slightly looser notion of regularity. Of course, this does not mean that the laws do not exist or take part in the explanation. Their presence is simply not obligatory. It should also be emphasized that the identification of entities can be made due to the spatial position, structure and hierarchy, while the ac-

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tivities can be identified due to the time organization, pace and duration. The diagram below represents mechanistic explanation: phenomenon (explanandum)

mechanism (explanans) This diagram illustrates the mechanistic explanation (adopted, see Craver, 2007, p. 7).

The top part of the diagram represents the explained phenomenon (explanandum). The ψ symbol indicates phenomenon, property or action explained by the mechanism. The S symbol denotes the mechanism as a whole. The lower part of the diagram represents entities (circles) and activities (arrows coming in and coming out of circles). They serve as explanans (what explains) in the process of explanation. X is a component (part) of the object, and ф symbolises action of the object in the mechanism. The explanation of Sψ phenomenon consists in the discovery and demonstration of structure of the objects {X1, X2, X3, ..., Xm) and actions {ф1, ф2, ф3, ..., фn}.

2. Norms of mechanistic explanation Comprehensive explanation requires a complete description of the causal structure thanks to which the explained phenomenon occurs.

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This calls for the identification of a constitutive level of the mechanism and for its decomposition, thus revealing entities and activities. The constitutive level is determined in scientific practice. For example, specific neural networks, specific subcortical and cortical structures or single neurons establish a constitutive level for the mechanisms examined within the framework of cognitive neuroscience, which studies the brain substrates of cognitive processes (Piccinini & Craver, 2011). Mechanisms considered in the context of cognitive neuroscience are usually hierarchical and multi-level. In case of basic neuroscience, single neurons or biological and chemical entities, constituting the neuronal level itself, become the constitutive level. It should be emphasized that in principle the constitutive level is not defined arbitrarily, but is identified pragmatically for each mechanism (Machamer et. al., 2000). Therefore, although mechanicists apply decomposition, which is de facto a form of reduction, they avoid highly reductionist statements about the fundamentality of the physical level for biological mechanisms (cf. Craver, 2007; Miłkowski, 2013b). However, the pragmatic nature of explanation in neuroscience does not mean arbitrariness. On the contrary, the process of clarifying demands compliance with certain standards. Craver (2006) notes that not every model constructed by scientists is an explanation. He distinguishes between: (i) the phenomenal models and explanations, (ii) sketches and complete explanations, as well as (iii) possibly models and actually models. Regarding the first distinction, it should be noted that phenomenal adequacy does not necessarily involve the explanatory force. The Ptolemaic model of the solar system, which properly predicts the positions of celestial bodies (and therefore is empirically appropriate, cf. Kuhn, 1957), but does not provide an explanation of their movements utilizing the causal mechanism (Craver, 2006), is an often cited example from the history of science. According to a mechanistic norm of explanation (N1), models explaining the phenomena are not only phenomenal models, as they allow for interventions and experimental manipulations, enabling control over the phenomenon. This allows for e.g. the

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determination of the causal relationships between objects described by the model (Woodward, 2003). The second distinction concerns the level of the detail of the description. Any explanatory model can be located on the continuum a mechanism sketch and a fully complete description of the mechanism (Craver, 2006). Although the mechanism sketches describe the structure of entities and activities, they are not complete explanations because they contain gaps identified in the diagrams by e.g. black boxes. Sometimes sketches also contain not defined precisely filler terms, to which Craver (2006) includes e.g.: “activate,” “encode,” “inhibit” or “represent.” While gaps in most cases are filled thanks to new discoveries, filler terms often pose an obstacle to the advancement of knowledge. A full description of the mechanism “includes all of the entities, properties, activities, and organizational features that are relevant to every aspect of the phenomenon to be explained” (Craver, 2006, p. 360). In fact, it is an ideal which very rarely, if ever, occurs in scientific practice. According to the mechanistic norm (N2), explanations should approximate fully complete descriptions of the mechanisms. However, the progress of science is often a result of simplifying and removing irrelevant factors from the model, and therefore explanations should be pragmatically complete. Models that are on the continuum between sketches and fully complete descriptions are referred to as mechanism schemata. Diagrams are simplified and abstract descriptions of mechanisms open to future descriptions of entities and activities (Machamer et al., 2000). The third distinction concerns the how-possibly models and howactually models. It refers to another norm of mechanicism (N3): in case of how-actually models “components described in the model should correspond to components in the mechanism (...). How-actually models describe real components, activities, and organizational features of the mechanism, that in fact produces the phenomenon” (Craver, 2006, p. 362). This means that the model constructed by scientists in order to explain the operation of the system should not only be adequate during the stages of input and output, but should also

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simulate actual behavior of internal sub-systems. A model that does not meet this requirement is the how-possibly model. Mechanistic models should not only be correct instrumentally, but also adequate in terms of scientific realism (Craver, 2007; Machamer et al., 2000). In summary, explanation (N1) cannot be a purely phenomenal model, (N2) should contain as few gaps and filler terms as possible, and (N3) should specify actual entities and activities, constituting operations of the system. Of course, norms (N1), (N2), and (N3) are not being fulfilled in accordance with the all-or-nothing principle. Craver (2006) applies these norms of mechanistic explanation to the Hodgkin-Huxley model (1952). This model mathematically describes how action potentials in neurons are initiated and propagated. Developing this model was a milestone in the development of neuroscience, and its creators were honored with the Nobel Prize in Physiology or Medicine in 1963. It characterizes the time-course of the action potential in following equation (cf. Craver, 2006, p. 363): I = CMdV/dt + GKn4(V − VK) + GNam3h(V − VNa) + G1(V − V1) I refers to the total current crossing the membrane. It is constituted by four factors: the capacitative current CMdV/dt, the potassium current GKn4(V − VK), the sodium current GNam3h(V − VNa), as well as the so-called “leakage current” G1(V − V1), which is a sum of smaller currents for other ions. Maximum values of conductivity for respective currents are symbolized by GK, GN and G1. V is displacement of Vm from Vrest. Differences between Vm and diverse ionic equilibrium potentials are maped by VK, VNa, and V1. Membrane’s capacity to store opposite charges on the outside and inside of a neuron is represented by CM (capacitance). Symbols h, m, and n, represent factors with different values in relation to voltage and time. Craver (2006) states that although the Hodgkin-Huxley model has many practical applications – e.g. it can be used in simulations of neuronal activity – it does not meet the norms of mechanistic expla-

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nation. He underlines that, “the authors insist that their model is not an explanation” (Craver, 2006, p. 356). He demonstrates that according to the Hodgkin and Huxley equations are merely empirically adequate description of the time course of permeability changes. Craver (2006, p. 364) also quotes one of Hodgkin’s and Huxley’s associates who thinks that this model “summarized in one neat tidy little package the many thousands of experiments done previous to 1952, and most subsequent ones” (Cole, 1992, p. 51). Equations do not contain a causal description of the mechanism of changing the membrane conductance. Hodgkin and Huxley model is therefore purely phenomenal, and thus does not meet the (N1) norm of a mechanistic explanation (just as the Ptolemaic model does). It is worth noting that this standard corresponds with the view of Aristotle, who claimed that describing via mathematical structures does not provide explanation – causal explanation is necessary (cf. Hankinson, 1998). Huxley’s and Hodgkin’s knowledge about action potential went far beyond the components of a basic equation. In other words, a mathematical model was supported by background knowledge of the facts that action potential is a result of the alteration of the permeability of a membrane; that ions move along the surface of the membrane in the direction of the equilibrium potentials; and that their movement results in a transmembrane current (Craver, 2006). Some entities and activities of the mechanism are distinguished. Inclusion of this knowledge leads to a partial explanation of how nerve cells generate action potentials. Nevertheless, it is not a complete explanation, but only a sketch. Thus, it does not satisfy the norm (N2), although Craver believes that this model may provide a basis for explanations: The equations include variables that represent important components in the explanation. And they provide powerful evidence that a mechanism built from those components could possibly explain the action potential. And the equations, supplemented with a diagram of the electrical circuit in a membrane, and supplemented with details about

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how membranes and ion channels work, carry considerable explanatory weight (2006, p. 365).

This example draws attention to an important practical consequence: mechanistic explanations in biological sciences – especially in neuroscience – often use diagrams. They not only help in understanding a particular phenomenon, but reveal its causal structure, actually taking part in scientific explanation (Abrahamsen & Bechtel, 2015; Sheredos, Burnston, Abrahamsen, & Bechtel, 2013). The last standard of mechanistic explanation (N3) is also not completely satisfied. The mathematical equation of Hodgkin and Huxley leads to a how-possibly sketch of the action potential (Craver, 2006). Multiple equations, other than the one proposed by Hodgkin and Huxley, can be used to predict the action potential’s time-course. Background knowledge behind these equations can be very different, as well as their empirical interpretation. In other words, different explanatory mechanisms may result in different mathematical models of phenomenon. According to the standards of mechanicism: (N1), (N2) and (N3), explanation of how action potentials in neurons are initiated and propagated, has become possible thanks to decades of scientific discoveries, among others concerning the structure and function of ion channels. Craver presents the transition from the phenomenal model of Hodgkin and Huxley to a mechanistic explanation consistent with this model: (…) it is now well-known that conductance changes across the membrane are effected by conformation changes in ion-specific channels through the cell membrane. Biochemists have isolated these proteinaceous channels, they have sequenced their constituents, and they have learned a great deal about how they activate and inactivate during an action potential. It is in this wealth of detail (some of which is discussed below) about how these channels regulate the timing of the conductance changes, as described by the H[odgkin]-H[uxley’s] equations, that explain the temporal course of the action potential (Craver, 2006, p. 367).

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In this section we presented standards of mechanistic explanation using action potential (a phenomenon fundamental to neuroscience) as an example. In the next section we will present a practical application of mechanistic framework using the example of the more complex phenomenon of spatial navigation.

3. Case study: spatial navigation Let us consider one of the examples presented by supporters of mechanistic explanation – spatial navigation in humans and animals (Craver & Darden, 2001; Craver, 2003). The issue of the mental representation of space has been present in philosophy at least since Immanuel Kant. At present it has become the subject of psychological and neuroscientific research and is a perfect test area for the mechanistic framework. At the end of the first half of the twentieth century, the results of experiments conducted on rats led Tolman (1948) to believe that animals store relations between experienced objects and events in form of maps in their memory. This view was not consistent with the stance of the behaviorist who explained spatial navigation by referring to the sensory-motor response relationships. Although Tolman showed that cognitive representations of space explored by the animal may resemble maps, he did not indicate their location in the brain. A year later, Donald Hebb (1949) suggested that to increase the strength of synapses occurs when the presynaptic and postsynaptic neuron are simultaneously stimulated. Hebb’s rule is often illustrated by the maxim: “What fires together, wires together.” It is assumed that the spatial representation is associated with learning processes that occur by an increase of the weight of synapses (as described by Hebb), which is referred to as long-term potentiation (LTP). It is known that this phenomenon can be divided into three phases: the induction, expression and maintenance. The induction of LTP depends on the activation of NMDA receptors, the increased re-

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lease of glutamic acid in axon terminals in CA3 part of the hippocampus, and the depolarization of membrane of postsynaptic neurons in the CA1 part of hippocampus (symbols of the hippocampal parts are derived from a different name of hippocampus: cornus ammonis). This means that the LTP is induced when the simultaneous activation of pre- and postsynaptic cells occurs, something which is consistent with Hebb’s rule. The opening of ion channels allows the transport of NMDA receptor-bound calcium ions inside the neuron, playing a crucial role in the induction of LTP. AMPA receptors are activated, resulting in the opening of the sodium channel – it brings an influx of Na+ ions and membrane depolarization (see e.g. Longstaff, 2007, pp. 387–420, for details). Cells associated with spatial navigation have been found in the mouse hippocampus thanks to research using implanted micro wires (O’Keefe & Dostrovsky, 1971). This conclusion has been confirmed in studies using pharmacological agents and gene knockout technique – changes in the production of LTP in the hippocampus are associated with deficits in spatial learning, but not necessarily learning in general (cf. Tsien, Huerta & Tonegawa, 1996). In contemporary neuroscience it is generally accepted that spatial maps (the existence of which was suggested by Tolman) are stored in the hippocampus (Derdikman & Moser, 2011). Honoring three neuroscientists – John O’Keefe, May-Britt Moser and Edward Moser with the Nobel Prize in Physiology or Medicine “for their discoveries of cells that constitute a positioning system in the brain” has been the crowning achievement of research on spatial navigation. This system – commonly known as cerebral GPS – consists of several subsystems: space cells located in the hippocampus and grid cells, border cells and head-direction cells activated in the parahippocampal cortex and the medial entorhinal cortex (see Derdikman & Moser, 2011, for a literature review). Derdikman and Moser (2011) claim that “space is represented in these structures by a manifold of rapidly interacting maps generated in conjunction by functionally specific cell types such as place cells and grid cells” (p. 42).

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Place cells are pyramidal neurons in the hippocampus which are activated when the animal is in a specific location. In other words, the specified location is a reception area for a given place cell. Quirk, Muller and Kubie (1990) showed that these cells are not specific only for the visual modality, as they are activated both in the light and in the dark. Although place cells are highly specific, apart from the location they also encode rules of learning with regard to scents. This is evidenced by the results of an experiment conducted by Wood, Dudchenko, and Eichenbaum (1999), in which rats were tasked with recognizing odors other than presented before, demanding the use of episodic memory. Studies in rats show that the brain navigation system is innate. Langston, Ainge, Couey, Canto, Bjerknes, Witter, Moser & Moser (2010) have identified place cells in the brains of rats, when these animals left the nest few days after their birth and began to explore the environment. In the same work, they demonstrated that the movement from one location to another, or a change in environmental conditions is accompanied by the update of the spatial map encoded by the same populations of neurons in the hippocampus. Langston et al. (2010) called this process remapping. Grid cells, which like place cells are used in animal spatial navigation in the environment, are also subject to it. Network cells are named after the fact that the open space around the animal is coded as discharges, which are visualized by a hexagonal network. These neurons have been located in medial entorhinal cortex (Fyhn, Molden, Witter, Moser & Moser, 2004) and in part of the hippocampus called the subiculum (Boccara, Sargolini, Thoresen, Solstad, Witter, Moser & Moser, 2010). These structures include border cells and head-direction cells as well. In summary, the contemporary approach to the hypothesis of cognitive maps is as follows (cf. Longstaff, 2007): the hippocampus is the key structure of the brain for spatial navigation. Optimal navigation of an animal in the environment is the function of cognitive maps. Cognitive maps are created thanks to episodic memory – through the process of association of sensory and motor guidelines representation of

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the animal’s location in space is generated. Maps are dynamic – they can be modified in the course of space exploration by the animal. Finally, mapping does not require reinforcements, which runs counter to the key assumption of behaviorism. Even such a brief description of discoveries in the field of neuroscience allows for reconstruction within the mechanical framework (Craver & Darden, 2001). Our approach to mechanism – adopted from MDC – requires first of all the distinction of entities and activities in spatial navigation. We distinguish the following entities: pyramidal neurons (e.g. space cells), molecules (e.g. glutamate molecules), receptors (e.g. NMDA receptors), brain structures (e.g. hippocampus), and whole organisms (e.g. rat). Activities include e.g. firing of pyramidal cells, increased glutamate release, depolarization of the cells, coactivation of pre- and postsynaptic cells, or exploration of the environment by the rat. Craver and Darden (2001) distinguish three aspects of the mechanism: spatial, temporal and hierarchical. The spatial layout of mechanism is associated with the location of its elements. According to most researchers, the hippocampus is an area of the brain which consists of part of the elements of the spatial memory mechanism. With this assumption, we can study the construction of the hippocampus in order to discover how it translates into its activity and how this is related to the functioning of a given mechanism. The temporal aspect refers to the direction and order of occurrence of activities within mechanisms. It carries information about how activities responsible for productive continuity are temporarily positioned with respect to each other. This helps to avoid the paradox of the asymmetry characterizing deductive-nomological model which hinders a correct indication of causal relations of explained phenomena. As far as the hierarchical aspect of the mechanism are concerned, four levels of biological complexity can be abstracted the above description, listed from the highest to the lowest: (L1) level of the whole organism; (L2) level of brain structures;

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(L3) level of synapses; (L4) level of neuroreceptors. The mechanistic explanation of the phenomenon takes place in reverse order to the above hierarchy. Activation of the NMDA receptors (L4) can result in LTP, which relates to the increase in synaptic conduction (L3); LTP enables the consolidation of spatial map in the hippocampus – the structure of the midbrain responsible for the consolidation of information (L2). It is due to the spatial map that an organism (L1) – e.g. a rat – can carry out the task given by the researchers, e.g. find its way in the maze. These levels are not completely independent of each other and their hierarchy is a nested one. The mechanistic explanation is therefore of an integrating nature: The elaboration and refinement these hierarchical descriptions typically proceeds piecemeal with the goal of integrating the entities and activities at different levels. Integrating a component of a mechanism into such a hierarchy involves, first, contextualizing the item within the mechanism of the phenomenon to be explained. This involves “looking up” a level and finding a functional role for the item in that higher-level mechanism” (Darden & Craver, 2001, pp. 118–119).

We encounter such a nested hierarchy only if the mechanism manifests a productive continuity between the stages at different levels. According to MDC it is this objectively existing productive continuity which enables explanation and understanding of the mechanisms. The discovery of a productive continuity between levels in the hierarchy is possible thanks to experimental interventions. Bottom-up and top-down experiments are distinguished within the framework of the mechanicism (Bechtel, 2002; Craver & Darden, 2001). In the first case, the influence of lower-level interventions on the functions of the higher levels of the system is studied. In the case of lesion studies, researchers are interested in the ways in which damage to parts of the system affect the functioning of the whole system.

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It is known that damage to the hippocampus results in spatial memory deficits, confirming the role of this structure in spatial navigation. One of the most famous neurological cases – of a patient with the initials H.M. – provides an example of bottom-up experiments. As a result of the surgical removal of the hippocampus, which aimed at the alleviation of seizures, the patient lost the ability to memorize new facts (anterograde amnesia). On the other hand, in the case of top-down experiments, intervention is performed at a higher level and its impact on the lower level of the system is examined. Studies using techniques of single-units recording, conducted by Nobel Prize winners – O’Keffe and Moosers – are cases of top-down experiments. It is similar in the case of research using neuroimaging techniques, i.e. fMRI. The consistency of the results of top-down and bottom-up recording provides arguments for the existence of relations between different levels of the mechanism. However, productive continuity is revealed by a third type of neuroscientific research – multilevel experiments. Studies on the influence of so-called gene knockouts on spatial memory elements located at different levels of the studied mechanism are an example of such experiments (McHugh et al., 1996; Rottenberg et al., 1996; Tsien et al., 1996). This bottom-up research is considered the first one to simultaneously test the functioning of the mechanism of spatial memory at all levels. It involved damaging the gene responsible for specific NMDA receptors in mice. During the study it has been found that these mice cope with the tasks far worse than animals in the control group. This phenomenon is explained by the fact that the failure of NMDA receptors in cells prevented LTP in hippocampal CA1 region (Tsien et al., 1996). What is important from the methodological point of view is the fact that intervention at the bottom level led to the detection of phenomena on multiple upper levels, an essential feature of a multilevel experiment. In line with what has been said in the previous section, explanation (N1) cannot be a purely phenomenal model, (N2) it should contain as few gaps and filler terms as possible, and (N3) it should specify the actual entities and activities constituting the system. As far as the norm

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(N1) is concerned, the abovementioned theory of spatial navigation is certainly not only a phenomenal model, but an explanatory one – it is not only a summary of empirical data. Norm (N2) is partially satisfied. First of all, the mechanism outlined above has gaps. The area of LTP expression is the subject of controversy among scholars (cf. Longstaff, 2007). Most scholars believe that the area of expression in the synapses of CA1 and CA3 hippocampal neurons is located in the postsynaptic membrane. However, some researchers claim that the expression of LTP occurs in the presynaptic membrane. Secondly, one can argue that terms such as “encode” or “represent,” appearing in the papers on spatial navigation, belong to filler-terms. Thirdly, with regard to the above-described experiments using gene knockout technique, Craver and Darden (2001) argue that their results lead to a possible mechanism. According to them “it is a sketch because we are not remotely in a position to trace out all of the mechanisms at all of the different levels” (p. 132). This results in a lack of the complete fulfillment of a norm (N3): although we know that, for example, space cells or hippocampus are entities of the mechanism, gaps result in the classification of this mechanism as possible, though very probable.

