Logiciel: Six Seminars in Computational Reason [1 ed.] 9783000715914, 3000715916

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Logiciel

r Logiciel Six Seminars on Computational Reason

AA Cavia

ISBN 97S-3-00-0-l 591--t © &&& c/o I he New Centre t o r Research N Practice, 2022 All rights reserved. No reproduction, copv or transmission of this publication may be made without written permission. Published by The New Centre for Research • ,-ry ' expressing a different aspect of compu-

rupture—a traumatic event in human rationality, the cleaving

unon in tnrn. White Gadd i ge„m,

recu„f functions^

of reason from mind, from which a novel image of thought emerges. This will lead us inextricably to the second question,

limit a mathematical domain, and Church's lambda calculus a fotmal language Turing's modeI, by ^ ^^

which can be succinctly reformulated as: what are its reasons?

break from both mathematics and logic, a departure which would lay the foundations for a new science of

as a figure which presents itself as mere procedure, simply a

These models represent distinct traditions in the development of computational idea^-broadly speaking, mathematical

perhaps unwittingly, cast its shadow over a fundamental rift

The computational emerges in twentieth century thought mechanical mode of calculation—but it has ultimately, and in modern mathematics. The origin of this schism lies in the

5

A Minimal Program

I. OCjluM

intuitionist philosophy of l.L.J Brouwer, a doctrine which

treatment of interaction comes by way of a critique of inter­ action grammars, concurrency, and game semantics, expand­ ing on the inferentialist model with a distributed account of

places undccidabiliry at the heart of mathematics, a heresy en­ acting a radical Break from J lilhertian formalism. Developing

pragmatic reasoning, taking its cues from Brandom's critique of AI. Symmetries in logic, exemplified by the proof trees of

an inruirionistic view of computing leads to new positions on computation and contingency, realizability and truth, interaction and language. While the cognitive elements of Brouwer s theory will lie shed in favour of a generalized no­

Gentzen's natural deduction, are traced to Girard's dialogical work on geometries of interaction, while parallelisation is dis­

tion of inference, the so-called two acts of intuitionism will condition a view I will defend as computational. Here, I will attempt to develop a unified perspective on reasoning, based

cussed with respect to interactive computing. I close the book with an attempted integration of computational reason, re­ marking on the challenges posed by the spectre of generalized

on a computationalism grounded in the principle oj univalence (Voevodsky), advancing a structuralist conception of mathe­ matics. Following recent developments in computer science, I synthesize the univalence axiom with the manifold hypothesis in machine learning, endorsing a topological account of com­ putational reason grounded in two fundamental operationsencoding and embedding. A geometric theory of representa­ tion is presented as a bridge between the inductive revision of beliefs and formal deductive logic, so-called blind models and symbolic A I. I distinguish this model of computational reason from mathematical thought, discussing critiques of the atter from Lautman, Zalamea, Macbeth, Dutilh Novaes, and Longo. This topological model is characterized as a form of computational mferentialism, and I situate it within recent debates in computational theory of mind, namely representa-

intelligence, and a discussion of the contemporary relation be­

t'^ fPh on ^su m 3S ^odor ^ Pytyshyn), neurophilosophy (Churchland), and predictive coding (Clarlc, Metlger, et al). The semantics of this topological theory is discussed with respect to various computationalist models, and its neu rogeometnc disposition is rendered in relation to the work of cognitive scientists Jean Petitot and Peter Gardenfors A

tween computation and philosophy. This short book calls for the reappraisal of two dogmas which are central to computationalist positions in epistemology and ontology. The first is the identification of computa­ tional with mental kinds, asserting the computational nature of minds, a belief which has both motivated and hamstrung the project of Artificial General Intelligence (AGI), tethering it to parochial notions of human reasoning, whilst simultane­ ously constraining models of human cognition to computa­ tional ideas. Commencing with the functionalism of Fodor and Putnam in the 1970s, and the ensuing discourse on mul­ tiple realizability, abroad consensus has emerged which shares the common view that computational explanations obtain in representational theories of mind. This in turn conditions outlooks on the nature of computation, best summarized by Fodor's remark that there is "no computation without repre­ sentation".1 From this a number of computational theories

1

Fodor, j. A., 1981. Representations: Philosophical Essays on the Foundations of Cognitive

Science. Brighton: Harvester Press, pp. 225-257.

7

A Minimal Program

I OtJK

lu\c emerged (e.g. I l.irm.in. block. (-halnicrs).' In these the­

and a related conflation of digitality with logical bivalence.

ories, computation is invariably invested with semantic con­

This finds its contemporary expression most explicitly in

tent, often expressed through .111 appeal to functional, con­ ceptual or inferential role. Alternative svn tactic accounts have

digital philosophy, which we can trace through figures like Bostrom, Chaitin, and Wolfram, to the earlier work of Zuse,

been presentee) by (.honisky. Stich, and others, as a rebuttal to

and further to the mathesis universalis of Leibniz. While cri­

such theories, and these largely leave the question of seman­

tiques of such forms of computational universalism are plen­

tics outside the bounds of computation, whilst still preserv­

tiful (e.g. Piccinini), digitality is a deeper assumption taken as a given by media theorists engaging in those very critiques

ing a role for computational explanation in cognition. A more

(e.g. Galloway). Here we can take Deutsch's remark that "the universe computes" as paradigmatic of a view in which

forceful rejection of this form of computationalism is whatl call luring orthodoxy, the conflation of the computational self, which finds its contemporary philosophical expression in

physics is describable as automata—in which the physical microstates of matter accord to computational states.3 This is

the work of Piccinini. I will argue that computation is under-

mediated by the lawlike Landauer principle, enshrining the

determined in Piccinini's account, failing to outline the req­

coupling of the computational and the physical via an appeal to a thermodynamic analogy at the heart of information the­ ory. Indeed, a binary encoding of form into 'bits' is posited

with the machinic, a dominant view in computer science it­

uisite degrees of freedom needed for a theory of computation w ich both constrains its ontological footprint and allows fihful expression of its intrinsic epistemic affordances.

as a fundamental property of information theory, and disen­ tangling logical bivalence from notions of computation will

n the process, I will trace the current debate on semantics tn computing discussing the work ofbgan, Ladyman, and

be another aim of these notes. I will approach this question, not from the dual of the digital and the analogue, but rather the dyad of computation and the real. I will try to show how

agnr. On this point my text will follow developments in

the hngu,stic tradition offered by Church i model of comput­ ing, namely the type theoretic version of the lambda calculus, tugtnng for „s irreducMiry ro Turing's model. The ,1m will

a constructive view of computing, following from the logic of Brouwer and the algebra of Heyting,can loosen the grip of the digital on computation, leveraging univalent foundations in mathematics to develop a view on the continuous and the dis­

be ,0 develop a d.stmer semantics that can clarify the relation of computation to reason, without adhering to either Turing otthodoxyotafullybloivn comp„t„,ona, tLry„fmilK( 8 e second dogma I w,u 0„dine Js c„ . , ,sm, namely the ontoiogiea, of the

crete which extends beyond the presuppositions of Boolean logic. In contrast to the dominant model-theoretic approach to logic, a topological treatment of types will be explored, as

2

Block, N„ 1986. Advertisement for a Semantics for Psycholoov he w Philosophy, 10, pp. 615-678. hology. Midwest Studies

3

Deutsch, D.E., Barenco, A. and Ekert, A., 1995. Universality in Quantum Computation. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 449 (1937), pp. 669-677.

