Talk:Philosophy of mathematics

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Former good article Philosophy of mathematics was one of the good articles, but it has been removed from the list. There are suggestions below for improving the article to meet the good article criteria. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.
August 8, 2006 Good article reassessment Delisted
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critique of formalism[edit]

I deleted the folowing sentence, wich I think was really misleading:

"While published proofs (if correct) could in principle be formulated in terms of these games, the effort required in space and time would be prohibitive (witness Principia Mathematica.)"

For what I've seen of maths, every book or paper is very careful about what hypothesis or axioms it uses, down to the simplest notions. So you actualy can trace the proof back to the axioms; sure pricipia mathematica didn't work, but it's from 1913! Bourbaki could at least be cited, and though it was pretty bad in a pedagogic sense, it does work.

Also removed this; I'm not even sure of hat it means, except that the motivations for proving a theorem does not come from the "string rules", and that is already in the article.

" In addition, the rules are certainly not substantial to the initial creation of those proofs. "

Aleph42 16:07, 1 December 2007 (UTC)

Phil of Math, a branch of Analytic or general philosophy[edit]

I'm reverting the change of phil. to anal. phil. in the first sentence again. Greek phil. of math. from 2500 years ago is still much studied, and is hardly an example of anal. phil. Ditto for Oriental philosophies. There is no reason that any phil. of math. that one might develop should have to follow the anal. phil. trend, even if many currently do. JJL 17:36, 29 November 2006 (UTC)

Again I'm reverting the removal of information about what kind of philosophy is primarily involved in the Philosophy of Mathematics listed on this page.

As you say above, much philosophy of Math is currently, Analytic, this is the issue, and I agree, much is analytic, certainly, and this page is too. The page is not from philosophy but from Analytic philosophy. As soon as this changes then change the revert. Now if you prefer the name Philosophy, well and good. However, just because Pythagoras is mentioned does not mean it is not almost entirely based on Analytic philosophy: Frege, Russell, Whithead, Hilbert etc. In non-analytic philosophy there is no such thing as the "philosophy of mathematics", the compartmentalising of philosophy into such a thing as the "philosophy of mathematics" in the first place, is Analytic, so even by the given name of this page it is analytic, this is why your argument about subsets does not work. --Lucas

Even if I were to agree that much modern phil. of math. is anal. phil., there's still much phil. of math. that predates it--again, the Greek tradition of phil. of math. comes to mind. I see your similar change at Philosophy of science is being contested too. Phil. of math. (and sci.) may be done in many different ways. Is there no Oriental tradition of it? I strongly disagree that info. was removed--to the contrary, I feel that your view is partisan. JJL 00:20, 30 November 2006 (UTC)
The argument you make here is that analytic phil of math, is only a subset of phil of math in general. You miss my point, my point is that there is no such thing as the "philosophy of mathematics" in non-analytic philosophy, or Greek or "Oriental" phil. This compartmentalising of philosophy is itself an effect of Analytic philosophy. And this is the information I claim is being removed by you. In other words the article without this additional information appears to accept the "Analytic" idea that there is a separate division called "phil of math", and with it that it is not fully in touch with philosophy in general. Given that the page itself is called this, the least it was doing was adverting to this problem by adding the adjective and link, thus enabling further investigation by a reader.
I have removed 'analytic'. If we Google "Kant's philosophy of mathematics" we get lots of hits, moreover it turns out that many courses of that name are being taught. Similarly for "Aristotle's philosophy of mathematics", or for many other non-analytic philosophers.
I certainly agree that calling something the 'Philosophy of X' is an analytic custom (or probably is). But that does not mean that what Plato, Aristotle, Kant &c were not doing philosophy of mathematics, or that they did not have a philosophy of mathematics, even if they did not call it that. The article is not about what philosophy of mathematics is called, but about the discipline itself. Therefore: remove the "analytic". Dbuckner 15:16, 1 December 2006 (UTC)
I agree w/ Dbuckner, the 'analytic' should go. Whether other philosophers or traditions used the name isn't relevant to whether they were engaged in the practice. --The Hanged Man 15:46, 1 December 2006 (UTC)
You seem to think that Plato, Aristotle and Kant are not Analytic and so because the article covers them, it is also not Analytic. All of Plato, Aristotle and Kant are read in the Analytic tradition. How much you can say they were practicing what is now called "The Philosophy of Mathematics" is only a testament to how successfully it has been sundered out from a mainstream philosophy in the Analytic school.
You say "it is not about what philosophy of maths is called but the discipline itself." My point is that this discipline, as a separate discipine, is only enforced in the Analytic school and for its own reasons, and so the word in the article only gave information on this. If you wish to ignore one half of what goes on in the world of philosophy, fine, I think some people might guess it anyhow (as you seem to), since the current article is written in English and is quite Anglo-centric.