4. Summary and further remarks In this article we have presented a mechanistic framework of scientific explanation. We argued that within biological sciences – especially neuroscience – it is more adequate than the neopositivist deductive-nomological model. We showed how different methodologists understand the key term to this approach – “mechanism,” focusing on MDC proposal. In view of the fact that the mechanistic framework is a special case of causal explanation, we explicated the concept of “causality,” referring to the approach of Woodward (2003). We presented the norms of mechanistic explanation, showing (on the basis of Hodgkin-Huxley’s model (1952)) that not every model constructed by neuroscientists has an explanatory character. Using spatial navigation

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as an example we showed that contemporary theories (Derdikman & Moser, 2011) approach the standards of mechanistic explanation. An example of spatial navigation also showed that the mechanistic framework is of a unifying nature – it integrates multiple levels of biological complexity (Craver & Darden, 2001). At this point we have to say that mechanicism demands a rethink of some issues raised traditionally in the general philosophy of science. Firstly, we wrote that mechanicism is more natural than the deductive-nomological model due to the fact that neuroscience is not a typical nomothetic science (its task is not to formulate laws). Although in principle it is true, it must be said that the very concept of “law of nature” is not self-evident (cf. Armstrong, 1983; Cartwright, 1983), and thus classifying strategies of scientific explanation on the basis of their use of laws of nature may raise objections. Moreover, the very concept of causality that appears in the mechanistic explanation refers to regularity (Craver & Kaiser, 2013; Miłkowski, 2013b). This means that with the right formulation of the concept of “law of nature” mechanistic explanation and deductive-nomological explanation may prove to be uncompetitive (Anderson, 2011; cf. Brożek, 2015). Secondly, there is a dispute within the philosophy of science about whether this discipline should rather be of a prescriptive or descriptive nature. In other words, the question is whether philosophy of science should only explicate actual scientific practice or rather determine the way for scientists. It seems that the mechanistic view incorporates arguments from both sides of the dispute. The very notion of “mechanism” is not artificial, but is derived directly from scientific practice. However, the mechanistic view formulates norms which distinguish explanatory models from non-explanatory ones. On the other hand, it is difficult to imagine a case of a completely accurate model of a phenomenon examined by neuroscience. To paraphrase a long discussion on the role of the concept of “truth” in science, perhaps we are doomed only to possibly models. This point is related to the last issue we would like to briefly comment upon.

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Machamer, Darden and Craver’s (2000) approach to mechanicism, which we have adopted in this paper, is marked by scientific realism. While within basic neuroscience it is possible to maintain such an approach, in the case of cognitive neuroscience the issue becomes more complicated (cf. Revonsuo, 2001; Brożek, 2011). Many explanations formulated by cognitive neuroscientists are very difficult to interpret in the spirit of scientific realism. This is because such theories operate not only at the level of observable brain structures, but they also refer to cognitive structures, which as a rule are treated as objects postulated by theory.

References Abrahamsen, A., & Bechtel, W. (2015). Diagrams as tools for scientific reasoning. Review of Philosophy and Psychology, 6(1), 117–131. Andersen, H. (2011). The case for regularity in mechanistic causal explanation. Synthese, 189(3), 415–432. Armstrong, D. (1983). What is a law of nature? Cambridge: Cambridge University Press. Bechtel, W. (2001). Decomposing and localizing vision: An exemplar for cognitive neuroscience. In W. Bechtel, P. Mandik, J. Mundale & R. S. Stufflebeam (Eds.), Philosophy and the neurosciences: A Reader (pp. 225–249). Oxford: Basil Blackwell. Bechtel, W. (2002). Aligning multiple research techniques in cognitive neuroscience: Why is important? Philosophy of Science, 69(S3), 48–58. Bechtel, W. (2005). The challenge of characterizing operations in the mechanisms underlying behavior. Journal of the Experimental Analysis of Behavior, 84(3), 313–325. Bechtel, W. (2008). Mental mechanisms: Philosophical perspectives on cognitive neuroscience. New York–London: Routledge. Bechtel, W. (2009). Molecules, systems, and behavior: Another view of memory consolidation. In J. Bickle (Ed.), Oxford handbook of philosophy and neuroscience, (pp. 1–40). Oxford: Oxford University Press. Bechtel, W. (2010). The downs and ups of mechanistic research: Circadian rhythm research as an exemplar. Erkenntnis, 73, 313–328.

Concept_AF.indd 231

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232

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Bechtel, W., & Abrahamsen, A. (2007). In search of mitochondrial mechanisms: Interfield excursions between cell biology and biochemistry. Journal of the History of Biology, 40(1), 1–33. Bechtel, W., & Richardson, R. C. (2010). Discovering complexity: decomposition and localization as strategies in scientific research. Cambridge–London: The MIT Press 2010. Boccara, C. N., Sargolini, F., Thoresen, V. H., Solstad, T., Witter, M. P., Moser, E. I., & Moser, M. B. (2010). Grid cells in pre- and parasubiculum. Nature Neuroscience, 13(8), 987–994. Brożek, B. (2011). Philosophy in neuroscience. In B. Brożek, J. Mączka & W. P. Grygiel (Eds.), Philosophy in science: Methods and applications (pp. 163–188). Kraków: Copernicus Center Press. Brożek, B. (2015). Explanation and understanding. In this volume (pp. 11–43). Cartwright, N. (1983). How the laws of physics lie. Oxford: Oxford University Press. Cole, K. (1992). Neuromembranes: Paths of ions. In I. F. G. Worden, J. P. Swazey & G. Adelma (Eds.), Neurosciences: Paths of discovery (pp. 143–157), vol. I. Cambridge, MA: The MIT Press. Craver, C. F. (2002). Interlevel experiments and multilevel mechanisms in the neuroscience of memory. Philosophy of Science Supplemental, 69, 83–97. Craver, C. F. (2003). The making of a memory mechanism. Journal of the History of Biology, 36, 153–195. Craver, C. F. (2006). When mechanistic models explain. Synthese, 153, 355–376. Craver, C. F. (2007). Explaining the brain: mechanisms and the mosaic unity of neuroscience. Oxford–New York: Oxford University Press. Craver, C.F. (2008). Physical law and mechanistic explanation in the Hodgkin and Huxley model of the action potential. Philosophy of Science, 75, 1022–1033. Craver, C. F., & Darden, L. (2001). Discovering mechanisms in neurobiology: The case of spatial memory. In P.K. Machamer, R. Grush & P. McLaughlin (Eds.), Theory and method in the neurosciences (pp. 112–137). Pittsburgh: University of Pittsburgh Press. Craver, C. F., & Darden, L. (2013). In search of mechanisms: Discoveries across the life sciences. Chicago: University of Chicago Press. Craver, C. F., & Kaiser, M. I. (2013). Mechanisms and laws: Clarifying the debate. In H. K. Chao, S. T. Chen & R. L. Millstein (Eds.), Mecha-

Concept_AF.indd 232

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On Explanation in Neuroscience: The Mechanistic Framework

233

nism and causality in biology and economics (pp. 125–145). Dordrecht: Springer Netherlands. Darden, L. (1991). Theory change in science: Strategies from mendelian genetics. Oxford: Oxford University Press. Darden, L., & Maull, N. (1977). Interfield theories. Philosophy of Science, 44(1), 43–64. Derdikman, D., & Moser, E. I. (2011). A manifold of the spatial maps in the brain. W: S. Dehaene, & E. Brannon (eds.), Space, time and number in the brain: Searching for the foundations of mathematical thought (pp. 41–57). Amsterdam: Academic Press. Descartes, R. (1983). Principles of philosophy. Dordrecht 1983 (originally published in Latin in 1644). Draaisma, D. (2000). Metaphors of memory: A history of ideas about the mind. Cambridge: Cambridge University Press. Fyhn, M., Molden, S., Witter, M. P., Moser, E. I., & Moser, M. B. (2004). Spatial representation in the entorhinal cortex. Science, 305, 1258–1264. Glennan, S. (1996). Mechanisms and the nature of causation. Erkenntnis, 44(1), 49–71. Hankinson, J. R. (1998). Cause and explanation in ancient Greek thought. Oxford: Oxford University Press. Hebb, D. (1949). The organization of behavior. New York: Wiley. Heller, M. (2011). Philosophy in science: An historical introduction. Heidelberg: Springer. Heller, M., & Życiński, J. (2014). Wszechświat: maszyna czy myśl?. Kraków: Copernicus Center Press. Hempel, C. G. (1966). Philosophy of natural science. New Jersey: Prentice Hall. Hempel, C. G., & Oppenheim, P. (1948). Studies in the logic of explanation. Philosophy of science, 5, 135–175. Hodgkin, A. L., & Huxley, A. F. (1952), A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117, 500–544. Hume, D. (1975). An enquiry concerning human understanding. Oxford: Clarendon Press (originally published in 1748). Juś, M. (2014). Spór o redukcjonizm w medycynie. Warszawa–Toruń: UMK. Kuhn, T. S. (1957). The Copernican revolution: Planetary astronomy in the development of western thought. Cambridge, MA: Harvard University Press.

Concept_AF.indd 233

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Kuhn, T. S. (1962). The structure of scientific revolutions. Chicago: University of Chicago Press. Langston, R. F., Ainge, J. A., Couey, J. J., Canto, C. B., Bjerknes, T. L., Witter, M. P., Moser, E. I., & Moser, M. B. (2010). Development of the spatial representation system in the rat. Science, 328(5985), 1576– 1580. Longstaff, A. (2007). Neuroscience. 2nd ed. New York: Taylor & Francis Machamer, P., Darden, L., & Craver, C. F. (2000). Thinking about mechanisms. Philosophy of Science, 67(1), 1–25. Marcum, J. A. (2008). Humanizing modern medicine: An introductory philosophy of medicine. New York: Springer. McHugh, T. J., Blum, K. I., Tsien, Z. J., Tonegawa, S., & Wilson, M. A. (1996). Impaired hippocampal representation of space in CAl-specific NMDAR1 knockout mice. Cell, 87, 1339–1349. Miłkowski, M. (2013a). Explaining the computational mind. Cambridge, MA: The MIT Press. Miłkowski, M. (2013b). Wyjaśnianie w kognitywistyce. Przegląd Filozoficzny – Nowa Seria, 2(86), 151–166. O’Keefe, J., & Dostrovsky, J. (1971). The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat. Brain Research, 34, 171–175. Pearl, J. (2000). Causality: Models, reasoning, and inference. Cambridge: Cambridge University Press. Piccinini, G., & Craver, C. (2011). Integrating psychology and neuroscience: Functional analyses as mechanism sketches. Synthese, 183(3), 283–311. Quirk, G. J., Muller, R. U., & Kubie, J. L. (1990). The firing of hippocampal place cells in the dark depends on the rat’s recent experience. The Journal of Neuroscience, 10(6), 2008–2017. Revonsuo, A. (2001). On the nature of explanation in the neurosciences. In P. K. Machamer, P. McLaughlin & R. Grush (Eds.), Theory and Method in the Neurosciences (pp. 45–69). Pittsburgh: University of Pittsburgh Press. Rottenberg, A., Mayford, M., Hawkins, R. D., Kandel, E. R., & Muller, R. U. (1996). Mice expressing activated CaMKII lack low frequency LTP and do not form stable place cells in the CA1 region of the hippocampus. Cell, 187, 1351–1361. Salmon, W. C. (2006). Four decades of scientific explanation. Pittsburgh: University of Pittsburgh Press.

Concept_AF.indd 234

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Scriven, M. (1962). Explanations, predictions, and laws. In H. Feigl & G. Maxwell (Eds.), Scientific explanation, space and time (pp. 170– 230). Minneapolis: University of Minnesota Press. Sheredos, B., Burnston, D., Abrahamsen, A., & Bechtel, W. (2013). Why do biologists use so many diagrams?. Philosophy of Science, 80(5), 931–944. Tolman, E. C. (1948). Cognitive maps in rats and men. Psychological Review, 55, 189–208. Tsien, J. Z., Huerta, P. T., & Tonegawa, S. (1996). The essential role of hippocampal CA1 NMDA receptor–dependent synaptic plasticity in spatial memory. Cell, 87(7), 1327–1338. Van Fraassen, B. (1980). The scientific image. Oxford: Oxford University Press. Wimsatt, W. (1972). Complexity and organization. In K.F. Schaffner & R.S. Cohen (Eds.), Proceedings of the 1972 Biennial Meeting of the Philosophy of Science Association (pp. 67–86). Dordrecht: Reidel. Wood, E. R., Dudchenko, P. A., & Eichenbaum, H. (1999). The global record of memory in hippocampal neuronal activity. Nature, 397(6720), 613–616. Woodward, J. (2003). Making things happen: A theory of causal explanation. Oxford: Oxford University Press. Życiński, J. (1988). The structure of the metascientific revolution: An essay on the growth of modern science. Tucson: Pachart Publishing House.

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The Epistemic and Cognitive Concept of Explanation in Economics. An Attempt at Synthesis

1. Introduction There are a few crucial terms in the contemporary philosophy of science around which most of the disputes at conferences and in academic papers circulate. Most of them (those terms) have their sources in the neo-positivistic propositions of the Vienna Circle members which were formulated at the beginning of the 20th century. Among others we find the notion of science itself and the problem of demarcation, the notion of scientific theory, meaning, understanding, causation and, last but not least, the concept of explanation. One can hardly write or speak about one notion without any reference to others, as they are so closely related and sometimes seem even to be confused or superseded. This problem, for example, refers to the notions of explanation and causation. If we claim that “to explain an event” means “to ascribe a cause to it,” we simply shift the problem of explanation to the problem of causation. If we then look deeper at what causation could be, we may find that at least some of the theories of causation surprisingly resemble the theories of explanation and it is hard to escape the impression of being entangled in a vicious circle. Therefore, any attempts at the systematization of concepts (which will also be the case below) are always very suspicious and susceptible to justified criticism. Nevertheless, no one acts in a conceptual vacuum. One of

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the commonly accepted methods of exiting the vicious circle is to outline at least the intellectual background of our thoughts and the main assumptions which we take as true. This basic strategy is also adopted in this paper. As the title suggests, I am going to tackle the concept of explanation, but solely concerning myself with the so called epistemic and cognitive interpretation. As both of them are vague, it is necessary at the beginning to clarify what is understood under those terms with reference to the contemporary, more or less recognizable ideas. And thus I start, presenting the set of conceptions, classified by Mayes and Thagard. Among them I distinguish the sub-sets of epistemic and cognitive concepts which can be characterized both by their essential features and by the examples given, both taken from the contemporary philosophy of science (Kitcher, Stegmüller and Hayek) and from macro- and microeconomics. I further claim that those concepts on the one hand look alike but on the other they seem to be fundamentally different. Comparing them with one another we may, however, find that they are two different stages of the on-going process of explanation, which by its nature has to engage both our cognitive apparatus and intersubjective, scientific “brainstorm,” occurring in the Popperian World 3. The account which I propose at the end is “weakly psychological” or “weakly pragmatic,” which means that it includes not only the description of the process but also the normative element which allows us (to a certain extent) to discriminate between a better and worse explanation.

2. Many concepts of explanation Historically (at least beginning with the ancient Greeks) to explain an event meant to indicate the cause. The Aristotelian account of causes was probably the first attempt to present the natural order and necessary relations (causation) to make nature ordered. However, until the beginning of 20th century no one really pondered the concept of explanation in terms of scientific inquiry. This transpired thanks to

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the precursory work of Hempel and Oppenheim (1948). Their main achievement at that time was to initiate a philosophical discussion over the subject issue. Although their account is still a good starting point in nowadays papers, probably no one consider it to be sound and accurate. There has been so much criticism in the meantime, so many objections presented and accounts proposed, that it can be treated only in terms of a historical curiosity. However, it must honestly be admitted that the main idea which underlies their account is still being developed by some philosophers of science. As well as having many different concepts of explanation, we have almost as many attempts at its systematisation. It does not make sense to present all or even the majority of them, but from this paper’s perspective it does seem to be reasonable to choose those which will be further helpful in my own partial classification i.e. the various concepts of explanations according to G. Randolph Mayes (2005) and the classes of theories (which are supposed to be the main tool of explanation) according to Paul Thagard (1988). Mayes begins with the main division of the theories of explanation into the realist and epistemic camps. The proponents of the former claim that “entities or processes an explanation posits actually exist.” Thus explaining an observed event usually entails some ontological claims. It especially refers to the unobservable entities which are postulates of the explaining theory. On the other hand, we have the epistemic set of theories which are defined by Mayes in opposition to the realists. “An epistemic interpretation holds (…) that such entities or processes do not necessarily exist in any literal sense but are simply useful for organizing human experience and the results of scientific experiments – the point of an explanation is only to facilitate the construction of a consistent empirical model not to furnish a literal description of reality” (Mayes, 2005). He further distinguishes five different groups of the concepts of explanation. These are: 1. Causal realism 2. Constructive empiricism

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3. Ordinary language philosophy 4. Cognitive approach 5. Naturalistic approach. In accordance with his main criterion of division he includes the first and the last of those groups in the realistic set and the others to the epistemic one. The most obvious seems to be causal realism. As mentioned above, it reflects the oldest, Aristotelian concept as well as a very common intuition of what explanation actually means. One of the most often quoted examples is a mechanistic account of the late Wesley Salmon. Salmon started his original struggles with the notion of explanation with the statistical-relevance account, which somehow responds to the objections raised against nomological conception of Hempel and Oppenheim and thus constitutes its creative development (Salmon, Jeffrey & Greeno, 1971). However, he soon radically changed his position and began promoting the causal concept, which was also narrowly mechanistic and reductionistic (Salmon, 1998; 2006). Studying the contemporary literature on the subject, we may conclude that this is the dominant view at present. Many philosophers and scientists, although in agreement with the causal model in principle, have shifted the discussion towards the problem of causation. The latter is interpreted very differently, beginning from the said mechanistic way, through the concept of counterfactuals, manipulability by Woodward (2003), Bayesian network by Pearl (2009) and a very original account of multi-causality by Cartwright (2007). A different approach, but also realistic, is the naturalistic one. Philosophers adhering to this trend often refer to the so called maximum coherence of those elements which are used in the explanatory sentences (theories). They could be a) observed objects and events, b) unobserved objects and events, c) nomological connections. Thus, they claim that a fundamental principle of scientific reasoning is the “inference to the best explanation” and within many various theories the best is the one which demonstrates the maximum level of coherence, which is measured (or assessed) in accordance with the prede-

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fined method. The realistic feature in those accounts discloses in the strong claim that the most coherent theory is the closest to the true description of the world and their unobservable elements are (most probably) truly existing entities. Causal realism underlies most theories in contemporary, mainstream economics, especially macroeconomics. It seems to reflect the very fundamental, Aristotelian tendency to see the world through “causal” glasses, trying to select out all the possible, simple, causal relations, which by default are considered to be real, actually occurring in the outer world. Causal realism also assumes that variables which are bound by the theory into causal interdependence are real features of the world and represent real facts or events disregarding their (those variables), sometimes very conventional character, especially in economics. One such example of a theory could be the IS – LM model proposed by Sir John R. Hicks in 1937 as the mathematical interpretation of the Keynesian work The General Theory (see Keynes, 2009 and Hicks, 1937). The model is considered somewhat outdated and simplistic at present, but it is nevertheless commonly lectured on at schools of economics and is a necessary content of any handbook of macroeconomics (see for example Mankiw, 2010), used to explain what happens in an economy (A working model, 2005). The model, very well known to almost any economist, constitutes an attempt at binding several measureable variables, constructed as a deemed representation of the market conditions into the set of equations describing two curves: both are supposed to represent the interdependence between the interest rate (r), and income (Y). IS is expressed mathematically by the following equation: Y = C(Y – T) + I(r) + G Wherein C stands for the sum of consumption dependent on income (Y) and taxation (T), I stands for planned investments dependent on the rate of interest and G stands for governmental spending. The LM

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curve is supposed to represent the supply and demand for real money and is expressed by the following equation: M/P = L(r,Y) wherein M stands for the supply of money and P for the price level, which means that the symbol M/P denotes the supply of real money balance. L stands for liquidity i.e. quantity of money demanded which depends on the interest rate (r) and income (Y). This model discloses several features which make it a good example of causal realism. Firstly, although it was developed almost entirely in abstraction from the real market and is based on very strong assumptions regarding market conditions, it claims to depict the actual reality of that market and is used both to explain the fluctuation within the bound variables as well as to create the best policy by manipulating some of those variables (interest rate, taxation, governmental spending). Secondly, the model is commonly interpreted in a causal and mechanistic way. It is claimed that there is a cognizable causal mechanism which is responsible for certain values of the variables measured on the market. This mechanism is sometimes described in general terms (like monetary transmission mechanism)1 and sometimes the interpreters use the examples of a reasonable individual’s behaviour (Monetary policy, 2012). Thirdly, although the model shows the interdependence between variables, it does not tell us about its strength, which can be estimated and formalized as certain multipliers, on the basis of the historical measured values of the variables. Those measurements are not used to possibly falsify the theory (what should be expected in the Popperian version of a demarcation between science and pseudo-science) but rather to “calibrate” the causal relations which are nonetheless taken for granted.   See the following quote: “The IS/LM model shows an important part of that mechanism: an increase -in the money supply lowers the interest rate, which stimulates investment and thereby expands the demand for goods and services” (Mankiw, 2010, p. 315).