9

L ogiue

A Minimal Program

a nniIti-\ allied regime characterised by a proof-theoretic se­

Any account of computational reason needs to acknowl­

mantics. In the course of this argument, I will examine ac­

edge the deep history of computational ideas in human

counts of the continuum and the relation of computation to

thought. Retrospectively, we can identify and assimilate

contingency, which ensues from an intuitionistic treatment

historical notions such as algorithm, automata, induction,

of the real numbers. Whilst it might suffice to point to quan­

recursion, and undecidability, into a broader history of com­

tum and analog computing to demonstrate the generality of

putation. It is not the aim of these notes to fully elaborate

computation above and beyond the digital, only through an

this history, and it will suffice to allude to certain aspects to

alternative logical foundation can this be developed fullyinto its own generative model. -

give the reader a flavour of both the diversity of conceptual tools and depth of historical development required to arrive

Novel interpretations become possible once computation

at a contemporary treatment of computation as its own lo­

is freed from these two central dogmas. One consequence,

gos. Taking undecidability as an example, it is perhaps first

traced through the chapters that follow, is a notion of com­

explicitly acknowledged as a formal issue by the Buddhist

putation as a distinct mode of reasoning, marking out its own

logicians around 100-300 AD—Dignaga, Dharmakirti, and

episteme

in Aristotelian terms, computation distinguishes

Nagarjuna, amongst others—leading to formulations such

elf from tekhne as a distinct logos, a world whose logic struc-

as the fourfold negation known as catuskotil This scheme

the conditions for the possibility of encoding reason,

allows for the possibility of a multi-valued logic which ad­

c , this book outlines a model of computation ground­ ed not tn static axiomatics-which I will trace to Hilbertian

mits dialethic statements as superpositions of multiple truth

nottons of formalism-but rather dynamic inferential acts,

while also allowing for the converse, statements which may

wh,ch follow from the constructive view. Proceeding from th. interpretation of log,c, thls modd *„ bt elaborated as a means of resolving , mnibcr ,b wrnUvm """"f m'"m- ^e imp,ica,ions of ,hi worldview, grounded rn rhe type theory of Mattin-Ldf will be explored terms of their accompanying and epistemic

values, a violation of the Law of Non-Contradiction (LNC), be neither true nor false, a clear violation of the Law of the Excluded Middle (LEM). These make up two of the three fundamental laws of western logic, acknowledged since Aristotle as immutable, and their rejection in any form would constitute a major threat to modern mathematics. Indeed,

commitments, and on this topic I will cover rh • • physics of Gisin as well as the computational • C mtUm°nlSt

Nagarjuna's treatment of the catuskoti, which he calls "the

Harper. A form of informational realism will Imer^ZZ"

to be referred to as the excluded middle in the west, finding

tentative to universale, a view which both admits the reali v of information as a matter of fact in the world, and commits to an irreducible contingency of the real neCessar,'>'

its recent reassimilation only via the intuitionism of Brouwer

middle way" in thought, is an early expression of what came

p"est.

10

G.. 2010. The Logic of the Catuskoti. Comparative Philosophy, 1(2), pp. 51-51.

Logiciel

and Hey ting and modern paraconsistent logics. We can now view these challenges to western rationality, evident also in ancient Chinese philosophy, as early expressions of the con­ cept of the undecidable in logic, a term which would only come into focus as a properly computational phenomena fol­ lowing Hilbert's formulation of the entscheidungsproblem in 1928. Similarly complex genealogies underlie many concepts fundamental to an account of computational reason—math­ ematical induction was deployed by Euclid in his proof of the infinity of the primes, circa 300BC, and further refined by Persian mathematicians such as Al-Karajl in the tenth centu­ ry AD. The indications are that we should not be blinkered into assuming we are dealing with a twentieth century de­ velopment the digital computer as a distinct technical arti­ fact—but rather a logical structure integral to the long arc of reason itself. Indeed, I will mention almost nothing of those computational devices, infrastructures, and technologies which seem to pervade our environment and infiltrate our p enomenological experience, focusing instead on the conceptual underpinnings of computation—for it is the nature ° t is concept which requires clarification at this point in time, lfcomputationalism is to bring its epistemology, seman­ tics, and ontology into a new phase.

A Minimal Program

mathematics, philosophy of mind, and not least my "home discipline", computer science. While I have attempted expo­ sition of technical concepts where necessary, invariably trade­ offs have to be made in the name of expediency when dealing with such diverse material. In the spirit of the seminars which bore this material, I have attempted to craft a text which is ap­ proachable by readers coming from a range of backgrounds, without alienating either experts or novices in the aforemen­ tioned disciplines—as such, no real technical background is assumed on the part of the reader, aside from an interest in contemporary philosophy. No doubt some simplifications will irk those with deeper knowledge in any given domain. All I ask is that you trust the discursive value of the overarch­ ing argument and the balance I have tried to strike between rigorous analysis and general accessibility—a configuration which is admittedly heterodox by academic standards, but which underpins the open and informal nature of this text.

Lastly, a note on methodology and style-this is a work of synt etic philosophy, blending references from the analytic and continental traditions, an open-ended mode of enquiry ic ana ogies, echoes, and resonances sit alongside isomorp isms, 1 entities, and formal correspondences, in which stances, doctrines, proofs, dogmas, worldviews, and speculaions ree y ming e. It is a multi-disciplinary text, navigating umber of discourses, including cognitive science, logic, 12 13

ONE

Computation & The Real

"To the question whether we need intuition for the solution to mathematical problems it must be an­ swered that language itself here supplies the nec­ essary intuition... the process of calculation brings about just this intuition." 1 - Ludwig Wittgenstein

Computation &The Real

1.1 The Excluded Middle For as long as axiomatic systems have existed, they have been plagued by the figure of the undecidable. Undecidability first expressed itself in the history of western thought geometri­ cally, via Euclid's fifth postulate in the Elements—a seeming­ ly innocuous observation that parallel lines on a plane will never intersect, a valid but nonetheless unprovable proposi­ tion within Euclid's axioms. An unresolvable state of affairs brought about by infinite quantification, statements like the parallel postulate have afflicted the consistency of formal sys­ tems ever since, but only in the twentieth century has western mathematics faced up to this issue at the heart of its disci­ pline. I aim here to outline the relation between intuitionism and computation, by taking such forms of undecidability as a thread with which to bind these two developments in reason­ ing. Our concern here is an overview of the historical devel­ opments that lead to the emergence of an alternative model of computation, founded on such attempts to confront the lim­ its of determinacy, an intuitionistic view of computing with its own distinct semantics, metaphysics, and ontology. Since

17

Computation & The Real

Logiciel

computation can be deemed a historical attempt to develop;

Intuitionism can be summarised in two acts. In the first,

formal account of contingency, and intuitionism represent

Bnniwcr outlines a conception of mathematics rooted in

the first serious western effort to accommodate such uncer

intuition,

tainty within logic, the influence of the latter on the genesis of computational ideas cannot be overstated.

