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Talk:Philosophy of mathematics/Archive 2

Template:philosophy

Older discussion: see /Archive 1.

Why so much thunder and lightning?
It is astonishing how this highly academic topic (and not a big one nowadays) has generated so much heat. This entry should mainly review some history of ideas, then the ideas of the major contemporary figures (Boolos, Putnam, Benaceraff, Rosen, Burgess, Field, Fine, Resnick, and so forth), with lots of links to other entries with more details. I warn all and sundry that a major contributor to this article of late, Jon Awbrey, does not have the academic qualifications needed to sustain his self-assurance.202.36.179.65 21:19, 15 July 2006 (UTC)

 * Please move your dicussion on each other's qualifications to talk pages unless you are willing to be specific about how it affects this article. 142.177.110.74

 * Perhaps, also, the reason this is "not a big topic nowadays" is because prejudices and ignorance guide so much comment on it. There are many unexamined beliefs, like some of those below. 142.177.110.74

More peaceful discussion

 * ... mathematics is studied in a markedly different way than other languages. The capacity to acquire mathematics, and competence in it, called numeracy, is seen as separate from literacy and the acquisition of language. Some argue that this is due to failures not of the philosophy of mathematics, but of linguistics and the study of natural grammar.

What thinkers are making these "linguistic" objections? Also, the half-implicit idea that human language and math are fundamentally the same and should be studied in the same way and by the same specialists is not NPOV and perhaps absurd.

 * Noam Chomsky's universal grammar theory has been extended to mathematics by some - especially to music. Not sure that failure to converge the fields is evidence that "the same specialists" should be involved - maybe a few genius unifiers though.  As for the issue of what is "half-implicit", it would be nice to try to define that term.  There are obvious similarities between math and other means by which humans discourse:  the use of symbols, acquisition by training and reinforcement, learning of grammatical forms that are correct vs. incorrect ways to use the symbols, and the way they are stored and recorded.  Given all this duck-quacking, some people see a duck.  It is not "absurd" and certainly no suggestion that this idea exists fails the NPOV test.  The statement is carefully qualified in a section named to emphasize the extensions and explorations that have gone beyond the 20th century "schools" (none of which have remained credible apparently, as there is not much going on in this field) so is not objectionable.  It would be objectionable up front in the very first paragraph maybe, but it could go there too with appropriate qualification.  And maybe it should, since looking at mathematics as language would be a way for neophytes to start to ask some serious and difficult questions about how mathematics differs. 142.177.110.74

While there are undeniably commonalities between natural language and math, there are many differences; perhaps the most fundamental is that people automatically learn to be extremely competent speakers of their native languages without any formal instruction, while even the brightest mind will not get very far at all if it has to start without any math education. --Ryguasu 07:28 Feb 21, 2003 (UTC)

 * Nonetheless, there is a capacity to assess reciprocity that is seemingly built into the human brain, which would require some means of assessing number at least intuitively. People's ability to acquire language has limits and so does that ability to acquire math.  but some people are prodigious acquirers, like idiot savants, so it is wrong to imply that no one has an intuitive grasp o at least some mathematics. 142.177.110.74

"Other languages"?? enough of that urban legend that "mathematics is a language"! this absurd misconception is an example of the infatuation that human beings have with language and their verbal abilities. No wonder Hilbert believed that the right language could crank out all of mathematics mechanically without recurring to any meaning or understanding at all. Instead of more of the same, I would like to encourage people to see examples of animal mathematicians like parrots, apes and monkeys: watch especially the episode of Scientific American Frontiers called "Animal Einsteins" (transcript available at pbs site). I think that the animals in the show are a refreshing piece of information from outside the formal system in which we find ourselves undeciding around in circles =) --209.129.16.122 01:21, 30 June 2006 (UTC)

 * Obviously people object to the theory, or it would be listed among the "Schools". It may however be a variant of social constructivism or logicism.  142.177.110.74

 * Definitely more material on "animal mathematicians" would be extremely welcome. Note the mention of embodied mind issues below.  142.177.110.74

-

I've just made a few additions to the page (and one embarrassing error, quickly corrected - I'm sure someone else can find any others). There is now a bit more on logicism, realism and formalism. I think logicism could do with alot more, I only put in the bare minimum.

Also, a little history at the start might be a good idea. Given kant's huge influence on the subject someone should really put something in about him. I put a brief bit in alluding to the paradoxes at the start of the 20th century, and problems in founding analysis since it was so important, and isn't really covered under any of the subsections.

IanS

 * The paradoxes should be explicitly listed, and also the fact that the 20th century schools did nothing to resolve them. There are great articles in Tymoczko's editions about the history of philosophy of mathematics, from many of the great names in the field.  A good place to mine for material. 142.177.110.74

[sorry for my bad English!] Quoting form the page:
 * The most accessible, famous, and infamous treatment of this perspective is Where Mathematics Comes From, by George Lakoff and Rafael E. Núñez. (Since this book was first published in the year 2000, it may still be one of the only treatments of this perspective.) For more on the science that inspired this perspective, see cognitive science of mathematics.

If I understand well, this may well be what the French biologist Jean-Pierre Changeux says in his book "Matière à pensée" -- which is a dialog with Alain Connes. I don't think it has been translated into English.

The book was translated into English in 1995, as "Conversations on Mind, Matter, and Mathematics". Changeaux's position is, AFAICT, "embodied". Connes keeps a more level head in that book.