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In opposition lie epistemic theories which include constructive empiricism, ordinary language philosophy and the cognitive approach. The first component is ascribed to Bas van Fraasen, who proposed the concept of explanation based on why-questions, always asked in a certain context (contrast class), justified by background knowledge. Thus, the direction of the explanation and its scope is always determined by the present level of scientific knowledge and human curiosity. The relevance to true reality is weak if any, therefore we cannot say that even possessing a set of good (consistent) answers to why-questions we actually have any access to the true reality (Van Fraasen, 1980). The explanation has similar features according to the language philosophy, which is simply “an attempt by one person to produce understanding in another by answering a certain kind of question in a certain kind of way” (Mayes, 2005). An example of a sort of explanation in economics which seems rooted in the epistemic approach will be given in the next section, after the accounts of Kitcher and Suppes are elucidated further. However, it is worth noting that almost every scientific theory can be interpreted in an epistemic way. Proponents of this approach are simply trying to tell us what we actually do when we claim to explain an event or a set of events with the use of a scientific method, and if we have several competing explanations, how to discriminate between them. Even in case of theories which seem to be strongly anchored in the causal and realistic foundations, those assumptions are considered to be an endogenous, ontological part of an explanation model, usually constituting an unnecessary and misleading burden. If we look again at the IS-LM model, elaborated upon above, we can easily abandon a mechanistic and realistic view and consider it as one of the possible instruments which could be helpful to introduce an ample monetary policy supportive of sustainable economic growth, a low unemployment rate and a satisfactory level of income. If any instruments appear which better meet those objectives, we should switch to them, without even asking the question about which of them better depicts the economic realm. An explanation, especially in economics, appears to

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be solely a problem solving oriented undertaking. However, if we assume that one theory can be better than another in fulfilling the given objectives, we must also find or propose criteria for such a judgement. The normative element of the epistemic approach is thus inevitable. From the perspective of the main subject of this paper, the cognitive approach distinguished by Mayes seems to be extremely interesting. As can be inferred from its name, the process of explanation in this approach is reduced purely to the cognitive activity of an agent. It consequently disregards any references to the true image of the world as well as to the normative aspect of the scientific research. The cognitive ability of the others and respectively their “instructions” in the system are of no relevance. This approach is further divided into two sub-sets. One, connected with the Artifical Inteligence trend which tries to make use of the terms coined by folk psychology, and the other is one originating from neuroscience and which tends to reduce the process of explanation to the particular functions of the neuronal system. Mayes mainly describes the first. However, the neuro-scientific approach seems to be more interesting and I will return to it in the second part of the essay by presenting an interesting example, namely the Hayekian theory of mind. A different classification, although also interesting, was presented by Paul Thagard. At the beginning, however, it should be noted that this author does not attempt to classify various concepts of explanation, as for him there is only one correct concept: “Explanation and problem solving are computational processes mediated by the rules, concepts, and problem solutions that can constitute theories” (Thagard, 1988, p. 2). For him explanation is an on-going process which is not directly aimed at explaining something but at constituting a theory. “…explanation is not an explanatory structure, nor something that explains, but a process of providing understanding. (…) Most generally to understand a phenomenon is to fit it into a previously organized pattern or context” (Ibidem, p. 44). The most important tool of an observed event’s explanation is a theory which must be formulated and which embraces an event (explanandum) as one of its im-

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portant elements. From this point of view, the reasoning presented initially by Hempel and Oppenheim, wherein an explanandum and law of nature were the most important part, in Thagard’s terms were rather an example of a theory, which constitutes that “previously organized pattern,” than a special concept of explanation. Thus a crucial problem for the philosophy of science seems to be: What is a theory? Actually within the list of central philosophical questions, this is on top, and this is the central answer: “Theories are complex data structures in computational systems; they consist of highly organized packages of rules, concepts, and problem solutions” (Ibidem, p. 2). This definition arises out of the computational philosophy of science offered by Thagard. At the foundation of that philosophy lies the so called “weak psychologism.” He noticed that one of the most distinguishable features of the different accounts was their psychological (pragmatic) or normative perspective. In other words, either we focus on the real processes which occur in the scientist’s mind, or we tend to construct a set of postulates, how this processes ought to look like to provide us with a sound and accurate scientific theory. Both extreme positions are wrong. “Weak psychologism uses empirical psychology as a starting point, since it presupposes an empirical account of the mental processes about which to be prescriptive. But it goes beyond mere description of actual mental processes to consider what sorts of inferential practices are normatively correct. (…) Knowledge is both private and public, inhabiting the brains of particular thinkers, but also subject to intersubjective communication and assessment. Weak psychologism aims to capture both these aspects” (Ibidem, p. 8). We may conclude at this stage that, by his computational philosophy of science Thagard proposed the first known attempt at the synthesis of the normative and the psychological approaches to the process of explanation. The synthesis which will be offered below is different, although it shares one common feature with Thagard’s: it is founded on the same presupposition of weak psychologism. As it has been mentioned above, the central notion in his philosophy is a theory, a tool for providing an explanation and thus an under-

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standing. However, even if we have a more or less vague but common understanding of what a theory is, there is a noticeable dispute in philosophy regarding its nature. Thagard proposes the following topology of theories with the use of their nature as a criterion: 1. According to logical positivistism, theory is a syntactic structure, reducible to the set of sentences, which are susceptible to the axiomatic formalization in a logistic system. Such a theory has nothing in common with the scientist’s brain activity or with the actual process of theory formation in the society of scientists. The theory in this account is highly normative. If the sentences are not susceptible to formalization or if they are not consistent, the theory is no longer valid and needs to be replaced. 2. A more modern, semantic / set-theoretical approach was proposed on the basis of Tarski’s model theory. The nature of the theory itself could be sought in the set of models and in the function of interpretation, which has the power to transform of the elements of the given formal system (constants, variables, relations, functions etc.) into another system (model) providing that the true / false values of the sentences will be preserved. As such, the approach is usually highly epistemic in Mayes’s terms, one of the examples will be described in more detail below. 3. The pragmatic approach, which was advocated mainly by Kuhn and according to whom theories are mere devices used by scientists in a particular context for problem solving. The structure of the theory is of no importance or is too vague to be formalised in any reasonable way. The core element, however, is a paradigm dominant in a given historical and social context. Such an account is narrowly pragmatic and descriptive and thus anti-normative. It provides us with no tools for discrimination between a good and bad theory.

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3. The Epistemic concept of explanation On the basis of this introductory part on classifications and typologies described therein, I would like to specify what will be understood under the notion of the epistemic concept of explanation. Partially following Mayes, I claim that reviewing numerous and various approaches of the contemporary thinkers on the subject matter we are in a position to distinguish between at least two different groups of concepts, which at first glance seem very different, namely epistemic and cognitive ones. I refer to Mayes only partially as he includes cognitive concepts into an epistemic set and posits them against realistic concepts. Thus we can distinguish epistemic concepts sensu largo and sensu stricto. Accounts which can be ascribed to the epistemic group sensu stricto share some common features with each other: 1. Explaining in those accounts is not reduced to the selective reasoning aimed at a particular phenomenon. It is rather aimed at formulating a theory, a previously organized pattern of sentences or propositions and of inferential rules, with the use of which the observed phenomena gain their meaning for humans (scientists). 2. They make abstraction from any true, ontological reality. We may even say that the proponents of those accounts are scientific agnostics. Explanation is not about looking for real phenomena in the outer world, which are either inaccessible or of no relevance. It is a method of an organization of our various experiences. They may, however, make use of the concept of semantic truth, especially to distinguish the best chosen order (see p. 4 below). 3. Experience plays an important part in an explanation. In the process of explaining, there must be room for setting our common sensory data against an explanatory theory. There may be various ways or methods of discriminating between the proposed theories on the basis of their confrontation with those sensory data like empirical adequacy or predictive power or others.

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4. Experience can be structured in various ways. The epistemic concept of explanation usually includes or should include the criteria for choosing the best order. 5. The most distinguishable feature of those accounts, especially if we compare them with the cognitive approach, is that explanation is not an individual, but an entirely social undertaking. It takes place in the discourse between members of society, and especially between scientists. What happens in the particular scientist’s mind is of no relevance for the account as long as it is not expressed in a form comprehensible and accessible for others. In other words, a process of explanation, a process of theory construction and its testing, takes place in Popper’s World 3 of the objective knowledge. Among the many concepts which meet the requirements specified above, we may, just for the sake of an example, select two accounts and make them more familiar. Phillip Kitcher offered his solution to the problem of explanation by the unification of the range of different phenomena. “…Successful explanations earn that title because they belong to the set of explanations, the explanatory store, and that the fundamental task of the theory of explanation is to specify the conditions of the explanatory store. Intuitively the explanatory store associated with science at a particular time contains those derivations which collectively provide the best systematization of our beliefs” (Kitcher, 1989, p. 430). This short introduction discloses the most important features of the epistemic approach, namely the main task is not to look for the true structure of the world but to pursue a better systematization of our beliefs, which doubtless arise out of our sensory experience. How can we systematize them? By constructing the explanatory store which is supposed to unify the set of beliefs accepted at a particular time in science (E(K)), and which is a set of argument patterns. The latter is an ordered triple consisting of a schematic argument (sequences of a schematic sentences, which employ symbols – dummy letters – instead of words), a set of sets of filling instructions (instruc-

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tions necessary for decoding the schematic sentences), and the classification of a schematic argument (to distinguish between the premises, conclusions and inference rules). The explanation is considered to be good if it belongs to the good or acceptable explanatory store. There are two crucial criteria for the quality of the explanatory store: the stringency of the argument patterns within the store (more restrictions means more stringency) and the number of derivations and conclusions. “…E(K) is a set of derivations that makes the best tradeoff between minimizing the number of patterns of derivation employed and maximizing the number of conclusions generated” (Ibidem, p. 432). Thus, the criteria for choosing the best, accessible order was introduced, too. As can easily be noticed, Kitcher’s account is very close to the syntactic proposals of Hempel and Oppenheim, at least in that part which reduces the operation within the explanatory store to the sentences or the set calculus. The difference, however, is in its pragmatic character. Unlike neo-positivists, Kitcher does not presuppose any inferential rules, valid or required objectively, regardless of time and place where the explanatory store is being constructed, but assumes that they may vary significantly between various stores. The most important feature is not whether the conclusions are “correctly” derived from the assumed premises but whether the store is “effective” at the current stage of human knowledge. Another characteristic example can be set-theoretic conceptions, which were proposed for the first time by Patrick Suppes (1967), and then developed by, among others, Sneed and Stegmüller. These concepts are in principle compliant with the view of Thagard at least in this part, in which their defendants claim that explanation is a process of providing understanding within the constructed schemata, the scientific theory. Therefore, like at Thagard the most important question related to explanation is about the nature of the theory. For Suppes a starting point to tackle this problem was a critique of the “standard sketch,” deeply rooted in the early neo-positivistic view. Pursuant to this “sketch” the scientific theory was considered to consist of two parts: abstract logical calculus and a set of rules that assigns an em-

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pirical content to the logical calculus. It sounds familiar. In Kitcher’s proposal we had schematic sentences and filling instructions plus classification rules, although not necessarily compliant with the inference rules of first order logic as was often the case of the early concepts. The problem which was noticed by Suppes was that this “standard sketch” is very schematic and not very pragmatic. In other words “…it is unheard to find a substantive example of a theory actually worked out as a logical calculus...” (Ibidem, p. 56). The richness and variety of the schema according to which the particular theories are constructed is so striking that philosophers should try to capture them and propose an account which better reflects the actual practice of the scientists and better suits the variety of sciences, having in mind that there is something more than physics and chemistry that should be taken into account. So the following directions were indicated: 1. A much better formalization of the theory than first order logic is mathematical set-theory combined with the model theory and concept of isomorphism between the empirical and mathematical models. 2. The theory can be characterised (and assessed) both from an intrinsic and extrinsic point of view, which means that both the intrinsic structure of the theory and the extrinsic tacit assumptions which are placed outside it and are often not even realized are of a great significance. 3. The theories have their own hierarchies. Not all of them are simply the sophisticated description of the outer world, but some are simply tools for scientific procedures. It especially refers to the theory of an experimentation or statistical methodology “…that intercedes between any fundamental scientific theory and any experimental experience” (Ibidem, p. 63). 4. In assessing the theories we should take an instrumental point of view. We should not focus on the establishment of their truth or falsity but on their usefulness in inferring new statements of facts. Like the typical statistical decision theory, scientific theories “are

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not regarded even as principles of inference but as methods of organizing evidence to decide which one of several actions to take” (Ibidem, p. 65). The further developments of this account went into the more formalized structure. The theory was understood as an ordered couple – structure [K, I]. K is a core of the theory consisting of the possible models MP (in Tarski’s sense) and their sub-set, picked class of models (M). I also stands for a set of models, which are an intended application of the theory. The important elements are the classification rules which decide which objects fall under the predefined predicates or not and are respectively ascribed to the particular set. Thus, we have the formalization built around the set-theory, models and their interpretation and the pragmatic approach expressed in the concept of I, models of theory application. As it has been mentioned above, an epistemic account claims to encompass the daily practice of scientists on the one hand, and introduces a few, weak normative postulates on the other hand regarding that practice which, if followed, is supposed to provide us with better understanding of our sensory experience and thus, with better accomplishment of our practical goals. Almost every theory can be interpreted in terms of this approach. Some of them, however, seem to follow those postulates more properly and thus constitute a better example of the epistemic concept. Most complaints about the alleged falsity of the economic theories and their inability to produce accurate predictions are connected with macroeconomics, like in the case of the IS-LM model described above. In fact, they are often constructed with very strong ontological assumptions in mind regarding the preexistence of certain entities and causal relations between them. Juxtaposing the model’s predictions with the contradictory sensory data makes its defence a hard task. The situation looks a bit different in the case of microeconomics, especially in the field of research on consumer behaviour. In some of this research economists, instead of building models of the behaviour which would be founded on the as-

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sumed features of human nature, interests, desired ends and methods of their accomplishment, simply dig through a tremendous amount of data and try to select the most distinguishable and repeatable patterns with the use of statistics, seemingly useful for the predictions of a future behaviour. This technique, usually supported by the sophisticated informational technology, brings about very prospective results. One of the examples is the activity of hiQ Lab, a company based in Silicon Valley, which renders its services to well-known Internet electronic services providers.2 The Internet is in itself an interesting living laboratory for such research mainly because of an infinite stream of data to be proceeded. IT professionals or rather data scientists have designed intelligent software which is able to “learn” from its own mistakes during the numerous repetitions of the predefined tasks. At the end, they are able to literally present a “theory,” for example, on the characteristic features disclosed by Internet website visitors who are more inclined to buy products offered in contrast to those who simply do website window-shopping, or the theory on the employees at risk of leaving their employer and alleged reasons for such leaving. The applied method of theory building seems to be rooted in the epistemic approach to scientific explanation, especially in the set-theoretic account. 1. Relations which are expected to be selected by the computer program or by the scientists are mainly mathematical and statistical functions. The reasoning is not based upon first order logic but on set-theory and statistics. Those functions constitute possible models of the customers’ or employees’ behaviour. 2. The “theories” proposed at the end can be characterized both from the intrinsic and extrinsic point of view. The latter is almost equally important. Data miners do not take the variables proposed by economics for granted, but they are extremely sensitive with   The case study was described and analysed by The Economist in the section Finance and Economics (Silicon Valley economists, 2015).

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regards the method of how they were constructed and what they are supposed to represent. “Silicon Valley economists obsess over how the numbers they use are collected, and would not accept something as old-hat as GDP” (Economics evolves, 2015). 3. The development of “data economics” goes both in the direction of the “end-theory” construction and improvement of the tooltheories, statistics, informatics, mathematics. They constitute the ordered hierarchy of theories. 4. The goal is purely instrumental and they indeed made an abstraction from any truth or real picture of the investigated world. Even if there are hypotheses on the possible causal relations between the observed events, they are formed purely for the sake of manipulation and they are constantly ready to be falsified and replaced by another, more suitable relation by the unbiased computer program. The expected result of the data mining should be another model, which depicts the possible application of the theory, and instructs the users whose variables could be manipulated in order to reach the desired goal.

4. Cognitive concept of explanation It did not require much time after the first, purely normative concept was proposed, to realize that if we take a more pragmatic approach and begin to ask questions regarding not how we ought to cognize and explain the world, but rather how we have been doing so thus far, we will end up in the human cognitive apparatus. This characteristic pragmatic and psychological turn in the philosophy of science, which occurred in the 1930-s and 40-s of the 20th century coincided with the development of early modern cognitive psychology as well as neurobiology and informatics. Those thinkers who were closer to the psychological current tended to envisage the problem of explanation in terms of Artificial Intelligence and started to construct various formalizations of our cognitive processes, with the extensive use of popular

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psychological notions. Another current of thinkers appeared simultaneously who proposed the neuro-scientific approach and claimed that tracking the real cognitive process requires comprehensive knowledge of the human neuronal system, its design and functioning. On the basis of the early neurobiological research, it seems justified to put forward a hypothesis that those processes are computational in their nature. The neuronal impulses which activate the relevant parts of the system, enabling it to perceive and structure the set of information from the outer world, look like the data processing in artificial, computational systems. If so, the possible models which were supposed to resemble those processes would have to be computational, too. The problem, however, is that data processing is strongly dependent on the computational frame within which it is run, i.e. on the set of instructions and the artificial language which is used to express those instructions. Mayes put it in this way: “…the simple philosophical question ‘What is explanation?’ is not well formed. If we accept some form of epistemic relativity, the proper form of such question is always ‘What is explanation in the cognitive system S?.’ Hence, doubts about the significance of explanatory cognition in some system S are best expressed as doubts about whether system S-type explanation models human cognition accurately enough to have any real significance for human beings” (Mayes, 2005). One of the early examples of such an approach is included in the Hayekian philosophy of mind, offered in his exceptional monograph The Sensory Order, which he himself considered to be his most important achievement (Hayek, 1992). The later version of this essay dates back to 1952, but regardless of its advanced age it is still surprisingly vivid. Alas Hayek was not very interested in the achievements of the neo-positivists connected with the notions of scientific explanation or theory, therefore in his papers we will not find any references to those conceptions. His own account seems to be detached from the contemporary philosophy of science, but on the other hand strongly rooted in the best knowledge of a human neuronal and cognitive system and which makes it exceptional and still inspiring. On

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the basis of this knowledge Hayek made several important ontological and epistemic assumptions. The world in which a cognitive apparatus is placed is threefold. It consists of the physical order, which is an order of the physical stimuli, and not an order of the deemed outer physical reality. That reality is accessible for the cognitive apparatus exclusively through neuronal sensations, therefore we can say something only about the part of a physical reality which is able to evoke the reaction of the neuron. The system is obviously blind to the processes which, in the course of human evolution or an agent’s individual development, have not made the neurons sensitive. Scientific knowledge teaches us that there might be plenty of such processes. The second one is the neural order, consisting of neurons, synapses and physic-chemical processes, which in fact is a sub-set of the physical order, but as it is responsive to the latter, it is shaped by it according to certain principles and instructions. The third one is the mental order of sensation, the most important from an agent’s perspective, which constitutes sensory qualities and representations which we perceive. Neural and mental orders can be interpreted as a set of various items combined with predefined rules of their discrimination. The system of neuronal connections and the instructions regarding its sensitivity and responsiveness, encoded in the biological structure, constitute the so called “map.” The map can be pictured as the set of co-ordinates necessary for the space and time orientation, and it reflects the regular relations occurring in the outer world imperfectly. That mapping is strongly dynamic and subject to constant reconstruction (semi-permanent), which is possible because the sensitivity of the particular neuron can be, to a certain extent, modified in an agent’s lifetime by regular and repetitive stimulation. The main function of the neural order is to discriminate and classify the sensations coming from the physical order. The mental order, the order of our mental representations, is isomorphic with the neural order, but both are not isomorphic with the physical order. The map is a foundation for the abstract model which emerges in the system. As the mapping of the outer world is imperfect and susceptible (although weakly)

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to changes, the model may from time to time disclose its inconsistency, therefore it is permanently tested by setting the predictions for future sensations generated within the model with the sensations actually perceived. If any contradiction is found, the system looks for the solution, which usually means the restructuring of the map. The important consequence of this idea is that the neural system requires a certain level of regularity and repetitiveness from the physical sensations to evoke the neural impulse. The system is “blind” (not responsive) to the irregular stimuli, which cannot be classified. Therefore, the map understood as the basic system of classification is a sine qua non condition of any perception. Contrary to popular empirical accounts, the abstract element of our cognition has absolute primacy over the sensations. What is then an explanation? It “…consists in the formation in the brain of a ‘model’ of a complex of events to be explained, a model the parts of which are defined by their position in a more comprehensive structure of relationships which constitutes the semi-permanent framework from which the representation of individual events receive their meaning” (Hayek, 1992, p. 179). Could the Hayekian account of cognitive explanation be of any value in economics? It is rather obvious that we cannot present any kind of explanation of economic phenomena with the use of it. By this model we rather focus on the processes occurring in the economist’s mind or in the mind of a market agent, which for us are inaccessible as long as they are not disclosed in the form of a theory presented in an intelligible way.3 And if they are eventually presented and discussed, we shift to the epistemic account. Yet the Hayekian account, although published almost 80 years after the foundation of the Aus-

  It needs to be emphasized that in the Hayekian account the commonly used term “scientific explanation” does not have any sense. If explanation is reduced to the sensory and mental processes, everyone is a scientist whose mind tries to organize sensory impulses and construct a “model” of the outer world. Both an economist and a market agent may have their “theories” which eventually are disclosed and discussed. The discrimination between scientific and layman theory becomes valid only on the epistemic stage of explanation. 3

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trian school of economics4, brought about the very comprehensive and well- grounded justification for the main methodological postulates of the school. Those were as follows: 1. A radically “subjectivist” strain in Neoclassical marginalist economics, 2. A dedication to a prioristic “pure” theory, with an emphasis on “methodological individualism,” 3. Subjective valuation of consumer goods, 4. A stress on uncertainty and information in the economy, 5. A concern with the psychology of economic actors, in particular, the supremacy of strategic, self-interested behaviour when facing each other and political and social institutions.5 The founders of the school which contributed the most to those postulates, Carl Menger, Ludwig von Mises and Eugen von BöhmBawerk, were trying to argue for them mainly on the basis of our intuitive knowledge of human nature, learnt by introspection. Hayek went much deeper, mining our brain and confirming those intuitions with the use of modern neurobiology and neurocognitivism. Both directions in contemporary economic research are still vivid. Needless to mention the recent book by Roman Frydman and Michael Goldberg, Imperfect Knowledge Economics(Goldberg & Frydman, 2009), whose methodological assumptions directly resembles those mentioned above, or neuro-economics, a very disputable field of research, but pursued by some scientists with promising results (Camerer, 2007).