"[as] a languageless activity of the mind having its origin

In 1912, LEJ Brouwer—an established topologist at

in the perception of a move of time. This perception of

the time—announced a new perspective on mathematical

.1 move of time may be described as the falling apart of a

thought which was to spark a major controversy. In a public

lite moment into two distinct things, one of which gives

lecture, Brouwer outlined the central tenets of what he called

wav to the other, but is retained by memory. If the twoity

intuitionism, a doctrine shaping his views on the nature of

thus born is divested of all quality, it passes into the empty

mathematics, asserting its claim to the status of a foundational

form of the common substratum of all twoities. And it is

system.2 He brazenly attacked the presuppositions underlying

this common substratum, this empty form, which is the

t e greatest open problem in set theory, known as the continu­

basic intuition of mathematics."3

um ypothesis, dismissing the entire conjecture as nonsensical upon its reliance on non-denumerable infinities, such as the cardinality of che real numbers, X, Indeed, Cantor's entire wort „„ transfinirude was

be rejected as a non-,,,,,1™,

W3S CaSt °n C°re tenets of what Hilbert called0"^' an > called Cantors paradise", namely the set theoretic found,

matiest.n b™""' Of me hod

Br°UWer'S

*ffirmati«

See° " *

S"'a'r d«"™blc

mously that of ninete.mh

e

J

'

gebra. I his edifice is rooted in an informal notion, an appeal to the faculty of intuition, which issues a challenge to a priori

comprised

conceptions of mathematical form, instead irrevocably cou­

objects sud]

pling itself to an inductive process, precipitated as it is by the

"

Kroneelcer, who had exclaimed"n" Hebe Gottvemacht all«

tol lowed by another—serving as the basis for the natural numIx rs. proceeding to encompass the whole of arithmetic and al­

that ma.lie

::b:rln this-b-«* Z tlC7T

1 nc nrst act ot intuitionism highlights the temporal nature of lope via a concept Brouwer calls "twoity"—simply one thing

^T," ganzen ZMen

perception of a move of time". In Kantian terms, intuitionism ^"doctrine can be taken to allude to both pure and sensible !ntuitl0n at once> engaged

hat der

in the genesis and transformation of

l»th concepts and percepts. For Kant, intuition is to be distin­

guished from sensation, just as it is to be treated as distinct from the conceptual, effectuating singular as opposed to general

Brouwer, L.E.J. 1913 nq7Ri i • . otMafftema,,cs.North.Ho||andnpU™^^3dnd formalism. In:

18

Philosophy and Foundavor*

Logiciel

Computation &The Real

representations. Intuition in um ^cnsc is not Simpy dwau!

Kant cites geometry as a domain of mathematical enti­

representing particular objects given to sensibility, which aim

ties with no empirical basis, for which not being grounded

be subsumed by general concepts, mirrored as it is by a real

in experience, they cannot, in a priori intuition, exhibit any

of pure intuitions, such as space and time, which are a prio

object such as might, prior to all experience, serve as ground

in matters of reasoning. In the dual cognitive system propose

for their synthesis."8Mathematics appears to have no objective

by Kant, the role of intuition spans both understanding an

grounding in the Kantian account, a stance which seemingly

sensibility; it is not a neatly delineated or modular faculty bii

endorses a claim regarding the irreducibility of the intelligible

rather an intermediary mode of representation, one which w

to the empirical. But as Longuenesse argues, Kant's position

could characterize, in modern terms, as "sub-symbolic". A

on this point is complex, and a convincing case can be made

Longuenesse points out, for Kant mathematical objects con

for the primacy of judgement, acting on a manifold of intu­

orm to the realm of the intelligible, they originate in pun

ition, over the prior application of pure categories, even in the

uition, but unlike the objects of metaphysics they do no

context of mathematical structures.9 By contrast, a more tran­

I rl ,tate. Cranscenc'ent:a^ categories.5 In Kant's transcend® uction, Longuenesse observes the "contrast between

scendental Kantian view is to be found in the Sellarsian no­

mathematical concepts, which do not require a deduction fo,

temically irreducible" representation imbued with categorical

eir a prion validity, and the pure concepts of the understand

form, providing predicative content to empirical judgements,

ch

'„W

°Se

Brouwen

and implicitly structuring our observational reports.10 Here,

^ Kan"an conception of mathematics as a realm priori judgements following from pure incelli-

no clear decomposition of intuition and concepts is forthcom­

°bjective

of svnrlT

tion of "conceptual intuition", which for Brassier is an "epis-

validity is difficult to

movementP7mg

a praCtice induced

movement.'Vhi116' * creativity itself k ^

justify."6

of form

by the dynamic

which is integral to that

eS'S' Whlth 'S tbe aCt of mathematical

constitutes inf^Mfe!?*"17 ">nCepCUal in nature'insofarasit e the second f • °wed with epistemic content,as C

S

e second act of mcuitionism will make explicit.

ing, and intuition is anchored by what Sellars calls "correct picturing", a form of structural realism conditionedby spatiotemporal priors,mediated via non-conceptual representings.11 These two perspectives serve to highlight the hermeneutic labyrinth offered by the slippery notion of intuition in Kant, emphasizing the stakes at play, namely a cognitive account of the genesis of form. Both views offer select points of alignment with Brouwer's doctrine, the former interpretation, centered

Engstrom, S., 2006 Unri £~,e. B„ 2020. C;^rnCaSenSibi,i,y- ln^^ PP-2-«. JUdge: Sensibility and Disou^^ ^nscenclental Analytlc of the CrittleTp'^

Immanuel, K„ 2018. Critique of Pure Reason. Charles River Editors. (A87-88/B 120) Longuenesse, B., Kant and the Capacity to Judge, pp, 199-209

The tens,on between

intuition.

PnnCet°n Un'VerSI,V PreSS'

a computational treatment of modaUonjc™ m

S8ynthe,lc

chapter 5.

^

a priori will be explored in terms of

Brassier, R„ 2016. Transcendental Logic and True Representings. Glass Bead Journal.

ibid.

