The origins of ideas fundamental in the mathematical and physical sciences

555 Pages·029.643 MB·English

by  Stephens, John Alan

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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This dissertation, written by
___ Mr . _ John A. Stephens
under the guidance of h.% Faculty Committee
s___
on Studies, and approved by all its members, has
been presented to and accepted by the Council
on Graduate Study and Research, in partial ful­
fillment of requirements for the degree of
DO CTOR OF P H ILO S O P H Y
Dean
Date..¥*I.}.8.’ 1950
Committee on Studies
I Chairman
\
.
TABLE OF CONTENTS
INTRODUCTION PACE
A* 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 KNOWLEDGE........ 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 thesis........ ". . .................. 4
PART II
THE TWO PRINCIPAL SOURCES OF THE IDEAS
FORMING THE SCIENCES OF MATHEMATICS AND
iv
CHAPTER PAGE
SECTION I ......... 46
The role of the Fundamental Human Re­
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 . 46
A. Geometry ......................... . . 46
B. Arithmetic . . . . . . . . . . . . . . 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
IV. THE EMPIRICAL AND PHYSICAL CHARACTERISTICS
OF BABYLONIAN AND EGYPTIAN MATHEMATICS
AND THEIR IMPORTANT CONTRIBUTIONS . . . 138
A. Introduction ....................... 138
B. The influence of extraneous interests . 145
C. The'outstanding achievements . . . . . 152
V
CHAPTER PAGE
V. THE GREEK ACHIEVEMENT AND HOW THE GREEK
CONCEPTION OF THE PHYSIS AFFECTED THEIR
MATHEMATICAL INTERESTS............ • . . 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 * ......................... 184
D. The three famous problems............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 .................... 309
E. Theory of functions of a complex variable 353
F. Differential equations ................. 378
vi
CHAPTER PAGE
G. Geometry . ................. . 403
H. Vector algebra and tensor analysis . . 413
VIII. IMPORTANT'EPISODES IN MATHEMATICAL AND
PHYSICAL SCIENCE FAVORING THE CONJECTURAL
THEORY OF INTUITION AND CONCEPTION . . . 476
A, Geometry and analysis ................. 479
B* Algebra ........................... . 491
C. Physical science . ..........* . . * 503
BIBLIOGRAPHY .......... . 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* This thesis is concerned with
the first problem* It tackles, as a major objective, the
question of Origins1 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.