Monthly Archives: August 2006

Notes on recent reading

I interrupted my chain of thought on the philosophy of mathematics by reading some published material on that topic. I had “told myself” that I should avoid doing this, in the interest of keeping my thoughts fresh and free of any confusion that could come from reading other possible interpretations which may be based on unsound principles. Nonetheless, I proceeded to order a few books through the public library, based only upon titles, and ignorant of who the authors were.

I wound up with three books, of which I bothered reading only one – The Philosophy of Mathematics, by Stephan Korner, published in 1960. I’ve read a string of books now which all suffer from a common problem – incredibly poor writing style – and this one just added to the sequence. Although it is possible that I am finding the subject just too complicated to understand, Korner’s writing seems a torture of partially and poorly defined words and phrases. I’ll also admit to reading this book far too quickly, not taking notes, and being deeply bored by the experience. Through this haze, he presents three major schools of thought on the nature of Mathematics: Logicism, Formal Systems Theory, and Intuitionism.

The Logicists (Frege, Russell, Whitehead, Quine) seek to derive Mathematics from the fundamental rules of Logic. This school is a direct derivative of the rebirth of rationalism after Kant. Mathematics is a product solely of the mind, and bears no relationship to Reality. Mathematical truth is a synthetic a-priori truth: existing in a realm beyond experience, but created by the human mind. Not surprisingly, this attempt is seen to fail.

Formal systems theory is the attempt to consider Mathematics as a language of symbols, and to use this language representation and a set of transformation rules (the rules of Logic) to attempt to show the completeness or incompleteness of a set of axioms. This approach culminates in Godel’s Incompleteness Theorem, which shows, for the specific field of Number Theory, that given any assumed set of axioms, one can construct a “true” statement relative to those axioms which cannot be proven starting from those axioms and using the transformation rules. This esoteric proof is then generalized (foggily) to apply to all of Mathematics, and indeed, to all of Logic. Basically, this approach to understanding Mathematics results in the conclusion that mathematical truth is by its nature incomplete, and therefore of questionable application.

Korner’s exposition of Intuitionism I found to be almost entirely opaque. This approach is strongly empirical, and requires of every truth that it be inherently provable, and constructable. This leads to the denial of the law of the excluded middle: A or not A, as to prove the existence of a thing, it is not sufficient, according to the Intuitionist, to prove the impossibility of its non-existence. All existence proofs are required to lead to a construction of an example. I cannot claim to understand intuitionism – once the excluded middle was denied, I decided to spend my time elsewhere.

All three of the schools Korner describes are derived from Kant’s disastrous epistemology, as Korner correctly indicates. The author describes each school, and then offers his criticism, ending the work by proposing some philosophical framework of his own. My interest diminished rapidly after the introductory chapters, in which he discusses Kant with obvious reverence.

The other books I picked up included a set of lectures on mathematics by Wittenstein – after reading up on Wittenstein a bit, and discovering he was a self-contradicting linguist, I abandoned all hope of getting anything out of his impossible to read lectures.

The final book is more modern (1990s), entitled “What is Mathematics Really?” by Reuben Hersh. I have not opened this book, but Hersh apparently presents his “humanism” theory of mathematics, in which Mathematics is primarily a social activity which resists a firm definition. “Mathematics is what Mathematicians do.” How insightful. Reviews indicate that the presentation is very poorly constructed, that the history he presents is inaccurate, and his arguments range in quality from silly to questionable. I believe I’ll return this one to the library unopened.

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