21

Computation & The Real

Logiciel

close compatibility with i

that the vague nature of Brouwer's pronouncement encour­

constructive notion of intuition, whereas the Sellarsian theory

aged the portrayal of intuitionism as an eccentric disavowal

of picturing can bridge an intuitive account of structurewith the real.

of classical mathematics, prone to offending the sensibility of

on immanent judgements, finds

For Brouwer, the intuitive basis of mathematics is ground­

many a mathematician along the way, unable to satisfy the demand for precision which is the norm in the discipline.

ed in a "common substratum", which can be interpreted as i

The second act sheds light on the intuitionist treatment

gesture to the continuum, a murky "empty form" recalling

of the continuum, insisting on the constructive nature of

Anaximander s apeiron, the boundless or without limit. That

mathematical objects, intuitionism admitting two ways of

which is not given to mathematical reason can have no logical

constructing said objects:

basis, but it nevertheless acts as its "basic intuition", a notion I will seek to clarify as we proceed. The naive sounding refer­

"firstly in the shape of more or less freely proceeding in­

ence to a "languageless activity" can be interpreted in a vari­

finite sequences of mathematical entities previously ac­

ety of ways, and I offer a Peircean view in which abductiveacts

quired... secondly in the shape of mathematical species, i.e.

play a role in proof networks in a later chapter. But another in­

properties supposable for mathematical entities previously

terpretation arises from the sub-symbolic nature ofgeometry,

acquired, satisfying the condition that if they hold for a cer­

which is implicit in the topological model of computational

tain mathematical entity, they also hold for all mathemati­

reason I will go on to endorse. In this view, non-conceptual

cal entities which have been defined to be "equal" to it..."13

(geometric) representings provide the basic intuition for the generation of (topological) structures which fall under con­

Mathematics is framed as a means of establishing identities

cepts in the form of types. But as we shall see, the decoupling

between objects produced via denumerable operations, or

o geometry and structure will be complicated by a close in­

in turn those between species (i.e. types) of objects. This

flection of contemporary mathematics. Brouwer's appeal to

marks out intuitionism as a form of constructivism, and

inguistic nature of intuition can also be conceived as

would in turn provide a treatment of the continuum as an

pt at staking out mathematics as a formally autono­ mous pract.ce, distinct from the natural language of speech acts, o re v e n f o r m a lc l a i • , , gIC~an InterPr«ation which accords with the proof th pr°°f~chcorct'c semantic tradition thatintuitionkm w I ltionism would put into motion Tn - - r • ion. in any case, it is fair to say

infinite sequence of free choices, rejecting the givenness of

Goldblatt, R.. 2014. Chapter 8: The Logic of In, - Analysis of Log,c. Elsevier °f

a pre-existing unity or totality in favour of a generative ac­ count of the real numbers. In this regard, Brouwer's doc­ trine resembles one of Simondon's three modes of intuition, namely the philosophical variety, which is "neither sensible

In: Topoi: The Categorial B'ouwer, LEJBrouwer's Cambridge lectures on intuitionism, p. 8.

23

Computation & I he Heai Logiciel

irments led to a major break with the

nor intellectual",14 but rather a systemic means of formation

Brouwer's two comm

or genesis, distinct from both concept and idea, and un­

existing consensus at t: e

Hilbert>s

>

program of formal-

^logidsm 0f frege,

moored from the unity of the real:

dominant school ofmathe-

"Intuition is not merely the grasping of figural realities, like the concept, nor a reference to the totality of the

Sllingwirh regards ,o foundadons. Bmuwe, summarized this eminently philosophical rupture as such,

ground of the real in its unity, like the idea; it aims at the real insofar as it forms systems in which a genesis occurs; it

"The question [of] where mathematical exactness does ex­

is the knowledge proper to genetic processes."16

ist, is answered differently by the two sides; the intuitionist says: in the human intellect, the formalist says: on paper."17

While Simondon s motivation is an account of the evolution of technicity, it shares with Brouwer's portrayal of mathemati­ cal reason a systematic role for intuition in the genesis ofform, a genesis which eschews the unity or totality of the real. As yman has remarked, modern logical frameworks descend­ ed from Brouwer often dispense with the first act, choosing to follow the constructive path whilst remaining non-committal on its intuitive

philosophical basis.111 In

a sense, these two acts

ecoupled in the mind of the working mathematician, U a philosophical position would need to develop an integ ated view in order to call itself intuitionistic, as opposed to :;^Tctr-1 wil1 defend such an intuiti°nisdc view temPoral substratum which yields twoity is foi E /SIS f 11 elib°",inS "» of computation! will defi. H C "" generates a dynamic notion of inference I will defend as integral to computational reason.

Simondon, G., 2017 On the M H Univocal Publishing.' p, 254 ° "

Existence

Ibid.

Ladyman, J, and Presnell, s. 2014 Formal Type Theory, p. 20.

24

Technical Objects. Minneapolis:

For formalism, mathematics is an analytical process of deduc­ tion performed on a set of axioms which are to be treated as logical givens, with the emphasis on the internal consisten­ cy of a symbolic structure governed by unambiguous rules. axiom

^iv"uexPressivity

a

system, a formalist must add

15 ™' mutable gloLlawrThUPPle!llent ^*Xistia& m°ns a very different outlook on creativity to the int -

extensibility will rest .Ultl°mstic

m°del

rul«

^^ormation

^ °n

which yield new t

'essness of the symbols i\P n0t'ng nothing in

thernsel

:;lytic;lpracdce.This

^

°f inferential

^orrnalist, the meaning-

^^ t^e'r 'ac'c °f referent, de-

as;:'

bUt.inSteadembod

yingaputely

T,,*i

Zuh""' ClMsic'1 Icgic i'nh ™°Uld be Pro,en unten-

rooted

in it, r

ide",lf>'

ntrast,

« n""'"8ase"'"'ticswhlch

woulH

Inguistic

in nature.

own semantics

A Pnmer

I will endorse, whose

on Homotopy Type Theory Part 1: The

' ^r0uwer [_ p , 0r^al

'sm.

0, igiDating

in a challenge

Computation & The Real

Logiciel

aimed at the heart of logical foundations. The three found­

objects, instead regarding proof construction an ontologi-

ing axioms of western logic under formalism, immutable laws

cally ampliative exercise which brings a truth into being. In

which have held since Aristotle, are the following:

other words, truth is only ever to be treated as a side-effect of the demonstration of a proof, leading Dummett to remark

1. Law of Identity: A := A

that intuitionism implores the mathematician to replace the

2. Law of Non-Contradiction: -i(AA-iA)

notion of truth, as the central notion of the theory of mean­

3. Law of the Excluded Middle: AV-iA

ing for mathematical statements, by the notion of proof".20 In this constructive view, computation is vital in providing a se­

Intuitionism would come to reject the third of these—the law

mantics for mathematical statements—as Dummett argues,

of the excluded middle (LEM)—as a general axiom, on con­

arithmetic expressions like 2+2=4 are meaningless without

structive grounds. The LEM simply poses that a statement

summoning an algorithm that establishes their identity.2'

must be either true or false, there is no middle ground in logic,

This has implications for the law of the excluded middle, giv­

a fundamental law which Brouwer attacks via a strictly phil­

en that if no proof currently exists for an arbitrary assertion

osophical argument. For constructivists, the evidence of the

A, then no guarantee can be made in advance regarding its

truth or falsity of a mathematical statement is only laid bare

decidability, and we will see in due course how various devel­

upon the construction of a proof, a process which Heyting

opments in logic come to evolve this idea into a fully-fledged

likened to an algorithm, in a symposium on epistemology

proof-theoretic semantics. Brouwer would turn this split on

held in Konisberg in 1930.18 At this renowned conference,

the LEM into an attack on formalism as a foundation for

eyting defended the foundational prospects of intuition-

mathematics:

^ m, while Carnap defended logicism, and Von Neumann put rWjlj^ C^C

^orma^st case. Proceedings were rounded off by endum in the form of Godel's presentation of his first

a

etCneSS t^eorem-

f

"The long belief in the principle of the excluded third in mathematics is considered by Intuitionism as a phenome­

It is this specific historical moment

non of history of civilisation of the same kind as the old-

°r ^autman> marks the transition from the "naive" to

. time belief in the rationality of pi... intuitionism tries to

P^ase ^Mathematical thought.19 Heyting pot e intuitionist stance against the classical Platonist

explain the long persistence of this dogma by two facts:

insistence on the independent existence of mathematical

an arbitrary single assertion; secondly the practical validity

T , one

Heyting A

firstly the obvious non-contradictority of the principle for

1930 The I

Putnam, H. (eds.), 1984 University Press

Philoso t,^°Un^a1:'ons Mathematics. In: Benacerraf, P. and P Y o Mathematics: Selected Readings. Cambridge

A., 2011. Mathers,

ldeas ^

^

phys/cgi

^

a&c

p

Mi

Dummett, M„ 1975, The Philosophical Basis of Intuitionistic Logic. In: Studies in Logic and the Foundations of Mathematics, Vol. 80. Elsevier, pp. 5-40.

