The origins of ideas fundamental in the mathematical and physical sciences

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THE ORIGINS OF IDEAS FUNDAMENTAL IN THE MATHEMATICAL AND PHYSICAL SCIENCES

A Dissertation Presented to the Faculty of the Graduate School The University of Southern California

In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

By John Alan Stephens May 1950

UMI Number: DP29621

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Pl\, O. P 'so S * 3-2 T h is d is s e rta tio n , w r it t e n by

___Mr ._John A. Stephens u n d e r the g u id a n c e o f h.%

s___

F a c u l t y C o m m itte e

on S tudies, a n d a p p r o v e d by a l l its m em bers, has been p resen ted to a n d a cce p te d by the C o u n c i l on G r a d u a te S tu d y a n d R e search, in p a r t i a l f u l ­ f i l l m e n t o f re q u ire m e n ts f o r the degree o f DOCTOR

OF

P H IL O S O P H Y

Dean

D a t e . . ¥ * I . } . 8. ’

1950

C om m ittee on Studies

I .

\

Chairman

TABLE OF CONTENTS

INTRODUCTION A*

PACE Statement of objective . • • • • • • •

vii

B . . Arrangement of Parts • • • • • • • • •

xiii

C♦

Summary

xiv PART I

THE CONJECTURAL THEORY OF KNOWLEDGE ♦ . . .

1

For which the sources of ideas are: distinct from the modes of veri­ fication ..........

1

CHAPTER I

A COMPARATIVE DESCRIPTION OF THE CONJECTURAL THEORY OF K N O W L E D G E ........

4

A critical description of the theories: of knowledge favored by the philo­ sophies of rationalism and empiricism,, and a description of the conjectural theory of knowledge presupposed by this t h e s i s ........ ". . .................. PART II THE TWO PRINCIPAL SOURCES OF THE IDEAS FORMING THE SCIENCES OF MATHEMATICS AND

4

iv

CHAPTER

PAGE SECTION I

.........

46

The role of the Fundamental Human R e ­ sources, i.e., physical structures, mental dispositions and basic experience, In shaping early ideas of physics and mathe­ matics, and the utility of the symbolic system of mathematics for physical science. II. THE FOUNDATIONS OF MATHEMATICS AND PHYSICS A. Geometry

.........................

B. Arithmetic

.

46

. .

46

. . . . . . . . . . . . . .

106

C. Physical science . . . . . . . . . . . .

114

III. HOW PHYSICAL SCIENCE IS ADVANCED BY THE USE OF MATHEMATICS

.

118

SECTION II The role of ideas in shaping the historical development of mathematics and physics

.

1ST

.

138

IV. THE EMPIRICAL AND PHYSICAL CHARACTERISTICS OF BABYLONIAN AND EGYPTIAN MATHEMATICS AND THEIR IMPORTANT CONTRIBUTIONS A. Introduction

. .

.......................

B. The influence of extraneous interests C. The'outstanding achievements

. . . . .

138 .

145 152

V

CHAPTER V.

PAGE THE GREEK ACHIEVEMENT AND HOW THE GREEK CONCEPTION OF THE PHYSIS AFFECTED THEIR MATHEMATICAL I N T E R E S T S ............

• . .

165

A.

Why the Greeks were geometers . . . . . .

165

B*

The work of Thales

177

C.

The work of the Pythagoreans and the

................ *

reason for their geometrical interests D.

* .........................

184

The three famous p r o b l e m s ............200

E. Archimedes, the classic example of our principal argument F. The character of Greek physical science VI.

THE MATHEMATICAL FOUNDATIONS OF CLASSICAL MECHANICS

...................................

A. Galileo Galilei

........................

B* Sir Isaac N e w t o n ............. ........... VII.

THE IMPACT OF CLASSICAL MECHANICS ON ANALYSIS, GEOMETRY AND ALGEBRA

....................... 285

A.

The calculus of variations

. • . . . . .

287

B.

Special functions and boundary problems

. 298

C.

Theory of functions or a real variable

. 302

D.

Potential theory

E.

Theory of functions of a complex variable

353

F.

Differential equations

378

.................... 309

.................

vi CHAPTER

PAGE G.

Geometry

. ................. .

H.

Vector algebra and tensoranalysis

403 ..

413

VIII. IMPORTANT'EPISODES IN MATHEMATICAL AND PHYSICAL SCIENCE FAVORING THE CONJECTURAL THEORY OF INTUITION AND CONCEPTION A,

Geometry and analysis

B*

Algebra

C.

Physical science

BIBLIOGRAPHY

.

.................

......................... ..

..........

.

.

..

476 479

.

491

..........* . . *

503 510

INTRODUCTION A,

STATEMENT OP OBJECTIVE

The twentieth century has witnessed a growing appre­ ciation of the exploratory function of perceptions and ideas for enterprises of a physical or logical order*

The theory

of knowledge most in harmony with this attitude has pros­ pered accordingly.

For the proponents of this burgeoning

theory, sense-perceptions of even the most concrete, sen­ sational variety are not ultimate revelations of fact, as they are for the more hide-bound empiricists, but prelim­ inary assessments or conjectures which the course of events may or may not bear out.

Similarly, rational perceptions

of even the most general and abstract kind are not infal­ lible or inevitable revelations of intuitive reason, as they are for the more rigid rationalists, but surmises which must be judged by the self-consistency of their log­ ical consequences.

Hence, intuitive perceptions, whether

of sense, instinct or reason, are not to be taken at their face value where prediction and deduction are the goals# They do not carry a blank certificate of validity and therefore their claims must be certified by an external standard or criterion.

If the purpose at hand is to con­

struct a scheme of relations which can be used for the pre­ diction of sense-experience then the making of good ore-

vlii dictions Is the proper criterion, and our scheme of rela­ tions will be a physical system*

If we wish to build a

scheme of relations which can be used for making system­ atic deductions then self-consistency is the key criterion and our scheme of relations will be a logical system* The theory which presents perceptions and concep­ tions as conjectures is confronted with two distinct problems:^ (1) what are the sources of the surmises, and (2) what are the appropriate standards of judgment and the proper methods of application?

The second problem has been

subjected to the most searching analysis by some of the keenest thinkers of the nineteenth and twentieth centuries* But the first problem, in spite of its profound Importance, has been largely ignored* the first problem*

This thesis is concerned with

It tackles, as a major objective, the

question of O r i g i n s 1 in two specific fields, mathematics and physics*

Hence our primary aim la to investigate the

origins of the fundamental conjectures which have entered into the development of mathematics and physics, ^ As Archimedes long ago proclaimed, and as the log­ ical empiricists are fond of repeating as though it were their discovery, the methods of discovery and the methods of verification are distinct* Since Archimedes was appar­ ently the first to perceive clearly the meaning of this distinction and the first to capitalize on it, e*g*, his discovery of mathematical propositions by mechanical methods, we shall refer to the conjectural theory of per­ ceptions and ideas implied by his distinction and prac­ tices as the Archimedean Theory.

ix Some advocates of the theory we have described take the view that the exploratory ideas are actually arbitrary definitions giving rise to systems of relations which, in some fortunate instances, happen to be applicable to the description of the world of sense-perception.

It is quite

true, on our view, that no necessary relation binds the objects of sense to the elements of conceptual systems. But surely it is too far-fetehed to assume that the evolu­ tion of the applicable systems was merely fortuitous.

Our

inquiry into the influence of mathematics and physics upon each otherls development is undertaken from the standpoint of the general theory that the flourishing of conceptual en­ terprises depends upon a proper division of labor between intuition and reason.^

According to this conception it is

---- *.rj....— ... -.

Perception, in general, is any process of being aware. Intuition, in general, is any perception character­ ized by a sense of obviousness or naturalness. In our discussion we shall be concerned only with the perception significant relationships. Ihe perception of signifi­ cant relationships varies by degrees from the kind for which, so to speak, the objects determine the relations to the kind for which the relations determine the objects. At one extreme is concrete sense-perception in which individ­ ual objects are seen to have certain specific relations, and at the other extreme is logical, abstract perception of general relations, as such, and their mutual implications. In between these extremes are the contextual perceptions. In contextual perceptions the perceived relations are en­ meshed in contexts and engaged with objects with varying degrees of generality. Instinctive associations have a cer­ tain limited fixity but they apply to definite objects more or less generally. Imaginative and rational associations of the contextual type are more versatile and display a wider and freer use of general relations.

the task of intuition to dig up surmises and conjectures, and the duty of reason to judge critically in the light of the criteria appropriate to the purposes at hand. Our theory is, more specifically, that ideas help­ ful in the construction of physical science have been drawn from (1) concrete sense-perception, (2) instinctive asso­ ciations, e.g., substance and attribute, (3) experienced associations of perceptions, (4) imaginative combination of relations and rational discrimination of general rela­ tions, and (5) intuitive insights of an analogical order. One of the most important sources of analogical insights suggesting fruitful developments in physical science has been mathematical ideas.

We plan to discuss the various

sources of the ideas utilized in physical science, but we shall oay special attention to the influence of mathe­ matical ideas on the formulation of physical systems. In a similar fashion we hold to the theory that ideas helpful in the building of mathematical systems have been drawn from (1) concrete sense-perception, e.g., cir­ cularity, (2) Instinctive associations, e.g., continuity, (3) associations learned or acquired by experience, e.g., rigid displacement, (4) perception of abstract connections, as such, and of their general relationships, and (5) in­ tuitive insights of an analogical type.

Ideas taken from

physical science have been an outstanding analogical source

xi of Important developments In the science of mathematics* Here, too, although we plan to consider the various sources of mathematical Ideas— including the analogical suggestive­ ness for each other of the various branches of mathematics, e*g*, analytic geometry— we shall devote special attention to the Influence of ideas from physical science on the evolution of mathematical systems* Since mathematical relations are utilized for the expression of physical relations, the effort to provide mathematical solutions of problems posed by physics has sponsored the Invention of much new mathematics, e.g*, boundary-value problems*

This more Indirect contribution

of physics to mathematics is noteworthy and is also dis­ cussed in the thesis.

Qu£. S.UTP-9S-C Ixk ih ls . IM s ia ,

is . M

sources of the basic ideas which have been used in the construction of physical and mathematical gysterns*

Sn

case of physical science we shall examine particularly the Influence of ideas drawn from mathematicsf and in the case of mathematics we shall study particularly the Influence of ideas and problems derived from physical science*

The

study we propose is carried out in Part II of the disser­ tation.

Part II is divided into two sections*

In the

first section the significance of what might be termed the fundamental human equipment, e#g*, physiological structures,

xii mental dispositions and basic experiences, for the shaping of the early physical and mathematical ideas is analyzed. With the beginning of historical traditions ideas begin to make their weight felt and in the second section the Impact of physics on mathematics and the contributions of math­ ematics to physics are historically discussed.. It must be borne in mind that the urgency of the problem we propose to investigate depends upon the conjXectural or Archimedean theory of perception and ideas. To the extreme empiricist and to the extreme rationalist the issue we are stressing is minor.

For, on the one hand,

if our ideas represent merely the learned associations of sensation then it is experience which automatically 'gen­ erates1 our ideas; or, on the other hand, if our ideas are fixed intuitions characteristic of the human species, then it is our natural mental endowment which bestows upon us the forms of our perceiving and reasoning.

If the per­

ceived connections of sense-objects and mental-objects are the result of a mental activity which has the gift of pierc­ ing through the gilded finery of sense to the 'real treasure1, or if they are the result of a mental activity which makes 'sense' of its-objects by connecting them in modes harmo­ nious with its own particular nature, then it will be the nature of the mind which is the sole source of our ideas.

xiil The criticism of our program implied by these views cannot lightly be cast aside*

Eminent thinkers, e.g.,

Milne and Eddington, have maintained these views, and they exercise a subtle fascination which none can deny.

What

is the witchery and where is the flaw in these ideas? Many who profess the conjectural viewpoint do not always speak consistently; partly, perhaps, because they have not troubled themselves to unravel the full implications of the conjectural theory for the realistic and idealistic posi­ tions.

For these reasons we have felt it necessary to

explain in some detail the character of the general view­ point.

This is the primary purpose of Part I and Part III

of the dissertation. B.

ARRANGEMENT OF PARTS

The thesis is divided into.three parts.

In the

first part the general viewpoint from which the study is made is explained against the background of a brief his­ torical discussion of outstanding epistemologieal theories. The primary thesis is expounded In the second part which Is divided into two sections.

In the first section the con­

tributions to mathematics and physics of sense-intuitions and Instinctive intuitions are analyzed, and in the second section the contributions of intuitive surmises of an analogical order are historically discussed.

The third

xiv part presents, briefly, certain trends in mathematics and physics which support the underlying theory of the thesis* These trends point up the conceptual and conjectural char­ acter of physical and mathematical systems, and clarify the relation of Intuition to the appropriate criteria for judging these systems* C.

SUMMARY

'

In view of the vast scope of the thesis it has seemed desirable to present a preview of the argument in the form of a brief summary of the dissertation itself* PART I The traditional theories of knowledge are criticized 1 on the ground that it is arbitrary and artificial to make certainty the prime specification of what is to be accepted as knowledge*

The claim is made that it is more sensible

to assess the results of sensing and reasoning in terms of the inclusiveness and accuracy of the predictions which can be made from them*

Thus the emphasis is placed on the pre­

dictive function of sensing and reasoning rather than on some supposititious revelatory function.

Hence, empiricists and

rationalists to the contrary, there is nothing ultimate or absolute about either sensing or reasoning*

When used for

making predictions they are conjectural processes and are subject to judgment by the criteria appropriate to such

J i XV

processes. fhe perception of significant relations varies be­ tween two poles.

At one extreme is immediate, concrete

sense-experience in which a specific object is directly seen to have or to enjoy such and such relations and properties. At the other extreme is abstract perception of the proper­ ties and mutual Implications of unembodied relations.

In­

termediate to these extremes are contextual perceptions. Contextual perceptions, exhibit varying degrees of generality ranging from the more specific instinctive perceptions to the freer and more general associations of reflective dis­ crimination.

Even though general relations are perceived

as such, the perception is termed contextualf if the rela­ tions are regarded as intimately involved with contexts and objects,

fhe perceiving and applying of general relations

with some regard for consistency is termed reasoning. Intuitive perceptions, in general, are marked by a peculiar forcefulness which,stamps the perceptions as valid and authoritative.

Of special significance to our study are

the intuitions of sense and instinct together with the dis­ criminative perceptions suggested by analogical intuitions. For, according to our theory, given a reporting sensation, the character of the executed action will depend upon the preliminary elaboration performed by sense-perception it­ self and upon the richer elaboration wrought by contextual

xvi perceiving or reasoning*

By and large the rough and ready

needs of immediate and local action are well served by the Intuitively formed associations we have just described* But for a more ambitious program embracing vaster and more precise actions it becomes necessary to call upon modes of connecting which Intuitive perception cannot directly sup­ ply or embody.

For example, there is now available in the

form of modern physical science a vast edifice of connec­ tions, incomprehensible to intuition, by means of which, on the basis of a given reporting sensation, the most farreaching and accurate predictions can be made,

fhe elab-

oratlve possibilities have been extended far beyond the scope of intuitive perception.

Nonetheless intuitive per­

ceptions have been of great value in sponsoring conjectures and surmises which helped to launch and guide the building of conceptual systems for predictive and purely deductive purposes*

This is especially true of the analogical type

of intuitive insights.

We are, of course, particularly

Interested in the analogical employment of mathematics in the formulation of physical principles and vice versa. PART.II fhe first section of Part II analyzes the role by intuitions of sense and instinct in forming basic concepts of Euclidean geometry and early cal science, fhe second section is a study of influence of such intuitions and of analogical

played the physi­ the in­

xvii sights, especially the latter, on the historical de­ velopment of mathematics and physical science* SECTION I The first chapter of Part II takes up the question of the origins and foundations of geometry*

Since we seem

to have a particular affinity for a three-dimensional Euclidean geometry, as evidenced, for example, by the fact that it was the first geometry, the discussion was opened by ashing three questions regarding the origin of this geometry*

(1) Poes three-dimensional Euclidean geometry

represent the hind of associations we have learned to make as a result of our physical experiences?

(2) Poes it rep­

resent a form innate in the kind of sense-perceptions we enjoy?

(3) Or is this geometry a characteristic property

of the peculiar way in which we perceive certain general relations, I*e., is It a property of the way in which human beings must think? The answers to these questions are Important for our inquiry*

If the third possibility is correct then physical

science could not have played more than an insignificant part In the development of geometry.

And the question of

g e o m e t r y ^ influence on the evolution of physical science would resolve into the problem of wshowing h o w ‘the data of experience had been inevitably molded into the fixed

xviii forms of geometrical thought.11 If the second interpretation is correct then geometry as a generalization of the spatial properties given in sensation is divorced as to origin from the physical properties of the world, i.e., those having to do with the structure of sensations. And whatever application it might have to the explication of the world's order will he adventitious. If the first theory is appli­ cable then the origin of geometry is intimately in­ volved with the structure of physical experience, and the history of geometry's services to physics would he a matter of adapting the relatively pure develop­ ments of geometry to the expression of ever more subtle connections of experience.4 The third possibility is disproved by the invention of non-Euclidean geometries and the construction of ndimensional manifolds.

At this point an extension of the

argument presents itself. In conceiving of a system of differences in which one thing is to be determined by n operations is the mind compelled to think of these operations as measurements? Is it only by measurement that the mind can uniquely differentiate the elements of a continuum? The properties of measurement would then be imposed upon the conception of the n-dimensional continuum.5 A negative answer is returned by analysis situs which abandons measurement and bases its theories on the Idea of order.

^ Infra., p. 92. 4 I k M . , pp. 92-93.

5 Ibid.. P. 70.

xlx The possibility that two-dimensional Euclidean space is a native form of perceptual sensing is eliminated by such imaginative constructions as those of Helmholtz and Beltrami*

It is not easy to dispose of the three dimen­

sional case*

Some, taking a hint from the conceptual

meaning of dimension, argue that perceived dimensions are actually unique sets of series of sensations and conclude that the number and properties of the series have been learned by experience.

Others argue that the perceived

meaning of dimension is qualitative not structural and maintain that dimension is a given qualitative feature of sensation.

Our conclusion is that although extension ap­

pears to have the earmarks of an irreducible quality its significance is primarily a function of experience and other dispositions.

Consequently, the actual foundation

of perpetual dimension is the interaction of the human being with the physical world.

It cannot be understood If

either is abstracted from the other* Although the arguments of Helmholtz and PoIncar^ to the effect that the property of rigid displacement or free mobility is a product of physical experience disclose the basic role of physical experience in the formulation of geometry it is not to be concluded that the propositions of geometry are therefore logically dependent upon ex­ perience.