The article is written in English because it is the English Wikipedia. Zero sharp 20:47, 3 December 2006 (UTC)

Yes, it is in English Wikipedia, but it does not mean it must exclude anything that is not local to the language. Being Anglo-centric was an issue I implicitly hint at correcting, it is not a problem necessarily, except when it is making global claims. --Lucas
They are read in the analytic tradition, but they are not members of the analytic tradition. Just because, e.g., French philosophers might not do much philosophy of math, or even mark it off as a major sub-discipline, doesn't make phil. math. a branch of analytic philosophy. In fact, I think it is probably a mistake to think of analytic philosophy as a discipline with branches, rather than a difference of emphasis. The analytic-continental distinction is highly problematic and probably, at this point in time, not very relevant. Furthermore, I'm highly suspicious of your claim that only analytic philosophy breaks philosophy down into any divisions (e.g., it sure seems like Aristotle and Kant both did so).
Also, again, just because someone doesn't consciously categorize their practice in a certain way doesn't mean that it doesn't qualify as a member of that category. --The Hanged Man 00:51, 4 December 2006 (UTC)
This is true for categorizing non-human things, but if in philosophy in general, most see it as erroneous to have a certain branch/discipline, it does not mean you can force it upon them. Granted in your own way of doing philosophy you might have created such a branch, and can consciously include what you like in it, but you cannot say that I must also have such a branch.
As to whether or not the continental-analytic divide is clear or relevant here is, I agree, perhaps another issue. And perhaps Analytic is only a different emphasis. I'm just saying that a large number (they are perhaps mainly continentals or eastern) do not want, or need, or agree with having this as a separate discipline. But that only the "analytic emphasis", as you call it, does suggest such a separated branch.
As to Aristotle and Kant, well, they too were entitled to consider branches as they saw fit, but, as far as I know, neither wanted Phil of Math as a separate discipline at their academies, even though both are also part of, and read in, the Analytic tradition and other traditions.
What evidence is there that they would be against such a division? --The Hanged Man 23:55, 9 December 2006 (UTC)
In the same manner that there is evidence that in Analytic philosophy there is such a division or branch (as given by the article). In the university departments of Analytic philosophy the "phil of math" can be taught as a separate module. A list of views can be presented, within the assumed question of "Is maths true or not?" And answers listed: math is a formalism, based on logic, empiric, fiction, social, etc.
In continental philosophy things do not go this way, it is not treated separately but as part of other areas (epistemology, ethics, etc.,) or as a part of the study of a certain philosopher. Mainly however, phil of math as epistemology and maths in general, right or wrong, is not considered to be an area that needs to be reassured or doubted by a philosophy of mathematics (as it might have been in the hayday of epistemology).
So says you. Where's the evidence? It seems that the burden of proof is on you at this stage. And why should a vague sense of how things are taught in various academic departments have so much weight? --The Hanged Man 07:06, 11 December 2006 (UTC)
I make an argument, it is not refuted and you ask for more. What I argue against is the removal of information from the article.
In fact, the burden of proof is for you. How to show a negative? Yet I will still give references, which is presumambly what you mean by evidence. One easy way for you to check, given that you are not familiar with academic philosophy departments, is to look at analytic philosophy, there see some of the different branches listed, now check continental philosophy. Now check some continental tradition philosophers who covered some mathematics Hegel and Husserl on geometry, you will find that without knowing Hegels system dialectic etc. you will not get his skinny chapter of Logic on "number". Similarly Derrida and Husserl will not provide much sense on geometry without first following what they mean by phenomenology and deconstruction. So you see phil of math is embedded in larger systems for continental and is not apparent as a separable discipline or branch.
Ah, now the argument ascends to insults? Very nice. Besides the fact that you have no idea what sort of familiarity with academic philosophy departments I have (a major gaffe on your part), on Wikipedia, information requires evidence in order to be included, not in order to be excluded. At best, your attempt to narrow the scope of phil math constitutes original research at this stage, and it seems to me (and several others) that it is either misleading or false. Citing Wikipedia doesn't provide much in the way of evidence for your claim, especially since none of the WP articles you refer to say anything helpful about philosophy of math, except perhaps making it obvious that Husserl had one. Further, your argument that Hegel or Husserl's philosophy of mathematics make little sense outside of their systems doesn't hold much water, as we'd have no problem saying that they had an epistemology, or a metaphysics, or an aesthetics, though they too can only be productively understood in the context of a systematic reading. Furthermore, I don't think you could properly understand Frege's, Russell's, or Quine's phil of math without a similar hermeneutic approach.
Look, there isn't a single person who agrees with you, and many who disagree with you. It seems to me that if you can find no evidential support and no one to support you, then you need to give it up. As far as I'm concerned, any more attempts to revert the article prior to consensus on your view being reached on this Talk page would constitute vandalism, and the appropriate actions ought to be taken, or at least needs to be dealt with using something from WP:DR. (Any suggestions about this? I've never had to deal with this kind of stubbornness before.) --The Hanged Man 06:52, 14 December 2006 (UTC)
Do we need mediation here? I'm getting dizzy watching the volley. Wikipedia does have its NOR policy to enforce. But, if the first sentence needs to be general, perhaps the 'Recurrent themes' list could include some reference to the 'analytic' specifics with a link. jmswtlk 14:49, 14 December 2006 (UTC)