26 27

T

Logiciel

Computation & The Real

of the whole of classical logic for an extensive group of sin

I - generated via a topological model of computation in due

pie everyday phenomena."22

course. The computational nature of undecidability would

k

become apparent following Hilbert's framing of the entschei-

While Brouwer admits that such a law can trivially hold to:

dt.^problem in 1929. Whilst Church and Turing would for­

single assertions, given the demonstration or refutation of

mulate definitive solutions to this challenge, demonstrating

proof, it cannot hold as a general axiom. This law had seemed

computation as a powerful means of modelling contingency,

so fundamental to the practice of mathematics up until tin-

it is Brouwer's logic which emerges as the key implication of

point, that Hilbert famously replied that "taking the prin­

the undecidability of the halting problem, rendering the un-

ciple of excluded middle from the mathematician would k

deeidable a symptom of a deeper indeterminacy of the real, a

the same, say as proscribing the telescope to the astronome

symptom exposed by computational acts in the form of deci­

or to the boxer the use of his fists."23

sion procedures.

It is precisely thisev

cluded middle which Brouwer sought to fold back in to th; mathematical domain, on account of a fundamentally tern poral characterisation of mathematics, elicited by twonessas; primary intuition yielded by a "common substratum", a form which we will come to equate with the spectre of time itself. The rejection of the LEM places undecidability at the eart of mathematics, insisting that mathematicians drop atonist notions regarding the 'discovery' of proofs, emphang instead a dynamic and resource sensitive search spaceot p

1.2 Three Canonical Models

tial proof construction. A search space which is notsim

I.ct us examine the development of computation in the shad­

ply conditioned by a priori forms, but is instead constrained

ow of intuitionist ideas with a semantic analysis of canonical

y those topologies which mathematical creativity is able to realize as novel structures in the world, and we will come to

models. The three canonical models of computation devel­

ese very spaces for the construction of proofs can

oped in the 1930s are Godel's general recursive functions, f hutch's lambda calculus, and Turing Machines. I will ar­ gue that each of these represent a different face or aspect of computation, and that the commonly accepted equivalence

Church, A., 1949. LEJ Brouwer C Proceedings of the Tenth I n t e m a t - n ' 1 3 ? ^ " 3 " 6 3 5 '

PhllosoPhy.

and Mathematics.

9 r e s s o f P h i l o s o p h y ( A m s t e r d a m , 1948), North-Holland Publishing Comnan ^ P a n y ' p p - 1 2 3 5 - 1 2 4 9 . T h e J o u r n a l o f S y m b o l i c Logic, P t PP- 1 3 2 - 1 3 3 .

Ewald, W . and S i e g , W „ 2013 D a w w and Logic 1917-1933 S p r i n g e r B e r l i n FcuicHi

Lectures on !he

Foundations ofArithme::

"f these models belies a more complex relation. The aim is to mike the ampliative nature of a type theoretic account of somputing based on Church's model apparent as we progtew

28

marking it out as irreducible to a Turing Machine (TM).

Computation & The Real

Logiciel

Let us distinguish the approach taken by eachmodel in turr

Primitives:

While Godel attempts to give an account of computabili:.

- Constantf:z(n) = 0

in terms of a mathematical domain of functions,

Church:

concerned with providing a calculus with which to represen: computational operations, while Turing in turn focuses or

- Successorf: S(n) = n+1 - Projectionf:pfk^kj = k.

the description of an automaton that can realize compu­ tations. We can broadly align these models with distinc: mathematical (Godel), linguistic (Church), and mechanistic (Turing) traditions in computational theory. Moreover, the; allow us to see computation as a historical attempt to mod el contingency, as a project to address equivalent notions o: incompleteness (Godel), inconsistency (Church), and undo-

Operators: - Composition:/0 £ - Recursion: 9(h): h(x,y+l) =g(x,y, b(x,y)) - Minimization: p.

cidability (Turing) in a formal manner. What these modelshare in common is not an identical definition of compuu-

Recursion is singled out as a key property and formalized

n, but rather a diagnosis of contingency in formal systems,

as an operation in which a function iteratively calls itself.

which we can view through the lens of intuidonism and it.

An unbounded search operator (p) can be introduced to

rejection of the LEM.It is this aspect of computation

extend the primitive recursive functions in order to han­

which

inherits and further extends intuitionistic ideas developed in years eading up to this flurry

of activity. Notably, each

model makes semantic assumptions regarding the nature ot computatKin, and it will be my focus in this section to idem ^these presuppositions. tunic; C

natura 1 n

^

S mOC*e''

Partial recursive functions take finite

Se^USnces^ ob natural

dle second-order recursion—functions which not only call themselves but which supply themselves as arguments, the so-called Ackermann function schema.24 The purpose of p is t» seek the minimum input that produces an output of zero, and to use that as an origin for an unbounded search, mak­ ing explicit the halting characteristics which Turing would

numbers and return a single

en on to formalize for TMs. The recursivity of computation­

function T 1?'' SUking °Ut 3 sub~sPace of all mathematical C°nSidered comP^a«onal, the definition o. th^orm ° this formal system outlined below:

al 'unctions is offered here as a founding property, and the runaway dynamics of recursive processes are not tamed, but ™cr accepted as intrinsic, embodied in the form of the un­ funded search algorithm, p. Godel would go on tolink this

30

c

* _< Introduction to

^ j JN, hj Q (jg Groot, J. and Zaansn A IQKO I nen' A-C" 1952Vol 483 NpwvJu New York. Van Nostrand. pp. 262-308.