XX

Geometrical propositions are certain and empirical generalizations are merely probable* Regardless of what points, lines, etc*, might be, we are certain that objects having the properties specified by the postulates will also have the properties deduced from those postulates* Is this then our answer? Is the function of the mind in all this business of mathe­ matics merely to develop definitions of purely abstract objects and then to deduce the Implied properties of the objects thus defined? The office of the mind in the realm of experience will not, therefore, be that of a king who dictates the laws according to which exper­ ience shall conduct itself; but rather that of the humble adviser who offers his definitions in the hope that at some point experience will see fit to Accept* them* The mind can never say that the logically de­ fined object will ever be R e a l i z e d 1, either in exper­ ience or in any other way; it can only say that if they ever a r e , then such and such will also follow*6 Three differing attitudes as to the nature of the conceptually defined objects, which might or might not find perceptual fulfillment, are to be noted.

Poincari be­

lieved that the mind has a direct intuition of the con­ ceptual object, i.e., the mathematical continuum, which validates the conceptual theory once and for all*

C* I.

Lewis takes the view that the mind has preferred attitudes which it actively attempts to use in the ordering of ex­ perience.

The mind, however, is not permanently married

for better or worse to these ideas, and if they are suffi­ ciently abused by experience they might be exchanged for others. 6

Ibid., pp. 93-94.

xxi Our conclusion is that the Primitive propositions are either definitions which the mind endeavors to make useful in the business of ordering experience or they are propositions founded on mathematical insights of a very general order, which, though of assured validity in the mind*s domain, do not have a guaranteed application in the world of senseperception* Prom an eplstemological standpoint * * * the difference between the two viewpoints is very pro­ found, but for our purposes the difference is not great* (1) In either case the mind shows a deep preference for the utilization of its definitions on the one hand, and of its ideas on the other, in the working out of the order of sense-experience* And a large part of the task of the historian of natural science who is forewarned as to these preferences will be to show how they have inspired the ideas and affected the choices which have gone into the building of physical science. (2) More­ over, if the primitive propositions are regarded as general insights then it is pertinent to ask why and how the amorphous, general ideas have come to take their various specialized forms. And if these proposi­ tions are regarded as definitions then it will be in order to ask why, of all possible definitions to choose for development, have the ones selected been hit upon. To answer this the historian of mathematics will have to consider respectfully the suggestions and requests of physical science.7 Essentially the same conclusions apply to arithmetic. Similarly the question can be asked: Are the lnaturalt forms for the organization of sense-experiencere.g*, subject and attribute, causal efficacy— categories of the mind, revelations of sense or generalizations of experience?

The answer proposed is that they are instinc­

tive perceptions which have been generalized and expanded

7 Ibid.. pp. 101-102

xxii because they have been found of value for the organization of sense-experience* In the concluding chapter of the first section of part two, the general function of mathematics with respect to physical science is analyzed.

The point is made that

progress in determining connections useful for making pre­ dictions depends upon the development of some scheme of symbolic representation which enables the making of pre­ cise, unique deductions*

Mathematics provided the neces­

sary symbolic scheme for the making of unique predictions. Furthermore, mathematics has played an all-important role in freeing the mind from modes of ordering sense-percep­ tions which were drawn from relatively circumscribed in­ stincts and from the limited conditions of local experience. nThe mind is given a new means of breaking through the blur of Immediately given connections to profounder and more significant relationships *H Q PART II SECTION II This section takes up the historical study of the effects of mathematical and physical ideas on the develop8

IM 4 . , p. 113

xxiii ment of physical and mathematical systems respectively* The potential suggestive value of mathematics and physics for each other*s development at any given time depends on the available resources of the mathematical and physical systems*

Hence, in the earlier period when the resources

were weak the more primitive intuitions assumed the leader-* ship in directing the course of development.

The mathe­

matics of the Babylonians and the Egyptians was purely empirical and intuitive.

Mathematical propositions depended

upon special cases and Isolated insights.

The Greeks, be­

ginning with Thales and Pythagoras, discovered that the true or proper dependency of mathematical propositions was not on particular, empirical cases but on deduction from prior premises.

Thus mathematics was placed on its own

proper foundations.

Though Greek mathematicians succeeded

in transcending sense-perception, they accepted instinctive Insights as genuine and therefore they regarded the postu­ lates of Euclidean geometry as self-evident and ultimate. Of the ancient physical practices having mathemat­ ical consequences, the most momentous was the application of number to the measurement of space.

From this arose

metrical geometry and the mathematical continuum. For the ancients the physical world was peopled with beings actuated by desires. understanding, were one.

Thing and motive, by their

Greek physical theory was founded

xxiv on a radical distinction between thing and motive*

This

separation of motive from thing was of capital importance for the future of physical science.

It paved the way for

a more discriminating use of human intuitions in the rep­ resentation and understanding of things and motives. Logical analysis, in which the Greeks excelledj of the notion of an abiding thing or substance quickly dis­ closed that perceptions of quality could not be consistent­ ly applied to permanent substances; only the perceptions of quantity proved to be logically compatible with the concept of an abiding thing. The Greeks did not display a comparable penetration in their treatment of physical motive.

The Greeks inter­

preted motivating forces as attributes of substance, in the traditional pattern of vital, inner forces.

Although

they blazed the way to new frontiers in the analysis of thing, they accepted the ancient view of motive force.

Had

the Greeks more profoundly grasped the notion of motivating force as a causal or functional relation they might have been led to the experimental determination of causal link­ ages between the elements of experience, and thus have be­ come interested in a more experimental science.

But their

concern was primarily with, the M a t u r e 1 of fact, and not with the 1causal connections1 of fact. We suggested that in this respect Greek physics was

XXV

a decisive factor in their choice of mathematics*

If the

idea of causal relations or conditions instead of natural inner forces had been uppermost in the physical thought of the Greeks it is altogether possible that they would have turned their attention toward the development of a func­ tional algebra instead of toward a metrical geometry.

As

it was, a metrical geometry was exactly what was required for the representation of a static world of atomic states and situations.

Democritus, for example, looked upon motion

as a given, unanalyzable state, rather than as something to be represented as a functional relation between space and time. Reversing the viewpoint, it seems probable that the Greek successes in metrical geometry seriously reinforced their conception of a physical universe of states and essences.

Had the Greeks made as impressive gains in

functional analysis as they made in geometry it is likely that their physics would have been radically modified. Under the impact of such a mathematics their attention would naturally have been turned toward the problem of finding constant, dependent relations instead of per­ manent, independent substances. It remained for Galileo and Newton to breach the ancient notion of force as a thing or attribute by their use of force as a functional relation.

The chapter on

xxvi The Mathematical Foundations of Classical Mechanics de­ scribes the role of mathematical ideas in the discovery and exposition of this conception and indicates the reper­ cussions in mathematics of their physical exploitation. This chapter carries out the central thought of the previous chapter by showing how Galileo*s ideas of a mathematical function led him to the mathematical definition of space and time in a manner suitable for the experimental deter­ mination of their functional relations in various dynam­ ical motions.

Conversely, the manner in which Galileo*s

structural use of force as a dependent relation encouraged the idea of a mathematical function is described. In the second part of this chapter the parts played by Euclidean geometry and the rules of arithmetic in determining the formulation of the principles of Newtonian mechanics are discussed.

The chapter on The Impact of Classical Mechanics on Mathematics is the longest and most detailed chapter of the dissertation. portant aspects

It is a study of

of the influence of

some of the more im­ mechanics on the

development of analysis, geometry and algebra. The hand of physics has not been equally heavy in all departments

of mathematics.

Of

arithmetic and algebra,

geometry and analysis, it is analysis which has prospered

xxvli most richly from the patronage of physics* accidental.

And this is not

Our preferences have had a hidden,'behind-the-

scene voice in the matter.

The realm of the continuous is

close to our hearts and we are at ease in dealing with it. But the developing quantum theory may have brought the hour when we shall be obliged to deal with the mathematics of the discontinuous*

If we accept the challenge, it will be

but another measure of our willingness to accept the neces­ sities of an objective standard rather than our own pre­ dilections or instinctive inclinations. Although the reciprocal contribution between physics and mathematics has been heaviest in analysis, it is pro­ found in the case of geometry and outstanding in the field of arithmetic and algebra. ANALYSIS The spirit which marks modern mathematics is a special concern for rigor and a flair for general­ ization. The germinating conditions of such a spirit are to be found in the work of the mathematicians of the eighteenth century. The deep concern of twentieth century mathematicians for a general theory of analysis was encouraged by the rank growth of special functions which had their origins in the physical problems of eighteenth century physics. The theory of functions of generalized variables, for example, is an Im­ portant feature of twentieth century analysis, and it was in the calculus of variations that functions of variables other than those of the regular calculus were first comprehensively Investigated. The growth of the calculus of variations up to and Including Lagrange was directly inspired by physical consider­ ations. Similarly it was Fourier1s use of trigono-

xxvili metric series in problems of heat conduction— they had already appeared in the analysis of the problem of the vibrating string— which created the need for an ade­ quate theory of functions of a real Variable. Lagrangefs use of the potential in his formulation of Newtonian gravitation formed the basis for the theory of function of one complex variable as developed by Cauchy and Riemann. . . . I n addition to the development of the calculus of variations and of Innumerable special func­ tions, the pursuit of mechanics in the eighteenth century was responsible for a theory of differential equations, the initiation of the theory of line, surface, and volume integrals, suggestions of multI-dimensional space, the beginnings of potential theory, the encourage­ ment of a theory of functions of a real variable and fundamental beginnings in the theory of functions of a complex variable.9 Under separate headings appear discussions of (l) The Cal­ culus of Variations, (2)Special Functions and Boundary Problems, (3) Theory of Functions of a Real Variable, (4) Potential Theory, (5) Functions of a Complex Variable, and (6) Differential Equations. GEOMETRY Differential geometry received noteworthy inspira­ tion from physical science, especially from cartography and geodesy#

And, hence, our brief discussion on geometry

stresses modern differential geometry. ALGEBRA In algebra the department to profit most from phys-

9 Ibid.

xxix ioal science was vector algebra.*

Physical algebra, how­

ever, was of great assistance for more formal developments. Hypercomplex numbers were readily obtainable as a formal generalization of Gauss* representation of a complex number as a number couple, but the nature of the algebra of such numbers was an ambiguous problem until the formulations made by Hamilton and Grassmann in constructing an algebra applicable to vectorial operations in space clarified it. Vector and tensor analysis are discussed in some detail because of the great Importance of tensor analysis as the mathematical basis for the formulation of the general theory of relativity. PART III In Part I we described the general theory underlying the Inquiry undertaken in Part II.

Part III is a histor­

ical study of certain major developments in mathematics and physics which support the general theory. The theory was presented in Part I that such con­ ceptual systems as physics and mathematics were conjectural enterprises which are free to draw inspiration from all manner of intuitive sources, and which answer only to criteria specified by the purposes in view, i.e., good predictions and consistent deductions, respectively.

The

theory was expounded in contrast with the more traditional

XXX

epistemologies• In Bart II the Intuitive origins of mathematical and physical formulations— with special reference to their reciprocal suggestiveness— were historically studied. Part III is devoted to the presentation of the more important developments In mathematics and physics which substantiate the thesis that the proper criteria of in: 'tuLtlon*s conceptual efforts are good predictions and consistent deductions rather than the intuitions them­ selves. ’Evidence* for the theory has arisen from many quarters.

The implications for such a theory of the bio­

logical theories of evolution and of Helmholtz*s psych­ ological theory of *interpretive? sense-perception were idrawn in the philosophy of pragmatism.

The Implications

contained in the speculations of Mach and Poincare on the conventional status of basic physical principles, and in the reasonings of Hilbert and Russell on the meaning of number, were drawn by the members of the Vienna Circle. The great diversity of the contributions from math­ ematics to the theory of conjectural concepts has not, in general, been sufficiently appreciated.

The contributions

of the logical analysis of the number-concept and of nonEuclidean geometry are widely appreciated, as are— to a slightly lesser extent— the contributions of the analysis

xxxi;of the real number continuum and of non-commuting algebra* But the contributions stemming from the work of such men as Gauss, Galois, Abel and Plucker are rarely recognized* The arithmetic of algebraic numbers, for example, is of profound importance for the modern abstract theory of concepts, but has been almost entirely overlooked* The discussion takes up in turn the important episodes for the general theory occurring in geometry and analysis, algebra and physical science.

-PART I THE C&NJECTURAL THEORY OF KNOWLEDGE For Which The Sources of Ideas are Distinct From the Modes of Verification

CHASTER I THE CONJECTURAL THEORY OF KNOWLEDGE A orltioal Analysis of the Theories of Knowledge Favored by the Philosophies of Rationalism and Empiricism, and a Description of the Conjectural Theory of Knowledge, Presupposed by this Thesis, which emphasizes the distinction between the Sources of Ideas and the Modes of Verification. The Methods of Knowing When the battle between the empiricist and the rationalist was first joined, the issue was not the nature and possibility of Knowledge but the method of acquiring it.

The empiricist maintained that men were in touch with

the world by means of the senses and that the connections which were repeatedly and normally revealed by the senses reproduced the connections between the things of the world. Reasoning in this fashion, Galileo was led to the classical distinction between primary and secondary qualities.

The

rationalist believed that sensing was a personal and ar­ bitrary activity which was not fit to serve as a measure of, or as a dependable means of insight into, an objective, independent world.

Thus Descartes, while following Galileo

in the distinction between primary and secondary qualities— reducing primary to extension and motion— gave this dis­ tinction a rationalistic twist by classifying objective

extension In space among the clear and distinct ideas;: pres *i

ent in the mind without benefit of experience,

and by de­

ducing therefrom the existence of matter on the ground that, 2 since extension is the fundamental property of matter, 3 extension without matter is inconceivable. Hence, for Descartes, it was reason, not experience, which certified the objectivity of primary qualities.

Descartes1 concep­

tion of the primacy of reason Is clearly revealed In his critical remark, BEverything that Galileo says about the philosophy of bodies falling in empty space is built with­ out foundation; he ought first to have determined the nature of weight.0

Only the clear light of reason could

be trusted to reveal the true order and connection of things.

1 HIn the first place, I am able distinctly to imag­ ine that quantity which philosophers commonly call contin­ uous, or the extension in length, breadth, or depth, that is in this quantity, or rather in the object to which it is at­ tributed. 11 Rene Descartes, Meditations on First Philosophy. Meditation V. E. S. Haldane and G-, R. T. Ross, translators, The Philosophical Works of Descartes. (Cambridge: At the University Press, 1931), Vol. 1, p. 179. 2 Rene Descartes, The Principles of Philosophyr Part II, Principle IV. 3 Ibid,. Principle XVI. 4 Sir James Jeans, The Growth of Physical Science (Cambridge: At the University Press, 1947) p." 179.

The Possibility of Knowing Be it noted again, however, that the point at stake was not the possibility of knowledge, which both the ration­ alist and empiricist were willing to assume— even if it meant, as for Descartes, dragging in the Deity as voucher7~ but the method of finding and testing knowledge.

Ensuing

developments brought to the forefront this other aspect of the problem:

Is knowledge possible?

For, when the employ­

ment of experience to reveal the universal and necessary connections of things was more rigorously examined, it came to light that no logically sound bridge could be found lead­ ing from experience to the world of things, the realities whose connections were supposed to be represented in ex­ perience and reflected in thought.

It became clear that the

mere experiencing of a connection even though endlessly repeated was not an adequate ground for determining the universality and necessity of such an experienced connection. Nor could any principle be found which was drawn solely from experience and which could be used to test or vindicate the permanence and necessity of experienced connections.

Since

experience could not lead to Hnecessaryn connectedness and since, for the empiricist, thought is barren, knowledge must not be possible. In the meantime, the rationalists were faring no

4 better.

How could the strictly necessary connections

rigorously deduced from principles founded on the ”pure light of reason” represent the true, inner connections of the things of the world and be so completely out of step with the order of experience?

Ho one could question the

intellectual integrity and genius of Descartes and yet the difference for experience between his methods and results and those of Newton were there for all to see.

If the mind

possessed special insights not drawn from experience and therefore not ansxferable to experience what was to be the criterion of their accuracy and truthfulness?

The great

Descartes had sought and found such a criterion, but when in his able hands it led to results which did not jibe with the course of advancing experience, as Hewton took pains to demonstrate,5 what could be expected from lesser men. But Descartes* misfortune, though disastrous, did not force an abandonment of the search.

There is surely a

special bit of irony in the fact that the system which Hewton boasted was inspired by facts of experience should have been transformed at the hands of Laplace into a SelfEvident system, from which could be deduced such scientific truths as the stability of the solar system.

But Laplace!s

self-evident mechanics had not been self-evident to Aris5

'

Sir Isaac Newton, Frlncipia, Book III, General Scholium.

totle.

Actually, a good part of the self-evidence of me­

chanical principles was a matter of basking In the re­ flected glory of the empirical results•

The criterion of

true ideas was still to be found. Into this breach entered a new thought which seemed to give hope of repairing the damage and reconciling the differences.

The function of reasoning— i.e., the appli­

cation of connections natural or self-evident to the human mind— was not to provide knowledge of the world of things but of the world of experience.

The connecting activity

of the human mind was not to be regarded as isolated in some royal realm of its own and as having nothing to do with the lowly activity of sensing and experiencing.

The

connecting activity of the human mind was to be thought of as vitally present in all acts of human perception.

Since

experiencing is therefore not some haphazard activity, since the mind is always vitally engaged in the forming of its character, we have a gilt-edged guarantee that it will always make sense, that it will always exhibit a certain characteristic structure.

It will do this, not because

there are constant rules of behavior of the world*s things which are being reported to us in our experiences, but simply because of the intimate engagement of the human mind with all perceiving; simply because we should be quite inincapable of perceiving any; object unless the organizing

activity of the mind has brought this object into some pro­ portion with its own permanent structure. has a characteristic structure* is involved in all perceiving.

The human mind

The activity of the mind Therefore all experience

will exhibit certain kinds of connectedness.

In this

fashion, Kant restored certainty and universality, i.e., the possibility of knowledge, to the connections betxi?een the objects of experience. But K a n t 1s harmony was bought at a great price.

For,

since experience must be forever limited to expressing the connection natural to the human mind, It could not be ex­ pected to represent the connections of the world of things. Outstanding developments in the various sciences— notably mathematics, physics and biology— encouraged many of the *

thriftier minded to refuse Kant's price. The Meaning; of Knowledge The discoveries in mathematics of non-Euclidean geometries and non-commuting algebras broke the iron grip of self-evident necessity which had adhered to the m l n d fs mathematical conceptions and operations.

Our spatial per­

ceptions seem to be of the Euclidean variety but the reason for this could no longer be attributed to the omnipresent functioning of a mind which was incorrigibly Euclidean. A strictly Euclidean mind would have, found contradictions

7 in any system containing a postulate denying any one of the Euclidean postulates* fact.

And this was not found to be the

The way was open for a return to empirical doetrines.

Evidently only experience could decide which system of geometry was valid for our world, from the standpoint of the realist, or which system could be most conveniently used, as the conventionalist would put it. Progress ih physics and investigations in the founda­ tions of logic and mathematics seemed to encourage one brand of empiricism and developments in biology encouraged another.

The former sponsored various kinds of empirical

positivism, and the latter promoted various kinds of empiri­ cal realism.

But in all cases, the issue at stake was not

merely the method of acquiring knowledge, nor the possibility of knowledge, but of the meaning of knowledge. The spectacular achievements of mathematical and experimental physics, climaxing in the General Theory of Relativity and the Quantum theory stimulated the growth of empirical positivism.