There is little of an insult in suggesting that someone is unfamiliar with departments of philosophy. Nor that phil of math is mainly a separated out discipline mainly in anglophone departments.

Most I think agree with me on this, Analytic does tend to have a different structure (branches disciplines etc.) compared to Continental. The issue of evidence is not forthcoming from you. There is no evidence for your case to allow you say that this is a "branch of philosophy" rather than a "branch of Analytic philosophy". To say it is a branch of philosophy in general is original research. What I am attempting here is to balance up the page and remove point of view. The entire article is almost a mirror image of the analytic philosophy story of mathematics and its attempts to prove it true or false or as fiction etc.. Your pretence that this is somehow basic philosophy and not the analytic type is perpetrated.

As to Hegel, Husserl, Derrida etc., as I said, to give justice to their ideas on math you need to see it as part of their entire philosophy and that includes thinking about ethics, metaphysics, their method, view of history etc. In approaching phil of math in the manner of the article these wider issues are not expected and are left out and it simply becomes a matter of showing things about math, as though maths were a given. This is analytic through and through. It may be a given for Russell etc. nor do I think you need to know much about Russell's phil of history or ethics to understand his attempt to prove math is based on logic.

By not highlighting these facts and that the main idea of a phil of math is Analytic you mislead the reader and pretend that all of contemporary phil sees maths in this way, as a separate branch, when in fact you have no evidence for this.


The followign coppied from Philosophy of science: Lucaas, can you find suitable sources that support your contention; or failing that, can you find other resources that list Phil of maths specifically as a branch of analytic philosophy. If not, your position is original research. Banno 20:00, 14 December 2006 (UTC)

(cf, talk for phil of science...

The Article had said correctly that it was a "branch of Analytic philosophy", this was then removed, and someone said it was a main branch of philosophy in general, without any backup or citation. If you can cite somewhere to say this is so that is ok.

To do the reverse is to try and prove a negative. For example I say it is not a branch in Continental, you say find evidence that it is not a branch there; just as if I said "there are no mallards in France", you would say "find me no mallards in France".

Other pages do at least say they are a branch of western philosophy, see epistemology, where it declares it as a branch of western philosophy.