''"""r :$

31

Computation & The Real

Logiciel

unbounded search operator to his first incompleteness theo­

set theory was extended with types in order to avoid tne ia-

rem in due course. The semantics of Godel's system are en­

mous Russell paradox at the center of the foundational proj­

tirely inherited from mathematics, that is to say, the domain

ect of the Principia Mathematica,27 These antinomies can be

and range of such functions are assumed to denote their inpu:

construed as paradoxes ofself-reference, and the types which

and output, as this denotational semantics was the accepted

extend Church's calculus into the simply typed lambda cal­

approach in the discipline at the time.

culus (STLC) can be considered a defense against infinite

By contrast, Church's lambda calculus is so minimali; can be summarized in just three expressions:

recursive regress arising from unconstrained function appli­ cation. This marks out the computational as the family of functions operating on the natural numbers, distinguished

L, M, N : : = x

terms

| (~kx. N)

abstraction

| (L M)

by the function type, N—>N. The STLC would become the basis for Martin-Lof's intuitionistic theory of types (ITT)

application and composition

which forms the foundations for the account of computa­ tion developed in Chapter 4, and this would go on to supply a more creative definition of types beyond a mere restriction

Here the pipe symbols (|) denote expression substitution a;

on a function or a set, introducing its own operational se­

is conventional in Backus-Naus Normal Form (BNF). So

mantics distinct from the denotational tradition.

called terms (L, M, N) can be assigned to variables (x), these

Lastly, the Turing Machine (TM) is a model of an autom­

ound to anonymous functions known as lambdas (a)

aton capable of computing any effectively computable func­

witch can then be applied in chains (composition).® This

tion a circular definition enshrined in a property known

^%^er"order functional calculus for computation­ al operations, one which has been highly influential in!

as turing computability.28 Turing's emphasis on the physical

CS a

evelopment of programming languages, such as Haski F r r in . , ° °&' yP g °f terms is required to to v ns'stency

K"l

,

pointed out by two students of Chu

osser> an issue which is related to the Rich . ox in ormal languages, and constitutes a form ofd

-gttment (Cantor).* This echoes the mannerinwl

H.ndley, J.R.,

1997. Bas,c

'bid. pp. 12-27.

32

^^

^ ^^

^

realizability of computations propelled the founding of com­ puter science as an empirical discipline in its own right, intro­ ducing key concepts such as determinacy, boundedness, and locality.® The model is strikingly simple, an infinite input/ output tape containing cells of symbols are read by a 'head' which can in turn overwrite symbols on the tape based on

"A u'miZl

LT °f Th°USht 0xtod

University Press, p. ,28

i-., ivioiwam, h. and Ullman, J.D. 2013 Antnmato Th* , IThird Edition], Pearson, pp. 3,5-377.' ^

Computation & The Real

Logiciel

transformations enacted bv a finite state automaton (FSA).

intensionality—should be introduced. The origin of modern

Notably, this architecture lacks much of what we associate

semantic theory is in Frege's Sinn (sense) und Bedeutung (ref­

with modern computers—such as stored programs, mem­

erence), and this is a common starting point from which to in­

ory, or even a CPU—since it would take decades for Von

terrogate the symbolic. Frege's innovation was to distinguish

Neumann, Zuse, and many others to develop these ideas, Of

the sign from its denotation, allowing for an account of the

the three canonical models, the TM in particular suffers from

internal structure of symbols and their relations, advancing a

a formalist bias, which makes little attempt to specify what

new denotational semantics for formal languages. Frege was

kind of transformations are distinctly computational,beyond

working in parallel toPeirce, whose semiotics located symbols within a trichotomy of the sign, alongside icons and indices.

an appeal to operations which "include all those which are used in the computation ofa

number".30

For Peirce, an indexical sign is characterized by its capacity

Specifically, TMsdo

not articulate the relation between computation, logic and

to refer to an atomic unit or individual, without exhibiting a

mathematics, other than stipulating their symbolic nature.

resemblance or likeness, andIwill elaborate on the computa­

One could be excused for thinking there was no logical basis

tional nature of indexicality via notions of addressability and

at all to computation based solely on Turing's account, as it

encoding in §4.4.31 With Carnap's development of intensional

permits arbitrary state changes and offers no further clarifica­

logic, denotation was formalized further, allowing logicians

tion. This has furnished the TM formalism with a universal­

to treat intensions as pliable functions and extensions as

ity which allows for its application in ontological

arguments,

fixed states, leading to the autonomous study of intensional

extending forms of computational realism into the natural

structures. It should be noted that the symbolic, in so far as

world, some of which are critiqued in §2.2. While the formal

it refers to a mark or inscription, is intrinsically geometric I

or syntactic aspect ofcomputation is emphasized with respect

draw a close relationship between topology and syntax in later

to an infinite tape containing symbols, the notion of symbol

chapters, situating a notion of geometric structure at the root

is however deeply contested, and the assumptions

by

of my definition of the symbolic. To define an inscription or

°n thi; froM ™rit closer scrutiny, as they speak to key issues regarding the semantics of computation

mark as symbolic thenhas three interpretations, it is to claimit

It is not my intention here to outline a broad semiotic the­ ory ofsigns, but rather to locate the role of the symbolic with­

(logical), and a topology (computational). It does not require

in computational reason, and in this regard certain concepts

without which the grapheme has no sense (Sinn).Iposit that

which are of particular Merest-denotation, indexicality and

an intension is a minimal condition for a symbol to exist qua

made

has an internal structure (mathematical), an intensional logic a denotation or referent (Bedeutung), but does imply a syntax,

31

Peirce, C.S., 1902. Logic as Semiotic: The Theory of Signs in Philosophical writings of Peirce. Courier Corporation.

35

Computation & The Real

Logiciel

symbol, and that this intensional structure has a geometric ba­

tradition proceeds from Heyting, via the work of Gentzen to

sis. I will attempt to offer this ultimately topological theory as a candidate for a unified perspective on reasoning in due course, resolving some of the semantic issues thrown up by Turing, but for now let us diagnose them.

modern practitioners like Martin-Lof, Abramsky and Girard, and finds its computational expression in a descendant of Church's calculus. Turing's model, by contrast, should be seen as a reflection of early formalism, a necessary but insuffi­

Early formalism can be accused of adopting an uncriti­ cal or naive attitude towards semantics insofar as its claims of establishing an axiomatic science of deductive analysis on meaningless signs have been shown to be untenable. Tarski was to show that no such purely syntactic regime can be es­ tablished when discussing any language descended from clas­ sical logic, but rather that a meta-linguistic apparatus is need­ ed to underpin meaning in these so-called formal languages.'For Tarski, any semantic treatment of such languages hinges on the identity of truth values, in other words, in how truth is stablished. In systems descended from classical logic, such as

cient account of computational reason, by virtue of its seman­

e Boolean logic underlying digital computers, truth tables equired to assign values to logical operations, and these xtrinsic to the specification of the language itself—it is not possible to articulate them axiomatically, leading Tarski to prove that no system of arithmetic could ground its own truth values. Tarski's approach would later develop into a Ig y influential body of model theory, whereby models

work on the general recursive functions in 1934. Godel s the­

Cta ;?SUistic Prarnes undergirding the semantics of po stble worlds in modal logic. Intuitionist semantics marks

for answering all Diophantine yes or no questions of a cer­

a nroofth tional

PartUu ^ thiS apPr°ach> instead CtIC

tic naivety. These two semantic traditions, model-theory and proof-theory, constitute distinct perspectives on meaning in formal languages, and we shall explore this interplay further in Chapters 3 & 4, as part of a project to develop a strictly constructive model of computation. In time, all three canonical models articulated their own relation to contingency. The first

of a proposition known as the Godel sentence, and it was Godel's attempts to later formalize this notion that led to orem struck a fatal blow to Hilbert's project for a fully axiomatized mathematics, an insurmountable challenge summa­ rized by Godel decades later in his personal correspondence. "The few immediately evident axioms from which all of contemporary mathematics can be derived do not suffice tain well-defined kind. Rather, for answering these ques­ tions, infinitely many new axioms are necessary, whose

bating

truth can (if at all) be apprehended only by constantly re­

,^SiS ^ estab^ing meaning, an opera - by Dummet in §1.1. This second

newed appeals to mathematical intuition...