According to this doctrine, the con­

nections of experience are a matter of ultimate fact which are to be discovered and established by critical, i.e., scientific, inquiry.

This "order” does not need to be

grounded in some ulterior and unknowable order of things, the feeling for which is merely metaphysical superstition. Our knowledge of the order of experience is safe and sane,

8

certain and positive.

Why not he satisfied with this?

The concept of a world of things cannot tell us anything more about the order of experience than we can find out from investigating experience itself.

Therefore, the notion

of a world of things is superfluous baggage, helpful to savages but a clutterance for civilized thought. It is clear that the flight from metaphysics has been aided and abetted by the profoundly human desire to •play it* safe1, and the tendency to associate knowledge with certainty.

What is it that we know for sure?

One

\.

whose ideas of absolute space and absolute time and immu­ table mass have been ground out of him by such an unfore­ seen catastrophe as the general theory of relativity is disinclined to be again caught out on the metaphysical limb. He has become gun-shy, and feels like staying at home where he belongs.

The risks of metaphysical speculation are to

be avoided b y vnot taking them.

This means sticking close to

6 MThere is perhaps no harm In such an assumption [.i.e., of a real world) — in fact, certain minds may find that it enables them more firmly to grasp and feel con­ fidence In physical theories; yet It must be stressed that the assumption is no necessary part of physics, and that in a logical development of the subject the safest course Is to omit it entirely.** Robert B.,Lindsay and Henry Margenau, Foundations of Physics (New York: John Wiley & Sons, 1936), p. 2. p3racketed expression not in the original3

9 experience, the closer the better*

The gradual emergence

of the quantum theory carred on the good work by rendering obsolete the tactual and visual pictorial!zation of mass and energy, particles "and waves, electrons and protons-— left over remnants of G-alilean and Lockian primary qual­ ities*

Deprived of these, we seem to lack the concrete

means of pinning down objective entities and this dis­ couragement, immeasurably deepened when the physical a p ­ plicability of our conception of the continuity of space and time is challenged, accelerates our willingness to sur­ render objective entitles altogether* Associated in the most intimate way with these views based on developments in the field of physics are the ideas concerning knowledge stimulated by inquiries in the field of logic and mathematics*

The ideas originating in this

field are more liberal than the older positivism, and more compatible with the empiricism inspired by biology in the sense that absolute certainty as the criterion of knowledge is surrendered*

The investigations in the foundations of

logic and mathematics brought to the front the question of the meaning of knowledge*

It was concluded— incidentally

with much fanfare and horn-too ting— that no assertion was meaningful unless it could be brought into some significant, i*e*, specific and definable, connection with actual ex­ perience, and that this reference, whether explicit or im­

10 plied, constituted its meaning.

The interpretation of

Hsignificant connection11 has varied over a wide range of specifications.

The *operational!sm* of Bridgman —

originally developed as a requirement that the conditions of producing and measuring all physical quantities he com­ pletely specified— reduced all concepts, all meanings, their observable terms* trous consequences.

to

But such a reduction led to disas­

What, for example, could be the mean­

ing of statements referring to the future?

And what could

be the meaning of general inductive assertions?

The pos ­

sibility of knowledge is to be based on what we mean by knowledge, and just when we arrive at a seemingly usable theory of meaning we stumble upon the old bugbear of empir­ icism, the Principle of Induction.

Notable efforts have

been made to meet this and other criticisms® by liberalizing V j?~. W . Bridgman, The Logic of Modern Physics.(New York: The Macmillan Company, 1927). The ’operational* viewpoint was first formulated by the German mathematician Kronecker in his repudiation of all mathematical concepts which could not be defined or expressed in a finite number of steps or operations. ® For example, **But if it [i.e., Heisenberg’s view that quantities which have no direct relation to experiment ought to be eliminated) is taken as many have taken it"to mean the elimination of all non-observables from theory, it leads to nonsense. For instance, Schroedinger1s wave function is such a non-observable quantity, but it was of course later accepted by Heisenberg as a useful concept.11 Max Born, Experiment and Theory in Physics. (Cambridge: At the University Press, 1944), p. 18. [Expression not in origin­ al 2

11 the interpretation of 11significant connection11 and by the point-blank conclusion that it is impossible to 11Justify" ' the principle of induction because the very meaning of Jus­ tification is the providing of inductive evidence; the only other possible kind of justification being deductive dem­ onstration which, by assumption, isn*t applicable here.

In

this way the Principle of Induction becomes not a principle of knowledge but a rule of procedure. If you wish to discover reliable laws, you must try, try, and try again to generalize from a maximum of past experience and as simply as feasible. Then, if there is an order in nature, not too deeply hidden or too complicated, you will find it.9 The general theory of relativity gave a powerful Impetus to these ideas by showing what revolutionary con­ sequences followed from the rigorous application of these criteria of meaning to classical physical concepts.

Such

simple concepts as space and time measurement, simultaneity and rigidity were found to involve serious ambiguities. Radical revision In the way of meaningful definition was necessary before they could be used for Einstein*s purposes, the construction of a system whose laws would be independent of the motions of any given frame of reference. ^ Herbert FeiglJ. "Logical Empiric!smn , Twentieth Century PhilosophyT edited by Dagobert D* Runes (New York: Philosophical Library, 1943), p. 389.

12 The logical empiricist thus links meaning to 11defin­ able connection with experience11 and knowledge to meaning* If he is now asked: 11What is the significance of logical and mathematical reasoning for knowledge?'1 he will answer: "Logieal reasoning, of which mathematical reasoning is merely one kind, has to do only with the formal and logical properties of relations; it is concerned exclusively with the conclusions which can be drawn from propositions in virtue of their formal properties*

It has nothing to do with mat­

ters of fact and can provide no information either positive or negative with respect to factual matters.

If facts have

a certain form, certain conclusions may be deduced, but this is because of the form and not the facts.

And as to

whether the facts have this form or that is no concern of logic but of science.*1

If we wish we may speak of knowledge

of form as knowledge but it is of an altogether different kind and has nothing to do with knowledge of fact.

It is

purely analytic.

The vast accumulation of biological fact, classified but uncorrelated, was brought to life by the theories of evolution. cism.

These theories revivified a realistic empiri­

Central in this theory was the conception of a

plurality of beings joined in mortal battle for the goods which they individually required to sustain and secure

13 their continued existence.

All present structures and

functions--irrespective of their originating causes— must therefore give an accounting of themselves in terms of their biological prosperity value*

What, then, is the

biological value of sense-perception?^ If sense-perceptual — and rational, too, for that matter— processes have flourished because of their service in’ promoting the w e l ­ fare of the organisms which enjoy such capacities, what could that service be other than the communication of sig­ nificant information about the activities of other beings? But what of’the logic which denies any necessary connection between experience and hypothetical beings? a question of relative plausibility.

It is clearly

If the biological view

rings true, then it will be necessary to modify our concep­ tion of the meaning of knowledge,

The classic gauges of

knowledge were logical and mathematical reasonings which, with their irrefutable certainties, served for over two thousand years as the archetypal model of all knowledge. This classical specification must be abandoned.

Sense-

perception does not provide certain knowledge; nonetheless, it has something important to offer and deserves the name of knowledge*

The conclusions and evidence of experience are

not to be condemned because they do not live up to the standards of certainty which apply to logic. It is worth noting, remarks the empiricist, that the

use of mathematical systems as the perfect model of all knowledge was by no means a matter of chance* instinctively reach out for certainty*

Human beings

And in mathematical

knowledge this instinct found its most positive vindication and satisfaction*

Here was tangible proof that the desire

for certainty was not a hopeless passion which could be satisfied only in a world of dreams. tainty and security has deep roots*

The thirst for cer­ Life is a battle.

What

is more natural than to wish victory, not here and there and by luck, but always and certainly?

Whether it was such

a motive, or vanity, or a spiritual desire for ethical guidance which fathered and nursed man*s love of certainty is incidental to the fact that few instincts have such a powerful clutch on human feelings*and thoughts*

Its sor-

eery beguiles the most ruthless critic into lowering his guard and casts a hypnotic spell over the most rebellious and otherwise unfettered of minds*

It was this profound

predilection for certainty which caused knowledge to be identified with permanent conclusions, which demanded that our sources of knowledge be totally and comprehensively infallible*

Always, It seems, man teeters on the brink of

surrender to his instinct for finality.

Always he must be

on his guard against the temptation to abandon the search for better ways of dealing with the world in favor of some absolute generalization or dogmatic dictum to which all

^

15 future experience must necessarily conform.

Indeed, the

history of philosophy can best be described as one long effort to show that propositions are not merely signifi­ cant but necessary.

And in attempting to demonstrate this

“necessity11 the philosophers have displayed the greatest ingenuity In devising arguments subtly appealing to ethical, logical, aesthetic, causal and substantive instincts. Poking fun at the philosophers has been rare good sport since the days of Aristophanes, and so one might readily assent to the thesis that metaphysicians galore have suf­ fered from the malady. What of the scientists?

But are others free from taint? What of Laplace for whom the prem­

ises of mechanism were as self-evident as the dogma of any religion?

What of Helmholtz10 who laid down the dictum that

the only truly valid physical theory was that of central forces? The truth Is that not one of us is immune to the tug of this instinct for certainty and we must be perpetually on our guard against letting it prejudice our conception of the meaning of knowledge. . Perceiving and reasoning are means by which the human organism acquires Information about its environment.

We must begin, says the empiricist, with this

fact. Now If one asks the empiricist how sensing and reason-

ing perforin their services he will find that the ideas on this subject vary all the way from the crudest kind of empiricism for which sensing is nothing more than a mech­ anism for response to a stimulus, which hardly deserves to be called knowing, to a shadowy rationalism for which mental operations have not merely great freedom of analy­ sis and recombination but also powers of synthesis and fabrication. Those favoring the ideas of the old associationalistic psychologies look upon sensing as a process where­ by contiguous happenings in space and time impress them­ selves on the sensing organism*

If these contiguous

happenings are relatively uniform, that is, if they re­ peat themselves in a given order, this order of sequence will be hammered home on the organism, with the result that the habits of association formed will duplicate the sense-given, physical order,

in those cases where the

organism is capable of entertaining images of senseimpressions, the images will mirror the contiguous asso­ ciations given in the sense-impressions.

Thinking and

reasoning are thus defined as processes of automatically following the associations between images which have been stamped by a repeated correlation of sense-impressions, these, in turn, having been stimulated by the repeated contiguities of external happenings.

It is in this fashion

17 that the structure of things and events shapes and molds the course of sense and thought and the very form of mind itself.

The mind of wax, which faithfully preserves the

-

tracings of the recording instrument, the world, more or less deeply in proportion to the number of retracings, is the happy metaphor which has carried home the meaning of this psychology from the time of Plato*s Theatetus onward. In speaking of the genesis of the elementary mental categories William James gives a curt summary of this older view: . . . the first account of their origin which we are likely to hit upon is a snare. All these mental a f ­ fections are ways of knowing objects. Most psycholo­ gists nowdays believe that the objects first, in some natural way, engendered a brain from out of . their midst, and then imprinted these various congenitive affections upon it. But how? The ordi­ nary evolutionist answer to this question is exceed­ ingly simple-minded. The idea of most speculators seems to be that, since it suffices now for us to become acquainted with a complex object, that it should be simply present to us often enough, so it must be fair to assume universally that, with time enough given, the mere presence of the various objects and relations to be known must end by bringing about the latter*s cognition, and that in this way all mental structure was from first to last evolved. Any ordinary Spencerite will tell you that just as the experience of blue objects wrought into our mind the color blue, and hard objects got it to feel hardness, so the presence of large and small objects in the world gave it the notion of size, moving objects made it aware of motion, and objective successions taught it time. Similarly in a world with different impressing things, the mind had to acquire a sense of difference, whilst the like parts of the world as they fell upon it kindled in it the perception of similarity. Outward sequences

18 which sometimes held good, and sometimes failed, natu­ rally engendered in it doubtful and uncertain forms of expectation, and ultimately gave rise to the disjunctive forms of judgment; whilst the hypothetic form, 11if a, then b,M was sure to ensue from sequences that were in­ variable in the outer world. On this view, if the outer order suddenly were to change its elements and modes, we should have no faculties to cognize the new order by. At most we should feel a sort of frustration and con­ fusion. But little by little the new presence would work on us as the old one did; and in course of time another set of psychic categories would arise, fitted to take cognizance of the altered world.11 The importance of our experiences of spatial conjunctions and temporal sequences in determining our habits of transi­ tion from one thought to another is hardly to be questioned. Things habitually seen together and things habitually fol­ lowing each other are so linked in thought*

That water

quenches fire, and skunks expel a potent perfume, are asso­ ciations impressed by experience.

The first case— water

extinguishes fire— is an experience of sufficient frequency and invariability that with most of us it is difficult to imagine a pouring-on-of-water which did not diminish a fire. In the second case, the association of skunk with scent is more casual because they are not invariably related.

But it

is one thing to insist on the Importance for thought of ex­ perienced spatial and temporal conjunctions and it is another thing to claim that all the mind's connections are thus formed*

How account for those instinctive joinings

11 William James, The Principles of Psychology. (New York: Henry Holt and Company, 1890), Vol. II, pp. 629-630.

19 which could not have originated from individual experiences? A certain school of thought has met this criticism by in­ cluding in the tracings of experience those which were so deeply wrought into the lives of generation after gener­ ation of our remote ancestors as to have become a permanent habit of human thought*

This use of the Lamarckian theory

of evolution is stated with a clearness leaving nothing to be desired by Mr* Spencer: On the other hand the supposition that the inner cohesions are adjusted to the outer persistences by accumulated experience of those outer persistences is in harmony with all our actual knowledge of mental phenomena. Though in so far as reflex action and in­ stincts are concerned, the experience hypothesis seems insufficient; yet its seeming insufficiency occurs only where the evidence is beyond our reach. Nay, even here such few facts as we can get point to the conclusion that automatic psychical connections result from the registration of experiences continued for numberless generations. In brief, the case stands thus: It is agreed that all psychical relations, save the absolutely indissol­ uble, are determined by experiences. Their various strengths are admitted, other things equal, to be pro­ portionate to the multiplication of experiences. It is unavoidable corollary that an infinity of experi­ ences will produce a psychical relation that is in­ dissoluble. Though such infinity of experiences cannot be received by a single individual, yet it may be re­ ceived by the succession of individuals forming a race. And if there is a transmission of Induced tendencies in the nervous system, It is inferrible that all psychical relations whatever, from the necessary to the fortuitous, result from the experiences of the cor­ responding external relations; and are so brought into h a r m o n y .wi th them.^ Herbert Spencer, The Principles of Psychology (New fork: P. Appleton and Company, 1895), 3rd edition, Vol. I, p. 424.

20 Thus, the experience-hypothesis furnishes an ade­ quate solution. The genesis of instinct, the de­ velopment of memory and reason out of it, and the con­ solidation of rational actions and inferences into instinctive ones, are alike explicable on the single principle that the cohesion between psychical states is proportionate to the frequency which the rela­ tion between the answering external phenomena has been repeated in experience*^ The universal law that, other things equal, the cohesion of psychical states is proportionate to the frequency with which they have followed one another in experience, supplies an explanation of the socalled 11forms of thought*’, as soon as it is supple­ mented by the law that habitual psychical successions entail some hereditary tendency to such successions, which, under persistent conditions, will become cumu­ lative in generation after generation. We saw that the establishment of those compound reflex actions called Instincts is comprehensible on the principle that inner relations are, by perpetual repetition, organized into correspondence with outer relations.-1*4 Let it be granted that the more frequently psy­ chical states occur in a certain order, the stronger becomes their tendency to cohere in that order, until they at last become inseparable; let it be granted that this tendency is, in however slight a degree, inherited, so that if the experiences remain the same each successive generation bequeaths a somewhat increased tendency; and it follows that, in cases like the one described, there must eventually result an automatic connexion of nervous actions, corresponding to the external rela­ tions perpetually experienced. 15 But the older empirical program of reducing all possible mental associations, and all mental relating, to the repetition of spatial and temporal conjunctions of

13 Ibid. . p. 460. 14 Ibid., p. 466. 15

Ibid., p. 439.

21 sense-impressions comes to grief when it attempts to a c ­ count for the mental connections formed as a result of acts of comparison.

The discrimination, of likenesses and dif­

ferences leads to relations which are Independent of the order in which outer impressions are experienced.

A study

of the relations formed by distinguishing likenesses and differences brings out these points and questions: (1) Since any given experience presents a new situa­ tion, the factors in the situation which are contiguously related merely ”resemble*1 other factors similarly related in former situations. How, then, are the elements or impressions to be contiguously related picked out, identified and discovered? (2) Invariable correlation of similar impressions does not of itself lead to like permanent cor­ relation in thought. In particular, the mind is suspicious of habitual correlations occurring under circumstances Identified as special. (3) The true function of reasoning is not the dupli­ cation of directly-given space-time correlations, but the analysis and recombination of experience in the effort to discover or apply significant general relations. All too often these are hidden rather than revealed by experience. ■ (4) How does reason find these significant rela­ tions? And what is their meaning? (5) What are the principles guiding the discrimina­ tion of significant relations, for example, those of likeness and difference; and what is the rela­ tion of perception and thought to them, i.e., is the mind free to use them or lay them aside as it pleases and must experience embody their pat­ tern? If perceiving and reasoning have as their function the discovery of factors and relations, or agents and modes

22 of action, pertinent to the needs and interests of human beings, then these activities— sensing and reasoning— are not primarily designed to reproduce the actual nature of an absolute nature or an absolute set of events.

Sensing

and thinking are geared for action, perpetually alerted to seize opportunity by the forelock and are therefore selec­ tive and interpretative in operation.

They are not after

a picture post-card reproduction of the whole truth and nothing but the truth.

They are hot on the trail of ends.

Interests and purposes and instincts, then, will guide reason in its selections, in its seareh for general relations, in its hunt for pertinent likenesses and dif­ ferences.

But let us disregard the motives and methods

which the mind uses to find the connections which are used as a basis for action, for dealing with the world, and let us ask— as the empiricist requests us to do— what is the meaning and what is the criterion of acceptance or rejec­ tion of these connections?

It is the answer to this ques­

tion which differentiates the dyed-in-the-wool empiricist from the incorrigible rationalist.

The empiricist main­

tains that at the core of experience is a factual structure to which the connections of thought must conform if they are to be ture.

The mind will find itself dealing with

litigating interests and varying modes of interpretation but in the end the issues will be decided by the structure

23 of the given facts, not by pet interests and not by a mind which is built so that it can see things in only a certain way . For the rationalist the mind does not have a clean bill of health.

It is not a noble judge deciding without

bias the issue of what connections shall be used and pre­ served .

The human mind has certain characteristic ways of

viewing or conceiving things and It is anxious to bring the structure of the facts of sense-experience into harmony with these ways of thinking.