What about conceptualism? I remember reading about it, but a quick search on the Web does not bring up any names (except for Brouer, but his is a special case). Still, people talk about admitting universals as being mentally constructed. Kind of like: numbers do exist in a Platonic sense because it is convenient for us to think of them as such, even though they really are figments of our imagination. Am I delirious? I could swear I saw it in Fraenkel's Foundations.

melikamp 04:09, 2 January 2007 (UTC)

The same argument threads through a number of the philosophy pages by now. Lucas is wrong about Husserl. His contribution to philosophy of mathematics was his first work, it pre-dates the development of phenomenology, and indeed it was criticised by Frege for being too indebted the psychologism of Mill and others. It was primarily in order to set himself apart from the psychologists that Husserl began the project which became phenomenology. That early phil of maths can indeed be considered separately from his work as a whole: indeed, if he'd dropped dead after writing it, we'd have no choice, because he would never have developed phenomenology. I don't know if Lucas is familiar with the intuitionists (Brouwer et al), but their work is precisely not analytic. For Brouwer, the truth of mathematical propositions is a function of the mathematicians internal intuitions, not the analysis of the terms in which it is stated. KD Jan 16 07
I suggest (a) we distinguished Analytical Philosophy from analytical philosophy. (b) we avoid ananchronism when applying branch names to texts when the latter predate the use of the former. --Philogo 11:04, 11 May 2008 (UTC)

Badiou again[edit]

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

Chaitin's remarks on Erdos sohuld be deleted[edit]

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.

Formalism is realism?[edit]

Why "formalism" is now a subsection of "realism"?--Pokipsy76 (talk) 17:40, 4 January 2008 (UTC)

This seems like a substantial problem for the article, or at least misleading for a reader that does not know better. Perhaps we can just move the 'formalism' section out of the 'realism' section? Taekwandean (talk) 13:51, 19 October 2010 (UTC)

Reference formatting[edit]

Just something to put on the to-do list for this article, and I put a template at the top to explain somewhat: The references need to have the


style in order to make this article more wiki-like, and I'll start things off, but if someone has more time, or I get around to it, the whole article could use it. Be warned, the static citation list at the bottom needs to be used for the reference list, so there'll be some copy-and-pasting to be done. Rhetth (talk) 21:39, 10 February 2008 (UTC)

Also, most citations will use the Template:Cite_book format (follow this link for a description of what to put where), so here it is:
{{cite book
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 | url = 
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Rhetth (talk) 21:54, 10 February 2008 (UTC)

Poor Tone[edit]

Few philosophers are able to penetrate mathematical notations and culture to relate conventional notions of metaphysics to the more specialized metaphysical notions of the schools above. This may lead to a disconnection in which some mathematicians continue to profess discredited philosophy as a justification for their continued belief in a world-view promoting their work.
Although the social theories and quasi-empiricism, and especially the embodied mind theory, have focused more attention on the epistemology implied by current mathematical practices, they fall far short of actually relating this to ordinary human perception and everyday understandings of knowledge.

This whole passage under Unfification is far too assertive for the evidence or information it presents. "some mathematicians continue to profess discredited philosophy as a justification for their continued belief in a world-view promoting their work". Discredited philosophy? I don't believe any school of philosophy of mathematics has been really discredited in this sense. If no one objects I think this should be changed to a less assertive phrasing at least. (talk) 23:48, 31 August 2008 (UTC)

OR from top to tail IYAM (talk) 20:23, 8 September 2010 (UTC)

A New Empiricism: Back to the Future[edit]

This section seems too long and digressive to merit a section on this page...perhaps it could be replaced by a link to another article that discusses the ontology of thoughts and minds? It's an interesting topic, but it's very deep and controversial, and only indirectly related to mathematics. Also, it looks like this idea is already addressed under the 'Embodied mind theories' section. Awoody87 (talk) 20:30, 24 December 2008 (UTC)

This section makes certain claims that don't seem to belong in wikipedia. Specifically the paragraph

A few decades ago such a proposition was unthinkable. Today, in light of neuroscientific research, a large number of observers consider it undeniable. It is admittedly disturbing, raising many questions about human values as well as contradicting most religious beliefs. But it is also liberating, relieving us of a vast library of puzzles and paradoxes that come with trying to conceptualize an immaterial world of the mind.

This seems to violate the norms that make wikipedia function. Who are these observers who claim its undeniable? Who is it liberating for? Does Wikipedia find it liberating? I don't think wikipedia is the sort of entity that can find an idea liberating.