  The foundation of the Austrian school of economics is usually associated with the publishing of Carl Menger’s opus magnum, The Principles of Economics (Grundsätze der Volkswirtschaftslehre) in 1871 (Menger, 2007). 5   The list is prepared on the basis of (The History of Economic Thought, 2003, p. 139). 4

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5. An attempt at synthesis By comparing both approaches to the concept of explanation, the epistemic and the cognitive one, we may find many features which make them congruent in major fields but at one point different. Let us start from the latter. It belongs to the ontological sphere. The epistemic account positions the process of explanation in Popper’s World 3, and in principle considers it to be a social undertaking. We merely investigate the inter-subjectively created products of human minds, which are somehow articulated and are comprehensible to other human minds. On the contrary, explanation in the cognitive approach takes place in Popper’s World 2. To understand the process of explanation we need to correctly copy or model the key functions of the human cognitive apparatus. Other features make both accounts likely. Both are strongly pragmatic. Their adherents merely focus on how the given phenomena are actually explained and which of the competing explanations win the contest and why. The normative aspect is at most secondary, and usually emphasized on the occasion of specifying the criteria for the contest’s winner. Both abstract from any ontological truth. To explain the phenomena we do not need to identify and deploy any concept of “truth” or “true reality,” “being” or processes like causation. It does not mean that there are no assumptions about the existence of something behind the sensational curtain, but we accept the strong imperfectness of those assumptions and the fact that the only reality which is given (perceived) is that curtain. We do not need to cognize this deemed and hidden reality, we simply need somehow to organize the curtain’s order. The consent for the imperfectness and temporality of this organization makes the process of explanation an “un-ended quest.” The most interesting point in both accounts is the so called weak normative aspect – the proposed criteria for comparing and assessing the process of explanation or its product which could be a scientific theory. In epistemic approaches they

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often specify two elements: consistency / inconsistency and complexity / simplicity.6 The explanation begins with the new observables which have not been explained yet, meaning that they have not been organized in any pre-existing order (theory) yet, or it occurs that the pre-existing order causes an inconsistency with those observables. If we transformed this into the language of the set-theoretic conception, we might say that we have new observables which are not recognized elements of any existing theory yet, which means that we do not have any possible model Mp comprising those observables which would comply with the intended application of the theory I (or theories I1 … In), or the classification of the observables results in inconsistency within the models. Almost the same happens in the cognitive account. However, instead of new observables at the very basic level (the neural order) we have new sensations (or their analogues in the cognitive model), which should be classified by the cognitive apparatus. Attempts at classification lead to inconsistency i.e. contradictory instructions (run and not run). On the more sophisticated levels, within the map and model in Hayek’s terms the said inconsistency is expressed in an inability to construct accurate predictions of the subsequent sensations. In the course of the consecutive attempts at organizing those observables or sensations, several different and competing orders may be proposed by the classification apparatus and all of them may occur to be consistent. Somehow they need to be discriminated between. Only one order can be applied at the moment, even though the other remain on stock as spare ones, to be possibly deployed in the future if a new inconsistency emerges. In the cognitive account it seems that   It should be noted that some accounts exploit the idea of coherency, specifying various conditions thereof. Whatever those conditions are, in major instances consistency and simplicity are among them. See for example proposal of Thagard in (Thagard, 2007) wherein among the principles of explanatory coherence we find the “principle of explanation” (the more hypotheses it takes to explain something, the lower the degree of coherence) and the “principle of contradiction” (contradictory propositions are incoherent with one another). 6

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one of the most important criteria for choosing a better order is its efficiency, understood in terms of quantity of energy consumption. This is one of the most important discoveries of neuroscientists that the human brain is an enormous energy consumer, requiring constant refuelling. This side effect of our highly developed cognitive abilities is a heavy burden which has meant that in the course of biological evolution the patterns of neuronal connections which needed less energy were favoured over others, providing that they were almost equally consistent. In fact the evolutionary mechanism seems to be strongly imperfect and it often chooses sub-optimal solutions, namely preferring imperfect patterns but more energy efficient then ideally consistent ones. It led to the creation of plenty of shortcuts and in the longer perspective the whole system of quick, strongly efficient but highly inaccurate solutions which seem to prevail in the daily cognitive practice of a human brain. The theory of two systems, their characteristics and the circumstances in which they are activated, was elaborated upon recently by Kahnemann on the basis of an extensive set of psychological experiments (Kahneman, 2011). If we applied this biological analogy to our scientific knowledge, we would have to conclude that our theories are likely very inaccurate, have very little in common with the true reality, but our pervasive existence suggests that they have at least been efficient enough to let us survive so far. How could this efficiency be interpreted? In a cognitive account it seems easier. If the efficiency is closely connected to the brain energy consumption, then the criterion can be translated into the requirement of less complexity, understood as the thermodynamic depth. It is one of the proposed definitions of the complexity measure. The system is more complex if it requires more of the total thermodynamics and informational recourses for its physical construction. As we assume that the cognitive process is at its foundations purely physical, the applications of this definition seem the most appropriate. However, the complexity can be measured in very different ways7, and thermody  See e.g. (Mitchell, 2009). She lists nine different definitions of complexity and relevant approaches to its measurement.

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namic depth seems useless in the case of epistemic account. We cannot measure the total thermodynamic resources used by all men (at present and in the past) involved in theory construction, and even if we did, it would be useless and it would not reflect the level of complexity. The particular theory might have been worked out through the centuries, engaging thousands of its contributors, but at the end we can have a very nice and simple set of equations. Therefore, it seems more reasonable to use the concept of complexity as “the algorithmic information content,” wherein the scientific theory in the epistemic account is compared to the computer program. The shortest one which is able to describe the theory’s subject, or which is able to represent the theory itself without losing any significant elements is the least complex.8 If it were possible to set the two different levels of complexity – different ways of their measurement, against one another, we could repeat after Hayek that the neural order in the cognitive account must always be of a higher degree of complexity than its products, the particular theories. There is one more analogue feature in both accounts. Organizing new sensations in the neural order is always done with reference to the existing system of connections. It has its quantifiable capacity for testing new sensations, which is limited and strongly determined by the existing order. If the encountered inconsistency is strong enough to endanger the existence of the order carrier (human being) it may require the deep restructuring in System 1 in terms of Kahneman (slower, more reflective and more energy consuming). However, in daily experience a brain tries to deal with the encountered minor inconsistencies by simply selecting the most appropriate pattern, a “shortcut” from within the already given ones of System 2. The same seems to happen within the scientific field. Before we demolish the leading theory, we try to fill it up with as many observables as pos  The algorithmic information theory was developed among others by Gregory Chaitain, and was first presented in (Chaitin, 1990). According to this theory, conceptual complexity of an object X is defined as the size of the most compact program for calculating X. 8

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sible, patching over the contradictions they may cause. The existing theories which are background knowledge are the only possible source for the “contrast class” for “why questions” in terms of Van Fraasen (1980). Either we find an answer within the existing theory or we have to restructure the background knowledge. This deep restructuring on the level of the neural order or on the level of background knowledge is a complex and therefore highly energy consuming process, to which humans are generally reluctant. On the other hand, both are very advantageous as they enhance the quantifiable set of alternatives or, in other words, quantifiable possible neural connections or possible models Mp, which can be deployed, if seemingly insoluble inconsistency occurs. The mental orders (and respectively the neural orders too) in the cognitive account from one hand and the possible models – possible theories in the epistemic sphere on the other hand, seem to be strongly connected in an explanatory feedback loop. The popular notion of “brain storms” accurately reflects the nature of the loop. The more or less spontaneous exchange of thoughts can be interpreted as the presentation of the possible models engineered within the mental order of the presenter, which in the pace of the storm can be re-engineered and adopted by other brains, mental orders. More models circulating in Popper’s World 2 and 3 enhance the capacity of background knowledge and thus the ability for deep restructuring or even the replacement of the already leading theory, as well as readiness for changes of patterns of the reflective System 2 of our brains, and subsequently the more energy efficient shortcuts in System 1. All in all, it must be emphasized that providing the starting point has been correctly assumed (i.e. the epistemic and cognitive approach to the process of explanation), the proper understanding of the concept of explanation requires thorough studies of both approaches, namely of how the brain works and how the theories are built and rebuilt. The context of scientific discovery is as important as the context of justification. Economics is a kind of science which is strongly policy oriented. Politicians and nations expect from economists a readyto-apply recipe for social problems, which, bearing in mind the im-

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perfection of this science and its relatively unsuccessful track-record, is highly ungrounded. Moreover, it seems to be extremely susceptible to any identified biases of our cognition apparatus as well as to social pressure. Therefore, this two-sided approach to economics explanation is even more necessary and there are chances for this as we have many different streams of contemporary economics, beginning from the mainstream, highly theoretical, extended model-building approach and ending on the behavioural and, already mentioned, neuroeconomics. They are all constructing models of the economic events in their minds, aimed at the explanation of economic phenomena, to be further presented in an eligible way, and possibly discussed, criticized and rebuilt. This un-ended quest for maximum coherence and simplicity (energy efficiency) can only be done through public discourse. If there are any social or mental obstacles against such discourse, we are losing (or at least diminishing) chances for constructing a better economic theory. To overcome any possible obstacles, their identification is a precondition.

References A working model. Is the world experience excess saving or excess liquidity? (2005, August 11). The Economist. Camerer, F. C. (2007, March). Neuroeconomics. Using neuroscience to make economic predictions. Economic Journal, 117, C26-C42. Cartwright, N. (2007). Hunting causes and using them. Aproaches in philosopfy and economics. Cambridge: Cambridge University Press. Chaitin, G. (1990). Algorithmic information theory. Cambridge: Cambridge University Press. Chaitin, G. (2015). Conceptual complexity and algorithmic information. La Nuova Critica, 61–62, 7–27. Economics evolves. A long way from dismal. (2015, January 10). The Economics. Goldberg, M. D., & Frydman, R. (2009). Ekonomia wiedzy niedoskonałej. (M. Krawczyk, Trans.) Warszawa: Wydawnictwo Krytyki Politycznej.

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Hayek, F. A. (1992). The sensory order. An inquiry into the foundation of theoretical psychology. Chicago: The University of Chicago Press. Hempel, G. C., & Oppenheim, P. (1948, Apr.). Studies in the logic of explanation. Philosophy of Science, 15(2), 135–175. Hicks, J. R. (1937). Mr. Keynes and the classics: A suggested interpretation. Econometrica, (5), 147–159. Kahneman, D. (2011). Thinking fast and slow. New York: Farrar, Straus and Giroux. Keynes, J. M. (2009). The general theory of employment, interest, and money. New York: Classic Books America. Kitcher, P. (1989). Explanatory unification and the causal structure of the world. In Ph. Kitcher & W. Salmon, Scientific explanation (pp. 410– 505). Minneapolis: University of Minnesota Press. Mankiw, N. G. (2010). Macroeconomics. New York: Worth Publishers. Mayes, G. R. (2005, 07 12). Theories of explanation. Retrieved 11 23, 2010, from Internet Ecyclopedia of Philosophy: http://www.iep.utm.edu/explanat/ Menger, C. (2007). The principles of economics. Auburn: Ludwig von Mises Institut. Mitchell, M. (2009). Complexity. A guided tour. Oxford & New York: Oxford University Press. Monetary policy. The broken transmission mechanism. (2012, October 3). The Economist. Pearl, J. (2009). Causality. Models, reasoning and inference. Cambridge & New York: Cambridge University Press. Salmon, C. W. (1998). Causality and explanation. Oxford: Oxford University Press. Salmon, C. W. (2006). Four decades of scientific explanation. Pittsburgh: University of Pittsburgh Press. Salmon, C. W., Jeffrey, C. R., & Greeno, G. J. (1971). Statistical explanation and statistical relevance. Pittsburgh: University of Pittsburgh Press. Silicon Valley economists. Meet the market shapers (2015, January 10). The Economist. Suppes, P. (1967). What is a scientific theory. In S. Morgenbesser (Ed.), Philosophy of science today (pp. 55–67). New York: Basic Books. Thagard, P. (1988). Computational philosophy of science. Cambridge, MA & London: A Bardford Book & MIT Press.

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Thagard, P. (2007, January). Coherence, truth and the development of scientific knowledge. Philosophy of Science, (71), 28–47. The history of economic thought. (2003, January 17). Retrieved February 27, 2011, from New School: http://cepa.newschool.edu/het/ van Fraasen, B. C. (1980). The scientific image. Oxford: Clarendon Press. Woodward, J. (2003). Making things happen: A theory of causal explanation. Oxford: Oxford University Press.

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Wojciech Załuski Jagiellonian University Copernicus Center for Interdisciplinary Studies

The Varieties of Egoism. Some Reflections on Moral Explanation of Human Action

1. Intentionalist vs. moral explanation of human action The explanation of human action may assume two basic forms. The first form is morally-neutral and consists in identifying the intention of the agent, i.e., the representation of the goal the agent wanted to attain by performing the action (the represented goal is either the stateof affairs to be effected by the action or the action itself if the action’s ultimate goal is the action itself). Of course, the intention of the action does not have to be morally-neutral but the ‘intentionalist explanation’ (as we shall call it) is not primarily focused on providing a moral evaluation of the action; it is primarily focused on identifying a specific mental state ‘generating’ the action, viz. its intention. The second form (which we shall call ‘moral explanation’) is essentially different, since it is explicitly morally oriented. It does not aim at identifying the representation of the goal of the action, but at ascertaining what moral motives (if any) stood behind the action. By moral motives we shall mean motives which directly entail moral evaluation (or, in other words, may serve as terms of moral evaluation) and therefore are crucial for the assessment of the moral character of actions to which they give rise. A simple but illuminating (though, as will turn out, requiring some corrections) classification of such motives will be presented in Section 2; at this stage of our analysis it will

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suffice to say that the classification was proposed by Arthur Schopenhauer and embraces three motives: egoism, empathy, and malice. It is worth emphasizing that the moral and intentionalist explanations are not mutually exclusive but rather complementary: a full explanation of human action (if an action has a moral dimension) should embrace both types of explanation. In this paper we shall not analyze the relationships between them, confining ourselves to making a rather obvious remark that the intentionalist explanation is more general than the moral one, because not all human actions have a moral dimension; we shall deal instead with a certain specific problem of the moral explanation of human actions.

2. Egoism as a motive of human actions The problem can be stated in the following way: in moral-psychological analyses aimed at providing a moral explanation of human action egoism is usually assumed to be a primitive – ‘unanalyzable’ – motive, i.e., not flowing from some other more basic psychological phenomena; but this apparently self-evident and widely accepted assumption is, in our view, deeply mistaken. We shall argue that: (1) egoism is not a primitive motive but, rather, a manifestation of some more basic psychological phenomena; (2) one can distinguish three different forms of egoism depending on its psychological basis, viz., cognitively-based, hybris-based, and instinct-based; (3) in each of its forms egoism can be regarded as a moral defect, although the degree of its moral wrongfulness differs depending on its psychological basis – we shall argue that it is highest in the case of hybris-based egoism, and lowest in the case of instinct-based egoism. Before proceeding to our analysis, some introductory remarks need to be made. Firstly, we assume a common-sense definition of egoism as an agent’s tendency to pursue her own interest to an exceedingly high degree, that is, without duly respecting the other agents’ interests. This definition presupposes that pursuing one’s own interest is not in itself

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morally wrong – it becomes morally wrong only when it is done ‘to an exceedingly high degree’, and the degree of the morally legitimate – non-egoistic – pursuing one’s interest is determined by morality, which says how much respect is due to other people’s interests in the course of pursuing one’s own interest. Egoism can be defined equivalently in at least two different ways. Schopenhauer’s defined egoism as “an urge to one’s being and well-being (der Drang zum Dasein und Wohlsein)” (Schopenhauer, 2009, p. 662; 2010, p. 196). This definition requires a slight correction, since, as we have already noted, not all types of care for oneself can be viewed as egoism. Egoism is an excessive “urge to one’s being and well-being” (an insufficient urge of this kind is abnegation) (cf. Wolniewicz, 1998, p. 109). Egoism can also be defined by modifying the Evangelical precept “love thy neighbour as thyself” (Mark, 12:31; Matthew, 22:39; Luke, 10:27), which arguably states the conditions of morally right conduct towards other people. Now, one can say that a person is an egoist if and only if she loves herself more than her ‘neighbour.’ Thus, we have three arguably equivalent definitions of egoism: as an agent’s tendency to pursue one’s interests without duly respecting the other agents’ interests, as an excessive “urge to one’s being and well-being”, and as loving oneself more than one’s ‘neighbour.’ Secondly, by assuming the common-sense definition of egoism we thereby do not subscribe to the contrived doctrine of psychological egoism, which asserts that all human actions are motivated by egoism.1 In order to make this doctrine proof against obvious counterexamples of non-egoistic actions, one needs to replace the common-sense definition of egoism with such definitions as, for example, “doing what one wants to do” or “acting in a way maximizing one’s utility function.” But the common defect of these definitions concocted to save the doctrine of psychological egoism is that they deprive the notion of egoism of its pejorative sense which is its es  A critique of psychological egoism was provided, e.g., by C. S. Broad in the article “Egoism as a Theory of Human Motives” (1950). On psychological egoism see also, e.g., Blum (2004). 1