For the most part— as far as

the modern rationalist is concerned— the m i n d 1s predilec­ tions are not imposed or forced upon the structure of im­ mediate particular experience— as Kant required— but upon our effort to determine the general structure of experience* The epistemological method of investigation leads us to study the nature of the frame of thought, and so be forewarned of Its Impress on the knowledge that will be forced into it. We may foresee a priori certain characteristics which any knowledge con­ tained In the frame will have, simply because it is contained In the frame. These characteristics will be discovered a posteriori by physicists who employ that frame of thought, when they come to examine the knowledge they have forced Into it. Procrustes again! These foreseeable characteristics are not by any means trivial; they are laws or numerical constants which physicists have been at great pains to deter­ mine by observation and experiment. As an example we may take the law of Increase of mass with velocity, which has been the subject of many famous experiments* It is now realized that this law automatically re-

24 suits from the engrained form of thought which sep­ arates the four-fold order of events into a three­ fold order of space and an order of time. When knowl­ edge is formulated in a frame which compels us to separate a time dimension from the four-fold order to which it belongs a component called the mass is cor­ respondingly separated from the four-fold vector to which it belongs; and it requires no very profound study of the conditions of separation to see how the separated component is related to the rest of the vector which prescribes the velocity. It.is this relation which is rediscovered when we determine experimentally the change of mass with velocity.15 For a scientific outlook I think the most funda­ mental of all forms of thought is the concept of analy­ sis. This means the conception of a whole as divisible into parts, such that the co-existence of the parts constitutes the existence of the whole.17 It is clear that the concepts of analysis as applied in physics must have been specialized according to some guiding principle; otherwise there would not be the same general agreement as to the products of analysis of the physical world, namely molecules, atoms, protons, electrons, photons, etc. There is another engrained form of thought which has selected the system of analy­ sis to be applied in physics. I will call this special­ ization of the concept of analysis the atomic conceptfr or for greater precision the concept of Identical structural units. . . . I will go farther and say that the aim of the analysis employed in physics is to resolve the universe into structural units which are precisely like one another.15 Granting that the elementary units found in our analysis of the universe are precisely alike intrin­ sically, the question remains whether this is because we have to do with an objective universe built of such

Sir Arthur Stanley Eddington, The Philosoohy of Physical Science, Tarner Lectures, (Cambridge: At the University Press, 1939), p. 9. 17 Ibid.. p. 118. 18 Ibid., p. 122

25 units,' or whether it is because our form of thought is such as to recognize only systems of analysis which shall yield parts precisely like one another. Our previous discussion has committed us to the latter as the true explanation. We have claimed to be able to determine by a priori reasoning the properties of the elementary particles recognized in physics— properties confirmed by observation. This would be impossible if they were objective units.. Accordingly we account for this a priori knowledge as purely subjective, revealing only the impress of the equipment through which we obtain knowledge of the universe and deducible from a study of the equipment. We now say more ex­ plicitly that it is the impress of our frame of thought on the knowledge forced into the frame.19 Clearly, then, for Sir Eddington the mind does not get at

the nub of things by emptying itself of preferred ways

of connecting, and by conforming to a HglvenH , ”factual" pattern.

The mind makes sense of the world indicated by

sense-perception and traces out its general patterns by bringing in its own techniques and systems for handling things. It is not easy to draw a fine line between modern rationalists and modern empiricists so that all are either on one team or the other.

There is a kind of pragmatic

rationalist with strong empirical sympathies who regards the mind as having certain engrained structures, native predispositions, which are not imposed upon experience willy nilly but are applied to experience because expe-

19

Ibid., p. 125.

26 rienoe "accepts" them.

This is the view of G. I.

L e w is

:^0

Our categories are guides to action. Those attitudes which survive the test of practice will reflect not only the nature of the active creature hut the general character of the experience he confronts* Thus, indirectly, even what is a priori may not he an exclusive product of "reason” , or made in Plato's heaven in utter independence of the world we live in. If philosophy is the study of the a priori, and is thus the mind's formulation of its own active attitudes, still the attitude which is the ob­ ject of such a study is one taken toward the content of an experience in some sense independent of and hound to he reflected in the attitude itself. What is a priori— it will be maintained— is prior to ex­ perience in almost the same sense that purpose is. Purposes are not dictated by the content of the given: they are our own. Yet purposes must take their shape and have their realization in terms of experience; the content of the given is not irrelevant to them. And purposes which can find no application will disappear. In somewhat the same fashion what Is a priori and of the mind is prior to the content of the given, yet in another sense not altogether independent of experience in general. Still another theory— even more empirical— looks upon the mind's Insights as "mental truth" as opposed to "real truth", using Locke's terminology.

The mind has

intuitions and preferential ways of thinking which are experimentally applied to experience.

If experience bears

out the application all is well, a happy accord between mind and nature has been found; if the fit is poor it is not a case of "so much the worse for experience", it is a

C. I. Lewis, Mind and the World Order (Hew York: Charles Scribner's Sons, 1929), p. 21.

21 Ibid. p. 24.

27 case of uso much the worse for the intuition11.

William

James was attracted by this way of thinking. Only hypothetically can we affirm intuitive truths of real things— by supposing, namely, that real things exist which correspond exactly with the ideal sub­ jects of the intuitive propositions. • go If our senses corroborate the supposition all goes well. But the intuitive propositions of Locke leave us as regards outer reality none the better for their possession. We still have to “go to our sense's** to find what the reality is. The vindication of the Intuitlonist position is thus a barren victory. The eternal verities which the very structure of our mind lays hold of do not necessarily themselves lay hold on extra-mental being, nor have they, as Kant pretended later, a legislating character even for all possible experience. They are primarily interesting only as subjective facts. They stand waiting in the mind forming a beautiful ideal net­ work; and the most we can say is that we hone to d i s ­ cover outer realities over which the network may be flung so that ideal and real may coincide.23 e

It is the true empiricist speaking:

The mind has its dreams

but the proof of the pudding is in the eating.

It is worth

noting that the logical empiricist adopts an attitude toward mental constructions which had several advantages over that of James.

The m i n d ^ constructions are a priori, i.e.,

■independent of experience, but they are purely analytic. There is thus no ornery question of how the mind has come to be shaped in some peculiar way, as would be the case if the b. priori ideas were synthetic.

22 23

James, o p . cit., Vol. II, p. 664.

Ibid., pp. 664-665.

And the

28 mystery of why a happy harmony between mind and nature should exist is also side-stepped. cist there is only one question:

For the logical empiri­ Are the relations between

the items of experience under consideration of this type or that type?

And this question is one to be decided by

actual observation? The notion of certain qualifying conditions or pre­ requisites which the mind*s interest sets up— but which may or may not be fulfilled— is nowhere so clearly stated 04

as by Helmholtz

.

in his famous paper on the conservation

of energy. Once for all, it is the task of the physical sciences to seek for laws by which particular proc­ esses in nature may be referred to general rules, and deduced from such again. Such rules (for example the laws of reflection or refraction of light, or that of Mariotte and Gay-Lussac for gas volumes! are evi­ dently nothing but generic-concepts for embracing whole classes of phenomena. The search for them is the business of the experimental division of our Science. Its theoretic division, on the other hand tries to discover the unknown causes of processes from their visible effects; tries to understand them by the law of causality . . . The ultimate goal of theoretic physics is to find the last unchanging causes of the processes in Mature. Whether all processes be really ascribable to such causes, whether, in other words, nature be completely intelligible, or whether there be changes which would elude the law of a necessary causality, and fall into a realm of spontaneity or freedom, Is not here the place to determine; but at any rate it is clear that the Science whose aim it is to make nature appear intelligible £die Natur zu

^ Hermann-von Helmholtz, On the Conservation of Energy (Berlin: Berlin Physical Society, G. Reimer, publisher, 1847), Introduction.

29 begreifen must start with the assumption of her i n ­ telligibility, an4 draw consequences in conformity with this assumption, until irrefutable facts show the limitations of this method. . . . The postulate that natural phenomena must be reduced to changless ulti­ mate causes next shapes itself so that forces unchanged by time must be found to be these causes. Now in Science we have already found portions of matter with changeless forces (indestructible qualities), and called them (chemical) elements. If, then, we imagine the world composed of elements with inalterable quali­ ties, the only changes that can remain possible in such a world are spatial changes, i.e., movements, and the only outer relations which can modify the action of the forces are spatial too, or, in other words, the forces are motor forces dependent for their effect only on spatial relations. More exactly still: The phenomena of nature must be reduced to [zuriickgeftihrtf conceived as, classed as] motions of material points with unalterable motor forces acting according to space-relations alone* . . . But points have no mutual space-relations except their distance, . . . and a motor force which they exert upon each other can cause nothing but a change of distance— i.e., be an attractive or a repulsive force. . . . And its inten­ sity can only depend on distance. So that at last the task of Physics resolves itself into this, to refer phenomena to inalterable attractive and repulsive forces whose intensity varies with distance. The solution of this task would at the same time be the condition of Nature*s complete intelligibility.25 Theoretical natural science therefore, if she does not rest contented with half-views of things, must bring her notions into harmony with the expressed require­ ments as to the nature of simple forces, and with the consequences which flow from them. Her vocation will be ended as soon as the reduction of natural phenomena to simple forces is complete, and the proof given that this is the only reduction of which the phenomena are capable.25 25 IbldTf translated by W. James, op. cit. . pp. 667-

668.

oc.

Ibid*, translated by J. Tyndall, ”The Conservation of Force” , Scientific Memoirs (Natural Philosophy), I, pp. 114 seqq. as cited by Leo Koenigsberger, Hermann von Helm­ holtz . (Oxford: At the Clarendon Press, 1906) p. 42.

30 In the light of this brief survey of the problem of knowledge I should like to define as clearly as possible the attitude from which I have approached the special problems posed in this thesis. There is a world of Interacting processes which are inter-connected in varying degrees of intimacy and effec­ tiveness.

A process Is any activity which has sufficient

vitality or unifying power to modify the behavior or character of the parts which make up any definably whole action.

Putting it more generally, a process is any hap­

pening in which the behavior or quality of any factor, such as thing or event, is controlled, directed or modi­ fied by some tendency, rule or agent.

This conception of

"process" is copied from our human experiences of striving and intending.

When extended to a "world of processes"

it is being used hypothetically and the manner and extent of its application must be determined experimentally. Physical actions are those which are primarily con­ trolled by the principles of the over-all physical proc­ esses*.

Living actions are those directed by some rel­

atively Individual and separate process.

Different

processes exhibit varying degrees of completeness or in­ dividuality, complexity and vitality; none is wholly in­ dividual in the sense that it is completely Isolated from

i

31 all other processes or in the sense that the control of the subordinate elements is perfect. The human organism is a complex of intimately a s ­ sociated processes*

One of these over-all processes is

termed the body process and another is termed the mental process or self*

The basic relation between the mental

process and the body process is that of expediting the interaction of both processes with the world-processes. This function Is expressed by acts and affections of the mental process termed sensing.and feeling.

While in­

timately involved with the mental process most sensings and feelings are vitally connected with other processes, di­ rectly with body processes, and more indirectly with other processes*

The most striking feature of the connection

between sensing and outer processes is its indicative and evaluative function*

Sensing points out the presence of

the other process and assesses its significance, which sometimes is its Immediate momentary value, but ordinarily its significance lies in that to which it may lead. This betokening and guiding activity termed sensing is not an act of absorption into the essence of the ex­ ternal process; it is not an intuition of its inner nature. Sensing is oriented in the direction of possible action and cannot be separated from a total mental process of which it is a part*

What, then, is the relation of sensing to

32 external processes and to the general mental process, how does it make its indications and its valuations? The pertinent aspects of the outer process are its presence and its structure, including its external rela­ tions or mode of behavior.

These are the aspects grasped

in sensing. Certain special body processes are directly con­ nected with special modes of sensing, fe.g., the eye to seeing.

As with outer processes we can determine the

structural relations of sensing to these body processes, but we are in the dark as to their nature.

The quality

of sensing is undoubtedly conditioned by the character of their functioning. The structure of outer processes is expressed in sensing by a correlation of differences in qualities sensed with external differences of structure.

The over-all men­

tal process is vitally involved in this connection, which s

is partly a matter of instinct, education, ideas and in­ terests, including purposes and physiological needs. Mormai response by a human being to an act of sensing is a relatively automatic adjustment along the lines indi­ cated by the pattern and quality more or less integrally given in the sensing.

Depths and relative sizes, for ex­

ample, are evidently a matter of education and yet they are integrally present in the sensing.

The savoriness of the

33 merely seen apple is less integral than_ its seen but not felt weight. When normal response is inhibited, when a problem is posed, the tendency is to break up or analyze the experiences or sensings at hand and by a process of dis­ crimination identify other situations pertinent to the present problem.

This process, in effect, leads to a new

kind of sensing.

It is a sense for general relations.

The

process of discriminating and applying general relations is called reasoning.

This kind of sensing is capable of

varying degrees of generality.

For the most part the rela­

tions are contextually, though not concretely, sensed. There is a third kind of sensing which has to do with the perception of the purely formal properties of relations and their logical Interlocking apart from their embodiment in contexts or objects.

This is logical or deductive sensing

and is characteristic of logical reasoning and of, at least some, perhaps all, mathematical reasoning. We have, in effect, distinguished three kinds of sensing.

Concrete sensing or sense-perception, contextual

reasoning— sensing or discriminating of general relations between objects— and abstract reasoning, sensing of the formal properties of relations and their mutual implica­ tions apart from any terms. In all three Instances the sensing is of significant

54 relations*

In the first two cases the sensed connections

are enmeshed with given objects*

In the third, only an

abstract object is involved, i*e*, the object having the structure required by the relations*

In the first case

the sensing of connections is intuitive;

this relation

naturally and inevitably belongs to this object*

In the

second case the sensing of connections is both intuitive and deductive, and in the third case it is deductive. The same questions we have raised with respect to the first kind of sensing are relevant to these other kinds. What is their function? processes?

What is their relation to external

What governs the kind of connections made?

What

are the (1) modes of discovery and (2) the criteria of truth or worth?

What is their relation to mental and

bodily processes?.

What are their relations to each other? As to Function

The Ingrown connections of concrete sensing— Euclidean space, for example— , the looser connections ac­ quired by experience, and insights of instinct and intui­ tion serve very well for the needs of local action.

They

are inadequate, as is to be expected Inasmuch as they are tied down to local conditions and objectives, for the pur­ pose of developing a system of connections which will en­ able more comprehensive and accurate predictions.

1

35 The great problem In the execution of this purpose is to break through the obvious structure of Immediate sense-experience, the form of the local a priori, to those subtle identities or relations which are the clues to usable, significant generalizations.

The function of con­

textual reasoning is to unearth and apply these generaliza­ tions.

The function of abstract reasoning is (1) to provide

or indicate the possible systems available for the expres­ sion of the structure of concrete sensing in ways leading to the best predictions and (2) to suggest, by the transla­ tion of various aspects of experience into its forms, hidden clues to general connections. Principle of Forming Connections (1)

4

bl

12. discovery.

Our theory is that general

relations are found by abstract reasoning on the properties of abstract relations arid objects,

e.g., Weierstrass1 con­

tinuous curve with no tangents, and by contextual reason­ ings and intuitive insights which are, for the most part, analogical in character.

The sensing of general relations

in this latter case.is a process of Intuitively perceiving, with more or less convincingness, that it is natural, a p ­ propriate, etc., for a thing to be so and so related and to have such and such properties. ' In this way the percep­ tion of general relations is governed and guided by the

36 m i n d !s dispositions, interests, and capacities.

Perhaps

the most elementary form of such insights are those gen­ eralizations issuing from a repetition of significant contiguities in space and time.

The mere happening

together of two events on several occasions tends to engen­ der in the human mind a feeling that they should always happen together.

The association of the two repeated events

becomes endowed with a sense of fitness and inevitability. A more important means of comparing and contrasting in ways leading to a better connecting of experience is by anal­ ogical insights based on the connections supplied by other experiences and other ideas.

Imagine the case of a

primitive savage who knew of no method for putting out fires either than that of letting them die out.

If he accidental­

ly were to spill his pot of water on the fire, thus ex­ tinguishing it, on several occasions, he would be led to conjecture that water would always put out fire.

This,

of course, is a low-grade form of association by contiguity. But supposing he forms the theory that the important rela­ tion between water and fire is not "water on fire equals dead fire", but one of mutual hate.

This analogical

generalization would lead him to look for other things which might hate water, such as sand, dirt, etc., a very Important result indeed. The development of abstract systems is also guided

37 by intuitive insights*

Numerous propositions were formed

be deductive analysis to be sure; but many propositions, particularly in the earlier stages of abstract reasoning, were developed by contextual or intuitive insights of an analogical character*

Consider, for example, the role of

imaginative spatial constructions in the discovery of geo­ metrical propositions* (2)

As. to. Acceptance*

generalization is good prediction.

The criterion of empirical

The criterion of log­

ical or abstract insight is consistency. Significance with Respect to External Processes The criterion of scientific generalization is p r e ­ diction, but according to the view we are advocating good predictions indicate a significant relation between the structure of external processes and the structure of the system interpreting experience.

And the more accurate and

comprehensive the predictions which the system enables, the more accurate is the correspondence between the two struc­ tures* Contextual insights lead to important connections for the purposes of prediction.

But the tendency, in these

cases, is to divert attention away from the true criterion, prediction, and the true significance,

structure, to the

intuited situation, image or analogy associated with the

38 generalization.

This leads to dogmatism.

The Daltonian

atom, for example, diverted attention from the structures which it was used to explain and predict.

Phlogiston per­

formed the same role in the theory of heat. The criterion of the insights of rational deduction is consistency. We have tried to indicate in the foregoing discus­ sion some of the more important views about the problem of knowledge, and we have endeavored to define our own postion in the matter.

We are now ready to ask:

“What mean­

ing will these various theories assign to those particular branches of knowledge called ematical Science‘t11

‘Physical Science1 and ‘Mat h ­

The answers can most simply be gotten

by asking of each in turn the following questions: (a) What is Physical Science? (b) What is the ultimate significance of physical propositions and generalizations? (c) How are these propositions discovered? (d) What is Mathematics? ( e) What is the ultimate significance of Math­ ematical propositions and generalizations? (f) How are these propositions discovered? Traditional Empiricism (a)

The subjects of physical science are the laws

of nature. (b)

The order of association of things in time and

space is duplicated in the order of sensations.

Therefore,

39 the spatial and temporal associations of sensation, repro­ duced in the connections between impressions, accurately represent the actual spatial and temporal conjunctions of things, i.e., the laws of nature, (c)

The means of discovery is passive observation

and passive reflection*

The constant repetition of spatial

and temporal conjunctions of things reported in sensation so drives home in our mind the external order of things that our trains of thought, our reflective processes of association, automatically reproduce them.

The connections

apparently discovered by our own mental Industry, our reflective and discriminative processes, have actually been molded in the image of naturefs own structure by nature herself.

Indeed all the connections present in the human

mind have arrived there in the same fashion, for those which have not been ingrained by nature from the time of birth— ice is cold, fire is hot— have been woven into the fibre of the human mind through incessant repetition over many genera­ tions.