There is a noted absence here of several schools of thought[edit]

Gödel is surely a Platonist as noted, but he also writes about the strong influence of Husserl on his philosophical thinking. Although I regard Phenomenology as a philosophy that has cross-cutting concerns with each of the philosophical schools represented here, rather than in contrast with any of them, I think that Husserl's work in the philosophy of mathematics deserves to be acknowledged. Likewise, I don't see represented here Structuralism, whether de re structuralism or the ante rem structuralism of Stuart Shapiro. I'm not the expert to do either of those, but would appreciate seeing the article extended. Perhaps there should be a Category: Philosophies of Mathematics page, to help help to break this article out into several other articles. Gene B. Chase 01:40, 7 December 2009 (UTC)

Discourse weakness[edit]

The section Platonism, introduces the topic with:

Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging.

(my underlines to elevate the discourse oddities) then later on some questions are posed, most of them justifiable and reasonable, but the odd ones being:

precisely where and how do the mathematical entities exist, and how do we know about them?

(my underlines here too) ... but where and how was defined in the initial sentence, why asking for them when the answers were provided for already in the initial definition? "How do we know about them?" seems reasonable enough, but "where?" is answered by "no spatiotemporal qualities implies: question meaningless!" and "how?" is answered by "eternally/unchangingly". The text must be fixed to make the questions where and how sound less like a fool's question. Rursus dixit. (mbork3!) 09:39, 4 May 2010 (UTC)


The Unification subheading under the "Beyond the Traditional Schools" section is chaos. It isn't clear at all what the section's point is, and it is less clear why it is regarded as unification. Maybe the point is best expressed by the fact that the section fails to use words like "unify" or "unification" at all. This is why I hate philosophy... so abstract as to be divorced from practical needs (of the readers). (talk) 20:38, 20 August 2010 (UTC)

I agree. The entire section on "Beyond the Traditional Schools" needs serious work to make it understandable. A few references would be helpful as well. --seberle (talk) 04:25, 22 August 2010 (UTC)

Introduction Too long[edit]

The introduction seems to be a bit too wordy, especially with the overly long list of bullet points. I would recommend removing the list and putting it in the main article and substituting a short prose summary. Is everyone okay if I do this? —Preceding unsigned comment added by (talk) 01:43, 8 February 2011 (UTC)

Further changes - suggestions[edit]

  • Introduce a separate arguments section? Many arguments are broad and do not favour (eg) Platonic realism over empirical realism. Indispensability argument needs mentioningo
  • Introduce a section dealing with relevant discoveries, eg. irrational numbers, non-euclidean geometry, transfinite numbers, russell's paradox, Godel.
  • Start with pythagoreanism? Already in history. However Platonism is both ancient and modern.
  • Forget trying to group the standard -isms as realist or anti realist, since in some cases it isn't clear. But do

mention in the text where it is clear.

  • The first part of the Platonism section looks in need of a rewrite
  • Quasi empiricism needs to be a school
  • More on category theory?

1Z (talk) 20:16, 14 February 2011 (UTC)

Husserl and Klein?[edit]

There needs to be some discussion of Jacob Klein in the body of the article; he carried on Husserl's project regarding mathematics and science. A serious article would also engage with Husserl's ideas in general (more than a couple of passing mentions); the man was only one of the most influential philosophers of the 20th century. JKeck (talk) 02:40, 25 April 2011 (UTC)

Wouldn't our page for Husserl be a more appropriate place for such an engagement? Tkuvho (talk) 17:44, 25 April 2011 (UTC)

new empiricism[edit]

I deleted an uncited paragraph in the new empiricism section full of speculation ('original research') and weasel, including claims and qualifiers like 'more basic' 'everything about' 'it all' 'infallible principles' 'not influenced by [external]' 'distorted' 'so to say'. It reads like an argument written from a single source known as the author. If you can say how this is encyclopedic and also cite it, please restore it. I hope you won't. (talk) 21:36, 2 January 2013 (UTC)

There are more problems with that section. Statements like "new empiricism shows", or "his explanation still nonetheless manages to demonstrate that there is no way around Kant's a priori logic." don't suggest objectivity. The "Social constructivism or social realism" section seems a bit suspect as well. Ssscienccce (talk) 03:13, 15 August 2013 (UTC)

Post structuralism?[edit]

The article could use a mention of George Lakoff and Rafael E. Nunez - that is it impossible to know the reality of mathematics beyond our capability to perceive. — Preceding unsigned comment added by (talk) 00:56, 12 November 2013 (UTC)