 * There should be more links to embodiment and embodied mind directly on the page, since the mathematical theory is an expression of those methods somewhat. 142.177.110.74

I often wonder why there is such a debate over whether or not mathematics originated as a tool of mankind or if it existed before humans. Clearly the answer is--both. Mathematics as language is Humankind's neuro-linguistic interpretation of geometric phenomena that existed in such forms as crystals before humans came into being. A human tool that arises from the hominid-style perception of common phenomenal activity in 1st to 4th dimensional space-time that occurs in apparently logical and interrelated patterns. Khranus

 * I personally agree 100%. If nothing else, the evidence from animal mathematicians is that when you make the problems accessible with isomorphic ones to solve, they solve them, and some animals (chimps notably) can learn the notation and use it - to get bananas, which is the same reason high school students learn it.142.177.110.74

On the absurdity of modern philosophy of mathematics (a rant)
Modern debates over both the nature of language and mathematics are very dumb, and shows why people hate philosophy and think of it as pointless semantic argument. I'm a double major in philosophy and cognitive science at Berkeley, and have had a class in which Lakoff taught; and I find many logical errors in his books. At any rate, nobody who is a realist, is suggesting that there is a number "one" floating out in some ethereal realm. Hopefully, those who are not realists would say that mathematics does have something to do with reality. Just like there are pseudo-sciences, cognitive science acts like a pseudo-philosophy and is philosophically naive. It talks as if there is no permanent relation between the "embodied mind" and what we call objective reality, that the mind doesnt come from reality (ie it talks about 'cognitive structures' in a mysterious way, as if the cognitive structures arent shaped by objective forces); as if other species with different metaphors couldn't approach the world using our metaphors preserving the truth of them (ie it doesnt realize language is the context its used in, and so it is meaningful universally); as if there is an "objective reality" at all (many philosophers since David Hume have tried to show this is wrong, there is nothing outside of experience; Hegel tried to argue Kant's conception of unknowable things-in-themselves that lied out of reality was wrong and subjectivity is objectivity); as if its possible for a species to exist compatibly living with us at the same time with a cognitive structure incompatible with ours (cognitive science professor Sweetzer at Berkeley went into some extravagant example that an observer the size of an atom would experience the world differently--as if its possible for such a thing to exist--[note, I hate thought experments in philosophy that assume things like radically different minds or worlds can exist, they always turn out to have a fundamental flaw]), etc. Hopefully, someone in philosophy will prove once and for all that epistemology is grounded in an ontology, and the relation between language and the world. This may mean returning to metaphysics and idealism. Brianshapiro

 * Who "talks as if there is no permanent relation between the "embodied mind" and what we call objective reality"? Lakoff?  Sometimes, but he's been corrected by others of his own "school".  For instance, when he tries to say "we can't possibly know" whether mathematics is objective "as if the cognitive structures arent shaped by objective forces" and especially "as if other species with different metaphors couldn't approach" mathematical problems (note especially the material on animal mathematicians).  142.177.110.74

 * Someone pointed out on an embodiment mailing list (which Lakoff subscribes to, but may not read) that animal mathematicians especially chimps using our Arabic symbols prove to anyone's satisfaction that there is at least a way to use the same methods of cognitive science to prove the objectivity of the constructs at least within our genus homo (which the poster considered chimps to be in - if they aren't, it's a wider family of relatives that understands). So Lakoff's own colleagues are taking your position, exactly.  Meanwhile there are attempts to "prove once and for all that epistemology is grounded in an ontology, and the relation between language and the world.  This may mean returning to metaphysics and idealism" but in a form based on, say, the accepted "reality" of the periodic table as advocated by fringe characters like Hubley and Pohl.  But it's also advocated by the more thoughtful Catholic and Sunni philosophers of science, and even the Dalai Lama.  Put more simply, a lot of these people, like Leonhard Euler, accept that integers might be real but not much else is - that you can count atoms and have chemistry, but can't get to an objective reality with complex transformations and analytics and therefore physics isn't underlying the periodic table, it's more a consequence of complex chemical critters like us trying to dig under the table.  What we find there is not likely to be all that objective and maybe it will get us all blown up as we argue about it.  ;-) 142.177.110.74

Questions about formalism (and its problems)
Hey. This article is actually pretty good. I don't know anything about the field and I found it interesting, at least.

 * It was on the good articles list for a reason. It will be back there soon once some of the material above is properly cited and integrated. 142.177.110.74

It seems to creep just a little away from NPOV against formalism. (Keep in mind I have no idea what I'm talking about...)

 * Formalism seems to have been crushed by, if nothing else, animal mathematicians. It was nearly as dead as logicism (quite dead) anyway. 142.177.110.74

It says: "The main problem with formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the minute string manipulation games mentioned above. While published proofs (if correct) could in principle be formulated in terms of these games, the rules are certainly not substantial to the initial creation of those proofs."  That just sounds like carping. It's really an attack on reductionism in general. When I as a programmer use a double, I generally don't have CMOS logic in mind, much less floating-point arithmetic, but that's hardly a "problem" for my reductionist "philosophy of how computers work". Or is it?

 * Formalism is based seemingly on greedy reductionism, trying to get more from looking at the givens than is there. Your programmer problems are only an issue if you are relying on your abstraction to investigate the circuit's problems with processing your abstraction.  In other words, if you are trying to fix your Intel chip's errors but can't see those errors because all your debuggers are running on the buggy chips, then, you will see no problem.  How do you think Intel made that error in the first place? 142.177.110.74

 * A separate article philosophy of how computers work would be hilarious as it would have to include naive views and how they arise. Fun. 142.177.110.74

The last sentence of that paragraph is: "Formalism is also silent to the question of which axiom systems ought to be studied."  This is a separate objection. I don't understand it. To the outside world, most of mathematics looks like basically unmotivated poking around; like pure science, it might yield rich practical applications at any moment, but if it doesn't, it's still considered valuable in its own right. Are all mathematicians that pursue random axiomatic systems purely for the hell of it formalists? Is John Conway a formalist? (This is a serious question; I actually don't know.) Is this really considered a "problem" in philosophical circles?