Experience is therefore the ultimate manufacturer

of that space and time structure we call physical science. (d)

The subjects of mathematics are the laws of

arithmetic and geometry. (e)

The laws of arithmetic and geometry are generali­

zations of experienced regularities. (f)

Number is an abstraction derived from our ex-

perlences of collections.

Geometrical figures are abstrac

tions derived from our experiences of shapes and sizes. The laws of arithmetic are generalizations derived from our experiences of handling units and collections*

The

laws of geometry are generalizations derived from our experiences in manipulating figures. Logical Empiricism (a)

The subject of physics is the laws of sense-

experience. (b)

The laws of sense-experience are the space-

time structures of sense-experience;

they have a prob­

ability value less than certainty, i.e., less than one. (c)

They are found by a process of intelligent

experimentation and observation using the method of induc­ tive inference.

Hypotheses are proposed and- then investi­

gated under conditions designed to give a high degree of probability if verified. (d)

Mathematics is a branch of logic.

The proposi

tions of mathematics can be deduced from purely logical concepts, by purely logical rules. (e)

Mathematics is analytic and a priori.

tautological.

It is

It is concerned purely with the formal

structure of a logically given set of relations.

It is

independent of experience, being, like logic, concerned

41 with form and not fact* (f)

Mathematical propositions are discovered by

analysis and by intuition, but all proof is analytic* Rationalism (a)

The subject of

physics isthe laws of nature*

(b)

The laws of nature represent the order of the

objective universe* (c)

The laws of nature are discovered by reason;

either directly by means of reason*s own insights or by inferences from experience;

these inferences being based

on principles of reason, e.g., cause and effect, substance and attribute, which are objectively valid, (d) properties (e)

Mathematics is

a study ofthe structure and

of our ideas of number and dimensionable order, Mathematics is synthetic and a priori as to its

origins and initial premises*

Mathematics is differentiated

from logic by ideas which are uniquely mathematical, number and the mathematical continuum.

e.g.,

These ideas cannot

be defined in any terms other than themselves. (f)

Mathematical propositions are discovered by

analysis and by intuition*

Proof may be either analytic,

i.e., deductive, or it may be intuitive*

In the former

case demonstration rests on a logical connection between the proposition demonstrated and certain premises.

In the

42 latter case demonstration consists of an appeal to innate . mathematical ideas or to such possibilities as that of mathematical Induction* The Conjectural or Archimedean Theory (a)

The purpose of physical science is unique and ..

comprehensive prediction of sense-experience.

Physical

science is any system of connections conscientiously used for this purpose. (b)

It is assumed that there are independent phys­

ical processes.

One of the basic functions of sense-

perception is to put us in touch with these processes*

It

is assumed that the system of connections enabling the making of comprehensive and unique predictions bears some significant relation to the structure of physical processes. Since the relation obtaining between particular senseperceptions and the conceptual system of general relations used for their prediction is not one of necessity but of conjecture, the probability value of a predictive pro­ position is less than certainty.

Consequently, the knowl­

edge of physical processes, based on the presupposition of a structural correspondence between the schemes of relations used for prediction and the systems of physical processes, is also conjectural. (c)

The connections utilized in physical science are

43 drawn from (1) concrete sense-perception, (2) instinctively formed perceptions and associations,

(3) experience, (4)

imaginative combination of relations and rational discrim­ ination of general relations, and (5) intuitive insights of an analogical order, e.g., those suggested by mathematical theories• (d) Mathematics is evidently either a study of the structure and properties of intuitive ideas of number and dimenslonable order, or a study of the logical coherence of systems derived from more or less arbitrarily pos­ tulated definitions. (e) Mathematical propositions, whether based on rational intuitions or arbitrary definitions, have no nec­ essary connection with sense-experience.

Hence, their

ultimate foundation is either in rational intuition or logical intuition.

Those favoring the theory of rational

intuition seels: to exhibit an ob.lect of Intuition having the properties required by the system in question.

Those

favoring the theory of logical intuition seek to reveal the self-consistency of the consequences logically Implied by the postulated objects.

But in either case, whether the

basic premises are revelations or conjectures, the govern­ ing criterion is logical consistency. (f) Mathematical ideas have been drawn from (l) concrete sense-perception,

(2) Instinctive associations,

44 (3) associations learned or acquired by experience,

(4)

perception of abstract connections, as such, and of their possible general relationships, and (5) intuitive insights of an analogical type.,

e.g., those suggested by the for­

mulations of physical science.

PART I I

THE TWO PRINCIPAL SOURCES OP THE IDEAS FORMING- THE SCIENCE OF MATHEMATICS AND PHYSICS SECTION I The role of the fundamental human resources, i*e., physiological structures, mental dis­ positions and basic experiences, in shaping early ideas of physics and mathematics, and the utility of the symbolic system of math­ ematics for physical science*

CHAPTER I I

THE FOUNDATIONS OF MATHEMATICS AND PHYSICS A. The Problem.

Geometry

Is geometry the work of the mind, of

the sense, or of physical experience?

Or does It have some

other basis not indicated by these possibilities?

If it is

of the m i n d fs essence then it is here that we should Introspectively look for the hidden artisan shaping the destiny of geometry.

The history of geometry, in this case, will

merely be the writing down of the master plan.

One In

possession of this plan could anticipate all those inven­ tions laboriously grubbed out of experience.

And, of course,

if used for the expression of physical relations, geometry will subtly endow these physical modes with Its own unique qualities* If geometry represents a form of sensation, then It will be the character and nature of our sensations which account for our geometry and which give content to its propositions.

Comparative inspection would be the appro­

priate method of inquiry for this theory. If geometry is the work of physical experience, then mathematical propositions will represent the more general laws of nature; and the proper method of inquiry will be

47 theoretical speculation and experimental questioning* Or is mathematics a relatively free conceptual enterprise which, being abstract and devoid of content, is not constrained to reflect any character, either of mind, sense, or physical nature?

If the foundations of mathe­

matics are arbitrary then the certainty of its demonstra­ tions must rest on its self-consistency, i.e., on its tautological character. The pertinence of our common and ancient heritage. Ordinarily mathematics and physics are not recognized as sciences until their devotees display some degree of deliberateness and persistence in their investigations.

The

achievements of the early trail-blazers, however, ought not tofbe discounted.

Their efforts, to be sure, show ^

little flair for generalization and not much heart for toeing the ideal mark in their respective disciplines, good predictions for the one and logical demonstration for the other; which, taken together, are the heart, as generaliza­ tion is the soul, of these two sciences. Although our primary interest Is in the reciprocal suggestiveness of these two sciences in matters concerning generalization after they have become subjects of a pur­ posive investigation marked by some awareness of their true standards, we do not believe that this program could be adequately carried through without taking into account the

48 character of our ancient heritage*

But even the ancient

heritage had a beginning, and the roots feeding this be­ ginning lie in some common element shared alike by the primitive and the sophisticated speculator.

This common

element fertilizing and perhaps molding each m a n ’s physical and mathematical explorations requires identification, for here is a factor which might be persistenly influencing the direction of m a n ’s search for physical and mathematical connections# It will be proper, therefore, to begin the main sub­ ject of our thesis with an analysis of this common element, which might at any moment in the history of mathematical and physical generalization be expected to guide the m i n d ’s eye to some overlooked possibility, some unilluminated connection.

We shall follow this analysis with a summary

of how the ancient methods of prediction, i.e., ancient science, repaid the favors they had received from mathe­ matics.

The main discussion opens with the official begin­

ning of science at the hands of the Greek scientists. Why is Soace Homololdal? The world in which we live and breathe and have our being is spatial and flat.

This is true for both the un­

tutored savage and the most sophisticated intelligence. Why is this?

(1).

Is it that our mental activities have a

49 puritanic cast which requires that things he connected in certain definite ways, thereby forcing upon our seeings and touchings a certain spatial frame?

(2)

Or are mental per­

ceptions free, leaving sense-perceptions yoked to the flat frame?

t>3)

Or, finally, is flat space a product of ex­

perience, a result of the kind of school we find ourselves attending? Now if flat space is a given form of mental intuition then the mind could not conceive of any other kind of space. And likewise if flat space is an organic form of senseperception then we should not find it possible to picture any other form of space.

But if flat space is a matter of

our learning how our sensations go together, i.e., the rules of their ordering and measuring, then it should be possible for us to represent alternative or contrary rules of con­ necting sensations, thereby imagining other kinds of spaces* What are the facts of the case?

Thanks to the work

of such men as G-auss, Bolyai, LobatchesWky, and Hiemann we can pass definitive judgment on the first case at once. The special features of flat space are given in E u c l i d ^ axioms: (1)

Through two points no more than one line can pass.

(2)

The straight line is the shortest path b e ­ tween two points.

50 (3)

Through a given point lying without a given straight line only one line can be drawn parallel to the first* Two straight lines that lie in the same plane and never meet no matter how far extended being termed parallel*

Other Euclidean axioms refer to the continuity of space and the number of dimensions.

A solid is bounded by

a surface, a surface by a line, and a line by a point* The point is indivisible*

By the movement of a point a

line is described and by the movement of a line a line or a surface is formed, and by the movement of a solid a solid and nothing else is generated. How for conception the criterion of possibility is consistency*

If a definition or a set of postulates taken

in conjunction does not lead to self-contradiction then for the mind these propositions are conceivable.

If any space

other than Euclidean is to be inconceivable by the mind, the denial of any one of the axioms defining Euclidean space should lead to contradictions.

The Italian Saccheri was

evidently the first to attempt to find such contraditlons by denying the famous axiom of the parallel.

Such was his

Euclidean prejudice that he succeeded in convincing him­ self that he had actually been led into absurd conclusions by denying this axiom.

It remained for Gauss and Lobat-

chewsky to show that the denial of the axiom of the par­ allels and the assumption of Its opposite, that through a

51 given point external to a given straight more than one parallel can be drawn to the given straight line, did not lead to contradiction, but on the contrary led to a new geometry in no way inferior to Euclidean geometry from the standpoint of logical consistency* Nor is this all*

Riemann developed a system of geo­

metry which abandoned both the axiom of parallels and the axiom that only one straight line can pass through two points* Now it might be objected that a sufficient expansion of the consequences of the non-Euclidean geometries would lead to contradictory theorems.

But Riemann1s geometry

for the two dimensional,case does not differ from spherical geometry which is a department of ordinary geometry. Beltrami, following much the same procedure, vindicated Lobatshewsky1s geometry by correlating its two dimensional case with ordinary geometry. The firm hold which flat space has upon our percep­ tual intuitions cannot, then, be laid at the door of a miserly mind which can conceive of only one kind of space* But different as are the three geometries they, nonethe­ less, have important features in common* dimensional. isotropic.

All are three

In all, space is continuous, homogeneous and For all, space is indifferent to motion, i.e.,

rigid displacement is possible*

Shall we ask the same

questions all over again?

Have we at last come upon in-

evadable necessities of thought or of perception, or have these features been capitalized out of the earnings of experience?

What it amounts to is this:

If we cannot

blame our preference for a special kind of space upon our mental disposition, might it still be true that our mind is responsible for the general fact that sense-experience is spatial?

But, first, let us carry on with our discussion of

Euclidean geometry. Is flat space a direct, integral character of senseintultions, in which case we will have hit bed-rock and can carry our analysis no further, or is flat space the result of correlating our various sense-impressions in ways we have learned from countless, perhaps ancestral,

experiences?

If the former case is true then we should not be capable of representing or imagining any other kind of space, for this would mean calling up or creating impressions which we had no means of constructing— it would mean turning the skin of experience inside out*

It is the old story of one who

had never seen the color red trying to imagine it. sense-intuitions would have no way of producing it.

His But

if the latter theory is true, then, unless the grip of habit is too vise-like, we should be able to conjure up different series of impressions from those to which we are normally accustomed— those which we have been educated to

55 make— and in this way represent to ourselves new spaces. The issue is serious and there are notable arguments for both sides.

Both seem agreed that sensible space is

to be characterized as having dimension and to be capable of quantitative comparison, i.e., measurement.

But for the

one side dimension is primarily a quality, a directly sensed extendedness, whereas for the other side dimension is a question of ordered impressions. C. D. Broad1 vigorously presents a modified version of the view that spaciousness is directly sensed.

He

argues that depth as well as surface is a directly sensed quality of sensations*

(Broad distinguishes between an

act of sensing and the object sensed, which he calls a sensum.

It is the sensum which is colored and extended

not the process of sensing.)

He freely acknowledges the

roll which experience plays in developing a mature p e r ­ ception of depths and relative distances, etc., but he is convinced that the mere correlation of other senseImpressions, kinaesthetic,

etc., would not have the power

to create a new quality of depth in a visual sensum originally barren of such a quality.

2

The connection of

^ C. D ; B r o a d , Scientific Thought (Hew York: Harcourt, Brace and Company, Inc., 1925), Ch. IX.

2 Ibid., p. 296.

54 visual sensations with other sensations by experience pro­ foundly modifies the visual perception of dimension; but where only two dimensions previously existed three dimen­ sions could not be called into being by mere association, with some other kind of sensing,

e.g., tactual.

quality of depth must be a native inhabitant. to Broad*s^

The According

theory the maturing and education of spatial

perception is to be accounted for by supplementing the physiological conditions of perception with certain mental traces which represent the summarized lessons of past ex­ perience.

For Broad,^ then, solidity and depth, like shape

and size, are given, primitive characteristics of visual sensa.

And the function of the traces left by past visual

and tactual experiences is to determine the special form of solidity and depth which any given perception is to have. Depth is not a perception of flatness combined with a judg­ ment about former or anticipated tactual sensations.

It is

not a sensation of flatness plus associated impressions of past or future tactual sensations.

The traces left by past

tactual sensations do not act upon the present visual per­ ception by calling up past tactual images, but by modifying the quality of the visual sensation.

^ Ibid.7 p. 294.

4 Ibid.. p. 293.

55

The contrary view has powerful advocates: among their number appear such names as Helmholtz, Wundt, and Poincar&. Helmholtz5 and Poincarfe® drew their arguments from three principal sources, (1) the general notion of a con­ tinuum and the conditions which must be fulfilled to d e ­ rive from this a spatial continuum.

(2) Physiological and

psychological facts and (3) Imaginative construction of the series of sensible impressions which xtfould be obtained by intelligent creatures dwelling on spheres and pseudospheres. I have already indicated the significance of alter­ native construction.

If our sense-intuitions have by

nature the properties of flat space, then it would be im­ possible voluntarily to change these intuitions so as to picture spherical or pseudo-spherical space.

But if our

so-called intuition of flat space is a result of the con­ nections we have learned to make among series of impre"k ‘ Hermann Ludwig Ferdinand von Helmholtz, Treatise on Physiological O p t i c s .translated from 3rd German edition, James P. 0. Southall, editor (Rochester, Hew York: The Op ­ tical Society of America, 1924), and Hermann von Helmholtz, Popular Lectures on Scientific Subjects (London: Longmans, Green and Company, 1898), two volumes. Henri Poincarfe, The Foundations of Science. George Bruce Halstead, authorized translator (New York and Garri­ son, New York: The Science Press.

56 sions, then we have only to imagine a different rule of connecting our impressions in order to produce an "intuition11 of another kind of space*

H e l m h o l t z ^ two dimen­

sional construction of spherical and pseudo-spherical space is classic and deserves quotation in full: Let us, as w e may logically say, suppose reasoning beings of only two dimensions to live and move on the surface of some solid body* We will assume that they have not the power of perceiving anything outside this surface, but that upon it they have perceptions similar to ours* If such beings worked out a geo­ metry, they would of course assign only two dimen­ sions to their space* They would ascertain that a point in moving describes a line, and that a line in moving describes a surface. But they could as little represent to themselves what further spatial construction would be generated by a surface moving out of itself, as we can represent what would be generated by a solid moving out of .the space we know. •

•

•

Our surface-beings would also be able to draw shortest lines in their superficial space. These would not necessarily be straight lines in our sense, but what are technically called geodetic lines of the surface on which they live; lines such as are described by a tense thread laid along the surface, and which can slide upon it freely. I will hence­ forth speak of such lines as the straightest lines of any particular surface or given space, so as to bring out their analogy with the straight line in a . plane* I hope by this expression to make the con­ ception more easy for the apprehension of my nonmathematlcal hearers without giving rise to mis­ conception. Now if beings of this kind lived on an infinite plane, their geometry would be exactly the same as our planimetry. They would affirm that only one straight line is possible between two points; that through a third point lying without this line only one line can be drawn parallel to it; that the ends of a straight line never meet though it is produced to infinity, and so on. Their space might be in-

57 finitely extended, but even if there were limits to their movement and perception, they would be able to represent to themselves a continuation beyond these limits; and thus their space would appear to them infinitely extended, Just as ours does to us, although our bodies cannot leave the earth, and our sight only reaches as far as the visible fixed stars. But intelligent beings of the kind supposed might also live on the surface of a sphere. Their shortest or straightest line between two points would then be an arc of the great circle passing through them. Every great circle, passing through two points,is by these divided into two parts; and if they are unequal, the shortest is certainly the shortest line oh the sphere betweenxthe two points, but also the other or larger arc of the same great circle is a geodetic or straightest line, i.e., every smaller part of it is the shortest line between its ends. Thus the notion of. the geodetic or straightest line is not quite identical with that of the shortest line. If the two given points are the ends of a diameter of the sphere, every plane passing through this diameter cuts semicircles, on the surface of the sphere, all of which are shortest lines between the ends; in which case there is an equal number of equal shortest lines between the given points. A c ­ cordingly, the axiom of there being only one shortest line between two points would not hold without a certain exception for the dwellers on a sphere. Of parallel lines the sphere-dwellers would know nothing. They would maintain that any two straightest lines, sufficiently produced, must finally cut not in one only but in two points* The sum of the angles of a triangle would be always greater than two right angles, increasing as the surface of the triangle grew greater. They could thus have no conception of geometrical similarity between greater and smaller figures of the same kind, for with them a greater triangle must have different angles from a smaller one. Their space would be unlimited, but would be 7 found to be finite or at least represented as such. 7

Hermann von Helmholtz, Popular Lectures on Scientific Subjects, pp. 34-37.

58 Now Beltramifs representation of pseudo-spherical space in a sphere of Euclid's space, is quite similar, except that the background is not a plane as in the convex mirror, but the surface of a sphere, and that the proportion in which the images as they approach the spherical surface contract, has a different math­ ematical expression. If we imagine then, conversely, that in the sphere, for the interior of which Euclid’s axioms hold good, moving bodies contract as they d e ­ part from the center like the images in a convex mirror, and in such a way that their representatives in pseudospherical space retain their dimensions unchanged,— observers whose bodies were regularly subjected to the same change would obtain the same results from the geometrical measurements they could make as if they lived in pseudospherical space. We can even go a step further, and infer how the objects in a pseudospherical world, were it possible to enter one, would appear to an ob­ server, whose eye-measure and experiences of space had been gained like ours in Euclid’s space. Such an observer would continue to look upon rays of light or the lines of vision as straight lines, such as are met with in flat space, and as they really are in the spherical representation of pseudospherical space, fhe visual Image of the objects In pseudospherical space would thus make the same impression upon him as if he were at the center of Beltrami's sphere. He would think he saw the most remote objects round about him at a finite distance, let us suppose a hundred feet off. But as he approached these distant objects, they would dilate before him, though more in the third dimension than superficially, while behind him they xirould contract. He would know that his eye judged wrongly. If he saw two straight lines which in his estimate ran parallel for the hundred feet to his w o r l d ’s end, he would find on following them that the farther he advanced the more they diverged, be­ cause of the dilation of all the objects to which he approached. On the other hand, behind him, their distance would seem to diminish, so that as he a d ­ vanced they would appear to diverge more and more. But two straight lines which from his first position seemed, to converge to one and the same point of the background a hundred feet distant, would continue to

59 do this however far he went, and he would never reach their point of intersection. Now we can obtain exactly similar images of our real world, if we look through a large convex lens of corresponding negative focal length, or even through a pair of convex spectacles if ground some­ what prismatically to resemble pieces of one con­ tinuous larg er 1 ens.8 Thus the conclusion to which we are brought is that w e can Imaginatively picture spherical and pseudo-spherical spaces just as well as we ean conceive them.

