Talk:Philosophy of mathematics

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

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)

I rewrote the lead. For the moment, the new lead remains too short and consists simply of a difinition of the subject. It must be expanded for listing the main questions considered in philosophy of mathematics. I'll list these questions in the lead, with links to Wikipedia articles where these questions are better treated than in the present state of the article. A subsequent task would be to expand these list items in corresponding sections. D.Lazard (talk) 16:26, 8 May 2024 (UTC)

Some care must be taken with selecting references and writing for the modern position. There are certainly some intuitionists (at least) who would disagree that mathematical logic is entirely divorced from the philosophy. SoundwavePS (talk) 21:05, 22 September 2024 (UTC)

Certainly there are some, but they seem very marginal. On the other hand most modern articles using intuitionistic logic are totally disconnected from philosophy. (These articles are generally related to proof theory, which is undoubtly mathematics.) D.Lazard (talk) 10:41, 23 September 2024 (UTC)

cf. Intuitionism. I don't think e.g. Brouwer is by any significant means outdated but I'm sure there's modern commentary as well as opposition. Choice of logic to conduct mathematics in and interpretation of results of mathematical logic (that is, the subfield proper not its application to mathematics broadly) are still germane to the philosophy of mathematics. SoundwavePS (talk) 10:52, 23 September 2024 (UTC)

By "modern", I mean, in this case "since the middle of the 20th century". D.Lazard (talk) 11:05, 23 September 2024 (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)

Short description

@D.Lazard, can you give an explanation as to why my description was deleted? It was a good faith edit, trying to improve the artice. Your edit summary "this is not the place for defining "philosophy"" is unclear. My description was an attempt to define philosophy of mathematics, the topic of the article Farkle Griffen (talk) 18:38, 28 August 2024 (UTC)

It was indeed a good-faith edit, and it was likewise reverted in good faith. Short descriptions are not meant to be definitions of an article's topic; they're primarily intended to aid navigation on mobile platforms, etc. Per WP:SDNONE, if an article's title already suffices to disambiguate it, a short description is not required. Remsense ‥ 论 18:45, 28 August 2024 (UTC)

Ah, I see. Thank you Farkle Griffen (talk) 19:00, 28 August 2024 (UTC)

Section "Relationship With Physical Reality" needs a thorough rewrite.

Can someone with better knowledge of the subject than me put this section into ordinary idiomatic English? At present, it reads like a bad translation of a foreign text.

Examples:

During the 19th century, there were active research for giving more precise definitions to the basic concepts resulting of abstraction from the real world;

These formal definitions allowed to prove counterintuitive results, which are a part of the origin of the foundational crisis of mathematics.

Since the existence of such a monster seemed impossible, people had two choices: either they accept such unrealistic facts, which implies that mathematics does not need to reflect the physical reality; or they changes the logical rules for excluding such monsters.

After strong debates, axiomatic approach became eventually a de facto norm in mathematics. This mean that mathematical theories must be based on axioms...

The whole mathematics has been rebuilt inside this theory. Except if the contrary is explicitly stated, all modern mathematical texts use it as a foundation of mathematics.

Even the title should be rewritten. Only human beings have relationships. "Relation To Physical Reality." would be grammatically correct. Redpaul1 (talk) 22:06, 9 September 2024 (UTC)