 * Mathematicians often don't care enough about meta-issues to have any real philosophy. Remember, P of M is a field that philosophers have to bug mathematicians to care about usually.  ;-) 142.177.110.74

I mean, the real problem with formalism (stop me if I'm missing the point) is that it completely ignores the relationship between mathematics and nature. Right?

 * Right. Which becomes hard to ignore when you have parrots doing arithmetic and chimps doing rudimentary algebra.  Where does "nature" begin, then?  Who's in charge?  Etc. 142.177.110.74

In general, nice work. Jorend 21:48, 4 Apr 2004 (UTC)

Logicism
Given that logicism has its own article (for good reason), the section on that topic in this article should be summed up in a sentence or two and the material in that section moved/referred to the article. B 00:56, Apr 26, 2004 (UTC)

 * There should be enough on logicism to compare it to other fields. And to explain how it died, and why it's mostly covered in an historical article.  Oh, and some mention that lawyerss still believe it and philosophy of law is still mostly logicism-like. 142.177.110.74

==Commentary on David Corfield's

The Philosophy of Real Mathematics Page==

The anonymous user who added the David Corfield link to the main page also added the following commentary and quotations :-

 * This page talks about to an event for the philosophy.
 * It is happening an event in mathematics, that is important for the philosophy.
 * In Internet (Báez...) and for example in the book of Corfield you will be able to see some track.
 * Talking about Higher-dimensional algebra: (:"Many important constructions may profitably be seen as the “categorification” of (...)".
 * "It provides a way of organising a considerable proportion of mathematics. (...)"
 * " These constructions have applications in mathematics, computer science, and physics. (...)"
 * "Higher-dimensional algebra blurs the distinction between topology and algebra. (...)"
 * etc.''

Hmmm. This article starts with some highly dubious stuff, if you ask me (first section after intro). Also, high time to archive some older discussion. Charles Matthews 09:36, 17 Jul 2004 (UTC)

 * Agree about the archiving. Personally, I like Corfield's blog and reasd it regularly...too much category theory for me, though. JJL 06:09, 24 May 2006 (UTC)

 * Issues with this page seem to take years to resolve. I don't favour archiving but would not mind summarizing the outstanding issues and positions perhaps.  But until the page is cleaned up the discussion should stay here. EXCEPT the JJL/JA discussion which is mostly noise and belongs on your user talk pages at best.142.177.110.74

Relation to Philosophy of science
Is philosophy of mathematics part of philosophy of science?

 * I say no...I have read the phil. of sci. page and see the varied opinions there. JJL 06:08, 24 May 2006 (UTC)

 * No. But they can't ignore each other.  The mathematical practice issues parallel the scientific method discussions, and both take note of the social factors that drive the mathematician or scientist to do what they do, and believe what they believe.  Those articles should reference each other more clearly with that in mind.  Likewise the parallels between phil of sci and phil of math should be mentioned. 142.177.110.74

 * But there was no relationship between the fields in the 20th century, seemingly. The "schools" were trying to ignore the empirical use of their work in the sciences, and no real dialogue was going on between people like Thomas Kuhn and people like Imre Lakatos.  Sadly.142.177.110.74

Nonesense
This article appears unnecessary in its current form. You start from axioms. You find interesting results. The discussion on the existence of numbers independent of sentient thought is interesting but does not warrant this outpouring of hogwash. I'd flag it for deletion if it was not apparent so many people cared.

 * Provide evidence that you understand then mock.

 * Someone who cannot spell "nonsense" and is so obviously a believer in one of two very discredited "schools", shoud have no status whatsoever in these discussions. Plus not caring to even sign an obvious troll name so others could challenge their beliefs and talk back to them.  Hm.   Does this person believe what they are saying?142.177.110.74

Embodied mind theories
Would it be poignant to mention Kant's suggestion that our mathematical system is based on our physical ability to percieve time and the universe? (umm... reference... something we discussed in my phil of math class, I know I'm not just making this up) .. That is to say, an intelligent being with a different perseption abilities would have an alternate view on mathematics (perhaps one based on the continuum rather than the naturally discrete way us humans tend to parse the world).

 * I don't know about poignant. It might be relevant.  (I don't think much of that suggestion, though: to me math seems gloriously independent of time.)  Jorend 05:28, 15 January 2006 (UTC)

 * It's poignant in any encyclopedia that claims to be able to define the universe rather than calling the article correctly our observable universe. Denying that the models are limited to the observations, or that we (humans, thus "our") create them, is a God's eye view problem.  Maybe the ultimate one. AS for the intelligent being issue, read Eugene Wigner on this, notably "the unreasonable effectiveness...". 142.177.110.74

Fictionalism
It is not accurate that Hartry Field's work on Newtonian mechanics proves that "mathematics...should be rejected as false." He does not argue against mathematics, or say anything about the truth-value of mathematical propositions per se. He is arguing that scientific theories need not depend on the existence of abstract mathematical objects, and demonstrates this by using nominalistically-acceptable (i.e. physical) replacements for mathematical objects in a common scientific theory. In fact, if you are a structuralist, his theory does use mathematics, because it uses something with mathematical structure, regardless of its ontological status. Someone should fix this; I am not an expert on this topic.

 * We could use more info. on this. I am not knowledgable about his work. JJL 06:10, 24 May 2006 (UTC)

This section is decidedly confusing. In what way are geometry and "betweeness" axioms less mathematical than linear algebra? I just can't get any feeling for what is meant. A Geek Tragedy 16:12, 7 September 2006 (UTC)

 * Some argue that the geometry is the only valid mathematics and that the reduction to symbol causes problems. But I'm not arguing that here.  ;-) 142.177.110.74

A minor inconsistency
I noticed that under the section on formalism, we have the following:

"Hilbert's program was dealt a fatal blow by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency."

This may not match up with the article Gödel's incompleteness theorems which states:

"Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's program towards a universal mathematical formalism which was based on Principia Mathematica. The generally agreed upon stance is that the second theorem is what specifically dealt this blow. However some believe it was the first, and others believe that neither did."