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sential element.2 All in all, we reject psychological egoism, thereby assuming, in accordance with common sense, that not all human actions are motivated by egoism: they may be motivated by other moral motives (i.e., motives directly entailing moral evaluation). A simple and useful classification of moral motives of human actions was proposed by Schopenhauer. The classification embraces three motives: egoism (Egoismus), which moves an agent to pursue her own good as an end in itself, empathy or sympathy (Mitgefühl), which moves an agent to pursue the others’ good as an end in itself, and malice (spite) (Bosheit), which moves an agent to pursue the others’ harm as an end in itself.3 Accordingly, following Schopenhauer and contrary the doctrine of psychological egoism, we assume that human actions may be motivated by different moral motives. But we deem it neces  The above arguments against psychological egoism are commonsensical and conceptual but there are also empirical arguments against this doctrine. The strongest experimental support for the rejection of psychological egoism comes from the research made by the social psychologist C. Daniel Batson on the so-called Empathy-Altruism Hypothesis according to which altruistic actions (i.e., benefitting other people) are not always motivated by egoism (as the defenders of psychological egoism maintain) but are oftentimes motivated by empathy. By ingeniously manipulating the experimental set-up, Batson has made a strong case for the claim that in many circumstances the altruistic behaviour of subjects can be better explained by the Empathy-Altruism Hypothesis than by any alternative hypothesis postulating the egoistic motivation. Batson took into account three such hypotheses: the Aversive-Arousal Reduction Hypothesis, according to which agents help others in order to reduce their own personal distress caused by the view of suffering persons; the Punishment Hypothesis, which says that altruistic actions are motivated by the fear of socially- or self-administered sanctions; and the Reward Hypothesis, according to which altruistic action is motivated by the expectation of some kind of reward – awarded by others or ‘internal’ (the expectation of joy to be felt after helping other people). An excellent summary of this research and their implications can be found in Batson, 2011. 3   Schopenhauer pointed out that the others’ harm is not an end in itself for a pure egoist; it can only be a means for realizing her own good; it is, however, an end in itself for a malicious person. He maintained that each person’s character can be characterized by describing a relative strength of these motives, i.e., by indicating how important role they play in shaping her actions. Thus, e.g., the ‘diabolic character’ would be the following system of the motives: Egoism = 0, Sympathy, Altruism = 0, Malice = 1; it is assumed here that the intensities of all these motives sum to 1 (see Wolniewicz, 1998, p. 111). For a further analysis of various aspects of altruism see e.g. Scott, Seglow, 2007; Williams, 1973; and Nagel, 1979. 2

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sary to introduce three corrections to Schopenhauer’s account: (1) as already mentioned, his definition of egoism is too broad; egoism is an excessive “urge to one’s being and well-being’, and not just “an urge to one’s being and well-being”; (2) unlike Schopenhauer, we shall argue that egoism may take three different forms; (3) it seems that Schopenhauer’s classification should be supplemented with two more moral motives (apart from empathy), viz. impartiality (justice) and masochism. An impartial agent does not put the other person’s interests above one’s own, but, invoking the principles of justice, tries to find a fair balance between her own and the other person’s interests.4 A masochist pursues her own harm as an end in itself. Thirdly, we wish to stress, contrary to the popular belief spread by the (unfortunate and misleading) slogan of ‘the selfish gene’, that evolutionary theory does not support either the view that egoism is the only motive of human actions or even the view that egoism is the basic motive of human actions. As Mary Midgley aptly remarked: (…) total egoism “pays” very badly genetically. If you richly fulfil yourself at the cost of destroying your siblings and offspring, your genes will perish and your magnificent qualities will be lost to posterity. A consistently egoistic species would be either solitary or extinct. Since, therefore, we are social and not extinct, we cannot sensibly view ourselves as natural egoists (Midgley, 1978, p. 94; see also Sober, 1994).

Evolutionary theory supports the thesis that human beings are at least narrowly altruistic. This claim is based on the fact that evolutionary theory predicts the existence of the tendency of individuals to display kin altruism and reciprocal altruism. The fact that evolutionary theory predicts the existence of altruistic behaviour may seem surprising only if one assumes the mistaken view that since natural selection is ‘the selfish gene theory’ (according to which natural selec  The idea that Schopenhauer’s classification needs to be supplemented with the motive of justice was advanced by Wolniewicz, 1988, pp. 115-119; we fully endorse this idea. 4

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tion operates on the level of genes and that genes’ basic goal is selfreplication), it implies ipso facto that our behaviours will be egoistic. This view is mistaken because what the theory of natural selection – the selfish gene theory – says is only that natural selection has tended to generate behaviours which have increased, on average, our inclusive fitness, i.e., which increased the representation of our genes in successive generations; it does not say that only egoistic behaviours can be fitness-enhancing. Therefore there is nothing surprising in the fact that natural selection could have favoured the self-replication of genes which produce phenotypic effects that we describe as altruistic.5 These brief remarks on the evolutionary contribution to the debate on egoism and altruism provides another argument for our earlier claim that the doctrine of psychological egoism is untenable. Let us now return to our main point concerning the necessity of distinguishing three different varieties of egoism.

3. Three varieties of egoism Egoism – a tendency to pursue self-interest in an exceedingly high degree – may flow from three different psychological sources: cognitive bias of egocentrism, hybris, biological instincts of self-preservation and reproduction. Accordingly, one can distinguish three forms of egoism: cognitively-based, hybris-based, and instinct-based. We shall now present these three forms in some detail.

3.1. Cognitively-based egoism

Cognitively-based egoism flows from overestimating the ‘reality’ of oneself as compared with the ‘reality’ of others (i.e., from egocentrism). For an agent exhibiting this form of egoism other people are   I have conducted a more thorough analysis of this problem in Załuski, 2009, pp. 32–40.

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substantially ‘less real’ than herself – or even ‘unreal.’ This conviction leads to the diminished capacity to take the perspective of other people and the tendency to interpret events with excessive reference to oneself.6 Egocentrism arguably arises as a result of the fact that we have a direct access to our ego, to our mental life, and lack direct access to other people’s egos, to their mental life; the other people’s mental states (beliefs, emotions, attitudes) can only be known indirectly. This indirect character of the cognition of the other people’s mental states is particularly salient when the cognition takes the form of inferring (by analogical reasoning) the other people’s mental states from their behaviour. But it is present also when the cognition proceeds via ‘Einfühlung’ (some kind of psychological insight): even this form of cognition does not possess the ‘directness’ that is in-built in knowing ourselves. Clearly, the fact that we have privileged access to our own mental life does not have to lead to egoism, although, as it seems, it always leads to the conviction that other people are somewhat less real than ourselves – we are all to some extent egocentric. This fact leads to egoism only if egocentrism assumes (for reasons to be ascertained by empirical psychology) a form a strong egocentrism, i.e., if it generates in her the conviction that other people are substantially less real than herself or even unreal – that, figuratively speaking, the ontological status of other people resembles that of ‘shadows’ (they are regarded more as objects than as real persons). Thus, it is strong egocentrism, not egocentrism simpliciter, that underlies a cognitively-based egoism. In summary, cognitive mistake lies directly at the basis of egocentrism rather than cognitively-based

  Our account of egocentrism is somewhat different from, though related to, the ordinary one, according to which egocentrism is an excessive preoccupation with oneself, which results in not paying enough attention or even completely neglecting the fact that other human beings have their own ‘inner worlds.’ Thus, on the ordinary account, an egocentric person does not necessarily downplay the ‘reality’ of other people (she is just excessively preoccupied with oneself), whereas on our account an egocentric person regards other people as less real and, in effect, is excessively preoccupied with oneself and thereby prone to manifest (cognitively-based) egoism. 6

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egoism; cognitively-based egoism would, then, be a consequence of (strong) egocentrism. It is worth mentioning that interesting insights into the phenomenon of egocentrism have been provided by social and cognitive psychologists who identified and analyzed in depth of many of its manifestations, e.g., a cognitive bias called the ‘spotlight effect’ (cf. Gilovich, Medvec, Savitsky, 2000). The ‘effect’ consists in that people believe that they are more noticed than they really are. Thus, they believe that they are in a ‘social spotlight’, overestimating their own importance for other people. This phenomenon is obviously egocentric in nature: since people are in the center of their own world, they tend to believe that they are also in the center – or close to the center – of the other people’s worlds. In the case of the spotlight effect the overestimation of one’s own ‘reality’ (characteristic for each manifestation of egocentrism) does not therefore lead to discounting the ‘reality’ of other people but to overestimating the ‘reality of oneself’ in the consciousness of other people. The spotlight effect may be more or less acute for different people. It seems that those who manifest it particularly strongly are more likely than others to be subject to strong egocentrism in the form of discounting the reality of other people and thereby to cognitively-based egoism.7 By way of a historical digression one may note that the claim that egoism may be due to the fact that we have a direct access only to our own mental life and thereby see other people as ‘representations’ was hinted at by Schopenhauer (2009, p. 630). He developed this in  We have discussed only one egocentric ‘effect’ discovered by social and cognitive psychologists. The other ones (some of them strictly connected with the spotlight effect) are, e.g., ‘the illusion of transparency’ (the overestimation of the degree to which one’s mental states are known by others), ‘the self-as-target bias’ (the agent’s belief that the course of events in the world is particularly ‘inimical’ to her own plans, intentions – more than to the plans, intentions of other people), the ‘false consensus effect’ (the overestimation of the degree to which other people share one’s beliefs, emotions), and the opposing effect – ‘the false uniqueness effect’ (the underestimation of the degree to which other people share one’s beliefs, emotions) (cf. Gilovich, Savitsky, Medvec, 1998; Fenigstein, 1984; Ross, 1977). All these ‘effects’ reveal egocentric proclivities of human nature and thereby well illustrate the background at which cognitively-based egoism may appear. 7

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sight in the spirit of the Upanishads, suggesting that the correct cognitive stance consists not in recognizing that other persons are real but in negating the reality also of one’s own ego. Cognitively-based egoism can therefore be overcome in two different ways. The traditional (‘Western’) ‘therapy’ against it consists in recognizing that other people have the same – full – reality as oneself, whereas the Upanishads therapy (the ‘Eastern’ therapy) consists in recognizing that other people have the same – none – reality as oneself, and that the only reality is one of some ‘cosmic ego’ (Atman-Brahman). It is a matter of dispute whether the practical consequences of the liberation from egoism is the same irrespective of which of these two anti-egoistic therapies one undergoes. One could argue that the acceptance of the belief (endorsed by the Upanishads and Schopenhauer) that the reality of the individual ego is apparent may discourage one from performing acts of benevolence, because if other people’s egos are unreal, it is unclear why one should at all care for the other people’s well-being. The fact that one’s own ego is also unreal is not a plausible answer; it begs the question. The only plausible answer could be that all people are “immersed” in the non-individual ego (Atman-Brahman). But, following Max Scheler, one could ask whether beneficial acts thus motivated are not in fact a ‘camouflaged egoism’ (Scheler 1999, p. 246): if one helps others knowing that the others are in fact oneself (in accordance with the famous teaching of the Upanishads summarized in the phrase ‘Tat-Twam-Asi’8), then in helping others one in fact helps oneself. The above remarks also provide an answer to the question of whether cognitively-based egoism is morally blameable. The answer is obviously in the affirmative, for the simple reason that this form of egoism is relatively easily avoidable: most people9 can by relatively   “That art thou”, i.e., your individual ego hides at its depths – is identical with – the cosmic ego. 9   The exception is to be made for those who suffer from some cognitive defect – some kind of a deficit of mental energy – that incapacitates them to overcome their impression of the ‘lesser reality’ of other people. A precise description of this kind of defect belongs to the domain of psychopathology. 8

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small mental effort overcome their impression of ‘the lesser reality’ of other people – in other words, they can easily overcome their egocentric proclivities or at least prevent them from turning into strong egocentrism and, consequently, into cognitively-based egoism. 3.2. Hybris-based egoism10 Hybris-based egoism does not flow from the impression that other persons are substantially less real or unreal but from the conviction that other persons are of lesser worth, i.e., from hybris. Hybris is the belief in one’s own superiority over other persons and in one’s having special rights and privileges flowing from this purported superiority. It therefore implies a blatant denial of a fundamental moral equality of human beings. Hybris manifests itself as self-confidence bordering on arrogance, insolence, overbearing pride, self-aggrandizement. It is the opposite of reverence and humility. Reverence is the virtue that keeps human beings from trying to act like gods, does let them forget that they are humans ...An irreverent soul is arrogant and shameless, unable to feel awe in the face of things higher than itself. As a result, an irreverent soul is unable to feel respect for people it sees as lower than itself – ordinary people, prisoners, children... You can forget your humanity in either of two ways: by taking the airs of god (i.e., by being irreverent, hybristic – W.Z.), or by acting like a beast of prey (Woodruff , 2001, p. 3 and 8).

As for humility, it was thoughtfully defined by Józef Tischner as “a specific ‘mode of being’ of a human agent – a mode of being in truth, a way of manifesting her good will, in which she does not put herself “above herself” but also does not “bend forward” pretending to be more pitiable than she really is” (Tischner, 1984, p. 112, transla10   The following account of hybris is a shortened version of the account presented in Załuski, 2015.

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tion – W. Z.). According to Aristotle, a particular sharp manifestation of hybris is the deliberate humiliation of the other for no other reason than the pleasure involved. The cause of the pleasure of the hybristic man is that he finds himself greatly superior to others when ill-treating them. Thus, hybris manifests itself in an especially acute form when one gives offence neither for profit nor in revenge, but simply because one delights in inflicting shame on others (cf. Elster, 1999, pp. 62–63 and 173–178; Fisher, 1992, p. 10; Gowan, 1975, p. 25). Undoubtedly, hybris-based egoism is the most wrongful form of egoism, precisely because it flows from hybris, rightly regarded by the ancient Greeks as the most serious moral depravity and assumed in Christian ethics (where it was called superbia) to be the most serious of the seven deadly sins. One can hardly find any mitigating circumstances for this form of egoism: it is neither a result of in fact natural egocentric tendencies (as cognitively-based egoism is) nor a result of natural instinctive drives (as – as we shall see – instinctbased egoism is), but a result of hybris, which is a malign product of the human spirit, of human freedom. 3.3. Instinct-based egoism Instinct-based egoism is an excessive manifestation of the biological instincts of self-preservation and reproduction. These instincts “move” human beings to undertake actions serving the realization of their basic biological goals even if performing these actions means violating other people’s morally justified interests. Of all the three varieties of egoism, the instinct-based one is most distinctly hedonistic (evolution ‘has motivated’ us to pursue our biological ‘goals’ by the promise of pleasure accompanying their realization). One can therefore say that a behavioural manifestation specific for this form of egoism consists in attaching undue importance to hedonistic values at the price of neglecting other – higher – values. We come to the world with this form of egoism and to overcome it is the fundamental purpose of education as well as of moral self-improvement. Clearly, these edu-

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cational and self-improvement efforts are not always successful: instinct-based egoism becomes a dominant motive of action of many people. It is worth noting that it is this form of egoism that Immanuel Kant had in mind when he wrote about egoism as the basic obstacle for our following moral maxims, and thereby as a conditio sine qua non of the ‘radical evil (das radikal Böse) of human nature. Thus, in Kant’s view, egoism is at the root of human inability to unwaveringly follow moral principles (which is the essential content of his doctrine of ‘radical evil’) (cf. Kant, 1986, pp. 21-56, and a deep commentary by Tischner, 1999, pp. 23-30). Since instinct-based egoism is not based on the wrongful feeling of superiority over other people, it is less morally reprehensible than hybris-based egoism, and since it is a result of the overabundance of instinctive drives (and the concomitant ‘enslavement’ to the pursuance of pleasure) which seem to be harder to control and overcome than egocentric proclivities, it also seems less morally reprehensible than cognitively-based egoism. At the end of this section it is worth reflecting on whether the three varieties of egoism can be displayed at the same time in an agent’s egoistic behaviour. Hybris-based egoism and instinct-based egoism presuppose that other human beings are as real as oneself, so these two forms of egoism arguably cannot go in pair with cognitively-based egoism, but, as it seems, they can go in par with each other. However, it should be noted that the faint (and thereby not generating cognitively-based egoism) impression of the lesser reality of other persons, may strengthen and intensify the hybris-based egoism and instinct-based egoism. Obviously, one person may manifest various forms of egoism at various times.

4. Final remarks Some philosophers (e.g., Kant and Schopenhauer) described egoism metaphorically as an expression of the ‘beastly’ part of human nature

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(the part we share with animals) and opposed this vice to the ‘diabolic’ vice of malice (wanting evil for the sake of evil). However, as we have argued, this account of egoism is incomplete and misleading. It is incomplete because it is apt only with respect to one form of egoism – the form we have called ‘instinct-based egoism’, not to its two other forms: hybris-based egoism is not an expression of the ‘beastly’ part of human nature, as it is not rooted in our instincts but flows from a purely ‘spiritual’ vice which is hybris, whereas cognitively-based egoism is neither an expression of the ‘beastly’ part of our nature nor is a ‘diabolic vice’ but, rather, is the result of a certain cognitive bias – the bias of strong egocentrism. It is misleading because it suggests that egoism is a primitive motive, not requiring – or not even amenable to – further psychological analysis. Our considerations were aimed at showing that egoism is analyzable and should be analyzed: it may take three different forms depending on what psychological sources it flows from. Clearly, in all these forms egoism remains a behavioural tendency to neglect the other persons’ morally legitimate interests (which explains why these three forms of egoism are forms of egoism), but since this tendency may flow from different psychological sources for each form, egoism may be more or less morally reprehensible. As we have argued, its most morally reprehensible form is hybris-based egoism and its least morally reprehensible form is instinct-based egoism. In brief summary: egoism can flow from three different ‘parts’ of human nature rather than from (as claimed by Kant and Schopenhauer) one – the ‘beastly – part, viz. from (the failure of) intellect (cognitively-based egoism), from (the perversion of) spirit (hybrisbased egoism), and from (the excessive force of) biological instincts, i.e., from the ‘beastly’ part of our nature (instinct-based egoism). A full-blown moral explanation of a human action should explicitly state what form of egoism motivates the action (if the action being explained is egoistic). To point at egoism as a motive for human action without indicating its form is to stop halfway in explaining the action’s moral character.

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References Batson, C. D. (2011). Altruism in humans, New York: Oxford University Press. Blum, L. A. (2004). Ēgoisme, In: M. Canto-Sperber (Ed.), Dictionnaire d’éthique et de philosophie morale (pp. 611–619), Vol. I. Paris: Quadrige/Puf. Broad, C. S. (1950). Egoism as a theory of human motives. The Hibbert Journal, 48, 105–114. Elster J. (1999). Alchemies of the mind: Rationality and the emotions. Cambridge: Cambridge University Press. Fenigstein, A. (1984). Self-consciousness and the overperception of self as a target’. Journal of Personality and Social Psychology, 47(4), 860–870. Fisher, N. R. E. (1992). Hybris: a study in the values of honour and shame in ancient Greece. Warminster: Aris and Phillips. Gilovich, T., Medvec, V. H., & Savitsky, K. (2000). The spotlight effect in social judgment: An egocentric bias in estimates of the salience of one’s own actions and appearance. Journal of Personality and Social Psychology, 78(2), 2000, 211–222. Gilovich, T., Savitsky, K., & Medvec V. H. (1998). The illusion of transparency: biased assessments of others’ ability to read one’s emotional states’. Journal of Personality and Social Psychology, 75(2), 332–346. Gowan, D. E. (1975). When man becomes God: Humanism and hybris in the Old Testament. Eugene, Oregon: Pickwick Publications. Kant, I. (1986). Die Religion innerhalb der Grenzen der bloßen Vernunft. Reclam, Philipp, jun. GmbH, Verlag. Midgley, M. (1978). Beast and man: The roots of human nature. Cornell: Cornell University Press. Nagel, T. (1979). The possibility of altruism. Princeton: Princeton University Press, Princeton. Ross, L. (1977). The false consensus effect: an egocentric bias in social perception and attribution processes. Journal of Experimental Social Psychology, 13(3), 279–301. Schopenhauer, A. (2009). Die Welt as Wille und Vorstellung. Vol. I. Köln: Anaconda Verlag GmbH. Schopenauer, A. (2010). Die beiden Grundprobleme der Ethik. Charleston: Nabu Press. Scott, N., & Seglow, J. (2007). Altruism. Berkshire: Open University Press.