S£mantlCS aIkded

Logic, 53(1), pp.51-79

Truth

and Logical Consequence. The Journal of

Symbolic

-

36

incompleteness theorem

(1931) was explicitly based on the primitive recursive nature

Wang H , 1990 Reflections on Kurt Godel MIT Press D 129

37

Computation & The Real

Logiciel

1 his expression of mathematics as an infinitely extensible set

mechanical processes underpinned all three classes. As such, it

of revisable rules, as opposed to a canon of immutable laws,

is impossible to claim that canonical models present computa­

is precisely what 1 have come to call the inferential view, and

tion as a purely formal notion, just as it is not possible to claim

it suffuses all three models of computation in distinct wars,

their consistency, a fact which would lead Godel to critique the

Church extended the lambda calculus with types in order

CTT as an unstable foundation, consigning as it does all three

to resolve the logical inconsistency of his model in 1936,

canonical models to a lack of formal grounding. My claim is

and finally luring proved the undecidability of the halting

that the precedence of a mechanistic interpretation of compu­

problem for computable functions using TMs in 1937. The

tation imposed by Turing is harmful in this regard, and I will

equivalence of these'three expressions of contingency is wide­

examine the semantic implications of the attempted grounding

ly accepted, but Turing's treatment distinguishes itself as na­

of computation as the effectively computable in §2.1.

tively computational insofar as his framing of the problemis not strictly mathematical, but stated in terms of decision pro­ cedures, a perspective first posed by Hilbert and Ackermann in 1928 in the form of the entscheidungsproblem. Indeed, itis only through the lens of undecidability that the intuitionistic nature of computing becomes apparent—counter to the Boolean tradition, we will come to view computation as a means of physically realizing Brouwer s logic, thus admitting indeterminacy into its foundations.

1,3 The Continuum

A number of explicit attempts were made to forge equivaences between the three canonical models. Church's Thesis is

Brouwer's doctrine for mathematical reasoning opened up a

a conjecture that effectively computable functions, those char-

sorbing a foundational challenge to the decidability of logic. For Brouwer, mathematical objects are mental constructions,

I rr-.j f,n'te mccbanica] procedures, coincide with the c ass o o e s general recursive functions. This was further expanded to show the equivalence of the three classes of com-

deep rift in the philosophy of mathematics, in the process a

they ultimately appear to the human mind as a form of intu­ ition. Hilbertian formalism, the dominant modern school, in­ stead emphasised consistent axiomatic systems, to be treated as

Turing6computable andTt The Church-Turing thesis (CTT) risked Church in Princeton

C°mpUtabk

* C°mp°Sed

hermetic signswithout external referents.34 The three canonical

when Tumg

dea of the effectively cornn, 7°^^^ anc* use it to bind all three nodels under an f nodels, unde^an unprovable but intuitive notion that finite

s ThenatureoUheax,omf, :s—

career, and a more complex rendering Danielle Macbeth.

Computation &The Real

Logiciel

models of computation plumbed the fault line occasioned by Brouwer with their formal accounts of contingency, rendering incompleteness in a new light. The interwoven nature of intuitionism and computation, bound by their admission of incon­ sistency, is most evident in their mutual accounts of the Real. Questioning Platonist claims regarding numerical realism, and formalist claims as to the atemporality of mathematical struc­ tures, the second act of intuitionism conditions the relation of mathematical reasoning to infinity and the continuum, with considerable ramifications for accounts of the computational, The following antinomy emerges at the heart of intuitionism: how can a constructive practice which accepts only denumerable methods nevertheless claim the totality of the continuum ' as the primary intuition" of mathematics? At stake in this question is an integrated account of Kant's three categories of quantity unity, totality, and plurality. To develop an integrated account of these categories will require a constructive treatment of the number line. In clas­ sical mathematics, the computable numbers form part of a hierarchy of countably infinite subsets of the real numbers (R), which include the natural numbers (N), the integers (Z) and the rattonals (Q). Beginning with the natural numbers, a computational account of plurality is to be located formaly in the Church numerals. These portray natural numbers terms of the repeated application of functions—Church's o ing of numbers into the lambda calculus renders nums no more than a by-product of function composition, a natural number n defined as a function which maps any other function f to its n-fold composition: f" = f ° f o ... o f

And the number line develops as follows: 0= OfX - x 1= Ifx=fx 2 = 2fx =f(fx) Numerals then represent an index of applications of func­ tions to values, and it is the indexical nature of encodings which marks this out as a computational definition. This processual approach to numbers can be considered a natively computational treatment of not just the natural numbers, but the whole of the rational number line, thanks to the expand­ ed Church encoding, which supplements the definition with additional properties for dealing with negative integers and fractions.35 In this view, number is synonymous with the rep­ etition of a self-referential process, which has as its limit case infinite regress. Such an operational account subsumes the numerical under the concept of encoding, a proposal which is discussed further in §4.4. Contrast this to the canonical set theoretic definition of natural numbers, defined as the finite ordinals.36 The sequence proceeds from the empty set, 0, via a process of recursive embedding:

0=0

1={O} ={ 0 } 2 = {0,1} = {0,{0}}

Pierce, B.C. and Benjamin, C„ 2002. Types and Programming Languages. MIT Press_ p. 60, Halmos, P.R., 2017. Naive Set Theory. Section

40

11.

Courier Dover Publications, p. Bb

41

Computation & The Real

Logiciel

While both definitions of N constitute a finite iterative pro­ cedure making use of recursive operations, only the former proceeds from functional foundations which are intrinsically computational. For now it is sufficient to note that in both accounts the domain of the rational (Q) appears to be rooted in recursion, which resembles a structural form of embed­ ding—a theme which will resurface when discussing a geo­ metric view of computation in later chapters. Having given a computational overview of the rational number line, let us turn to the thornier terrain of the con­ tinuum, R. The construction of the Real marks the disconti­ nuity between the countable and the uncountable - crossing the threshold which marks the domain of the irrational, the ranscendental, and the incomputable. Whereas the rationals represent all whole number fractions, the real number line includes those irrational numbers, including surds such as \[2, as well as the transcendental numbers, such as n, which are assumed to be uncountably infinite in nature. Famously, set theory would conjecture the incommensurability of these wo domains, the continuum hypothesis asserting an unridgeable gap between the cardinality of the rationals and Tl^