 * Either way, it's dead, and Hilbert's program should not get such respect in this article. There are more modern issues to cover, and Hilbert's program can be covered in an historical article as recommended above. There is WAY too much attention paid to the discredited 20th century schools and not enough to the problems discussed on this talk page.  If anything the sections on language, embodiment, aesthetics, other animals doing math, all must be drastically expanded.142.177.110.74

Aesthetics
I added the OR tag ... The stuff about "self-sampling" of what's deemed beautiful by the community is interesting, but I'm curious about references. Thanks! --M a s 01:56, 9 May 2006 (UTC)

I have removed and placed this section here for further discussion and rewriting, on the assumption that some of it may be useful to future readers with some careful thought, research and citation. If there is a genuine issue about "self-sampling" and lack of an objective aesthetic-checks-and-balances system within mathematics, I'd prefer to have a citation from a credible source. Here it is as I just found it... Kenosis 03:18, 9 May 2006 (UTC)
 * === Aesthetics ===

 * Another criticism is that mathematics can be seen very narrowly as the science of measurement and as a vast number of trustworthy shortcuts to reduce the need to measure directly, and simplify calculation. Some of the schools have assigned rather more significance to mathematics than this mere utility -- even seeking sometimes moral guidance, or aesthetics of truth and beauty, in its abstractions. Some consider this a symptom of scientism. It is as inappropriate in this view as having, say, a philosophy of weapons or of war, separate from that of the larger social and species and planetary context of it.

 * This question is usually rejected by working mathematicians as "irrelevant", but of course they are exactly those people whose aesthetics of proof and of rigour have been already accepted -- they may thus be practicing self-selection of a particular aesthetic, and propagating it with few constraints, especially in those fields where mathematics is not immediately applied to life.

Thanks Kenosis. I definitely think Aesthetics should be a section here. Two or three references that I think are pertinent that come to mind include (sorry no wikiformats):

A. One from Davis & Hersh, wherein they provide two proofs of the irrationality of sqrt(2) - one is the standard (Hellenistic) proof, and another states something like if p^2 = 2 q^2, and if p and q can be uniquely factored, then there's a "pairing" of primes between p and q, but 2 is left over. Sorry I'm destroying their wording. They then argue that most mathemeticians would say the second is more "beautiful," because it gets closer to the heart of the matter (the real reason why sqrt(2) is irrational.) They bite the bullet and say or imply that beauty in Mathematics is universal (i.e. most mathemeticians would agree as to what's a beautiful proof and what's not.)  (According to the book Four Colors Suffice, the initial rejection of Appel / Hanken and the 4CT comes from the fact that it's not beautiful- because it didn't get to the heart of the problem.)

B. In contrast, in Meta-Math, G. Chaitin rejects Erdos and his Book: he provides three proofs of the infinity of primes. The first is Euclid, the second is Euler with his zeta function, and the third is Chaitin's own complexity based proof. Chaitin argues that each of the three is beautiful in its own way, because they all show different aspects of the same theorem.

C. Instead of "irrelevant," I think another is the standard quip (I don't know who said it first) that "Mathemeticians want to leave philosophy to the philosophers, and they just want to get back to doing math." The implication is that mathemeticians wouldn't do something that's not beautiful, and their sense of aesthetics is what drives them to do it. Are sculptors or painters critiqued about their sense of aesthetics? I'm not sure, but I think the argument about self-sampling would be applicable with them as well. I don't know too many painters but I think they would want to get back to painting too.

Thanks! --M a s 16:04, 9 May 2006 (UTC)
 * I agree math has a definite set of aesthetic qualities short of an official aesthestics Court of Appeals. Please rewrite and replace, and I feel sure thoughtful editors will back up the general thrust....Kenosis 01:31, 12 May 2006 (UTC)

 * The aesthetic view needs more coverage, and logicism (discrecdited) much less. 142.177.110.74

Subsection on "Ethics" removed and placed here for reconsideration
As "ethics" often implies something different to the general public, and also to modern medicine, to genetic researchers, and to nuclear physicists, to name just a few other potentially relevant disciplines, I am placing this section here for further consideration and a reasonably proper analysis of its content... Kenosis 01:35, 12 May 2006 (UTC)

 * It's becaues ethics implies different things in different fields that it must be re-interpreted from the perspective of each field. Removing this is a bad move, and it must go back in, in some form.  No one would argue today (given cryptography and child porn, if nothing else) that there are no ethical implications to doing advanced mathematics.  The views of medical, social, genetic, nuclear researchers all using mathematics should be integrated and linked, not ignored. 142.177.110.74

 * === Ethics ===

 * As well, there is little or no consideration given to the ethics of doing mathematics. In a technological culture, mathematics is seen as an absolute necessity whose value cannot be questioned and whose implications cannot be avoided - even if particular branches have no known purpose, or are considered useful primarily or only to enable conflict, e.g. cryptography, steganography, which are about keeping secrets, or the mathematics involved in optimizing nuclear fission reactions in bombs. While most would accept that physicists bear some moral responsibility for these activities, few, if any, have been willing to also criticize mathematicians.

 * Some of these criticisms have been explored in the sociology of knowledge, but in general mathematics itself has evaded the scrutiny often applied to the sciences of genetics, physics, economics or medicine. Which is interesting in itself, as mathematics is necessary to enable those and other sciences.

 * Evolutionary psychology for instance has embraced the idea that "the mind is a computer" in the sense of a Turing Machine. What are the implications of adopting an abstraction originating to explain computers formally, to explain the mind? ... 01:35, 12 May 2006 (UTC)

Mathematical Philosophy
I am aware of the fact that people think this 'should' mean something different from "Philosophy of Math." (in the sense indicated here and associated with Bertrand Russell's "Introduction to Mathematical Philosophy"), but in practice it is used interchangeably with "Philosophy of Math."; e.g. Mathew Mount's "Classical Greek Mathematical Philosophy", etc. My experience has been that only those interested in Russell's work draw the distinction. At the least, the term should appear in this article.