Flat space

is neither a form of mind nor a form of sense but a product of experience.

Is experience, then, the creator of space

and therefore of geometry?

Here certainly, if this be true,

would be the most capital of all physical contributions to mathematics*

It would appear, indeed, that the laws of

geometry are merely the most general laws of our series of sense-impressions, thus making geometry a branch of physics* But assessment of this conclusion must await further dis­ cussion. But, It will be urged, Helmholtz!s images were drawn in two dimensions.

What about the representation of

dimensions higher than three?

Have we reached at three

the natural ceiling of our conceptual powers?

And, if con-

ceputal fancy can soar beyond, are perceptual powers for­ ever locked behind In three dimensions?

0.. Ibid.. pp. 59-61.

Finally, are the

60 principles connecting our sense-impressions tied to a physical world and physiological mechanism which limit the possible dimensions to three? The proper analysis of this fresh problem calls for a clear understanding of the meaning of dimension*

There

are two meanings for dimension, one conceptual. the other perceptual.

Conceptually we think of a continuum in which

the individual elements are specified by a certain number of unique operations or measurements.

The number of unique

measurements required to specify one of the elements is called the number of dimensions of the continuum.

Per­

ceptual dimensions are ordinarily understood to mean sensed extension in different directions.

For those following in

the steps of C. D. Broad the perceptual meaning of dimen­ sion is perfectly clear

because it is a quality directly

given in visual and tactual sensation and simply means directed extendedness.

Length, breadth and thickness are

God-given qualities of sensation, mainly visual, and there is little more to be said about them; the little more being said by the traces of tactual and kinaesthetic sensation. But those who Join Poincarfe

,

a

n

d

Helmholtz in insisting

that dimension is not given in sensation but put there by education must find some novel meaning for dimension. Perceived dimension, they boldly maintain, is not a quality of sensation but the series of sensations which experience

61 has taught us to link together*

Both sides have the problem

of relating perceived dimension with conceived dimension, the physical continuum with the mathematical continuum. Their solutions to this problem are as radically different as their ideas on the perceptual meaning of dimension.

But

these questions can be more fruitfully investigated after we have probed more deeply into the meaning of the mathe­ matical continuum.

This inquiry will also serve to illum­

inate the other half of our primary problem*, namely

are

the common features of the three dimensional geometries, Riemannlan, Euclidean, and Lobatchewskian, necessities of geometrical thought?

What, then, is the meaning of the

mathematical, spatial continuum? The origins of the problem are prehistoric, begin­ ning with the first application of number to express the results of comparing sensed extensions.

That daring genius

who made the first measurement inaugurated the most fruit­ ful and the most paradoxical correlation between a con­ ceptual order and a perceptual order ever devised by man. Subsequent progress, slow but inevitable, was such that by the time of Euclid quantitative determinations had entered into the very heart of the science of form, a p ­ pearing in the very axioms themselves.

11The straight line

is the shortest path between two points.M

The axiom of

parallels asserts that nif two parallel lines are inter­

sected by a third line, the alternate angles, or the cor­ responding angles, are equal. and the sum of the angles of a triangle will be equal to two right a ngles",

Geometry

is the science of form and size. With size already expressed as number it was only a short step— though long in the taking— to the representa­ tion of shape, too, by number.

This involved the notion

of some frame of reference which could be used to establish a reciprocal correlation between numbers and distances to points in space in such a way as to make possible the ex­ pression of spatial relations, e.g., relatione of size and shape, as relations between numbers instead of relations between points*

The "locus of points equidistant from a

given point," becomes "(x-h)2 +

(y-k)2

s rS,f *

The invention of analytic geometry drove home the distinction between the conceptual order and the percep­ tual order.

There are two different continua and two dif­

ferent meanings of dimension.

There is the smooth, u n ­

broken extended curve of intuition on the one hand and the granulated, numerical series on the other.

Dimension for

the one is a unique direction of extension, for the other it is an independently variable number.

But if the mind

regards three dimensions as a matter of associating three Independent variables, why cannot the mind entertain the thought of many independent variables associated together?

63 Indeed It can.

But the question immediately presents it­

self., if the mind can thus entertain a multi-dimensional continuum what is the relation of this continuum to the particular three-dimensional continuum of normal senseactivity? Riemann^ was the first to tackle this problem in its full generality, asking (1) what constitutes a dimensional manifold, (2) what differentiates a spatial manifold from other manifolds, and (3) what differentiates our space, Euclidean, from other spaces? (1) A system of differences in which one thing can be determined by n measurements, Riemann calls an "n fold extended aggregate” or an "aggregate of n dimension". There is a certain exciting inclusiveness about this defi­ nition.

To appreciate the full sweep of it one needs only

to remember that a remarkable range of our sensations lend themselves to quantitative comparisons and arrangements* As several have remarked— including Helmholtz— the system of simple tones forms a two dimensional manifold if d i f ­ ference of pitch and intensity alone are considered and differences of timbre omitted.

Colors are resolvable Into iv*

fixed proportions of the three primary colors and hence may be regarded as a three dimensional aggregate. -

B. Riemann, Ueber die Hypo the sen welche der G e o metrie zu G-runde liegen. Habilitationsschrlft vom 10 Juni 1854, cited by Helmholtz, op. clt., p. 45.

What, then, distinguishes a geometrical manifold from all these other manifolds?

The east and west distance

interval between two objects can be compared with the north and south distance Interval of two other objects♦

We can

Hmeasure offH the interval in the first case and repeat the operation for the second.

But how can we take a measurei

ment of the interval between two tones of equal pitch and different intensity which can be compared with the interval between two tones of equal intensity and different pitch? Biemann was thus led to formulate the theory that any sys­ tem of geometry is characterized by the expression it gives for the distance between two points lying in any direction toward one another, beginning with the infinitesimal inter­ val.

Since it was Riemann*s Intention to show how the

general geometrical manifold was differentiated into the various special spaces by the differences in the specifi­ cations for measurement, he began with the most general form of expression for distance, one which leaves unspeci­ fied the actual kind of measurements *by which the position of any point is determined.

(For the square of the dis­

tance of two infinitely near points the expression is a homogeneous-quadrie function of the differentials of their co-ordinates.) Before proceeding with R i emann!s analysis let us recall a very important point lying at the heart of Euclid!s

65 method of geometrical demonstration.

The essense of

Euclid*s method of proof is the super-position of lines, angles, plane figures, solids, and so on.

If congruence is

to he established the only ultimate way of doing it is by juxtaposing the figures so that the congruence becomes self evident.

This is a process involving the transportation

of the figures and how can we be sure that the shape and size of the figures remain unaltered by this movement? Is it a necessity of thought?

Hardly.

Imagine an egg-

shaped body with a triangle drawn upon it at the sharper pole of the body.

How imagine the triangle without change

of length of sides moved to the blunter pole.

The sum of

the angles of the two triangles, with three pairs of sides equal, will not be the same.

All surfaces, then, are not

indifferent to the motion of figures lying on them*

If a

figure can be transported along the surface on which it lies without change of any of its lines and angles as measured along it, this is a fact which signifies a certain hind of surface; it is not something which pertains to the nature or properties of all surfaces* What property must a surface have in order to admit rigid displacement?

10

Gauss^*0 showed that if the curvature

K. F. Gauss, Werke, Bd. IV, p. 215, cited by Helmholtz, op. clt., p. 39.

of a surface were such that the reciprocal of the product of the greatest and least radii of curvature was equal over the whole surface then the surface would permit the transposition of figures without change of form or size* If the surface is bent or twisted without stretching or shrinking then the measure of curvature given by Gauss is not changed and figures may still be freely transported to all points of the surface without damage to angles and dimensions*

Thus a sheet may be curled into a cone or a

cylinder without affecting the measure of curvature and, hence, the movement of figures; but a sheet— it would have to be a flexible sheet, say rubber, paper would not d o — worked into a cornucopian horn of plenty would not have a constant measure of curvature.

Consequently, figures

sketched on the original surface would not recognize them­ selves after the transformation of the sheet into the horn* How we eagerly follow the question:

f,What charac­

teristics must a surface have if a figure is to be moved about in all directions without change of form or size?11 because this is the kind of surface with which we are so intimately acquainted in the work-a-day world of experience and which we studied in our school-boy days* h a v e n 1t reached home base*

But we still

Of all the surfaces with a con­

stant measure of curvature what distinguishes the Surface of our fatherland, the one with which we have grown up?

67 If the measure of curvature of a surface is everywhere zero, the surface is f l a t .

If the measure of curvature of a

surface is everywhere positive,

the surface Is spherical*

If the measure of curvature of a surface is everywhere negative, the surface is called pseudosoherical. For all of these surfaces It is possible to l!f i t n any selected portion on any other portion of the surface, but It will not be true of the spherical surface that there is only one shortest line between any two points and neither will it be true that through a point external to a given straight one and only one parallel can be drawn.

The sum

of the angles of a triangle will be greater than two right angles and proportional to the size of the triangle; hence, a geometry of similar figures will be impossible.

Figures

of different sizes will always have different shapes* The pseudospherical surface, so-named by E. B e l ­ trami^1 because it is a curved surface which is a sort of geometrical twin of a sphere, is unlike a sphere because on it there is only one shortest line between any two points. Here too the axiom of parallels fails.

Through the given

external point an infinity of straight lines may pass no one of which will ever, no- matter how far extended, Inter—

------------------

E. Beltrami, Sagglo di Interpretazlone della G-eometria Non-Euclidea, Napoli, 1868, and E. Beltrami,. Teoria fondamentale degli Spazii di Curvatura constantef Annali di Matematica, Ber.II,Tom.II, pp. 232-55, cited by Helmholtz, o p . cit... p. 40.

68

sect the given straight.

Furthermore, the sum of the

angles of a triangle will be less than two right angles by an amount proportional to the size of the triangle.

Here,

again, figures of different sizes will have different shapes. The flat surface will have only one shortest line between any two points on this surface.

Through a point

external to a given straight only one parallel can be drawn. The sum of the angles of a triangle will be equal to two right angles and figures of different sizes may have the same angles, i.e., the same shape.

This is the Euclidean

surface where we instinctively feel that we belong. In the spherical surface we recognize the Biemannian surface; in the pseudo-spherical surface of Beltrami we recognize the Lobatehevskian surface; and in the flat sur­ face we come upon our own Euclidean surface.

Consequently,

to distinguish our plane from all others, we only need to specify that (1) it have only one shortest line between any two points, (2) that the axiom of parallels be satisfied, and (3) that rigid displacement be possible.

Expressed

analytically the specification is, the measurement of cur­ vature is everywhere equal to zero. We are now ready to return to the program of Riemann.

Starting with the most general analytic expression

for the distance between any titfo points, direction and kind of measurement unspecified, Riemann developed cer-

69 tain algebraic expressions, made up of the coefficients of the terms in the expression for the square of the distance of two contiguous points and from their differential quo­ tients, which must be everywhere equal to zero if space is to admit unharmed displacement of bodies*

The quanti­

ties appearing in these expressions become Identical with those appearing in G-auss 1 measure of surface-curvature when applied to two dimensions*

For this reason, Riemann calls

these quantities, when they are equal in all directions at some point in space, curvature of the space at that point* When the value of this measure of curvature for a given space is zero for all points of this space then the axioms of Euclid are applicable to this space*

And, draw­

ing the analogy between spaces and surfaces, we may refer to Q-s flat space. A positive value for the measure of curvature at all points in the space indicates a spherical space, again following the nomenclature for surfaces, in which there is no unique shortest line and no parallels.

And a negative

value for the measure of curvature at all pointsin the space specifies a pseudo-spherical space in which there are many parallels* Thus we have arrived at a conceptual answer to the question:

wWhy is space homoloidal?11

Space is homoloidal

if its measure of curvature is everywhere zero.

But this

does not answer the question why perceptual space is homo-

70 loidal*

In what respects, then, has the foregoing discus­

sion advanced the arguments w i t h which we set out?

In the

first place it has broadened the previous conclusion that the mind could not conceive of any space other than Euclidean.

Hot only can it conceive of other three dimen­

sional spaces but it can also stretch out to n-dimensional manifolds*

A further question presents itself*

In con­

ceiving of a system of differences in which one thing is to be determined by n operations is the mind compelled to think of these operations as measurements?

Is it only by

measurement that the mind can uniquely differentiate the elements of a continuum?

The properties of measurement

would then be Imposed upon the conception of the ndimenslonal continuum.

Evidently not.

For analysis situs,

going protective geometry one better, discards measurement and develops its theorems exclusively on the idea of order. Step by step we have been forced to abandon the view that here at last was some characteristics way in which the mind thinks spatially.

Are there no properties, then, which

the mind necessarily imposes or uses when it orders things geometrically or spatially?

Is all thought cut out of one

piece of cloth, is it all purely analytic or logical? Let us turn back, for the moment, to the second way in which the analysis of our various problems has been helped by the previous discussion.

We were following the

71 question: is our space of vital activity three dimensional and Euclidean (1) because our mind can only think in this way,

(2) because this happens to be the given form or qual­

ity of perceptual sensing or intuition, or is it Euclidean and three dimensional (3) because of the way in which we have learned to connect up our sensations? sibility has been definitely ruled out.

Ihe first pos­

In order adequately

to debate the issue between the remaining alternatives it has been necessary to delve into the conceptual notion of dimension, for, though both theories of perceptual dimension are obliged to show how the perceptual continuum is related to the conceptual continuum— a'problem which the first theory would not have had to face— the third theory of perceptual dimension has a novel ring to the ears of com­ mon sense and can best be approached through a study of the conceptual continuum. We have demonstrated that neither Euclidean space in particular nor three-dimensional space in general was a necessary form of thought, by showing that the mind could conceive of other spaces and other dimensions.

Using the

same method we proved that Euclidean plane geometry was not a necessity of perception.by showing how perception could adapt itself to non-Euclidean geometries.

But can we

show that three-dimensions is not irrevocably cemented to the essence of perception by producing a representation of

72

a perception of four dimensions? C. D* Broad would return an emphatic no. It seems to me perfectly clear that, whatever may have been true of my infancy or of my remote a n ­ cestors, solidity is now as genuine a quality of some of my visual sensations as flat shape or red colour. A sphere does_look different from a circle, just as a circle looks different from an ellipse . . . Now I can quite well believe that the particular form of solidity possessed by a certain sensum may be in part due to traces of past experience of touch and movement . . . But I find it very hard to believe that experiences:of touch or movement could create a third dimension in visual sensa which originally had only two.1 ^ Now it does seem to me clear that visual solidity is in itself as purely visual as visual shape and size . . . It is a matter of plain inspection that the experience of visual solidity is as unitary an experience as that of visual shape In two dimen­ sions, and that it is impossible to distinguish it into a visual and tactual part.^-3 The conclusion is obvious.

Dimension is an organic

feature of visual or tactual sensations.

In order to rep­

resent four dimensions perceptually we should have to have an impression having four-dimensional character.

This we do

not have. Although Helmholtz takes the view that the special form of spatial perception, i.e., Euclidean, is a function of experience, of the way we have learned to associate our sensations, he is in agreement with Broad on the impossibil­ ity of perceptually transcending three dimensions.

12 Broad, op. clt., p. 290. 13 Ibid. . p. 291.

'73 But they could as little represent to themselves what further spatial construction would be generated by a surface moving out of itself, as we can represent what would be generated by a solid moving out of the space we know. By the much-abused expression "to represent" or " to be able to think how something happens11 I * understand . . . the power of imagining the whole series of sensible impressions that would be had in such a case. How no sensible impression is known relating to such an unheard-of event, as the movement to a fourth dimension would be to us, or as a move­ ment to the third dimension would be to the inhabi­ tants of a surface, such a "representation" is as impossible as the "representation" of colors would be to one born blind, if a description of them in general terms could be given to him.3-'* And a g a i n , It is different with the three dimensions of space. As all our means of sense-perception extend only to space of three dimensions, and a fourth is not merely a modification of what we have,, but something per­ fectly new, we find ourselves cy reason of our bodily organization quite unable to represent a fourth d i ­ mension.15 It is Poincar& who carries through the doctrine of "perceptual dimension is a structural percipitate of ex­ perience"

to its bitter end.

When it is said that our sensations are " extended" only one thing can be meant, that is that they are always associated with the idea of certain muscular sensations, corresponding to the movements which enable us to reach the object which causes them, . which enables us, in other words, to defend ourselves against it. And it is just because this asso­ ciation is useful for the defense of the organism, that it is so old in the history of the species and that it seems to us indestructible, nevertheless, it is only an association and we can conceive that

Helmholtz,

op

15 Ibid., p. 64.

. cit., pp. 34-35.

74 it may be broken; so that we may not say that sensa­ tion can not enter consciousness without entering in space, but that in fact it does not enter consciousness without entering in space, which means, without being entangled In this association.16 Different associations would have given rise to different numbers of dimension and, with a little work, one might actually grasp perceptually a fourth dimension.

To a p ­

preciate this possibility consider the following line of reasoning. And yet most often it is said that the eye gives us the sense of a third dimension, and enables us in a certain measure to recognize the distance of objects. When we seek to analyze this feeling, we ascertain that it reduces either to the consciousness of the convergence of the eyes, or to that of the effort of accommodation which the ciliary muscle makes to focus the image. . . . But it happens precisely that experience teaches us that when two visual sensations are a c ­ companied by the same sensation of convergence, they are likewise accompanied by the same sensation of accommodation. . . . it will thence result that C* has one, C two and the whole visual space three d imensions. But would it be the same if experience had taught us the contrary and if a certain sensation of con­ vergence were not always accompanied by the same sensation of accommodation? In this case two sensa­ tions affecting the same point of the retina and a c ­ companied by the same sense of convergence, two sensations which consequently would both appertain to the cut C 11, could nevertheless be distinguished since they would be accompanied by two different sensations of accommodation. Therefore C 11 would be in its turn a continuum and would have one dimension (at least); then C 1 would have two, C three and the

16P o i n c a r o p . cit., p. 274.

75 whole visual space would have four dimensions.