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Sober, E. (1994). Did evolution make us psychological egoists? In: idem, From a biological point of view: Essays in evolutionary philosophy (pp. 8–27). Cambridge: Cambridge University Press. Tischner, J. (1984). Etyka wartości i nadziei. In: D. von Hildebrand, J. A. Kłoczowski, J. Paściak, J. Tischner, Wokół wartości. Poznań: W Drodze. Tischner, J. (1999). Spór o istnienie człowieka. Kraków: Znak. Williams, B. (1973). Egoism and altruism. In: idem, Problems of the self: Philosophical papers 1956-1972 (pp. 250–265). Cambridge: Cambridge University Press. Woodruff, P. (2001). Reverence: Renewing a forgotten virtue. Oxford: Oxford University Press. Wolniewicz, B. (1998). Z antropologii Schopenhauera. In: B. Wolniewicz, Filozofia i wartości. Vol. I. Warszawa: WFiS-UW. Załuski, W. (2009). Evolutionary theory and legal philosophy. Northampton: Edward Elgard. Załuski, W. (2015). The psychological bases of primitive egalitarianism: reflections on human political nature. In: J. Stelmach, B. Brożek, Ł. Kurek (Eds.), The emergence of normative orders (pp. 83–106). Kraków: Copernicus Center Press.

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Is Explaining Religion Explaining Religion Away?

Religion and other symbolic systems designed to motivate action have a beauty and power that seem to wilt when explained in functional terms. (Wilson, 2002, p. 231)

Is explaining religion naturalistically undermining its beauty and power, as David Sloan Wilson seems to say? We need to read carefully: he writes that they seem to wilt. He goes on to argue that upon an evolutionary perspective, it is to be expected that our aesthetic preferences correlate to features or behaviours that increase fitness. Though religious beliefs and practices may thus be beautiful and powerful even when understood, there is a challenge implied in this view: ‘Some of the most beautiful and moving elements of religion come not from cosmic struggles and invisible gods but from the vision of a better life on earth’ (Wilson, 2002, p. 231). We should have ‘profound respect for symbols embodied in the word “sacred,”’ but these should be understood as effective rather than as descriptive (Ibidem, p. 233). Thus, the key question is: Are explanations, whether evolutionary ones by D.S. Wilson and various others, and the more recent wave of naturalistic explanations that draw on the cognitive sciences, undermining religious faith by offering a secular explanation for those beliefs? Do explanations of religious belief and practices explain it away?

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Theologians and other believers may argue that explanations of origins do not decide on truth; to identify causes and reasons would be to commit the ‘genetic fallacy,’ a fallacious form of reasoning which has nothing to do with genes. An account of the origins or causes of a belief or practice is distinct from a justification of that belief or practice as approximating truth. In the opening words of his The Natural History of Religion (1757), David Hume thus rightly distinguished two questions: As every enquiry which regards religion is of the utmost importance, there are two questions in particular which challenge our principal attention, to wit, that concerning its foundation in reason, and that concerning its origin in human nature.

This passage from Hume forms the opening of the Preface of a great book on ideas about the origins of religion (Schloss & Murray, 2009, p. vi). In that volume, various authors discuss the consequences of particular views of the origins of religious beliefs – thus, making transitions from explanations of origins and persistence to arguments about reasonability and truth. Here we will not consider the rich palette of approaches considered in the volume edited by Jeffrey Schloss and Michael J. Murray, but rather concentrate on a basic issue: To what extent can successful explanations of the origin and persistence of particular beliefs undermine the rationality of holding those beliefs. Clarifying concepts such as ‘explanation’ and ‘religion’ is essential to an assessment of the impact of explanatory proposals. Thus, in this contribution I will address the following questions in subsequent sections. 1. What is explanation? How does it relate to elimination? 2. What is explained? That is, what is religion taken to be? 3. Is explaining religion different from explaining perception, morality, or math? 4. What are the losses and gains if religion were to be explained?

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1. Explanation, elimination, and integration ‘I experienced a tree’ can be said descriptively: ‘I experienced a tree, but then I realized I was mistaken.’ However, experience is also used as an achieve­ment word. ‘This second sense includes a judgment on the part of the observer about the accuracy of the subject’s understanding of his or her experience’ (Proudfoot, 1985, p. 229). For an honest subject, the report as to what was experienced is to be accepted; the report describes how the event was experienced. Nonetheless, it may be appropriate to understand an event different from the subject’s own experience and understanding; ‘you thought you experienced a tree, but it was a street lamp.’ This is what may happen when we offer an explanation. It can be perceived as a form of ‘reduction,’ as the subjective experience is presented in other terms. ‘Explanatory reduction consists in offering an explanation of an experience in terms that are not those of the subject and that might not meet with his approval. This is perfectly justifiable and is, in fact, normal procedure’ (Ibidem, p. 197). Accepting a subject’s experience in the descriptive sense as authentic need not imply the judgement that the self-description is correct. We will consider different situations in more detail. Firstly, we will consider explanations that affect the experience itself. After learning of a different explanation, the subject will ‘see’ the event differently, and hence the original experience might no longer be available to us. Secondly, we will consider situations in which explanations do not eliminate the experience of the phenomenon considered, but rather integrate the phenomenon in a more encompassing understanding of the world. This pattern is typical of the natural sciences. Though embedding an initial understanding in a larger web of theories about the world may suffice to sustain the experience, this new context of the experience may make it into an experience of something else than thought previously.

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Explanations may undermine the experience I enter a dimly lit room. I notice a snake in the corner. The light is switched on; it turns out to be a rope. Once I know that what seemed to me to be a snake in a dark corner was actually a rope, the original experience with its emotional components, such as fear, will fade away. ‘It was only a rope.’ In this well known (though in practice uncommon) example, the explana­tion liberates from unnecessary fear by providing a very different understanding of the situation. Once you have seen it is a rope, the original experience is no longer a life option. An explanation may also bring with it the loss of the innocence of childhood, and diminish joy and spontaneity. When Santa Claus or his Dutch original, Sinterklaas, are unmasked, and gifts are ascribed to parents or grandparents, the associated experiences such as surprise, fear, and gratitude disappear, or may be redirected. For older Dutch children, their newly acquired understanding of ‘Sinterklaas’ gives a new opportunity, to join with the adults in arranging this family event. However, they will never be able to return to the naïve, immediate experience that used to come with the presence of Saint Nicolaas. Newly acquired understanding need not always undermine experiences as clearly as it does when the unintended illusion of a snake is dispelled or Sinterklaas is understood as intentional human play. Let us consider a third example. As an insensitive male, I may have said to a woman on a day she was quickly irritated: ‘Hormones; it is your period.’ The woman likely takes offense, as the explanation seems to challenge the genuine character of her feelings. I implicitly suggest that her annoyance was not caused by me; her mood is just a matter of chemistry. We agree on the mood displayed, but I change the understanding. This case resembles the case of the snake that turned out to be a rope, except that here the new explanation invokes chemical balances inside her body, rather than objects that are for both of us equally accessible. Besides, I even pretend to understand her condition better than she does herself. We do not come to agree on the proper perspective.

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Changing the terms is part of what it is to be an explanation. Explaining why opium puts people to sleep by saying that it has a virtus dormativa, a sleep inducing power, as happens in Molière’s play Le Malade Imaginaire, is no explanation at all. An explanation explains in other terms. However, this need not be accompanied by the disappearance of the experience. Opium still puts people to sleep, even if one does not consider the statement that opium has a ‘sleep-inducing power’ an explana­tion. When we deal with explanations of religion, part of the discussion should be about the nature of the explanation. Does the explanation undermine the experiences, beliefs and practices, or does it offer an understanding without undermining it? Rather than elimination, explanation can be integration One might worry that a reductionist explanation eliminates a phenomenon. However, there are good arguments for the opposite effect. Finding a physiological basis for a trait affirms its reality. Genes are not less real for being understood as strands of DNA. Pain is no less real if physiologically understood. Rather the opposite: if the doctor can locate the physiological process underlying my pain (‘a trapped nerve’), my friends will take my complaints more seriously. The specific issue at hand is explained by being integrated in a more encompassing understanding; my pain is understood in the context of human neurology. The understanding of explanation may be illustrated in terms of scientific theories. In an extremely successful reduction, all empirical conse­quences of the first theory, T1, can be derived from a more encompassing second theory, T2, which is at least equally specific in its predictions and explanations. Hence, to explain the phenomena (which were at first understood as empirical conse­quences of T1) one does not need T1 anymore. However, if a theory is superfluous, it is not thereby wrong. Rather, if one could derive the superfluous theory T1 from the more fundamental theory T2 the first theory would not be auton­omous, as it would still be an excellent

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theory for the domain with which it deals. Such a form of reduction is also conservative with respect to reference; it identifies entities, properties, relations, and questions rather than eliminating them (Schwartz, 1991, p. 210). Most scientific reductions, or successor theories, do not reproduce the earlier theory but allow the earlier theory as a useful approximation under certain conditions. In physics, classical mechanics is not reproduced by relativity theory; for low velocities, classical mechanics is an excellent approximation, but for velocities close to the speed of light predictions from relativity theory are different – and those predictions match observations better. The understanding provided by the new theory corrects our view of the world, even though it matches predictions in the domain where the old theory was successful. Such a reduction of theories (T1 as a limiting case of T2) is corrective, and so, presumably, would be a reduction of psychology to neuroscience. But a corrected psychology ‘would be a psychology still. Insofar as correction and improvement “threaten” our current folk psychology, we should welcome such threats’ (Ibidem, p. 212), as these are moments we acquire further insight. In the context of this essay, a relevant worry may be that revisions are not ontologically conserva­tive. When identification is possible (water is a large collection of H2O molecules), there is no problem, but if there are inconsistencies, e.g. between scientific chemistry (with the concept of H2O) and folk-chemistry (with the concept of water), those elements of the folk notion that do not survive experimental tests and theoretical integration need to be abandoned. A radical example has been provided by Paul Churchland (1981, p. 67), who argued that ‘our commonsense conception of psychological phenomena consti­tutes a radically false theory, a theory so fundamentally defective that both the principles and the ontology of that theory will eventually be displaced, rather than smoothly reduced, by completed neuroscience.’ An expectation that is three decades later still more speculation than reality.

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Do we always have to choose between a common-sense view and a scientific one, that is between the ‘manifest image’ and the ‘scientific image,’ as Wilfred Sellars (1963, p. 5) called them? The compatibility of a common-sense ontology and a scien­tific one can be defended in the case of ‘Eddington’s two tables.’ The physicist Sir Arthur Eddington once distinguished between ‘two tables’ in his room: the ordinary table, a massive, brown object, and the ‘scientific table,’ which is mostly empty space, dotted with the nuclei of the atoms and the surrounding electrons. Though these descriptions refer to a single table, these ‘two tables differ in properties, indeed in essential properties, so they cannot be identical’ (Schwartz, 1991, p. 213), or so it appears to be. The fallacy with the argument about the two tables, according to Schwartz, is in the understand­ing of substance, which is both a common-sense notion (I can lean on it, I cannot put my hand through it etc.) and a notion which carries various philosophical commitments, for instance that the presence of substance excludes empty space. The common-sense notion of substance, say solidity, is explained by the scientific account; for all practical purposes, the table remains solid, even though we may have to give up some ontological notions attached to substance, and hence be open to changing our philosophical commitments. Any scientific description of the table will have to incorporate the fact that I cannot put my hand through the table. ‘If all we mean by commonsense solidity is the functional notion, we have a reduction of solidity which preserves the main features of the folk notion by identifying it with its physical microstructure, or showing how it is constituted’ (Ibidem, p. 217). Both tables are equally real; they are the same table. Such a reduction is moderately conservative rather than radically eliminative. However, in as far as anything is eliminated, it is not the phenomenon of a reliable, solid table but our ontology of substance. One can distinguish various forms of ‘reductionism,’ ranging from identification (water and H2O), where elimination would be incoherent; constitutive, as when genes are discovered to consist of

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DNA, where the reduction embeds the original notion more strongly in our understanding of reality; approximative, where the original theory remains useful and relevant under certain conditions, even though that theory has been replaced by a different and better one; and moderately revisionist, where the effect is not so much elimination as revision, for instance of our ideas about solidity. ‘In all these, reduction is an alternative to, not a form of, elimination’ (Ibidem, p. 218). The new understanding is more encompassing; the earlier view is integrated into the new one. Thus, to summarize the conclusions of this brief tour, explanations need not be eliminative. However, they can be eliminative or corrective in important respects. The question is whether explanations of religion resemble any of the previous examples: • explanations of the apparent snake or Santa Claus, where our relation to the phenomenon is affected; • reductionist understandings of genes, pain and water, where the reality of the phenomena and the approximate adequacy of our understanding is affirmed by the enlarged picture of reality, while our understanding is modified and improved as well; • explanations of the solidity of the table, where our ‘philosophical’ idea about the ontology is modified drastically, though in human practice nothing needs to change.

2. What is explained? That is, what is religion taken to be? I will briefly consider a few contributions to naturalistic explanations of religion, to see how the authors understand religion, in relation to the explanation of religion they offer.

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Dean Hamer Dean Hamer wrote on the penultimate page of The God Gene (2004, p. 214) the following. The fact that spirituality has a genetic component implies that it evolved for a purpose. No matter how selfish a gene is, it still needs a human being as a carrier to perpetuate itself. There is now reasonable evidence that spirituality is in fact beneficial to our physical as well as mental health. Faith may not only make people feel better; it may actually make them better people.

Religion is based on memes, traditions of culture that are passed

on not by DNA but by learning, instruction, and imitation. As a result, memes may or may not be beneficial to the people who hold them. What determines their survival is how well they are transmitted from one person to another. The deep and abiding appeal of religious memes is based not on logic or even benefit to our species, but on their ability to evoke altered states of consciousness and spirituality that are so important in spirituality.

According to Hamer, if spirituality is partially genetic, it has been beneficial for the carrier; he even makes the further step that it thus evolved for a purpose. That is an overstatement; any purpose evolved spirituality has, may be an accidental side effect, drawing on human abilities and inclinations that provided other benefits. Besides, it is odd to say that it evolved for a purpose, as if the purpose preceded the process. Whereas his view of genes is positive, his view of memes (postulated units of cultural selection) is more critical. Memes do not need to be beneficial to those who carry them; the focus is only on their transmission. This is an odd way of making the difference; a meme that would undermine health would also undermine its own transmission. The language of memes has been developed by Richard Dawkins (1976) and Susan Blackmore (1999) to have a parallel conceptual vocabulary for units of physical and cultural evolution, genes

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and memes; the concept has been criticized precisely for the suggested parallelism and the implied cultural atomism. Though Hamer uses the language of memes, he moves away from the evolutionary perspective (benefit); memes are only useful as means to evoke spiritually useful altered states, which brings him back to the genetic dimension he gives priority. Hamer makes a difference between religion and spirituality. The biological explanatory project is not about belief in God or gods but about spirituality (Hamer, 2004, pp. 8–9). We have a genetic predisposition for spirituality, while particular religious forms are cultural, just as bird-song may be hard-wired though the songs themselves are cultural (Ibidem, p. 8). Scholars in the scientific study of religion will question whether this distinction between religion and spirituality is viable. It seems to me to serve an apologetic purpose in advocating a liberal and universal ‘spirituality’ that transcends all particular religions with their organizations and doctrines, while leaving all beliefs that conflict with science to ‘religion’ rather than spirituality (Ibidem, p. 210). Furthermore, Hamer considers himself agnostic on methodological grounds with respect to truth. ‘This is a book about why humans believe, not whether those beliefs are true’ (Ibidem, p. 16), the distinction argued for above. The truth question, whether there is a God, is beyond science, whereas the question why we do believe in God is potentially within our understanding (Ibidem, p. 14). There is nothing intrinsically atheistic or theistic about postulating a genetic and biochemical mechanism for spirituality; if God does exist, he would need a way for us to recognize his presence (Ibidem, p. 211). Lastly, Hamer considers knowledge of the natural processes underlying spirituality and religion to be purifying. Not only does he suggest that spirituality may make people better, but also that understanding the difference between spirituality and religion gives us another tool in offering hope for the future. For while our spirituality may be engraved to a degree in our DNA, we can change, reinterpret,

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and reconsider the memes written on the scrolls of our religious history. (…) Some of religion’s least desirable memes, such as the condemnation of pagans, of non-believers, of outsiders, can be difficult to erase or reinvent. But they can be altered – and in the case of religious memes that prove themselves to be destructive to peace, understanding, and compassion, they must be (Ibidem, p. 215).

A moral injunction on the final page is not uncommon. In the final sentence of his book The Selfish Gene (1976), Richard Dawkins called upon us to rebel against the selfish genes. However, while Dawkins thus called upon us to make culture (understanding, imagination) override genetic impulses, Hamer encourages us to alter culturally determined religious beliefs while remaining true to our genetically determined drive towards spirituality. Andrew Newberg and Eugene d’Aquili Andrew Newberg and Eugene d’Aquili have a similar interest in individual experience and spirituality in relation to the structure of the brain. Two decades ago, D’Aquili spoke about the experience of ‘Absolute Unitary Being’ (AUB), in which reality is viewed as a whole, arising ‘from the total functioning of the parietal lobe on the nondominant side (or at least certain parts of that area)’ (1987, p. 377). The dominant hemisphere deals primarily with our perception of the external world in all its variety; the non-dominant one might deliver ‘the perception of absolute, unitary, atemporal being’ (Ibidem, p. 378). Why God Won’t Go Away is a more recent presentation of this research programme that uses non-invasive brain scanning technologies to explore the neurological basis of mystical experiences. They have come up with particular proposals regarding neurological pathways that are involved in such experiences. They have argued that such capacities of the mind may have beneficial effects in terms of individual health and bonds between members of groups, and thus have been favoured in the course of evolution.

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Whatever the details, mechanisms that inhibit certain pathways and thereby inhibit certain forms of sensory input and of conceptual analysis, may well turn out to correlate with mystical states or other experiences considered religious. If we can describe such neurological processes, does this support or undermine a naturalist view of religious experiences? They seem to give two arguments for an affirmative answer. Firstly, the authors suggest that we should treat the workings of both hemispheres with equal seriousness. If we trust that we are in touch with reality via the dominant hemisphere, why would we not trust, so the argument goes, experiences governed by the other side? I think this argument faces various problems. The two hemispheres have coevolved, but they may well function in complementary ways rather than in similar ways; certain structures are more involved in the perception of the world, others more in emotions or in creating (rather than perceiving) an encompass­ing perspective which allows for balanced action. Secondly, they argue that the perception of material reality depends on the self-world distinction, and that this distinction develops over time in the young brain; in that sense, the brain/mind exists before the personal self. In the neurological processes that arise when external sensory input is no longer involved, leading to experiences of Absolute Unitary Being, this process of distinguishing self and world, subjective and objective reality, is undone. Awareness is freed of the usual subjective sense of self and of the spatial world in which that self could be, thus resulting in ‘a state of pure awareness, an awareness stripped of ego, focussed on nothing, oblivious to the passage of time and physical sensation’ (Newberg c.s., 2002, p. 151). This is not a proof of the existence of Absolute Unitary Being, but ‘when the mind drops its subjective preoccupation with the needs of the self and the material distractions of the world, it can perceive this greater reality’ (Ibidem, p. 155). The external world and the subjective world of the self are both ‘contained within, and perhaps created by, the reality of the Absolute Unitary Being’ (Ibidem, p. 155).

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They also indicate how they understand anthropomorphic ideas about God or gods from the traditions. Such personifications are ‘symbolic attempts to grasp the ungraspable’ (Ibidem, p. 161). When the images are taken too serious, ‘we create a God who leads us away from unity and compassion, towards division and strife’(Ibidem, p. 162). ‘History suggests that religious intolerance is primarily a cultural phenomenon, based in ignorance, fear, xenophobic prejudice, and ethnocentric chauvinism’ (Ibidem, p. 163). However, they see it as more than just narrow-mindedness; ‘the presumption of “exclusive” truth, upon which religious intolerance is based, may rise out of incomplete states of neurobiological transcendence’ (Ibidem, p. 164) – thus, exclusivist forms of religion are based on incomplete forms of experience and ‘perception.’ Their perspective is clearly pluralist and universalist. ‘All religions, therefore, are kin. None of them can exclusively own the realist reality, but all of them, at their best, steer the heart and mind in the right direction’ (Ibidem, p. 165). In this universalist context, we see also their expectations as to what their own understanding of religion may bring forth: The neurology of transcendence can, at very least, provide a biological framework within which all religions can be reconciled. But if the unitary states that the brain makes possible are, in fact, glimpses of an actual higher reality, then religions are reflections not only of neurological unity, but of a deeper absolute reality’ (Ibidem, p. 168).