JA: I don't as a rule champion many usages of BR, but the distinction in question hardly depends on him. No doubt the two are used by some people in some contexts as synonyms, but more often in practice, as I said, "mathematical philosophy" means two other things (1) a project of doing philosophy more geometrico — Leibniz and Spinoza, if not indeed Llull, come to mind here — or (2) an individual practitioner of the dark art in question, and his or her particular "mathematical philosophy". I have no objection to mentioning the term, if only to dispell the confusion, but I'm not sure it's worth waking the dogs at the top of the trip. Jon Awbrey 15:46, 12 May 2006 (UTC)

Also, what was the objection to the Hersh comment? JJL 15:03, 12 May 2006 (UTC)

JA: I came in here, and don't recall a comment by Hersh. Jon Awbrey 15:46, 12 May 2006 (UTC)

 * It was in the same post as the math. phil. comment but I didn't call attention to it in the edit summary, so it was reverted too. I've put it back in; he is guilty of being a "mathematician doing (bad?) philosophy" but he and Davis are still alive which adds to the impression that this is still an area of interest. Hersh just put out a collection of math. phil. essays and spoke about his social views in conjunction with its appearance, for example. Anyway, I'll add something back in about math. phil. as a starting point for others to move/modify/etc. JJL 16:04, 12 May 2006 (UTC)

 * I haven't seen any objection to Hersh. The section on Aesthetics was put here so it could be reasonably rewritten and reinserted.  I could as easily have left it up as a stub or empty title, but it's here until it contains something worthwhile that's not just a speculative POV with a complaint about lack of an internal aesthetics police, or whatever the earlier editor was attampting to say.  As to the section on Ethics, I don't see a ligitimate place for it in this article, though i'm always open for correction if I missed something signigicant.  How about the ethics of carpentry, because carpenters were needed to build Los Alamos and Auschwitz?... Kenosis 15:29, 12 May 2006 (UTC)

 * I was referring to the Hersh comment and ref. under social approaches that was reverted with the math. phil. terminology change. I agree that the ethics section doesn't work well as written, but wasn't referring to that (hence the new topic). JJL 15:41, 12 May 2006 (UTC)
 * Got it...Kenosis 16:08, 12 May 2006 (UTC)

Social constructivism
There's a lot of good activity lately. I'll try and take Kenosis' comments about aesthetics and add it in. But I'm only aware of mathemetician's own ideas about aesthetics, and not some philosophers dissociated from maths proper.

In the meantime, what is meant by the following?

 * Social constructivists see the process of 'doing mathematics' as actually creating the meaning, while social realists see a deficiency of either human capacity to abstractify, human cognitive bias, or collective intelligence as preventing the comprehension of a 'real' universe of 'mathematical objects'.

Any ideas? It doesn't seem to fit any grammatical structure that I'm aware of. The first part makes some grammatical sense but the second sounds fractured. (Sigh.. Postmodernists are their own worst enemies when it comes to presenting useful and constructive ideas in an understandable manner.)

 * Attempted rewrite: Social constructivists see the process of 'doing mathematics' as actually creating any meaning the symbols retain.  However, social realists see a deficiency of some human capacity as preventing the comprehension of a 'real' universe of 'mathematical objects' - one that actually exists.   Because of limits of the human ability to abstract, human cognitive bias, or collective intelligence, humans can do no more than approach, but never be certain of, the reality of their symbols.   142.177.110.74

Beyond the schools
I did some digging and it appears that all of the "Beyond the "Schools"" stuff was added by our now-hard-booted friend user:JRR_Trollkien, aka 24. See []. Separating the wheat from the chaffe is very difficult. Anyone up for starting off with a complete elimination? Thanks. --M a s 19:10, 17 May 2006 (UTC)

 * No. Removing good material because you don't like the author is just vandalism.  It seems obvious that this material was all valid, as it has stood very many hostile edits.  Let the cleanup take its course without resorting to ad hominem, please.  And JRRT restored a lot of material written by others, unclear who wrote what because of the various misdirections that are an inevitable result of ad hominem vandalism.  By the time a bunch of zealots have removed stuff and another bunch restored it, you can't tell anything about authorship.142.177.110.74

Relation to mathematics proper
I don't see the point(s) at which this section drives. It seems to consist of two unconnected distinctions, one of which is, AFAICT, mostly a justification for including a quote by Peirce. I'm a Peirce fan too, but I don't see what it adds. I would also say that I don't find the revert of a well-intentioned rewording to be m minor, nor do I see that "this is a wiki, remember" is a reason for doing so. (The condescending nature of this remark makes me wonder if the editor is feeling territorial about this article.) I felt it was not a useful section but attempted to reword it in hopes of getting toward a clearer point(s). Also, putting it prior to the Relation to phil. proper section seems to me to undercut the statement that this is an area of phil., not of math. JJL 20:17, 18 May 2006 (UTC)

JA: The point is this. Philosophy of mathematics did not spring fully armored from the heads of philosophy departments in the early 1900's. It arises initially from the critical examinations of reflective practitioners, and only much later becomes the speculations of non-practitioners. Sometimes an outside view adds perspective. We all understand that. But in a lot of cases, it's just uninformed imaginings. I realize that philosophers feel somewhat territorial about phil of math and phil of sci, to the extent that someone who actually has an acquaintance with hard-knocks research and the sweating out of proofs can hardly get a word in edgewise. But I think that philosophy suffers for that, so I will persist in trying to end the suffering. It's the least I can do. Jon Awbrey 04:33, 19 May 2006 (UTC)