of the rationals in the real instead suggests a thick consistency to the continuum—above all a viscosity which would thwart any attempt at drawing a clean boundary between the ratio­ nal realm and the seeming unity of the real—a state which motivates an intuitionist critique of classical mathematical accounts of the real. Classically, the continuum has been associated with ge­ ometry and the domain of continuous functions in metric spaces, and this has been contrasted with the symbolic do­ main of algebra, such that finding adequate algebraic de­ scriptions of the real has been riddled with difficulties. Of the numerous methods suggested for the elaboration of the real number line, two are generally accepted in modern mathe­ matics as canonical. The first, Cauchy sequences, stems from the calculus of Leibniz and Newton of the late eighteenth century, defining a sequence of rationals which converge to a limit, resting on the notion of equivalence classes to provide a criterion for identity. By formalizing convergence, Cauchy located the Real at an infinitesimal limit point. Later, a mod­ ern set theoretic treatment came via Dedekind, proposing instead a cut in the Real to partition the continuum into two sets of rational numbers, one of them necessarily an open-ended ordered subset with no specifiable upper bound. Badiou characterizes this set theoretic conception of the real "as a fiction at the void point of a cut , a real number comprising the "minimum matter situated exactly between two sets of dyadic rationals".38 Despite Badiou s reassuranc­ es that there is no mystery in the status of a number whose

t'1C rca's>

virtue of Cantor's diagonal method.37 is raming exposes an underlying complexity to the numer me which is not adequately captured set theoretically e re ation between rational and real is far from a trivial inon re ation rationals are said to be dense in the real, that , ationa exists between any two real numbers—indicating the deeply intertwined nature of these domains. The density

Kleene, Introduction to Metamathematics. pp 3-14.

42

8

Badiou, A., 2008. Number and Numbers, Polity, pp. 175-177.

43

Computation & The Real

Logiciel

"form and residue are infinite", his concession that number

only via a creative process. This would lead Brouwer to out­

has been replaced by an "operational fiction" should alert us

line a theory of the Creating Subject, the domain of the ra­

to the gnomic nature of this account of number, wherein

tional rendered in terms of an ideal mathematician enacting

point.3'

free choices ranging over digits (natural numbers). Above all,

For Badiou, the cut is a procedure which yields a "Active

free choice sequences highlight the intrinsic undecidability of

point" with no a priori ontological status, but its claim to

any attempt to establish identities in the domain of the Real,

existence is hardly merited as the operation cannot be de­

aided by their amenity to computable treatment as decision

fended as constructive.40 From the intuitionistic view, these

procedures.42 As Bauer notes, this gives an intuitive charac­

treatments of the Real—the concepts of the limit and the

terization of the Real in terms of canonical models of compu­

cut—are hampered by appeals to the totality of uncountable

tation, in that the "decidability of reals is real-world realized

Dedekind's cut appears to occupy an Archimedian

infinitesimal or infinite mathematical objects.

if, and only if, we can build the Halting oracle for Turing ma­

Brouwer's rejection of both Cauchy sequences and

chines."43 Notably, the interpretation of free choice given by

Dedekind cuts originates in his insistence on the construc-

Brouwer is broader than any computable notion, it could de­

tibility of mathematical objects, since neither infinitesimal

note a form of incomputable contingency at each time step or

limits nor open sets satisfy the constraints of intuitionist

else a deterministic algorithmic procedure. However, in both

thought. Indeed, Brouwer would posit a continuity principle

accounts the totality of the Real is rendered a mathematical

which would render the continuum theoretically indivisible.

impossibility, absent an oracular authority, and the construc­

In their stead, Brouwer suggests free choice sequences, empha­

tive viewleads to objects such as unbounded continuous maps

sizing the process of construction in the elaboration of a real

on closed intervals, constructs which classical mathematics

number, selecting a digit at a time to iteratively generate the

is fundamentally unable to accommodate, exposing its in­

sequences are com­

complete treatment of continuity.44 Moreover, it follows that

posed into spreads, which are akin to a time bound version of

determinate transcendental numbers, such as 7r and e, exist

an open set, and these can in turn spawn fans by way of in­

only insofar as they can be computed: they are to be afforded

real in a non-terminating

process.41 These

finite branching. This algorithmic portrayal of a real number

no a priori ontological status. In the next chapter, we will see

in the form of a generator amounts to a proto-computational

an interpretation from the quantum cryptographer Nicolas

approach to the continuum, the free choice at each time-step underlining the Real as a temporal phenomenon approachable Heyting, A., 1912. The Intuitionist Foundations of Mathematics. In: Benacerraf, P. and

Putnam, H. (eds.), 1984. Philosophy of Mathematics: Selected Headings. Cambridge Ibid. Ibid. Heyting, A. ed„ 1966. Intuitionism: An Introduction, Vol. 41: Chapters. Elsevier.

University Press. Bauer, A., 2013. intuitionistic Mathematics and Readability in the Physical World. In: A Computable Universe: Understanding and Exploring Nature as Computation, pp. 143 157. Ibid.

45

TWO

Logiciel

Gisin which advances an intuitionist physics stemming from this account. First, let us come back to the antinomy that motivated this discussion, as it will allow us to take stock of the constructive stance. Brouwer implies that the continuum must be regarded as a "primary intuition", suggesting a realist view of the Real number line, but proceeds to suggest that uncountable infinities are outside the grasp of mathematical thought, casting doubt on their existence as true mathemati­ cal entities. Ultimately, the unity of the continuum is not giv­ en to reason in the intuitionist account, and Real analysis is rejected as a mathematical practice tout court, as it treats the real numbers as a complete ordered field. The totality implicit in the category of continuity is dismissed, and in its place lie

The Two Dogmas of Computationalism

algorithmically mediated free choices ranging over sequences which are time-bound, open-ended, and denumerable.

"The m o d e l s o f m o d e r n p h y s i c s a r e c o n c e r n e d , therefore, b o t h w i t h c o n t i n u o u s a n d d i s c r e t e v a l u e s . It w o u l d s e e m a p p r o p r i a t e t o c o n s i d e r a h y b r i d s y s t e m . It will b e e x t r e m e l y d i f f i c u l t t o f i n d a t e c h ­ nical m o d e l o f a h y b r i d c o m p u t e r w h i c h b e h a v e s according t o t h e l a w s o f q u a n t u m p h y s i c s . - Konrad Zuse

Z.oe K . 2013 5(9701 Caicularnc space iRechnenaer Raumi. In A Computa&le J-„es= LAoe-a-anclmg and Exploring Nature as Computation (pp. 729-/861.

The Two Dogmas of Computationalism

Computationalist positions come in two broad flavours, ac­ tive in epistemology and ontology: computational theories of mind and assertions of computational realism. The former identify computational with mental kinds, while the latter assert computation more broadly as a matter of fact in the world. In their extreme form, these lead respectively to the multiple realizability of mind hypothesis, and the view that "the universe computes" (Deutsch). I will argue against these two central dogmas as speculative claims arising from unten­ able applications of computational theory, seeking in turn generative accounts of computation loosened from the grip of these tenets. As such, what follows is a diagnosis of the is­ sues raised by these dogmas with a specific focus on the role of computation within the related discourses, rather than a comprehensive overview. Later chapters will seek to resolve various difficulties arising from them, in the process iden­ tifying the manner in which they have hamstrung a critical philosophy of Artificial General Intelligence (AGI). Such a philosophy treats computation as a historical rupture in hu­ man thought, a cleaving of reason from mind, which yields an 49 i

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