
Category: Philosophy: Metaphysics
and Epistemology Category: Philosophy: General Category: Ayn Rand And Objectivism:
Philosophical
Theory Source:
Unavailable
Foundations Study Guide: Philosophy of
Mathematics
by David S. Ross, Ph.D. A
mathematician at Eastman Kodak Research Labs, David Ross has taught
mathematics at New York University and the University of
Rochester.
The philosophy of mathematics is the philosophical study of the
concepts and methods of mathematics. It is concerned with the nature
of numbers, geometric objects, and other mathematical concepts; it
is concerned with their cognitive origins and with their application
to reality. It addresses the validation of methods of mathematical
inference. In particular, it deals with the logical problems
associated with mathematical infinitude.
Among the sciences, mathematics has a unique relation to
philosophy. Since antiquity, philosophers have envied it as the
model of logical perfection, because of the clarity of its concepts
and the certainty of its conclusions, and have therefore devoted
much effort to explaining the nature of mathematics.
This study guide will recommend sources that provide an
introduction to the major issues in the philosophy of mathematics,
and the historically important views on these issues. Some
familiarity with mathematics is a prerequisite for thinking about
these issues. The book What is Mathematics?, by Richard
Courant and Herbert Robbins, is a brilliant exposition of the topics
and methods of modern mathematics. The book is intended for laymen,
but none of the essence of the mathematics has been omitted; it is
not a simple book, but it is rewarding.
Historical Views
Most philosophers have presented their views about mathematics in
works on more general topics. The anthology Philosophy and
Mathematics by Robert Baum contains selections on mathematics
from most major western philosophers, from Plato through Mill. The
selections include enough material to provide a context for each
philosopher's views on mathematics, and Baum's introductory essays
trace the philosophical influences on each thinker.
The most influential views have been those of Plato and Kant, and
Baum has a section on each of them. Interested Objectivists may want
to supplement Baum's section on Aristotle with a look at Thomas
Heath's Mathematics in Aristotle. Baum's book also contains
some modern essays, of which Max Black's "The Elusiveness of Sets,"
a criticism of the epistemology of set theorists, is worth
reading.
Analysis
Newton's theory of mechanics, and his invention of the integral
and differential calculus in support of it, are among the greatest
achievements in history. The central idea of limit is logically
subtle (this subtlety is what makes Zeno's Achilles paradox
perplexing), and Newton failed to treat limits rigorously. His
detractors—most notably Berkeley—made much of this flaw. Cauchy,
Weierstrass, and other 19thcentury mathematicians developed a
rigorous theory of limits, which provided an unassailable foundation
for Newton's theory and is a cornerstone of modern mathematical
analysis. This epistemological success story is well told in Carl
Boyer's The History of the Calculus and Its Conceptual
Development.
Another logical gem that is a central feature of modern
mathematical analysis is the idea of a wellposed problem, which was
introduced by the mathematician Jacques Hadamard. When a new
mathematical problem is proposed, the first order of business for
mathematicians is to establish that the problem has a solution, that
it has only one solution, and that the solution depends in a
reasonable way on the data (e.g., if the equation relates voltage to
illumination in a light bulb, a tiny increase in voltage should
result in only a small increase in illumination).
A problem that has these properties is called "wellposed." When
mathematicians establish that a mathematical problem is wellposed,
they are ensuring that it is a reasonable question to ask before
they try to answer it. Investigators in many other fields would be
well advised to adopt such careful epistemological habits.
Unfortunately, there is no philosophical introduction to this
topic.
Modern Issues
The popular current view is that mathematics has passed through a
series of logical or epistemological crises that have done it severe
damage. For a history of these "crises" (e.g. the invention of
noneuclidean geometry and the discovery of the settheoretic
paradoxes), and a thorough survey of the issues in modern
mathematical philosophy, see Morris Kline's Mathematics: The Loss
of Certainty. Kline was a mathematician; this book accurately
reflects the sort of attitude that one encounters among
practitioners, and it is well documented with pertinent
mathematics.