 * When I try to edit this for consensus, you simply blindly revert it. That isn't working together. How about working with the edits toward a middle view? I don't know why you think I must respect your views when I think they're factually wrong or stylistically inappropriate. JJL 04:25, 19 May 2006 (UTC)

JA: It's not blind reverting. I read all the changes in the edit history. But when you mix up content changes with structure changes, it's just too hard to evaluate. As far as I know, everybody marks reverts as minor. The edit line says revert, so that's what it is. Jon Awbrey 04:40, 19 May 2006 (UTC)

 * Agreed. Keep structure changes that lose text separate from content changes.142.177.110.74

 * Also agreed though that bald reverts (those without comment) are an insult. Avoid. 142.177.110.74

It droppeth as the gentle rain from heaven
JA: JJL and M a s, please be patient, as I tend to work very incrementally, and need a nap now and then. I will probably root out most of the awks and other infelicities myself, but there's a point to this that will have to be said one way or the other. Naptime ... Jon Awbrey 22:56, 18 May 2006 (UTC)

 * Good night. --M a s 23:53, 18 May 2006 (UTC)

 * I've cut out some of the rambling that might be more appropriate for a book than an encyclopedia. I still don't see how the Peirce stuff fits in--what's the point that you're trying to make? Perhaps it would be best to work it out for yourself in a text editor before changing things here. This article has unnecessarily many edits, it seems to me. Then you can work out the problems you see before submitting it where others will likely modify it further. JJL 04:08, 19 May 2006 (UTC)

JA: I have already worked with what was there, and some things are simply unacceptable, so I rewrote them. PoM and MP are simply not interchangable. I rewrote it to reflect the actual usage as accuately as possible. Jon Awbrey 04:45, 19 May 2006 (UTC)
 * We disagree on what the actual usage is. It is factually correct that some people use them as synonyms. (I'm one of them, though I recognize that not all do as I do.) Suggesting as in your current edit that people like me are "confuse(d)" is simply rude and contentious. The terms mean what they mean. Do you distinuguish between political phil. and phil. of politics? JJL 04:54, 19 May 2006 (UTC)

JA: Are you really saying you don't see the difference? Jon Awbrey 05:22, 19 May 2006 (UTC)

JA: The terms are simply not used interchangeably. To say that and to say it first is simply to invite confusion. More importantly, there is a very real distinction between Phil.Dept. PhiloMath, which few mathematicians find worth their attention, and the sorts of reflective guidelines that really do inform their work. That diff is real, no matter what names you choose to attach to the sides of it. But the fact is that mathematicians who find Phil.Dept. PhiloMath sometimes amusing but seldom apt, still have something that they would be more likely to call a "mathematical philosophy" that is a real force in their work. That is just how things really are. The pertinence of the Peirce quote is that he is one who knows how things really are in the doing of mathematics, and was able to articulate it far better than the average mathematician can. And there is real information in that, which is what we owe to the reader, not the conventional fictions of Phil.Dept. PhiloMath. Jon Awbrey 05:22, 19 May 2006 (UTC)

Given the obvious vandalism of intentionally mislinking phil. of math. to Big Muddy in this version, the annoying use of WP as a text editor leading to a huge edit history, the misleading labeling of edits (including what is 'minor' or a 'correction'), the persecution complex (academic philosophers are out to get him), the refusal to work with others as evidenced in the "No Mercy" title of this section, and a perusal of his User Talk page, I can now see why this happened:. I don't see any point in continuing to try to cooperate with someone whose knowledge of the subject is hit-and-miss and who seems to want to make the article his personal blog, with flowerey language and a rewritten history and present of the subject. I intend to continue editing the last version not by Jon Awbrey, whether that is from me or another, and ignore what is written by the troll. I suppose that WP needs its own Archimedes Plutonium or Alexander Abian--perhaps Mr. Awbrey has elected himself to this position. JJL 20:44, 20 May 2006 (UTC)

JA: Oh, lighten up. It was a transient bit of humor, on account of the fact that I found myself using a "stream of thought" metaphor in that particular paragraph, and the thematics of the "Father of the Waters" AKA Big Muddy, with memories of Huckleberry Finn's famous riverquest, and all, got engrained in my subconsciousness at an early age. It's moot now, fun's over, back to the grinstome. Jon Awbrey 23:36, 20 May 2006 (UTC)

JA: The rest of your remarks strike me as rather extreme, so I'll put off deciding whether to respond in any way until Monday morning or Tuesday afternoon or so when we've all been a littke better R&R'd. Jon Awbrey 00:00, 21 May 2006 (UTC)

JA: To User X: In the mean time [sic], I suggest that you review the official WP policy against personal attacks, WP:No Personal Attacks. As a personal rule, I do not respond to ad hominem attacks, especially with plaintiffs who write under the cover of pseudonyms. Jon Awbrey 14:58, 21 May 2006 (UTC)

Begin cleanup, phil of math ≠ mathematical philosophy, which is 2 other things
JA: My first edit on this article was 12 May 2006, with the edit line given in the heading above. Nobody in the world gives a RA how JA or X or the Man in the Moon uses these terms. But there is no evidence in the mathematics literature or the philosophy of mathematics literature that "mathematical philosophy" and "philosophy of mathematics" are widely regarded as equivalent terms. Indeed, much to the contrary. So consider a {citation needed} placed next to the assertion of their equivalence, as there will be one soon.