The if-then sentence seems trivial (if they are glimpses of reality, religions are reflections of reality), but it emphasizes their interest in spiritual realism and in focussing on the common denominator. This returns in the epilogue, where Newberg speaks of neurotheology as ‘a way of approaching religion that focuses on the universal elements that all religions seem to share’ (Ibidem, p. 175). It is to be noted that there hardly is any interest in the situated character of religious ideas and practices or the particular content of beliefs. The focus on religion is on the experiential side and on the re-

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ality suggested by mystical experiences, a reality of pure awareness beyond the limits of subject and object. David Sloan Wilson Quite different from Hamer and from Newberg and D’Aquili is David Sloan Wilson’s approach to religion in Darwin’s Cathedral: Evolution, Religion, and the Nature of Society. Central in his argument is an ‘organismic concept of religious groups.’ Thinking in terms of selection operating at various levels (a genome is a group of genes; a community a group of different individuals) has been prominent as well in Wilson’s earlier work on the evolutionary conditions for altruism, Unto Others: The Evolution and Psychology of Unselfish Behavior (Sober & Wilson, 1998). In Darwin’s Cathedral, he presents the following thesis. The hypothesis I am seeking to test proposes that religion causes human groups to function as adaptive units. This hypothesis has many rivals, including religion as a tool of exploitation, as a cultural parasite, as a by-product of cost-benefit reasoning (Wilson, 2002, p. 96).

A belief system does not only recommend behaviours, but offers also justification and possible rewards and punishments, thus curtailing cheating. Whether the belief system is fictional or realistic is not relevant; a fictional system can be more economical and effective. For belief systems, their ‘adaptedness must be judged by the behaviors they motivate, not by their factual correspondence to reality’ (Ibidem, p. 100). The other-worldliness of religion is not detached from reality. Rather, it [religious belief] is intimately connected to reality by motivating behaviors that are adaptive in the real world – an awesome achievement when we appreciate the complexity that is required to become connected in this practical sense. It is true that many religious beliefs are false as literal descriptions of the real world, but this merely

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forces us to recognize two forms of realism; a factual realism based on literal correspondence and a practical realism based on behavioral adaptedness (Ibidem, p. 228).

Similarly, Philip Kitcher argued in his ‘pragmatic naturalistic’ book The Ethical Project that religion may have emerged within groups of hominids as the beliefs and communal practices served the group by suppressing violations of in-group collaboration, so called ‘altruism failures (Kitcher, 2011, p. 111 ff). According to D.S. Wilson, we always face a trade off between factual and practical realism. Practical realism may be supported effectively by fictitious accounts. Science is an exception in that it is explicitly committed to factual realism; more limited but thereby enriching practical realism in the long run (Wilson, 2002, pp. 229–231). Religion need not be factual to be significant. As quoted at the beginning of this chapter: ‘Some of the most beautiful and moving elements of religion come not from cosmic struggles and invisible gods but from the vision of a better life on earth’ (Ibidem, p. 231). Different naturalistic explanations of religion focus on different aspects of religion. Wilson attaches importance to beliefs as they motivate behaviour and to institutional, social organizations; Newberg and Hamer emphasize individual experience and spirituality. They also differ in how they treat the diversity of religious particulars. For Hamer, Newberg, and D’Aquili, cultural variation is relatively unimportant. Besides, differences count as indicators of the cultural and human packaging of the real experience, which is in underlying structure universal. For Wilson, adaptedness is always to be understood in relation to an environment, whether social or cultural. Hence, diversity is unavoidable upon his approach.

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3. Is explaining religion different from explaining perception, morality, or math? To say that religions might be explicable as functional social symbols or as natural products of healthy brains, is not necessarily to deny that their central terms refer to realities. However, if a religious claim purports to be about a supernatural reality, such as one or more gods, one might raise the question of whether the claim may be right or wrong. We will consider this in relation to perception. That analogy gives rise to problems for transcendental religious references. I will briefly consider mathematics and morality as domains that might provide analogies that allow for some form of transcendence. Let us start with the example of observing trees. On an evolutionary view, the adequacy of our understanding of trees, with notions such as bark, leaves, and firewood is intelligible since our vocabulary has been fine-tuned in a long history of interactions of humans with trees and with each other in conversations about trees. Such a web of causal interactions lies behind the adequacy of our language about trees. If one came across a culture with no past experiences with trees or with other humans referring to trees, it would be a very surprising coincidence if they had an adequate vocabulary for trees. We are able to refer adequately to trees, because our language has arisen and been tested in a world with particular ostensible trees. And the claim that I know there is a tree in the garden behind my house, rests upon earlier experiences with that tree – its shade, the leaves it dropped, and much else. Philosophers engaged in epistemology have considered situations in which people have true beliefs, apparently for good reasons though in the end the justification turns out to be mistaken; these are called Gettier examples. The exceptional and artificial nature of such examples show the adequacy of the common sense view: generally speaking, knowledge about entities in our world is justified if properly based on perception, that is, on some appropriate form of interaction with the object of knowledge.

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Now back to religions and reference. For a naturalist there is no locus for particular divine activities in a similar ostensible way. Thus, it is extremely unlikely that our ideas about gods would conform to their reality (Segal, 1989, p. 79). Hence, an evolutionary view challenges religions not only by offering an account of their origin but thereby also by undermining the credibility of religious references to a reality which would transcend the environ­ments in which the religions arose. One may also return to the previous considerations about explanation and elimination. If an explanation of pain, in terms of a particular nerve firing, would not deny the reality of pain (but rather affirm it), why would an explanation of religious beliefs undermine their credibility? It seems to me that the problem has been captured well by Robert Segal (1999, p. 158). Unfortunately, God is not, like pain, a reality to be explained but it is rather, like atoms, an explanation itself of reality. The reality to be explained is religion, or its object. Where God is the explanation offered by nonreductionists, nature, society, and the psyche are among the explanations offered by social scientific reductionists. Those explanations, as rival ones to God, do challenge the reality of God, so that Eliade is justified in fearing them, even if he is not justified in rejecting them. These explanations may not refute the existence of God, but, if accepted, they may well render his existence superfluous – and in that sense threaten the reality of God (Ibidem).

Thus, even though explanations of the origins of religious beliefs are not direct evidence that the beliefs themselves are false, the absence of the referent of religious beliefs in the causal explanation does challenge the adequacy of those beliefs. Of course, from a prior theological commitment, one could always argue that even without explicit reference to God in the explanatory account, God is involved. As the Creator and Sustainer of reality, God has set the initial conditions and upholds the laws of nature,

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and thus is involved in any causal process that would have brought about religious belief. However, such a God would have been equally involved in any process that brought about unbelief, and all other varieties of belief that one might want to dismiss as superstition. Besides, the rationality intrinsic to natural processes might be an incentive to argue for a rational Creator (e.g., Coyne & Heller, 2008). However, this regards philosophical arguments, far removed from contemporary explanations of religious beliefs and practices. One might propose a different analogy, not between religion and sense perception but between religious claims and claims in mathematics and ethics. There one can make a stronger case that an account of origins, how we have come to a certain conviction, indeed does not decide on the truth or value of that conviction. However, there are relevant differences between the status of mathematics and ethics, and the status of religious ideas, for most believers. Mathematics may be seen as a second-order activity, growing out of the analysis of human practices such as counting and trading by continuing abstraction. Similarly, ethical considerations involve a second-order reflection, upon procedures or standards that may be fruitful in resolving conflicts of interests with reference to an (unavail­able) impartial perspective. As second order activities, they aim at norms of universal validity, but these universal, `transcendent’ claims may be construed without reference to the existence of a realm of abstract objects apart from the natural realm with all its particulars (e.g., Kitcher, 2011). Similarly, one might understand theism as an articulation of ‘the possibility of a view of human affairs sub specie aeternitatis,’ as if seen from an eternal, impartial perspective (Sutherland, 1984, p. 88; see for a more developed reflection on mathematics, morality and theism Drees, 2010, pp. 85– 134, and Drees, 2013, pp. 309–335). In contrast to mathematics and morality, and also in contrast to an abstract and apophatic (agnostic) theism, religious beliefs and practices that are object of contemporary explanatory attempts are first-or-

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der phenomena in which there is in most cases a claimed reference to transcendent realities, denizens of another realm. Whereas such references in morality and mathematics may be reconstructed in terms of procedures for justification, religions are much more tied to an ontological view of those realities: gods are supernatural realities or they are non-existent. Hence, an account of the evolutionary origins and adaptive functions of religion is a much stronger challenge to the truth of some religious doctrines than is a similar understanding of the origin and function of arithmetic or morality, since mathemat­ical and moral claims are not so much seen here as truth claims about reality, say about causally efficacious entities, whereas religious claims are often taken to be of such a kind. Thus, whereas an account of ethics that avoids reference to a non-natural realm would not affect morality, a similar account in theology would have more radical consequences, as it would undermine the referential character of statements that purport to be about a non-natural God who is in causal interaction with the world. A different ontological concept, not uncommon in the theistic traditions, of God as the timeless ground of being, and a different epistemic attitude in theology, acknowledging the limitations of human ways of imagining God and thus more apophatic (as in traditional ‘negative theology’ – we know that God is not like our human concepts and images), would not be affected by such naturalistic explanations in the same way, as these explanations tend to target beliefs about an intervening God.

4. Concluding remarks: losses and gains if religion were to be explained Is explaining religion naturalistically explaining religion away, undermining its beauty and power? Such was the question at the beginning of this paper. Let us try to take stock and reflect upon the situation.

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1. Explanation need not be elimination of the phenomenon. a. Reductionist explanations may be understood as affirming the reality of the phenomenon observed (e.g. pain/nerves; genes/DNA). b. Reductionist explanations may transform the experience (e.g., the snake/rope; Santa). c. Reductionist or naturalist explanations may change the terms, often the ontology. Though some of the more radical critics of religion seem to suggest that the challenge is of the snake/rope type (b; I am thinking of Dennett, 2006; Dawkins, 2007), the real challenge is in changing the ontology, and thus the understanding of the nature of religious belief and of God. 2. Explanations such as those of D’Aquili and Newberg, which seek to stay within the scientific fold, but also suggest that we are in touch with a higher reality, merge two options, namely that of treating ‘higher realty’ as distinct from natural reality and of treating ‘Unitary Being’ as a different understanding of natural reality. This raises questions as to how the two descriptions of a single reality relate. Methodologically speaking, Hamer and Newberg/D’Aquili are apologists for spirituality (or religion) by placing all that they consider valuable in the core of their research program. The diversity of religions and the universality of exclusivist elements is not appreciated, and hardly addressed. Wilson suffers less from this, as he can more easily explain why awful behaviour towards others may be beneficial for the group. Wilson’s approach is more consistently ‘naturalist’ by treating religious symbol systems as functional and therefore valuable, but this approach is most likely to affect religious attitudes and practices of informed participants.

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3. The main difference between the two approaches considered is that Hamer nad Newberg & D’Aquili make experience central, but with it, also the putative ontology associated with these experiences. Wilson makes behaviour central, and allows for the difference between an insiders account of motivation (God’s commands) and an outsiders account. In that sense, he accepts the way reductionist explanations may have consequences for the ontology assumed. If one accepts a more completely naturalistic approach (such as D.S. Wilson), religious beliefs and claims regarding revelation may serve as normative or motivational terms of praise, recommending behaviour and beliefs, but not as sources of information about a transcendent reality. 4. Whatever naturalistic explanation of religion might come available, the result would only be that religion would be among the human phenomena such as languages and political systems; religion itself would no longer be among the gaps that religious explanations would be called upon to fill. However well human religiosity is explained, one can always assume that there is a creator of natural reality, a ground of existence and norm of the highest good, the primary source of all natural, secondary processes. But that is philosophical theological discourse, at a great distance from the empirical and historical study of human religious practices and beliefs.

Acknowledgments A draft of this paper was presented at a symposium on explanation, held March 15, 2013 in Cracow, Poland, organized by the Copernicus Center in Cracow. An earlier version has appeared in Omega: Indian Journal of Science and Religion, 8(1), 2009, 7–23; see also Drees, 1996, pp. 189–194 and 213–223.

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References d’Aquili, E. G. (1987). The myth–ritual complex: A biogenetic structural analysis. Zygon: Journal of Religion and Science, 18, 247–269. Blackmore, S. (1999). The meme machine. Oxford: Oxford University Press. Churchland, P. (1981). Eliminative materialism and the propositional attitudes. Journal of Philosophy, 78(2), 67–90. Coyne, G. V., & Heller, M. (2008). A Comprehensible universe: The interplay of science and theology. New York: Springer. Dawkins, R. (1976). The selfish gene. Oxford: Oxford University Press. Dawkins, R. (2007). The God delusion. New York: Houghton Mifflin. Dennett, D.C. (2006). Breaking the spell: Religion as a natural phenomenon. London: Penguin Books. Drees, W. B. (1996). Religion, science and naturalism. Cambridge: Cambridge University Press. Drees, W. B. (2010). Religion and science in context: A guide to the debates. London: Routledge. Drees, W. B. (2013). God as ground? Cosmology and non-causal conceptions of the divine. In G. F. R. Ellis, M. Heller, & T. Pabjan (Eds.), The causal universe (pp. 291–321). Kraków: Copernicus Center Press. Hamer, D. (2004). The God gene: How faith is hardwired into our genes. New York: Anchor Books. Hume, D. (1757). The natural history of religion. [various later editions]. Kitcher, Ph. (2011). The ethical project. Cambridge, Mass: Harvard University Press. Newberg, A., & d’Aquili, E. (2002). Why God won’t go away: Brain science and the biology of belief. New York: Ballantine Books [orig. 2001]. Proudfoot, W. (1985). Religious experience. Berkeley & Los Angeles: University of California Press. Schloss, J., & Murray, M.J. (Eds.) (2009). The believing pimate: Scientific, philosophical and theological reflections on the origin of religion. Oxford: Oxford University Press. Schwartz, J. (1991). Reduction, elimination, and the mental. Philosophy of Science, 58, 203–210. Segal, R. (1989). Religion and the social sciences: Essays on the confrontation. Atlanta: Scholars Press.

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Segal, R. (1999). In defence of reductionism, In R. T. McCutcheon (Ed.), The insider/outsider problem in the study of religion (pp. 139–163). London: Cassell. Sellars, W. (1963). Science, perception and reality. London: Routledge and Kegan Paul. Sober, E., & Wilson, D. S. (1998). Unto others: The evolution and psychology of unselfish behavior. Cambridge, Mass: Harvard University Press. Sutherland, S. (1984). God, Jesus and belief: The legacy of theism. Oxford: Basil Blackwell. Wilson, D. S. (2002). Darwin’s cathedral: Evolution, religion, and the nature of society. Chicago: University of Chicago Press.

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Olivier Riaudel Catholic University of Leuven

Explanation in Christian Theology: Some Points in Common with the Human Sciences

1. The context Before asking ourselves about the concept of explanation in theology, we would do well to specify a few contextual elements which increase the effect of scepticism on knowledge which is neither technical (which has long been noted), nor quantifiable, nor mathematizable, particularly what one might consider to be a crisis of affirmation, as a crisis of confidence in the world and in life. The discourse of suspicion has spread well beyond those who are conveniently called the “Masters of suspicion.” Modern sciences, and not only the exact sciences, work at once in this crisis and at this crisis: they seek to provide new bases and new principles for old truths placed in doubt, and then once again place this new base in doubt. New reasons for placing the newly acquired certainty in doubt appear ever afresh in research: ‘De omnibus dubitandum est’ is how Kierkegaard summarized the Cartesian ideal. This Cartesian ideal is undoubtedly a bit naive from an epistemological point of view, and that much more so since in Descartes there is the idea that this doubt can be surmounted. Everyone knows the beginning of the second Meditation: The Meditation of yesterday has filled my mind with so many doubts, that it is no longer in my power to forget them. Nor do I see, mean-

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while, any principle on which they can be resolved; and, just as if I had fallen all of a sudden into very deep water, I am so greatly disconcerted as to be unable either to plant my feet firmly on the bottom or sustain myself by swimming on the surface. I will, nevertheless, make an effort, and try anew the same path on which I had entered yesterday, that is, proceed by casting aside all that admits of the slightest doubt, not less than if I had discovered it to be absolutely false; and I will continue always in this track until I shall find something that is certain, or at least, if I can do nothing more, until I shall know with certainty that there is nothing certain. Archimedes, that he might transport the entire globe from the place it occupied to another, demanded only a point that was firm and immovable; so, also, I shall be entitled to entertain the highest expectations, if I am fortunate enough to discover only one thing that is certain and indubitable (Descartes, 1901).

Whatever may be said about current debates on foundationalism (see particularly Armstrong, 1973; Goldman, 1979, pp. 1–23; Goldman, 1986; BonJour, 2000; Williamson, 2000), the fact remains that doubt has power, a scientific productivity, and we might even define modern sciences as amounting to means of making doubt productive; no knowledge can be affirmed which has not been through the school of doubt. The exact sciences are particularly sensitive to this insistence: new explanations are sought constantly: more general, more elegant and, of course, for understanding new data, or for better understanding the available data, we have to start by questioning what has been admitted until now, seeking new paradigms and new ways of tackling the question. This never involves defending what was already believed, and from that viewpoint theology is already in double jeopardy insofar as it fits into a tradition whose elements of continuity it prefers emphasizing, and in being handed down a position on the world and man that it holds as true.

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Nowhere in modern sciences is a discourse developed based on what “reality” might be, on what would “really” be; on the contrary, what is presented as, or what one believes to be reality, or what really is… is strategically called into question. On the contrary, positively affirming things from the outset passes as the best way of blinding yourself on what doesn’t correspond well in the theory so far held up as adequate, what “doesn’t stand up” in the context of the theory affirmed up to now. Theology has certainly known evolutions, but hasn’t it rather adapted itself to doubts cast, coming from outside, rather than on questions emerging from its own object? The doubt marking the modern conception of science is certainly not an end in itself; that is what remains right about Descartes’ position: this involves what allows us to be more adapted to reality, more relevant; and that this scepticism is not a goal is what the argumentation itself shows, advancing good or better arguments, better founded, more efficient, more elegant, with greater heuristic force; with this argumentation seeking as broad an agreement as possible. In other words, there is not just insistence on doubt; there is also an insistence on affirmation. In this context we would also need to see how this crisis of affirmation plays out when it is not only a question of studying what is, but also what should be, that is when we are not dealing with descriptive, but rather prescriptive research. The crisis of affirmation is not just a step in the logic of research. In its various forms, the hermeneutics of suspicion, or deconstruction, develop a critique shedding light on hidden interests or hidden logics, all hidden forms of affirmation which can have an effect of oppression, reaching the descriptive dimension of knowledge. But ultimately there too, any ethical procedure aims at an affirmation too, whether that be in the form of an ideal, transcendental norm which obliges, or an invitation to defend what allows partial, but real, achievements of justice, structures of the good life. We can immediately see the stakes involved in this context for theology.

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2. What is an explanation in the social sciences? The necessity of models Let’s start with a hypothesis that should be discussed. The class of science nearest to theology (if theology is a science), is that of the humanities. At my University, which has various “sectors,” theology belongs to the sector of the human sciences, that is the social sciences, law and the humanities, and our Vice-Rector is the one for the humanities. What is an explanation in the social sciences and humanities? An answer to this question necessarily has two aspects. A descriptive aspect and a normative aspect are involved. A descriptive aspect: comparing and analyzing the works of science as they actually exist, not as we wish they were. And it is from this description that we can clarify the sense that the notion of “explanation” assumes in their practice. But the answer to this question has a normative aspect too. Because once you leave the relatively well defined field of physical objects, you might be tempted to relax your requirements and recognize every systematized body of knowledge as a science. To even a highly organized intuitive knowledge – such as that of a developed body and codified technique, such as fencing or judo – we cannot give the name science without condemning ourselves to being unable to understand some thing or other involving the specific traits of optics, thermodynamics, or biochemistry. With that presupposition, we would even be led to refuse to recognize as sciences whole sections of physics or biology that do not attain the required systematicity. An important criterion might be: scientific knowledge presupposes a split between a lived reality, which is the actual experience of knowing, and an image, more or less abstract, of what is known. In theology, this split can be formulated as follows: between the language of faith and the language of theology, between a confessional and a more philosophical or conceptual language.