To determine whether there are flaws in the foundations of a
subject, one must first answer the more basic epistemological
question of what constitutes a proper foundation. The Objectivist
position that all knowledge must be grounded in perception, and
grasped and organized conceptually, has played virtually no role in
the historical development of the philosophy of mathematics. The
primary task of an Objectivist approach is to ground mathematics
objectively. An important secondary task is to explain how other
epistemological presuppositions have brought about the sense of
crisis and doubt that has characterized the field.
Stephan Korner's The Philosophy of Mathematics, an
Introductory Essay, is a less historically and mathematically
detailed treatment than Kline's, but it is more philosophically
sophisticated. Korner dedicates two chapters apiece—one expository
and one critical—to each of the three main modern schools of thought
on mathematical philosophy: the formalists, the logicists, and the
intuitionists. Korner's presentation is clear, concise and
unbiased.
Logicism
The logicist school, whose central figures are Bertrand Russell
and Gottlob Frege, had as its purpose to "reduce mathematics to
logic." Russell's Introduction to Mathematical Philosophy is
a nontechnical introduction to the logicist program. The logicist
conception of logic is radically different from the Objectivist, or
more generally, the Aristotelian conception of logic; and it is a
view of logic presupposed in most modern mathematical philosophy.
Russell's Introduction is an exceptionally clear exposition
of this conception of logic and its application to mathematics. It
is valuable as a guide to the premises that an objective approach to
the foundations of mathematics will have to challenge.
The works of Henry Veatch, notably Intentional Logic,
criticize Russell's conception of logic from an Aristotelian
perspective. Veatch argues from a tenet with which Objectivism
agrees—that consciousness is intentional, that it is always of or
about a world that exists and has identity independently of
consciousness.
Formalism
The formalist school was founded by the mathematician David
Hilbert. Formalists seek to express mathematics as strictly formal
logical systems, and to study them as such, without concern for
their meaning. (This is in contrast to the logicists, who seek to
establish the meaning of mathematical notions by defining them in
terms of concepts of logic.) Their primary motivation was to justify
the mathematics of infinite sets, which had been developed by Georg
Cantor in the late 19th century. The formalists hoped to express the
mathematics of infinite sets in such a system, and to establish the
consistency of that system by finite methods. If they succeeded in
this, they thought, they would have justified the use of infinite
sets without having to address the thorny question of just what such
sets are.
The formalist approach is explained and illustrated in Godel's
Proof by Ernest Nagel and James Newman. This short book is a
masterpiece in making sophisticated material accessible to
nonexperts. The book starts with an exposition of formalism, and
concludes with a very readable outline of the proof of Kurt Godel's
incompleteness theorem. This theorem showed, on the formalists' own
terms, that their program was untenable.
Intuitionism
The intuitionists, whose leader was the mathematician L.E.J.
Brouwer, are best known for their conservatism regarding
mathematical infinitude. They are opposed to the application of the
law of excluded middle to statements involving mathematical
infinitudes, as in a proof that takes the following form: either
there is a number with the property P or there is not; if not, a
consequence follows that is known to be false; therefore there
exists a number with the property P. Such proofs do not tell us what
the number in question is, or why it has the property. Constructive
proofs, by contrast, do provide this information, and intuitionists
require constructive proofs of mathematical theorems.
The intuitionists find their philosophical roots in Kant. Yet
their caution regarding the infinite should appeal to Objectivists.
Their position on the law of excluded middle may be interpreted as a
demand that a statement be established as meaningful before the laws
of logic are applied to it, a demand that Objectivism certainly
endorses. Their insistence on constructive proofs may be seen as a
means of specifying what is meant by the existence of a number.
Unfortunately, intuitionists are not always clear about the
meaning and philosophical foundations of their positions; they
attend to mathematical details at the expense of philosophical
exposition. There is no introduction like Russell's or Nagel and
Newman's. There are several pieces by intuitionists— Brouwer,
Heyting and Dummett— in the collection Philosophy of Mathematics,
Selected Readings, edited by Paul Benacerraf and Hilary Putnam.
The introduction to this volume also contains a clear discussion of
intuitionist principles.