17

Why has It been said that every attempt to give a fourth dimension to space always carries this one back to one of the three? It is easy to understand. Consider our muscular sensations and the ’series1 they may form. In consequence of numerous exper­ iences, the ideas of these series are associated together in a very complex woof, our series are classed. Allow me, for convenience of language, to express my thought in a way altogether crude and even inexact by saying that our series of muscular sensations are classed in three classes corresponding to the three dimensions of space. Of course this classification is much more complicated than that but that will suffice to make my reasoning understood. If I wish to imagine a fourth dimension, I shall sup­ pose another series of muscular sensations making part of a fourth class. But as all my muscular sensations have already been classed in one of the three preexistent classes, I can only represent to myself a series belonging to one of these three classes, so that my fourth dimension is carried back to one of the other three. What does that prove? This: that It would have been necessary first to destroy the old classifica­ tion and replace it by a new one in which the series of muscular sensations^should have been distributed into four classes. The difficulty would have dis­ appeared.**-® Obviously Poincar& is committed to the hilt to the ’group theory’ of perceptual dimension, ing the purely visual

even to the point of deny­

!localization' of exterior objects

as the following quotation illustrates. Will the difficulty be solved if we agree to refer everything to these axes bound to our body? Shall we know then what is a point thus defined by its relative position with regard to ourselves? Many persons will answer yes and will say that they localize' exterior objects.

17 Ibid., p. 254. 18 Ibid.. p. 275.

76 What does this mean? To localize an object simply means to represent to oneself the movements that would be necessary to reach it. I will explain myself. It is not a question of representing the movements them­ selves in space, but solely of representing to one­ self the muscular sensations which accompany these movements and which do not presuppose the preexistence of the notion of space.*9 Undoubtedly there is much truth in Poincar^ line of reasoning, certainly ther'e is much cleverness. much,will appear in the sequel.

How

Let us assume for the

sake of argument that the debate is a draw, that the issue over the meaning of perceptual dimension has not been d e ­ cided by demonstrating or failing to demonstrate the pos ­ sibility of a perceptual fourth dimension.

How will the

case then stand? The attractiveness of the more domesticated theories of ‘i n n a t e 1 extendedness over the wilder ideas of an ’intellectual1 extendedness of perceptions is hardly to be denied.

The view of Broad and William James may be more

prosaic than those of Helmholtz, Wundt, and Poincar&

but

they seem more straightforward. Voluminousness, as William James prefers to call it,

19 I b i d . p. 246

77 is an essential quality of each kind of sensation*

20

By

processes of discrimination, association and selection 21 operating under the principles of economy the extended­ ness of the various individual sensations are correlated --------- g?r—

—

C. D. Broad is not inclined to put much faith in the extended quality of sensations other than those of sight and touch. He takes the view that sight is the preeminent contributor to the perception of space. The sensations of touch and movement serve to qualify the quantitative features. E . g . , “My view is that nearly all the general concepts that we use in dealing with Space, e.g., distance, direction, place, shape, etc., come from sight, whilst the notion of one space and the particular quantitative values which these general concepts assume in special cases are duemainly to touch and to movement. Series of kinaesthetic sensations are not, as such, experiences of distance, di ­ rection, etc.; and 1 do not see how they could ever be in­ terpreted in such terms unless the necessary concepts had already been supplied by sight.11 Broad, o p . cit. . p. 300. William James, however, goes whole hog in the matter, a t ­ tributing extensity to all sensations. “ ‘Extensity*, as Mr. James Ward calls it, on this view, becomes an element in each sensation Just as intensity is.“ James, o p . c i t . . p. 3^35

21 “In the first place following the great intellec­ tual law of economy, we simplify, unify, and identify as much as we possibly can . . . . They become in short, so many properties of one and the same HEAL THING-. This is the first and great commandment, the fundamental ‘a c t 1 by which our world gets spatially arranged. “In this coalescence in a ‘thing1, one of the coales­ cing sensations is held to be the thing, the other sensa­ tions are taken for its more or less accidental properties, or modes of appearance. The sensation chosen to be the thing essentially is the most constant and practically im­ portant of the lot; most often it is hardness or weight, . . . In all this, it will be observed, the sense-data whose spaces coalesce into one are yielded by different senseorgans," James, o n . cit. f p. 183

78 to form objects in one and the same space.

If kinaesthetic

or tactual or visual sensations were to be cut off from each other and allowed to develop under the guidance of their respective geniuses, each would have developed its characteristic quality of extendedness, i.e., of space. Even yet, in mature and thoroughly Integrated perception, each sense retains its characteristic quality or flavor. The best that experience can do is to modify that quality, sharpen its differences and enrichen its vital associations, tone down the exaggerations and fortify the weaknesses. Thus,

since it is through tactual contact that we most fun­

damentally act upon the world, the sensations of touch take priority over all others in the relative scale of values; consequently visual extensions are assessed in terms of tactual sensations. in vision.

Even the sense of weight is anticipated

But the visual sensation Is not transformed into

the tactual sensation, nor did it by association with the tactual sensation acquire a dimension which it lacked. After making the claim that length, breadth and depth are inherent in visual sensation as a matter of fac­ tual inspection, the proponents of this view turn upon the opposition with the charge that their ideas are based on purely theoretical arguments.

The theories of Helmholtz

and his followers, they argue, are based on the principle that the connection between neural processes and sensations

79 Is uniform,

so that the same neural process should invariably

give rise to the same sensation.

If, then, the same process

in the sense-organ is found correlated with what seems to be a difference in sensation, the only consistent way to a c ­ count for, or ‘regularize*, this difference is to attribute it to the intervention of an intellectual activity.

The

image on the retina of a six-inch stick six feet away is the same as a parallel, 12-inch stick twelve feet away. There is no factor in the retinal process uniquely cor­ related with distance.

Shall we conclude, then, that the

perception of distance is purely an intellectual affair? Or, considering a different type of example,

each of us

while under the influence of some idea has seen objects which, when the idea was dispelled, could no longer be evoked.

Such intellectual illusions seem to have a dire

import for the sensationalistic theories of perception. William James summarizes it thus: Can the feelings of these processes in the eye, since they are so easily neutralized and reversed by intellectual suggestions, ever have been direct sensations of distance-at all? Ought we not rather to assume, since the distances which we see in spite of them are conclusions from past experience, that the distances which we see by means of them are equally such conclusions? Ought w e not, in short, to say u n ­ hesitatingly that distance must be an intellectual and not a sensible content of consciousness? And that each of these eye-feelings serves as a mere signal to awaken this content, our intellect being so framed that some­ times it notices one signal more readily and sometimes another?^2

Ibid.f p. 217.

80 And Helmholtz confirms it in the most positive terms: No elements in our perception can be sensational which may be overcome or reversed by factors of demonstrably experimental origin* Whatever can be overcome by suggestions of experience must be r e ­ garded as itself a product of experience and custom. If we follow this rule it will appear that only qualities are sensational, whilst almost all spatial attributes are results of habit and experience.^3 How do the sensationalists answer these objections?

Their

first line of defense, as I have indicated, is that they are sticking to facts and theorizing on the basis of facts, whereas their opponents are adjusting facts to theories. As a second line of defense Broad advocates the theory that along with the physiological conditions of per­ ception must be included mental traces left by former experiences and William James takes up much the same posi­ tion with his view that association and selection modify space-sensatlons but do not produce them. We have native and fixed optical space-suggestions; but experience leads us to select certain ones from among them to be the exclusive bearers of reality: the rest become mere signs and suggesters of these. The factor of selection, on which we have already laid so much stress, here as elsewhere is the solving word of the enigma.24 How does the score now stand?

The sensationalists

have been forced out of their old rut and have been

23

Hermann von Helmholtz, Physiological Opt i c s . p. 438, cited by James, op. c i t . f p. 218. James, o p . cit. , p. 237.

81 obliged to make room in their theories for the impact of ex­ perience upon perception.

They would no longer be admitted,

certainly, to the inner councils of their purist bretheren. But as matters now stand it would seem to me that the reformed sensationalists have the better of the argument. One point, however, remains.

Exploited to the full

by Poincarfe, it strikes at the very heart of sensationalist doctrines.

It has to do with the problem of spatial dis­

placement versus qualitative change* Broad and James maintain that motion,

;i.e., change

of relative position as opposed to qualitative change, is directly given in visual sensations.

Poincar6 claims that

all change in a visual manifold is qualitative and that the only way we have of distinguishing genuinely qualitative changes from mere displacement is by taking such steps as will reestablish the original sense-impression, the original visual manifold.

The visual impression received from an

object in one corner of a sensed field is not the same as the visual impression received when the object is in some other corner of the field.

Nor will it do to say that we

have a pure sensation of motion in the continuous passage of an object across the visual field.

What we have is a

continuous change in an aggregate of Impressions and there is nothing in this purely visual change to distinguish between growth or change of state and displacement.

82 Poincar& expresses it in this way: We see first that our impressions are subject to change; but among the changes we ascertain we are soon led to make a distinction. At one time we say the objects which cause these impressions have changed state, at another time that they have changed position, that.they have only been displaced. Whether an object changes its state or merely its position this is always translated for us in the same manner; by. a modification in an aggregate of Impressions. How then could we have been led to distinguish between the two? It is easy to account for. If there has only been a change of position, we can restore the primitive aggregate of impressions by making movements which replace us opposite the mobile object in the same relative situation. We thus correct the modification that happened and we re­ establish the initial state by an Inverse modification. If it is a question of sight, for example, and if an object changes Its place before our. eye, we can ’follow it with the e y e ’ and maintain its image on the same point of the retina by appropriate move­ ments of the eyeball. These movements we are conscious of because they are voluntary and because they are accompanied by muscular sensations, but that does not mean that we represent them to ourselves in geometric space. So what characterizes change of position, what di s ­ tinguishes it from change of state, is that it can always be corrected in this way.25 . . . I t follows from this that sight and touch could not have given us the notion of space without the aid of the ’muscular sense1. Hot only could this notion not be derived from a ------------

Poincar&, o p . cit., p. 71.

83 single sensation or even from a series of sensations. but what is more, an immobile being could never have acquired it, since, not being able to correct by his movements the effects of the changes of position of exterior objects, he would have had no reason w h a t ­ ever to distinguish them from change of state* Just as little could he have acquired it if his motions had not been voluntary or were unaccompanied by any sensation.26 Polncar& next asks: How is a like compensation possible, of such sort that two changes, otherwise independent of each other, reciprocally correct each other? A mind already familiar with geometry would reason as follows: Evidently, if there is to be compen­ sation, the various parts of the external object, on the one hand, and the various sense organ§,on the . other hand, must be in the same relative position after the double change. And, for that to be the case, the various parts of the external object must likewise have retained in reference to each other the same relative position and the same must be true of the various parts of our body in regard to each other.27 Compensation may take place, in other words, if both the ex­ ternal object and our own bodies are displaced as rigid solids.

But if there is change of form as well as change of

place then we cannot get our sense-organs in the same rel a ­ tive position and we shall find it impossible to restore the original totality of impressions.

“Therefore,11 P o i n c a r ^ ®

concludes, 11if there were no solid bodies in nature, 26 Ibid. . p. 72 27 Ibid. , p.

72

28 I b id., ,p..73

there

would be no geometry*” It frequently h a p p e n s : that we experience changes which,

though radically different, are said to he the same

displacement.

How can this be, Poincar& asks?

It is be­

cause they can all be corrected by the same correlative movement of our body. 1Correlative movement1, therefore,

constitutes the

sole connection between two phenomena which otherwise we 29 never should have dreamt of likening. Poincar6 is thus led to this summary of his argument 1. We are led at first to distinguish two categories of phenomena: Some, involuntary, unaccompanied by muscular sensa­ tions, are attributed by us to external objects; these are external changes; Others, opposite in character and attributed by us to the movements of our own body, are internal changes; 2* We notice that certain changes of each of these categories may be corrected by a correlative change of the other category; .3. We distinguish among external changes those which have thus a correlative in the other category; these we call displacements; and just so among the internal changes we distinguish those which have a correlative in the first category. Thus are defined, thanks to this reciprocity, a particular class of phenomena which we call dis­ placements. The laws of these phenomena constitute the object of geometry.5° 23 •Ibid., p.- 74 50 I b i d . p. 74.

85 At once we grasp the significance of P o i n c a r e s argument.

The sensationalists argue that spaciness is

an immediately given content of visual sensation; hut would the spaciness he there for an eye and a hody paralyzed and Immobilized from hirth?

There is nothing in our intuition

of any given impression to tell us that when this impres­ sion has changed it can he restored.

Only experience can

teach us that some changes can he inverted or cancelled out thereby leading to the distinction between change and motion. Otar previous inquiry into the metrical continuum had brought into the spotlight this all important feature of spatial perception which could never have arisen from any 'given1 quality of sensation. rigid displacement.

This was the feature of

The condition of free mobility lies

outside the range of pure space-sensing or spape intuition; it could only arise from experience.

Free mobility is

self-evident neither to rational intuition nor to senseintuition.

It cannot be deduced or logically demonstrated.

It cannot be seen; it must be found.

Free mobility is an

hypothesis. It is Just here, in the treatment of motion vs. change, that the discussion of the sensationalists is most marked by paucity.

William James, Indeed, seems to miss the very

point at issue: Now I firmly believe, on the contrary, that one of

86

R e i d ’s Idomenians would frame precisely the same conception of the external world that we do, if he had our intellectual powers. Even were his very eyeballs fixed and not movable like ours, that would only retard, not frustrate, his education. For the same object. by alternately covering in its lateral movements different parts of his retina, would determine the mutual equivalencies of the first two dimensions of the field of view; and by exciting the physiological cause of his perception of depth in various degrees, it would establish a scale of equivalency between the first two and the third.51 But, Poincarfe would challenge, how do you identify it as the same object?

Not only has the total visual complex

qualitatively changed, but also the physiological condi­ tions have changed;

the moving object affects different

points of the retina*

C. D. B r o a d ’s customary clearness,

like J a m e s 1, seems to desert him when he comes to this very same point of motion vs* change*

After making various ob­

servations regarding the relation of sensa characterized by having

’sensible movement*

to such conditions as head-

motions and homogeneous media he concludes: There is no doubt that sensible motion and rest are genuine unanalysable properties of visual sensa. I am aware of them as directly as I am aware of the red­ ness of a red patch, and I could no more describe them to anyone who had never sensed them than I could describe the color of a pillar-box to a man born blind. It is quite true that change is directly sensed, but the point at issue— and to which Broad, so far as I know, gives

James, o o * cit . . p* 214.

32

G. D. Broad, op. cit., p. 287.

87 no answer— is:

How is change distinguished from motion?

But the final k e y .of Poincar& beautifully arched theory falls into place when w e ask: “What is it that is achieved, what is discovered, when we carry through suc­ cessfully the operation of correcting changes by cor­ relative movements?”

Have we discovered that the space

which our senses reveal to us permits rigid displacement?

"

No, indeed, replies Poincar&, we have discovered space itselfJ

It is not a case of the senses revealing to us

an absolute, physical, box-like space which is a sort of theater of action; nor is it a case of the mind imposing a three-dimensional box upon the contents of sensation. In the mind lies the conception of an ‘amorphous continuum* and when experience reveals repeatable operations this continuum comes to life, so to speak, i.e., finds its a p ­ plication.

A continuum Is an ordered aggregate, an

ordered set of differences, and the meaning is essentially the same whether the members of the aggregate are con­ ceptual objects or sensations. this continuum on experience.

The mind does mot impose There was no reason to sup­

pose before all experience that an order amenable to such an interpretation could be found In experience.

But as a

matter of fact experience has provided the sort of ordered conditions under which this particular gift of the mind--

88 the continuum— *could flourish.

Hence it is that the con­

ceptual continuum and the perceptual continuum are es­ sentially the same*

And therefore perceived dimension is

not a sensed quality of extendedness but a series or order of sensations.

The proper representation of extension is

not a sensation but a series. Poincarfe reinforces the foundation of his theory by showing how the mind*s idea of continuity, once pricked into life by experience, guides the development of a p h y s ­ ical continuum which, in turn, spurs the mind on to the creation of the mathematical continuum. The failure to distinguish between two sensations and an intermediate third when the two are distinguishable from each other forced the mind to invent the mathematical continuum.

For example, a sensation A of 10 grams is in­

distinguishable from a sensation B of 11 grams which in turn is indistinguishable from a sensation C of 12 grams, but A is distinguishable from C. but A - C.

Therefore A - B and B - C,

The mind solves the paradox by intercalating

differences which the senses cannot discover.

We are led

to the notion of a several dimensioned physical continuum in this manner*. A system of elements— each element being an aggregate of sensations— will form a continuum if we can pass from any one of them to any other by a series of con­ secutive elements such that each is indistinguishable from

89 the preceding*

If such a continuum can be broken into a

system of several distinct continua by the removal of a finite number of elements all distinguishable from one another,— and thus forming neither a continuum nor several continua--the continuum is one dimensional*

If

the original continuum can only be broken by the removal of systems of elements which are themselves continua then 'the original continuum will have several dimensions. If cuts which are continua of one dimension suffice, we shall say C -has two dimensions; if cuts of two dimensions suffice, we shall say C has three dimensions, and so on. Thus is defined the notion of the physical continuum of several dimensions, thanks to this very simple fact that two aggregates of sensations are distinguishable or indistinguish­ able*33 But the mind is not limited to the resolution of the paradoxes of experience in its creation of a con­ tinuum.

Once launched on such ah enterprise it comes to

feel that there need be no limit to the intercalating of additional terms between two consecutive terms of a series*

The leap from the physical continuum to the math­

ematical continuum is easily made.

The elements or points—

aggregates of sensations— are defined in the physical con­ tinuum by the number of cuts required to isolate them. the mathematical continuum of n dimensions a point is

33

Poincare, op. cit*, p. 53.

In

isolated by the n distinct magnitudes or coordinates, required to specify it* There is no question but what Poincarfe has stolen a march on his adversaries in this matter of the mathe­ matical continuum versus the perceptual continuum.

Since

they start with the idea that extension is a given quality of sensation they are led to the conclusion that the dimen­ sions of geometrical space are obtained by a process of abstraction from the particular sensations of extendedness* From their standpoint we arrive at the geometrical notion of an empty, homogeneous, isotropic,

three-dimensional box

by a process of imaginatively abstracting the contents of visual sensation.

E. W* Hobson describes the process this

way: To give a systematic scheme descriptive of the rela­ tions in physical space is the first object of the Science of Geometry, although in some of its develop­ ments the Selence:.has so extended itself as to tran­ scend this primary object. In order to attain this object, the spatial relations are idealized, and transformed into a precise form, by means of a system of definitions and postulations* By this process of abstraction and idealization, a conceptual space, the space of abstract geometry, has been created. It is in this conceptual space that all the ideal objects of Geometry are regarded as situated, and as subject to a scheme of relations specified by a system of postulates.34 The simplest regularities and uniformities observable

E. W. Hobson, The Domain of Natural Science (London: Cambridge University Press, 1926), p. 128.

in the shapes and spatial relations of actual bodies were singled out and then conceptualized. Thus points, straight lines, planes, rectilineal figures, circles, spheres, pyramids, and other objects transformed and idealized from percepts into concepts, are the geo­ metrical objects with which Euclid deals in his a b ­ stract Geometry*^5 This account of the origin and meaning of geo­ metrical space bequeaths upon the sensationalists a very trying problem*

How do we get back from the breadthless

surface, the ghost-like line and the ghostly point to the substantial bodies of sense-perception?