JA: Moving along, but on a related note, the article as it stands is woefully inadequate in citations for its statements. To source statements is not simply to praise famous names and mention the title of a book here and there. It means precise paraphrase, preferably verbatim quotation, followed by source, date, page numbers, stuff like that. This, too, shall pass, as the {verify} tag already on one section will be moved to encompass the whole. CU there. Jon Awbrey 02:07, 21 May 2006 (UTC)

 * Someone should look up all the names of all the people named in the discussion.142.177.110.74

To cite or not to cite — Where's the question?
JA: User X, please review the WP policy WP:VERIFY and the WP guideline WP:CITE. WikiPedia is sourced research. Failing to provide citations and references for statements is defective scholarly practice — deleting citations and references that others provide is simply inexcusable. Jon Awbrey 14:40, 21 May 2006 (UTC)

Citations Needed
JA: Due to the fact that User X repeatedly removes or reverts the insertion of {citation needed} tags, I will create this place for documenting those requests for citations. Jon Awbrey 01:44, 22 May 2006 (UTC)

 * Assertion 1. "The terms philosophy of mathematics and mathematical philosophy are often taken to be synonomous".

JA: Assertion 1, though inessentially mutated in its phrasing from time to time, continues to go undocumented despite repeated requests for a supporting citation. Let me say that I have studied mathematics and philosophy of mathematics since youth, eventually achieving a Master's degree in the former, and the notion that these two phrases are as the Morning Star and the Evening Star is simply unheard of by me. Now, I am poignantly aware of not having heard of ∀ ∃ 2b heard of, so if the asserter-inserter has a reputable source for this claim of generally accepted synonymy, then I would like to augment my information by hearing of it. Otherwise, it will soon be necessary to quit shilly-shallying around and remove it as an unsourced assertion. Jon Awbrey 02:40, 22 May 2006 (UTC)

JA: Here is another statement in the text for which I have repeatedly requested a source, and for which User X has repeatedly deleted or reverted the request template:

 * Assertion 2. "The original foundations problem in mathematics ('which branch of mathematics is the one from which others are derived?')"

JA: Assertion 2 amounts to a parenthetical remark that the "original foundations problem in mathematics" is a problem about "which branch of mathematics is the one from which the others are derived". This sounds fishy to me, so I think that the reader deserves a source for the assertion. Moreover, the statement links to a stubby article in WP which itself has no sources and provides no explanation of why it should be distinct from the article on foundations of mathematics. Jon Awbrey 04:42, 22 May 2006 (UTC)

Undocumented Reverts
JA: As I understand the matter, it is extremely poor WikiPractice to revert edits without notification in the edit lines. Whether this is done by returning to a previous version and then beginning one's own new edits there, or whether one simply reverses in the current version all of the edits made by others, it is effectively all the same thing. This is what has occurred in the following transitions, where multiple edits of mine were effectively reverted by User X without notification of a revert in the edit line. Jon Awbrey 02:14, 22 May 2006 (UTC) JA: I request that User X cease and desist this practice. Jon Awbrey 02:14, 22 May 2006 (UTC)
 * Example 1: 8 edits by JA reverted without notice by X.
 * Example 2: 7 edits by JA reverted without notice by X.
 * Example 3: 7 edits by JA reverted without notice by X.

 * Funny, you find it a justifiable practice when done by you but not when you feel it is being done by others. You revert nontrivial edits and also mischaracterize the edits in the history. I've tried to discuss this on this page and via the edit history. I served notice concerning 'this practice' above. When you simply reverted all edits, misused the edit history, etc., I tried to be reasonable and engage you in discussion. When you began vandalizing pages by mislabeling links ("Big Muddy"), I ceased trying to compromise with you and to engage on the Talk page. While it's amusing to now see you citing WikiScripture for your purpose, it doesn't change things. I've seen other vandals take this tack before. Goodnight, Irene. JJL 03:08, 22 May 2006 (UTC)

JA: On a related note, when User1 mispells a word, and subsequently corrects that spelling on three separate occasions, each time having that correction reverted by User2, then the error becomes the intellectual impropriety of User2. Jon Awbrey 02:20, 22 May 2006 (UTC)

JA: It's a long-and-often-tedious work day in the Wiki City, and I normally try to keep it lite. But I understand that this isn't always possible. I don't know where you're from, so it was my mistake to think you'd know that Big Muddy is a term of endearment where I come from. Anyway, that was days ago, and I did not know the local situation then. I know the local situation better now. I believe that the last time I reverted the page was here, at 16:03 on 20 May 2006, and this was just a revert of one of your many mass deletions of new text. I try to observe a zero-revert rule in these sorts of hot dispute situations, and that is what I have been doing since shortly after 16:03 on 20 May 2006, when I realized what sort of situation it was. That doesn't mean that I quit trying to improve the text, or restore selected blocks of text that are deleted without explanations by others, well, so far, almost exclusively by you. Jon Awbrey 06:48, 22 May 2006 (UTC)

Doomed To Repeat It
JA: There appears to be some uncertainty about the secular ordering of developments in logic, set theory, the foundations of mathematics, and metamathematics during the century last. By way of helping to remediate this situation, I'll be adding some text on historical matters. It being my normal practice to get all the annoying busywork of assembling sources out of the way first, angels dancing about this locus may notice a few historical works showing up in the Bib (Further Reading) first, each of which will be moved to Refs as text is added that cites it. Watch this space $$\longrightarrow$$ " ". Jon Awbrey 03:14, 22 May 2006 (UTC)

 * Agreed, the historical stuff needs to all be cleaned up.142.177.110.74

POV Tag
JA: I have placed a POV tag on the article. User X continues to obstruct any attempt to add information to the article that does not meet with his/her personal approval, right down to the most minor details of wording and the types of citation data that are considered positively standard in all forms of sourced research. Until these issues are resolved, this is no longer a WP Article but a personal essay by X. Jon Awbrey 05:56, 22 May 2006 (UTC)

JA: A few additional thoughts on the POV problem. Although, with luck, the situation is changing, the article as it stands is written mainly from a particular POV, one that might well be thematized as follows: JA: As I have indicated, this is an extremely debatable position. I further realize that this POV is so implicit in certain approaches to the philosophy of mathematics that it is not even cognized, much less re*cognized to be a POV. This form of obliviscence is, of course, the natural human condition, but it is critical to the regulative ideal of NPOV, or BACPOV, to bestir ourselves against the deadly torpor that it so invitingly invites. I'll do what I can. You can too. Jon Awbrey 13:20, 22 May 2006 (UTC)
 * Resolved, that the purpose of mathematics is to provide philosophers of mathematics with something to philosophize about.