Objectivism
A proper understanding of abstraction is a prerequisite for
explaining mathematical concepts. Historical theories of
mathematical concepts have tended to embody the worst aspects of
historical theories of universals; Platonic realism, Kantian
idealism, and extreme nominalism dominate the subject.
Ayn Rand's identification of the nature of universals and her
analysis of the process of abstraction have much to contribute to
the philosophy of mathematics. There is, however, no Objectivist
literature on this topic. An indication of an Objectivist approach
to the subject is given in the essay "The Cognitive Basis of
Arithmetic" by David Ross. Comments by Ayn Rand on various
mathematical topics are contained in the appendix to the 1990
edition of Introduction to Objectivist Epistemology.
Objectivism recognizes a deeper connection between mathematics
and philosophy than advocates of other philosophies have imagined.
According to Ayn Rand's theory, the process of conceptformation
involves the grasp of quantitative relationships among units and the
omission of their specific measurements. It thus places mathematics
at the core of human knowledge as a crucial element of the process
of abstraction. This is a radical, new view of the role of
mathematics in philosophy. As Leonard Peikoff has put it in
Objectivism: The Philosophy of Ayn Rand,
Mathematics is the substance of thought writ large, as the West
has been told from Pythagoras to Bertrand Russell; it does provide
a unique window into human nature. What the window reveals,
however, is not the barren constructs of rationalistic tradition,
but man's method of extrapolating from observed data to the total
of the universe...not the mechanics of deduction, but of induction
(p. 90).
Thus, an area that an Objectivist philosophy of mathematics must
address is the meaning and structure of measurement in the
measurement omission theory; this subfield of the philosophy of
mathematics might be called the mathematics of philosophy. For the
Objectivist view, see Rand's discussions in Introduction to
Objectivist Epistemology, Peikoff's Objectivism: The
Philosophy of Ayn Rand, and David Kelley's "A Theory of
Abstraction."
Bibliography
Many of the titles in this listing are available at Amazon.com.
If you use this link, or the search box below, then IOS will
earn a commission from Amazon.com on each book purchased.
Robert Baum. Philosophy and Mathematics. San
Francisco: Freeman, Cooper, 1973.
Paul Benacerraf and Hilary Putnam. Philosophy of
Mathematics, Selected Readings. Cambridge: Cambridge
University Press, 1983.
Carl B. Boyer. The History of The Calculus and
its Conceptual Development. New York: Dover, 1949.
Richard Courant and Herbert Robbins. What is
Mathematics? Oxford: Oxford University Press, 1978.
Sir Thomas Heath. Mathematics in Aristotle.
Oxford: Clarendon Press, 1949.
David Kelley. "A Theory of Abstraction,"
Cognition and Brain Theory, Volume 7, 1984. Reprinted
by the Institute for Objectivist Studies, 1994. (Available
from Principal
Source)
Morris Kline. Mathematics, the Loss of
Certainty. Oxford: Oxford University Press, 1980.
Stephan Korner. The Philosophy of Mathematics, an
Introductory Essay. New York: Dover, 1986.
Ernest Nagel and James R. Newman. Godel's
Proof. New York: New York University Press, 1958.
Leonard Peikoff. Objectivism: The Philosophy of
Ayn Rand. New York: Penguin Group, 1991. (Available from
Principal
Source)
Ayn Rand. Introduction to Objectivist
Epistemology. New York: Penguin Group, 1990. (Available
from Principal
Source)
David Ross. "The Cognitive Basis of Arithmetic."
Poughkeepsie, N.Y.: Institute for Objectivist Studies
(forthcoming) .
Bertrand Russell. Introduction to Mathematical
Philosophy. New York: Simon and Schuster.
Henry Veatch. Intentional Logic. New Haven:
Yale University Press, 1952.

Foundations #3
Foundations is a series of Study Guides designed to help
individuals and discussion groups who wish to gain an overview
of a field from an Objectivist perspective. Each Study Guide
is prepared by an expert who selects and comments on readings
that either reflect an Objectivist viewpoint or are valuable
for other reasons. Specific works mentioned in this or other
Study Guides should be read critically; their inclusion does
not imply any endorsement by the
Center. 
PrinterFriendly  Send
to a Friend 