How can a dimen-

sionless point generate or be the limit of a dimensionable line, and how can the gossamer line generate the equally transparent surface?

The Herculean effort to close the

gap was made by A* N. Whitehead in his Theory of Extensive Abstraction. W h i t e h e a d

endeavored to define a point,

etc., b y ’means of a structure which would both satisfy the logical requirements of the geometrical postulates and which would find a satisfactory interpretation or application to concrete perception. What is the up-shot of these debates? clusion has our long argument brought us?

To what con­ We opened the

problem by asking, in effect: uWhy was our first geometry

35 Ibid. p. 131. '36 Alfred North Whitehead, The Concept of Nature (London: Cambridge University P r ess* 1920), Sapter IV*

92 Euclidean?*1 beings?

Why is Euclidean space N a t u r a l 1 for human

Why do we take to it like a duck to water?

In the

first place, we prefer self-congruent spaces of three di­ mensions and in the second place, of all the possible threedimensional,

self-congruent spaces we prefer a space which

is neutral, not merely with respect to the sizes and shapes of moving bodies, but also, with respect to change of size. We like a space in which shape is independent of size; a space in which the form may remain the same even though the magnitude of the figure, may vary. Here is a common— even preferred— feature running through all human experience. is this?w

And our question was:

HWhy

Is our geometry, and perhaps perceived space

too, Euclidean because our human mind is built so that it can only think in Euclidean ways?

Is it a given form of

our sensibility or is it a product of pur vital experiences? If the first answer Is correct then physics, liberally in­ terpreted as any set of techniques for the prediction of sense-experience, would have had only an insignificant part to play in the development of geometry.

And the problem of

analyzing the influence of geometry on the development of physics would be merely a matter of showing how the data of experience had been inevitably molded into the fixed forms of geometrical thought.

If the second interpretation is

correct then geometry as a generalization of the spatial

93 properties given in sensation is divorced as to origin from the physical properties, i.e., those having to do wit h the structure of sensations, and whatever application it might have to the explication of the world's order is adventi­ tious ♦

If the third theory is applicable then the origin

of geometry is intimately involved with the structure of physical experience, and the history of geometry's services to physics would be a matter of adapting the relatively pure developments of geometry to the expression of ever more subtle connections of experience.

Let us analyze each solu­

tion, In turn. First Case W e have flatly rejected the first alternative.

The

mind does not even impose the Euclidean frame upon itself, let alone upon experience.

Are w e to conclude at once,

then, that the propositions of geometry are drawn from and lean upon experience or sensation.

Not at all.

Geometrical

propositions are certain and empirical generalizations are merely probable.

Regardless of what points, lines,

etc.

might be, we are certain that any objects having the prop­ erties specified by the postulates will also have the prop­ erties deduced from those postulates. answer?

Is this then our

Is the function of the mind in all this business

of mathematics merely to develop definitions of purely

94 abstract objects and then deduce the implied properties of the objects thus defined?

The office of the mind in the

realm of experience will not,

therefore, be that of a king

who dictates the laws according to which experience shall conduct itself; but rather that of the humble adviser who offers his definitions in the hope that at some point ex­ perience will see fit t o . A c c e p t 1 them.

The mind can not

say that the logically defined objects will be 'realized1, either in experience or in any other way; it can only say that if they ever are, then such and such will also follow* The mind cannot command experience to 'pony up* to its abstract notions, but if it ever does then all these other things will also be true of experience. At this point it is necessary to draw a distinction between two types of k priorists*

One type is called an

analytic k priorist and the other a synthetic k priorist, to give them rather ancient tags.

Those lining up under

the flag of 'analytic k priori* are differentiated from each other primarily by their ideas about the attitude of a g ­ gressiveness which the mind adopts toward experience. Lewis, ^

for example,

takes the position that the mind has

no control whatsoever over the ■"

~ " r 1

C. I.

'given*

element in experience.

.

0. I* Lewis, Mind and the World Order (New York: Charles Scribner's Sons, 1929), p. 128.

95 (There is also, to he sure, the interpretative element.) The mind, however, has authority, and exercises it, to demand that experience live up to such and such specifica­ tions if experience is to be intelligible and meaningful, etc.

The attitude of the mind toward experience is not one

of passive inquiry.

The mind has cherished, fundamental at ­

titudes with which it approaches experience, in the hope that through the harmonizing of experience with these a t ­ titudes the order of experience will be made useful and intelligible.

The attitude of the mind toward experience

is that of one who boldly says: while I beat some sense into you. with my fists I'll use a club*11

“Hold still, you rascal, And if I c a n ft do it For C. I, Lewis the out­

standing k priori feature of the mind is precisely this singleness of intent.

But the mind has no inner vision

of the necessity of the ideas which it endeavors to apply. It has no guarantee that its definitions have an object, its ideas a content, order.

either of a mental or a perceptual

A turning of the way may come at which, to advance,

the mind may have to abandon a pet idea, and though this may be tortuous, the mind is not so irrevocably committed to any one idea that it cannot endure the torture.

The syn­

thetic k priorists are differentiated by the manner in which these objects are guaranteed and as to the nature of the objects.

For the older rationalists these ideas were

96 intuitively revealed and the object was the world. Kantians these ideas were

For the

d e m o n s t r a t e d 1or 'logically

deduced* by showing them to be the 'preconditions* of ex­ perience.

The object would therefore be, not the world,

but experience.

This procedure of 'revealing* the cat­

egories is subject, however, empirical generalization.

to the same difficulties as

It is one thing to show that

some principle is a sufficient condition of a certain state of affairs, but it is another thing to show that it is also the necessary condition. Poincare shrewdly avoids the difficulties of both the Kantian 'deduction* and the older rationalism by adop­ ting the view that the mind has a direct, intuitive knowl­ edge of the 'actualization* or ideas.

'realization' of certain

But this actualization is not in experience: it is

in the mind.

The objects are not the world and not experi­

ence but the mind's own activity.

It is not a case of the

mind laying down certain definitions which it can never be sure will be fulfilled or objectified.

The mind cannot .

be sure that experience will fulfill them— this can only be decided by experience itself— but the mind has a sure in ­ sight into a fact or concrete object of mind which changes mere definitions Into actualities.

Poincare's position,

it should be remarked, has a very strong strategic a d ­ vantage over that of the analytic theorists.

The only kind

of warrant which the latter have for the systems evolved from their abstract definitions is the mutual consistency of the propositions deduced from the primitive postulates, which act as definitions of the object.

But might it not

be that no objeeteould consistently have the properties which all the definitions taken- together attribute to the object?

The object is a function of the postulates or

definitions but it might be inconsistent to have a function with the properties given in the particular system of postulates defining the function.

The only crite­

rion which the proponents of the analytic theory have is that of developing the logical consequences of the p o s ­ tulates in order to find out if they are led to proposi­ tions which are mutually contradictory.

But since the

total number of implications are infinite they can never be sure, no matter how far they carry their deductions, that their confidence would not be betrayed by a further unravelling of the skein of implications.

Poincar& is

saved from this predicament because his system is not founded on abstract definitions which may or may not have actual fulfillment.

He begins with an intuition of the

possibility of the concrete fulfillment of the object, which serves thenceforth as the validation of the con­ sistency of the postulated or, rather, predicated properties.

98 What is this a priori intuition.

Poincar&’s shining

example is the m i n d ’s intuition of its power to repeat indef­ initely the same act when once this is possible.

In reason­

ing by recurrence we prove a theorem for n - 1; then it is demonstrated that if this theorem is true of n - 1, it is true of n; thus leading to the conclusion that the theorem is true for all whole numbers. Why then does this judgment force itself upon us with an irresistible evidence? It is because it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it . . . Induction applied to the physical sciences is always uncertain, because it rests on the belief in a gen­ eral order of the universe, an order outside of us. Math­ ematical induction, that is, demonstration by recurrence, on the contrary, imposes itself necessarily because it is only the affirmation of a property of the mind itself.38 Thus all arithmetic Is founded upon the m i n d ’s intui­ tion of its power to repeat indefinitely certain operations upon

u

n

i

t

s

.

39

All geometry is founded on a similar intuition;

Poincarb , o n . cit., p p . 39-40. 39

P o incare’s opponents maintain that the principle of Mathematical Induction can be reduced to the principle of contradiction and that therefore the principle of Mathemat­ ical Induction cannot be regarded as a characteristic form of thinking which distinguishes mathematical reasoning from log­ ical reasoning, e.g., uIt has however been shewn above that it is not necessary to state the principle [of Mathematical Induction] in this form, [i.e., as an unending chain of syl­ logisms] but that the truth of the principle follows by a p ­ plying the ordinary Logic to deduce the consequences of the possession by the infinite ordered aggregate of certain pro p ­ erties, in accordance with the principle of contradiction.” E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of F o u r i e r ’s Series. 3rd edition (Cambridge: At the University Press, 1927), Vol. I, p. 8. {Brackets not in original^

99 in this instance the generating insight is of the limitless possibilities of interjecting another term between two con­ secutive terms of a series. All happens as in the sequence of whole numbers. We have the faculty of conceiving that a unit can be added to a collection of units; thanks to experience, we have occasion to exercise this faculty and we become con­ scious of it; but from this moment we feel that our power has no limit and that we can count indefinitely, though we have never had to count more than a finite num­ ber of objects. Just so, as soon as we have been led to intercalate means between two consecutive terms of a series, we feel that this operation can be continued beyond all limit; and that there is so to speak, no intrinsic reason for stopping.40 G-eometry . . . has for object certain ideal solids, absolutely rigid, which are only a simplified and very remote image of natural solids. . . . The notion of these ideal' solids is drawn from all parts of our mind, and experience is only an occasion which induces us to bring it forth from them. . . . The object of geo­ metry is the study of a particular ‘g r o u p 1; but the general group concept pre-exists, at least potentially, in our minds. It is Imposed on us, not as form of our sense, but as form of our understanding.^1 Arithmetical operations and the mathematical continuum or the group concept are visions of the mind awakened by experience.

But the mind has no way of foretelling whether

or not experience will bear out these visions.

C. I. Lewis

and Poincar^ are together in their belief that the mind ap ­ proaches experience with certain fixed attitudes with which it endeavors to bring experience into alignment. 40 Poincar^, op. cit., pp. 47-48. 41

Ibid., p. 79.

Both are

100 agreed that the mind does not predetermine the given element of experience and that therefore the fitting on of these ideas is not a foregone conclusion but a question of ex­ perimentation.

But for G. I. Lewis,

though the mind has

its pet ideas, there is no sense of an ultimate, inner valid­ ation of these ideas which would mean that the mind could never abandon them in the face of the stormiest denials from experience.

For Poinear&, arithmetical and group concepts

are securely entrenched in the inner citadel and would weather the heaviest armament experience could bring against them. Now although the mathematical continuum is an intui­ tion of the understanding and not a merely abstract defini­ tion, this is not true of the special forms of the continuum. For the purposes of ordering our sense perceptions we must choose some particular metric, and this.choice will determine our geometry.

This choice is not dictated by the mind and it

is not imposed by experience;

ther choice is determined by

considerations of convenience. Only, from among all the possible groups, that must be chosen which will be, so to speak, the standard to which we shall refer natural phenomena. Experience guides us in this choice without forcing it upon us; it tells us not which is the truest geometry, but wh i c h is the most convenient.42

42"

Ibid., p. 79.

101 In our minds pre-existed the latent idea of a certain number of groups— those whose theory Lie has developed* Which group shall we choose, to make of it a sort of standard with which to compare natural phenomena? And, this group chosen, which of its sub-groups shall we take to characterize a point of space? Experience has guided us by showing us which choice best adapts itself to the properties of our body* But its role is limited to that*43 The gad-fly which drives the mind to its labors is experience*

But once under way the mind comes to the

realization that it has a natural birthright, namely, the genius to repeat endlessly the addition and intercalation of terms*

But the mind is not arbitrary in its Imposition

of the general idea of the continuum upon physical exper­ ience*

It can persuade, but it cannot Impose.

The mind

is not a grim Puritan who thrashes his moral code into his sons; rather, the mind is like the wise father who holds loose rein and who presents his sons with various possibili­ ties proportioned to their needs and interests* We are convinced that the issue lies between Lewis and Poincar&; they are the jousters for the championship. The propositions of mathematics are not the generalizations from experience of John Stuart Mill and they are not the minutely engraved universal intuitions of Kant*

The p r i m ­

itive propositions are either logical definitions which the mind endeavors to make useful In the business of ordering -------------

£3----!------Ibid. p* 91.

102 experience or they are founded on mathematical insights of a very general order, which, though of assured validity in the m i n d ’s domain, do not have a guaranteed application in the world of sense-perception.

From an epistemological

standpoint— i.e., the meaning and origin of the foundations of mathematical knowledge— the difference between the two viewpoints is profound, but for our purposes the difference is not great.

(1) In either case the mind shows a deep

preference for the utilization of Its definitions, on the one hand, and of its ideas, on the other, in the working out of the order of sense-experience.

And a large part of the task

of the historian of natural science who is forewarned as to these preferences will be to show how they have Inspired the ideas and affected the choices which have gone into the build­ ing of physical science.

(2) Moreover, in either case, it is

pertinent to ask why and how ’amorphous1 general ideas have come to take their various specialized forms, why of all possible definitions to choose for development have the ones selected been hit upon?

And to answer this the historian of

mathematics will have to consider respectfully the suggestions and requests of physical science. This second question takes us back to our basic problem: Why do we prefer Euclidean geometry?

Though the

mind may be responsible, as Poincar£ affirms, for the idea of the mathematical continuum,

the mind cannot be the sole

agent behind our vast exploitation of Euclidean geometry* Euclidean geometry has deep roots in perceptual experience, but are its essential features drawn from the quality of sensations or from the structure of sensations?

If they are

founded on the qualities of sensation then all that can be said 1st MIt is the nature of the human sensing process to sense sensa characterized by homoloidal spatial: properties,” But if these spatial properties are based on the way we have learned that our sensations can be most effectively put together, then— according to our principle that the tried and tested structures of sense-preception have something to say about processes other than the process of sensing— w e might infer a little something about the character of the inter-action of the body-process with the world-process; namely,

that the structure of the body is such and the struc­

ture of the world is such, that the character of their inter­ action is conveniently expressed in Euclidean terms.

From

this standpoint Euclidean geometry is founded on the bodyin-interaction, with a world. It seems clear that Helmholtz and Poincar^ are correct in insisting that the property of free mobility is not a given content of sensuous intuition^ that this property is founded on exploration, on comparing and discriminating different sensations.

It also seems clear, however,

that

this property is an integral feature of spatial perceptions,

104 i.e., of the perception of dimension.

We do not sense.mere

extensions or directions but paths of action, and an inde­ pendent direction, i.e., a distinct dimension, is a:, unique path of action.

The space of ordinary experience, is primarily

a theater of action.

And part of its essential character or

feeling is derived from those features of each kind of sensing which pertain to action.

The world of vision is not

a world of shapes and colors only, it is a world of potential action.

It is a world transformed and readied for business

w hich we actually see. muscle-sight. action.

Eyesight is actually body-sight and

Touch and sight and sound are orienters for

Thus we believe that dimension is more intimately

Involved with a volitional series of sensations than the sensationalists admit.

Furthermore, not only is dimension

linked w ith a possible series of impressions, but the indif­ ference of size to position and the indifference of form to growth are also linked with free motion. as we move,

In the first case,

eyesight gives us the lie about the indifference

of position to size; but we are able to discover this lie by reconnoitering; whence, thanks to the rigid frame of r e f ­ erence w e carry with us, we are able to say that the visual or tactual changes are changes of position and not changes of form or quality.

In the second case, the seen independence

of form to size is confirmed by transportation and super­ position.

105

than a

But w e

do not "believe that dimension is nothing more

series

of Impressions and we do not "believe that the

independence of form from position and size is entirely a matter of series. sensations♦

Direction and form are qualities of visual

Taken alone they are empty and meaningless, they

come to life when coordinated with possible action.

Nonethe­

less they make their own unique contribution to the total spatial perception.

There is, in mature perception, a direct

visual comprehension of the permanence of form, of the sim­ ilarity of two figures differing in size but with respective angles equal, and of the similarity of form of the distant object which increases in size as we approach. The basic function of sense-perception is (l) to discover, (2) for action.

to locate, and (3) to identify in readiness Since the range of the body's actions is normal­

ly at arm's length,

the Identifications of shapes and sizes

are instinctively evaluated in perception against this stand­ ard.

Our spatial perceptions are therefore an outgrowth of

the interaction of our bodies with other processes.

Euclidean

geometry, as an outgrowth of these perceptions applies,

there­

fore, neither to the world's structure nor to the body's structure but to the interaction of these two. assured, therefore,

We may rest

that given the present body and the

present world the Euclidean space will suffice very well for

106 the needs of local action.

But it should he obvious that

if we w i sh to base our plans on a more comprehensive and less limited foundation than the body*s immediate action we ' should bear in mind that it would be well to consider other geometries than those which have grown out of the limited conditions of the body-in-action.

It should also be clear,

however, that when the problems of physical science are up for consideration, our first thought and preference will be for Euclidean geometry.

We have actually given up Euclidean

geometry in the interest of more general and precise predic­ tion, but ,who will say that it was easily done?

And who will

deny that our Euclidean preferences have left their mark upon the history of mathematics and physics? B.

Arithmetic

The very same questions we have asked about geometry may be asked anew of arithmetic.

What is the origin and

meaning of the natural integers and laws of arithmetic? Why are the properties of integers hereditary? 2 4 2 = 4

Why does

and why does a + b » b + a?

Are they w ork of the mind, 1" ' ^ J

*

Multiplying the equations of motion by a

(

The symbol 51 signifies summation over all the particles of the system; which can be an integration if the particles are rigidly united or a summation if the system is a discrete aggregate of particles. Considering first the left hand member of (2) we note that **1--x a *fr

c/ or

and consequently

~~ | f or

319 Then */

* ^ jr CX( ^

= «

.

" 7 7 “

?r

g, + . ^ f r 6 ' 3^ r

i

i i } oV«' ' W t r h ^ q r

^r- 1 * ^r+ 1 * * * * *^n

and t remain—

(being holonomic this is possible without

violation of the constraints) general-particular particle

yr+^fi'4?r ,

then the coordinates of the become

;/t-+13 ' 4

,

jt-+ |_y'4^r *

qi>qg*••*^n***> and where the work done by the external forces in any arbitrary displacement given by

a

qi, A

A %

ls %

4 qi +- Q 2 4 P).

Now

yr (Q,P) is harmonic inside of S, because

it is a potential caused by a surface distribution on S, and ~y (Q,P) is always zero,

since the surface S is grounded.

Consequently G(Q,P) vanishes when Q, is on S and the desired harmonic function. Therefore, given a function

$ (P) which is harmonic

and continuously differentiable throughout a closed region encompassed by a surface S then,

r

i*

c 0C