Talk:Philosophy of mathematics

Badiou again

I see from Archive 2 that a paragraph on Badiou was deleted. I'd argue for a reference on grounds of notability. He is one of the better known living philosophers, mathematics is central to his work, and although he has stated that his main positions are not to be thought of as contributions to the Philosophy of Maths, he does have some papers on central PoM topics like the meaning of number. Unfortunately, I can't produce a paragraph with my current level of knowledge of his work. KD Jan 16 07 —Preceding undated comment added 19:31, 16 January 2007

His work is not commented upon by reliable, mainstream academic sources. This effectively means that his work does not merit inclusion here. --Omnipaedista (talk) 19:02, 20 August 2020 (UTC)

Chaitin's remarks on Erdos sohuld be deleted

I feel strongly that Chaitin's remarks about Erdos and his "book" should be deleted. First, they are undocumented, but I assume they are taken from Chaitin's book "MetaMath" (where such remarks do appear). This book is, at best, a popular account of some ideas in theoretical computer science. Gregory Chaitin is somewhere between a mathematician and a computer scientist -- he is certainly not a philosopher of mathematics. The general view on his work in the mainstream mathematical community is that his theorems are correct and are of some interest (note: we do not regard him as "one of the foremost mathematicians", a claim which appears on the back of the book) but that the philosophical conclusions he draws from them are at best wholly unjustified and at worst so sloppy and vague so as not to count as philosophy at all. In his books, Chaitin often phrases things so as to make clear that he has not done much research on a topic and is just giving his impression, as much as a personal statement as anything else. I do not know that Chaitin met Erdos or had any real idea of Erdos' ideas about mathematics. As regards "The Book", a form of it was published (first in 1998, several years before Chaitin's book). The book records beautiful proofs of theorems, often more than one. In fact, as regards the infinitude of primes, "Proofs from The Book" gives neither one nor three but six, including one which is is similar to (more sophisticated than but with stronger consequences) than Chaitin's "new" algorithmic information theory proof. In summary, what you are reporting is a hearsay opinion of one person on the thoughts and ideas of another, and that this opinion is unjustified is well documented.— Preceding unsigned comment added by 216.24.161.3 (talk) 22:30, 11 March 2007

Apparently OR argument under social constructivism

The following blockquote is currently the last two paragraphs under Social constructivism, for your convenience:

A criticism of this approach is that it is trivial, based on the trivial observation that mathematics is a human activity. To observe that rigorous proof comes only after unrigorous conjecture, experimentation and speculation is true, but it is trivial and no-one would deny this. So it's a bit of a stretch to characterize a philosophy of mathematics in this way, on something trivially true. The calculus of Leibniz and Newton was reexamined by mathematicians such as Weierstrass in order to rigorously prove the theorems thereof. There is nothing special or interesting about this, as it fits in with the more general trend of unrigorous ideas which are later made rigorous. There needs to be a clear distinction between the objects of study of mathematics and the study of the objects of study of mathematics. The former doesn't seem to change a great deal; the latter is forever in flux. The latter is what the social theory is about, and the former is what Platonism et al. are about.
However, this criticism is rejected by supporters of the social constructivist perspective because it misses the point that the very objects of mathematics are social constructs. These objects, it asserts, are primarily semiotic objects existing in the sphere of human culture, sustained by social practices (after Wittgenstein) that utilize physically embodied signs and give rise to intrapersonal (mental) constructs. Social constructivists view the reification of the sphere of human culture into a Platonic realm, or some other heaven-like domain of existence beyond the physical world, a long-standing category error.

These paragraphs appear to be, respectively, somebody's personal reasoning on the social constructivist view, and somebody else's counterargument. The especially egregious telltales to me are the hedged language ("So it's a bit of a stretch"), vague allusions to famous mathematicians and philosophers but not specific works of theirs, and more than anything, the phrase "Platonism et al. My instinct is to remove the paragraphs in question completely and the more citations needed template with them. Thoughts?

--A Lesbian (talk) 09:59, 22 January 2020 (UTC)

Concur. Do remove. Zezen (talk) 19:13, 19 August 2020 (UTC)

+1. I like the ideas expressed in these paragraphs as "to be discussed/thought about" - but this is obviously almost the opposite of an argument for inclusion in the WP. --User:Haraldmmueller 19:21, 19 August 2020 (UTC)

Article tagged as non-neutral point of view and incomplete

I have tagged the article with {{npov}} and {{incomplete}}. The rationale is as follows.

The article is blatantly non-neutral, as it ignores completely the point of view of mathematicians who have addressed the subject of this article. A good resource for mathematicians’ point of view is given by Armand Borel with references to and quotations of G. H. Hardy, Charles Hermite, Henri Poincaré, Albert Einstein and other authoritative mathematicians.

Also, section § Contemporary schools of thought is pure WP:OR, by providing a classification that does not reflect the thoughts of the authors that are classified this way.

The article is also incomplete, as missing several fundamental questions and most achievements of 20th century.

The main fundamental question that is not addressed is the relationship of mathematics with other sciences. This is presently sketched in Mathematics § Relationship with science, and deserves to be expanded here.

One of the achievements of the 20th is that the axiomatic method is presently a standard in mathematics, which is universally accepted by mathematicians. Another one is that mathematical logic is no more a part of philosophy, but is a part of mathematics. In particular, a logic is a mathematical object that can be axiomatized. The main logic is Zermelo–Fraenkel set theory (ZFC), and the other logics (such as intuitionistic logic and constructivism) can be defined in the framework of ZFC. This allows using several logics in the same work. In particular, the theorem prover COQ uses intuitionistic logic for proving theorems of classicl mathematics.

Another question that is not really addressed by the article is the main subject of Borel's article: Is mathematics a science, an art, a game, or all together?

References

D.Lazard (talk) 10:55, 19 November 2022 (UTC)

Greek understanding of the number "one"

The discussion in the History section of how the Greeks understood the number "one" could probably be sourced with this book:

Greek Mathematical Thought and the Origin of Algebra by Jacob Klein (MIT Press, 1968; Dover Publications, 1992).

Unfortunately, a search in the Google preview for "multitude" returns numerous results, but no whole pages. 50.47.142.96 (talk) 20:29, 29 October 2023 (UTC)