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We would say that given these conditions, to scientifically explain a phenomenon is to establish a concept, an abstract model, and show that that scheme fits into a more comprehensive scheme, as one of its parts (as a partial model) or as one of its special cases (sub-models). The more “complete,” means the more the explanation will be inclusive and the more we will estimate that the explanation serves its purpose, providing reasons for what we observe, and the more we can trust it. Although this definition is very broad, it can serve as a starting point. It already has two consequences. First, an explanation can only be relative. There is no radical explanation, no total and definitive explanation of a phenomenon. A professor of physics at our university, Vincent Lemaître, who took part in the discovery of the Higgs boson, recently said this: highlighting the Higgs boson “is not a goal in itself. All this is an endless quest. Discovery or non-discovery puts us on the way to other novelties. That’s life. Static understanding, understanding what is, ultimately, doesn’t interest me. What’s fascinating is the evolution of knowledge. If it were to stop, can you imagine what a tragedy that would be (…) And even if they find the Higgs, it would open the doors to other fundamental discoveries; we must nevertheless say that we’re still nowhere.” If every explanation is relative, then every explanation has a story because it is related to a reference system, which has been provisionally adopted as a structure for attaching the scheme. On the other hand, an explanation’s being relative and relational implies that not every explanation is necessarily a reduction. Explaining is not necessarily reducing one scheme to another. The covering structure is not necessarily a cause or factor.

3. This definition and the deductive, nomological model The former formulation, “to scientifically explain a phenomenon is to establish a concept, an abstract model, and show that this scheme

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fits into a more comprehensive scheme,” is a sort of weak form of a deductive nomologic model of sciences. According to the deductive nomological model defended by Hempel and Oppenheim (1948, pp. 135–175), scientific research proceeds as follows: the initial conditions of the experiment are connected to general laws (that is the nomological aspect), and from that set particular facts can be deduced and verified by the experiment in question (that is the deductive aspect). “The event under discussion is explained by subsuming it under general laws, i.e., by showing that it occurred in accordance with those laws, by virtue of the realization of certain specified antecedent conditions” (Ibidem, p. 152). An explanation inserts a fact into the regularity given by scientific laws, a regularity which allows confidently predicting the fact, as long as the particular initial conditions are known. The quality of the explanation will be estimated by measuring the quality of the prediction: if my explanation fails to predict what can be observed, then the explanation is falsified. If an explanation fails, it may be due to a lack of analysis or perception of the phenomenon or that the phenomenon is not referred to the relevant determinants, or that there is a co-action of several determinants that has not been clarified. In short, falsification identifies weaknesses and gaps in the network of determinants. It should be noted that this description of falsification already uses a broad or modified concept of explanation. The explanation of an empirical phenomenon seeks to describe the close interaction of several factors according to the regularity of universal laws. A description is inadequate when a phenomenon cannot be thus reduced to an existing system, and we then seek a new explanation that can take into account what the former presuppositions did not allow us to do. It is a broad concept of explanation in that it is a functionalist or pragmatic one. And this notion marks the death of the depiction that an explanation should present things as they are. The best explanation is the one that is most useful, most elegant, most predictive and most inclusive. To what extent do the humanities satisfy these requirements?

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4. From a phenomenon to a scientific fact: syntactic and semantic concepts If it is possible to ask how social phenomena may be considered to be scientific facts, it doesn’t make sense in theology (nor in philosophy): this reduction is difficult in the case of social phenomena because they are immediately given with a meaning: we can immediately perceive the meaning or else we feel the lack of meaning despite our search for one. This applies at once to the evolution of prices, to a religious rite, or to a revolutionary movement. We may also ask the extent to which it is possible to describe a social phenomenon without the idea of a standard. Finally, the description of these phenomena implements concepts and abstractions which are not necessarily objects of verification. But we should be aware that something that is controllable is not necessarily observable. Any scientific system includes both types of concepts: “semantic” concepts which intend to describe some aspect of a phenomenon and “syntactic” concepts, which are functors linking them to other concepts. This distinction between syntax and semantics is related to a state of knowledge. For Carnot, entropy was first a syntactic concept for calculating a thermodynamic state, before acquiring a semantic status with Boltzmann and Planck. In contrast, the concept of inertia weight plays a semantic role in classical mechanics, and a rather syntactic role in relativity. Falsification in theology Even if in some cases it is impossible to transform social phenomena into scientific facts, to be admissible an explanation in the social sciences must admit a possible refutation. It is an obvious criterion, but many assertions in the social sciences are formulated so that they cannot be refuted by any observation. Can we however consider that the principle of falsification is applicable in theology? When we say that “God exists” or “Jesus is risen from the dead,” are such statements

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potentially falsifiable? And if that is the case, should we consider that the statements that express the faith are merely hypothetical? Would faith be a number of hypotheses awaiting corroboration? Of course, the practice of theology has never been devoid of a number of criteria for rational judgment, which can be interpreted in terms of falsification’s criteria. Logical consistency, absence of contradictions, and the persuasiveness of arguments and demonstrations are the clearest examples. The Lutheran theologian Wolfhart Pannenberg proposes four criteria to falsify (or verify) theological statements which are, according to him, specific to theology and its object (Pannenberg, 1973, p. 348). Theological statements should be considered unproven if: They are understood as hypotheses about the Judeo-Christian faith, but cannot be identified as implications of biblical traditions (even in the light of experience that made things other than they were before); If they do not have a relationship with reality as a whole, a relation which can be translated into the words of present experience and which is identifiable by its relations with the state of affairs in philosophy (in that case theological statements will be critically rejected as mythical, legendary or ideological); If they cannot be integrated into other areas of related experience, or if such integration is not even attempted (for example, a reflection on the Church and its relationship to society); If their ability for elucidation remains below the state of affairs already reached in theology; if they cannot interpret the assumptions already available, and if they do not overcome the limitations of those assumptions already highlighted in the discussion. The second criterion establishes a link between “present experience,” “reality as a whole” and philosophy: it is the possibility of a philosophical expression that provides the possibility of linking the theological statement to “reality as a whole.” This is not to say here that the theological proposition can be reduced to a philosophical proposition: it is only to provide it its first formal universality: the

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theological statement can be understood by other forms of discourse and can enter into discussion with them. The requirement for integration into “other areas,” the third criterion, expresses the need for an integration of theology into a debate with other forms of knowledge: because faith is based on an indirect self-revelation (through history and human experience) as one of its moments, theology has to reflect on this reality through which God reveals himself, a reality also studied by philosophy, the humanities and the physical sciences. If the statements of these various forms of knowledge can inform one another, if this or that historical analysis can, for example, better understand this or that sociological observation, then that requirement must also apply to theology. Theology does not, of course, develop knowledge about a specific area of reality: close in that sense to philosophy, theology makes hypotheses about reality, its meaning and logic – considered as a totality. There is no regular law in theology. We are dealing only with events. One of the questions we have to answer in the conclusion involves clarifying the notion of divine liberty that we use in theology. It is clear that if we are reasoning using only the concept of God’s absolute freedom, potentia absoluta, the only explanation theology may have is this one: why is it so? because God has willed it so. Is there any form of prediction in theology (or even in the social sciences)? The most difficult criterion is of course the ability to predict something. We should clarify this issue. And first of all introduce a remark. In many cases, in the natural sciences, prediction only rarely applies to events, but rather to classes of events. In the social sciences, an explanation has a predictive value if its hypotheses are clear enough to be controlled, and if they have a probability greater than other hypotheses. Even that is difficult to achieve for a sociologist or economist. If economists really knew how to get out of the current crisis, we would be out of it... We can at least consider this: the form that prediction can

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take in the social sciences is the relevance of a theory for new phenomena, or the fertility of a theory or system. This is the form that the requirement of prediction theology can take. It is not to predict what God will do tomorrow. Now economists themselves cannot predict how the stock market will evolve tomorrow. And sociologists too are unable to describe society in 10 years. But we can speak of an ability to predict when a theological theory developed for a question provides a better understanding of other issues. The same applies when a theory allows us to understand a new question or interpret a new phenomenon. As you might imagine, this hypothesis assumes that theology agrees to bind its speech to conceptual hypotheses, to philosophical and, perhaps, ontological hypotheses. To conclude I would like to say a few words about a distinction that is often made to ensure the scientific nature of the humanities’ dealings with the natural sciences: the distinction between explaining and understanding.

5. Explaining and understanding Wolfhart Pannenberg (1973, pp. 136–224) highlights a structural analogy between explaining and understanding: in both cases, singular elements are related to something more general in order to understand those individual elements by placing them in this more inclusive set, which more inclusive set can then be transformed in order to include all those singular elements. Is this a reason to consider this distinction between explaining and understanding as an expression of a myth of two cultures (which Pannenberg does not do)? A myth of two cultures would be a naturalistic, conceived distinction between the world of nature and the world of spirit, which would in turn expel the humanities from the field of veritable sciences. Even before making appeal to this distinction between explaining and understanding within an explanatory process, two approaches can be selected: a nomothetic approach or an idiographic approach (Windelband, 1904).

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The first tries to infer an average model, or a certain number of generic models, starting from a studied group, and in that case the fluctuations between individuals of the same model tend to be considered as random and insignificant. While the second is centred on the study of individuals whose function is to be characterized without seeking to generalize the results, in order, secondly, to seek what those individuals may have in common. Beyond a possible distinction between explanatory processes, the major difference is this: understanding incorporates the singular element into a whole. It reports immediately on the interpreter’s world, while explanation refers to a coherent description of the world: disturbance in explanation refers to an inconsistency in this world description, while disturbances in understanding refer to what may be false or one-sided in the interpreter’s relationship to his world. There are certainly structural similarities: as to properly judging what is given or related to a totality of meaning, but there is in understanding a dimension which is immediately prescriptive, about how the subject should understand his relationship to the world, whereas explanation ends in an exclusively descriptive dimension. Since it will enter the relevant data into a relationship to the world, understanding is also a decision and (in the moral sense of the word) should gather interests, values, and faults which interfere in judgment. Basically, explanation by a system of formal and quantifiable laws is a special case of the understanding process. Its intent is to develop explanations where understanding is defeated, in order to situate an element in the broader context of global understanding. And this particular case of understanding, explanation by a formal or quantifiable system of laws, is only possible by a reduction of the world to its formal and quantifiable aspects. Reducing all knowledge to this deductive nomological aspect would be quite inconsistent with human existence, where knowledge is not limited to that single aspect. It would basically remove the human from the field of knowledge. This is a tension between two rationalities, not between two areas of knowledge that could co-exist in being separated from one another.

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Any explanation can potentially refer to the understanding the individual has of himself: what is the result of this explanation for me or for humanity? Any explanation may become the theme of a self-understanding: how can human beings understand themselves through this explanation? And vice versa the focus is regularly placed on the reality of what is proposed to be understood: should these messages be ultimately interpreted or should they be explained from functions and processes which are not intentional? You can always suspect an expression of being a symptom, an epiphenomenon, a neurotic construction or whatever, rather than a sensible proposal. This requires switching from a first person commitment to the distancing of a third person observer, from understanding to observation and explanation. Suspicion arises from the hypothetical identification of a mechanism already known elsewhere. So these statements should not be heard in the language they use, but in another language, which reveals and unmasks that other one. Understanding them would amount to decoding them. They may also not be clearly distinguished, since sensible statements may have symptomatic or functional features, and vice versa. Interpretation must have both in sight. Hence potentially, in the words of Paul Ricœur, a conflict of interpretations occurs. Conflict may exist between an explanation driven by suspicion, which is both archaeological and reductionist, and a teleological and amplifying understanding. A conflict, or rather tension, which is a necessary moment of interpretation: “To explain more is to understand better,” as Ricœur (1985, p. 2) often said. An archaeological explanation can certainly show how a text, for example, is profoundly related to a given context, but that does not exclude that same text’s also enabling us to answer to some radical questions which are still vibrant and important today, in another time and another culture. Hermeneutic certainly measures the distance between the production and the context of interpretation, but also emphasizes the continuity, the common humanity between the writer and the reader, a world com-

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mon to both of them. It tries to take the measure, never complete, of the conversation, already started, in the midst of which we intervene, “and in which we try to orient ourselves in order to be in turn able to make our contribution there” (Ibidem, p. 48). It is only in a conflict of rival hermeneutics that we see something of the being interpreted: a unified ontology is as inaccessible to our method as a separate ontology (…) But this coherent figure of the being that we are, in which the rival interpretations will implant themselves, is given nowhere else than in these dialectical interpretations (Ricœur, 1969, pp. 23–27).

Thus the distinction between hermeneutical sciences and nomological and deductive sciences certainly allows us to clarify the question; but it is misleading where it is a question of distinguishing sciences. It rather involves a polarity of scientific processes. Globally, understanding takes precedence: it is when I don’t understand something that I seek to explain it. And when I explain why it is so, which hitherto I didn’t understand, this explanation will provide the framework of a new understanding. Thus on the origin of the world, or the evolution of life, these scientific explanations make the Christian understanding of the world and life, of their origin, of creation and providence, evolve. In its approach, theology finds itself faced with this great variety of scientific positions and methods. In this context, what is its properly theological position? Properly theological, whereas it takes part in psychological, sociological, anthropological and ethical approaches; and is affected by the sciences of nature, history, philology, etc, all disciplines where it can, moreover, itself give its opinion and express itself in those languages. But when it approaches those languages, it does not thereby cease having a theological aim. As a hermeneutics of the Christian biblical faith, theology does not just interpret: it must assert the truth of the faith it interprets in social, historical, scientific and other contexts. And it has to make itself understood.

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Its goal is never just internal; in order to understand itself internally, it cannot shrug these questions and problems off; were it only to identify what it calls the heart of the Christian message. By definition, theology is involved in the conflict of interpretations. It must take functionalist deconstructions of its discourses and attempts at delegitimization into account, and take positions with respect to them in a responsible and communicable way. And that is a theological journey, not a preliminary. It is of course animated by confidence, even assurance. In short by the faith that it is possible to respond to these difficulties and objections and affirm the Christian message as a possible and legitimate message today. That assurance should not be opposed to its scientificity. On the contrary, we should be wary of theologians who with the most advanced weapons of criticism end up by making the point that no criticism has the last word, which saves the Christian message, but which amounts to nothing more than a hidden form of immunization strategy. This assurance should not be opposed to its scientificity: it may, on the contrary, be an aid in not running away from objections, and a kind of equilibriating element, so as not to leave doubt more room than it deserves in a sane faillibilist and post-foundationalist epistemology. Because it deals with explanation and understanding, theology is unceasingly led to transgress its borders. It is particularly important that it ask itself what religious convictions bring to psychism, society, and even science and knowledge. For theology itself, it is certainly worthwhile measuring these effects since these religious convictions are intended to provide a fullness of life; this may be an argument of legitimization but in no way a principle or a basis: for Christians, only the word of God has the function of ultimately ensuring its legitimacy and founding the faith, and theology too. There is endless rebirth in theology, and moreover precisely in a university theology, of a tension between distantiation and identification: distantiation seems to be a minimal requirement of scientificity and, from that point of view, the science of religions seems more

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adapted to the criteria of scientificity than are theologians who know a priori from the start what they have to defend; but we still have to make sure that this distantiation provides them a better understanding of what they have to describe. A double distantiation, with regard to its object and with regard to its values, in itself no way forms a scientificity; reductionism is only more scientific upon the condition of better corresponding to what it analyzes, but it is precisely that which is in question; while every refusal of reductionism immediately situates one in a form of belonging to that which is in question; which is not in itself a criterion of non-scientificity either. The question is particularly clear where religion is concerned: approaching it as the human sciences do, exclusively in its anthropological and social dimension, enters directly into conflict with what religion, whichever one it might be, understands about itself: it claims to be determined by the divine, by God, by the divinities; it presents itself under the priority of the divine, whereas the sciences of religions only have a method of access to the human and are apt to take what is a response in man to God’s call for a simple human construction. While there are certainly common procedures of explanation shared by the human sciences and theology, the points of rupture should not be forgotten either. This involves the question of truth in particular. Unlike approaches which seek a complete distantiation, hermeneutical approaches to religious phenomena or texts revindicate a certain participation in what is said by those texts, or done in practices; an interest for the reception – or even theories – of taking a certain benevolent standpoint in order to reach an understanding of the text; thus in pondering whether these meaningful proposals testify to authentic modes of relationship to the world, real and fruitful possibilities of life, we will ask whether there is something authentic which may be important for my own understanding of myself and the world. Faced with these approaches, theology owes it to itself to ask the question of truth. These texts and practices should not only present relationships and ways of dealing with the various aspects of life and existence in the world: theology must ask itself whether they are

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indeed what they after all claim to be, which permits the absolute and the infinite to have a place in this world and in this history. More generally, what philosophy and theology have in common in their explanatory procedures and which distinguishes them from the other sciences, including what are called the “human sciences,” is not the formulation of hypotheses or the hypothetical dimension of the concepts they use to account for what they observe; it is not even the empirical verification (for one can quite easily object to a philosopher or a theologian in making him assert facts that his discourse does not account for). What distinguishes them from the human sciences, or from an aspect of the human sciences’ work, for they too also have a philosophical dimension, is an approach which does not consist in delimiting an object, but in taking it in its totality (which is not the same thing as an addition of singular cases). Philosophy, like theology, does not proceed by partial results and progress by stages: each and every time we are dealing with total revisions (Pannenberg, 1973, p. 221) of hypotheses on the total sense of human existence.

6. Before concluding Before assembling some general conclusions, let us take a concrete example: God! What does “explaining the Trinity” mean? If the preceding hypotheses are exact, we should consider that theology explains faith in the Trinity when it proposes a model which provides an account of the statements of Scripture. Not in the sense that it would transpose the language of the Bible into another language. But a theology will work out a model whose relationship to the Bible it tries to highlight. Proposing such a model is necessarily explaining the conceptuality employed (people, substance, nature, relation…). It is thus showing the coherence of the historical route which has resulted in employing these concepts, or else it is contesting that route and proposing other concepts. To take another example, a theologian might, for example, dispute the influence of the

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Platonic model of Eros on the definition of charity in Augustine, and on his Western successors.  Explaining in theology is exposing what you say and why you say it, and proposing a model which makes it possible to better understand what you say; it is also putting what you say into relation with other fields of theology, even with other disciplines: one thus implements a criterion of coherence and a criterion of elucidation. In theology, can one go so far as to claim to explain why it is so in the sense that Christian theology would claim to show, despite what it asserts about divine liberty? Why, for example, is God triune? As strange as it may appear, that questioning is not absent from a part of theology since it seeks to translate the language of faith into an ontological language – when it, for example, from Richard of Saint Victor to Richard Swinburne (1994) seeks to provide a priori reasons in favour of the divine trinity. But if we contest (and it seems to me that we should) these forms of reasoning, we are forced to admit that even in theology explaining is always shedding light on a form of necessity.

7. Conclusion Let us summarize our reflection. 1. As we have discussed it so far, explanation is fundamentally based on the construction of a model. The internal relations between model elements, or the relationships between different models, constitute scientific explanation. It happens in different ways for nature sciences, for social sciences, and for philosophy and theology. This has two consequences. First, one of the tasks of theology is to build hypothetical models and to relate those models to other models, philosophical, scientific or social models. Second, in this way, I don’t think that we have to accept the separation between natural sciences, which would explain, and social sci-

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ences, which would understand. Explaining and understanding are rather two moments of knowledge. 2. It seems to me that we can consider that there is an explanation in theology if the theological statement is open to falsification, in the form we have exposed. The first two points have a consequence: in order to build models, models that may be falsified, even by other sciences, theology must use a philosophical language – to allow conversation and translation between the sciences. 3. The interest in theology of an explanation for a special point is related to that model’s interest for elucidating other subjects. This is the usual form of prediction in social sciences, in philosophy, and in theology. Remark: this is a pragmatic argument.

References Armstrong, D. (1973). Belief, truth and knowledge. London: Cambridge University Press. BonJour, L. (2000). Toward a defense of empirical foundationalism. In M. Depaul (Ed.), Resurrecting old-fashioned foundationalism. Lanham, Mass.: Rowman and Littlefield. Descartes, R., (1901). Méditations touchant la première philosophie dans lesquelles l’existence de Dieu et ma distinction réelle entre l’âme et le corps de l’homme sont démontrées. Second meditation (trans. John Veitch) 1901. Goldman, A. (1979). What is justified belief? In G. Pappas (Ed.), Justification and knowledge (pp. 1–23). Dordrecht: Reidel. Goldman, A. (1986). Epistemology and cognition. Cambridge, Mass.: Harvard University Press. Hempel, C. G., & Oppenheim, P. (1948). Studies in the logic of explanation. Philosophy of Science, 15(2), 135–175.

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Pannenberg, W. (1973). Wissenschaftstheorie und Theologie. Frankfurt: Suhrkamp. Ricœur, P. (1969). Le conflit des interprétations. Paris: Seuil. Ricœur, P. (1985). Du texte à l’action. Paris: Seuil. Swinburne, R. (1994). The christian God. Oxford: Clarendon Press. Williamson, T., (2000). Knowledge and its limits. Oxford: Oxford University Press. Windelband, W. (19043). Geschichte und Naturwissenschaft. Strassburg: Heitz.

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