Verify Tag
JA: I have placed a Verify tag on the article. The reason given in the edit line is this: "Not only is the article poorly referenced, but User X actively obstructs the use of proper citations." Jon Awbrey 06:20, 22 May 2006 (UTC)

JA: A few additional remarks on the problems with regard to WP:VERIFY. The reason for citing sources in specific detail is to facilitate the efforts of any user who desires to check the source of a statement. This demands details about the copy-in-hand that the citer is citing, including edition and page numbers when the citer is citing a particular statement and not merely referring to the circumstance that such-&-such a work exists. For this reason, it is the standard practice, dictated by all reputable journals in every field of inquiry for the citer to cite the copy-in-hand data. Unless the citer happens to be a collector of rare incunabula, the copy-in-hand is also likely to be one of the more reader-accessible copies, and so there is an added practical benefit to the reader in doing this. And even when the citer is a collector of rare incunabula, it is considered polite to provide references to more generally accessible reprints whenever it is possible to do so. Of course, if one knows of an online Eprint, then it's nice to toss in that link, too. Service to the reader is our motto. Jon Awbrey 13:44, 22 May 2006 (UTC)

On suffering a red link to live
JA: When I was just a WikiPiddler, not so many moons ago, I used to go about "getting the red out" wherever I went, and I'm certainly not talking about my eyes. Then some old hand WikiPelted me upside the head, and told me that WP was once very largely redlinks, and that's just the way that it knue where to grue. So please leave 'em be, we are not the judge, jury, and x-ecutioner of who or what will be needed in future or not. Jon Awbrey 21:10, 23 May 2006 (UTC)

 * Understood and agreed, up to a certain point. There still is the "redlink" tag for articles with too many.  As a third party reviewer on some articles that I have no stake in, if I see one that has too many redlinks, and too many irony quotes, too many prefaces of "so-called" I get the vibe of arrogance and/or crackpot-ness.  --M a s 00:20, 24 May 2006 (UTC)

JA: Yes, of course, but that's hardly the case in hand here, unless there is some "irony" in linking an author, editor, or translator of a cited source. I'm always a bit pleasantly surprised when I blindly link one of those and he/she turns out to be a Wiki-Imported-Person already. Jon Awbrey 01:00, 24 May 2006 (UTC)

Hardy
JJL, there was a discussion a while ago on G. H. Hardy about just this- how "dated" is the apology. Some mathemeticians have argued that Hardy focused on maths being a young man's game (counter with Erdos,) or as mentioned in the article pure maths being good precisely because it's useles (counter with cryptography.) If the sentiment is "pure mathemeticians love their job because it's useless now," rather than "pure mathemeticians love their job because it's useless" then maybe some wording would work. Thoughts? --M a s 00:20, 24 May 2006 (UTC)

 * I looked at the Hardy Talk page. What I want to communicate is that his distaste of math. being applied to war in particular and useful purposes in general wasn't just a distaste for number theory being applied to war. I feel that number theory was a good example for him then (plus, his own interest); perhaps today he'd talk about the uselessness of category theory. Does pointing out that his views are dated--I daresay, quaint--help advance the point regarding aesthetics that is being made here? What do we want to say about how Hardy's book relates to math. aesthetics? (Edit: Incidentally, what you have there now is fine by me. JJL 00:56, 24 May 2006 (UTC)) JJL 00:48, 24 May 2006 (UTC)

Now I know what Henry Ford meant
JA: This entire section on "History of the Philosophy of Mathematics", aside from the ever-persisting lack of grounding in standard sources, is yet another striking example of the world as seen from a single POV. History of the Philosophy of Mathematics Western philosophizing about mathematics has a history that goes at least as far back as Plato, who considered the ontological status of mathematical objects, and Aristotle, who considered logic and issues related to infinity (actual versus potential). Greek views of quantity strongly influenced their views of other areas of mathematics. At one time, the Greeks held the opinion that 1 (one) was not a number, but rather a unit of arbitrary length (so that 3, for example, represented 3 such units and truly was a number). At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair. Of course, this was well before 0 was considered a number. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: The first line drawn had unit length, and numbers represented multiples of it. Greek ideas of number were upended by the discovery of the irrationality of the square root of two, showing that the diagonal of a unit square was incommensurable with its (unit-length) edge: There was no number that represented how much longer the diagonal was than an edge. This caused a significant re-evaluation of Greek philosophy of mathematics, as non-Euclidean geometry would do to European philosophy of mathematics two millenia later. Beginning with Leibniz, the focus shifted strongly to mathematics-as-logic. This view dominated the philosophy of mathematics through the time of Frege and of Russell, but was brought into question by developments in the late 19th and early 20th century.

JA: Trying to claim that Leibniz was a logicist is only the most egregious of the errors in the above manifesto. His actual stance is more correctly a thesis of logic-as-mathematics.

JA: Naturally, all of the sourced material that I added several days ago which illustrated the historically documented presence of alternative views was automatically deleted or reverted by JJL. Jon Awbrey 02:45, 24 May 2006 (UTC)

 * Logicism was not 'brought into question', it was eradicated, by those developments. 142.177.110.74

Article Size
"This page is 45 kilobytes long. This may be longer than is preferable; see article size." The article doesn't actually seem overlong to me, but much more is needed regarding pre-20th century work (at the least). Any thoughts on how to cut it down? Most of the space seems to be going to the Schools, but each gets only a brief treatment here. I wouldn't want all that shifted off to another page. JJL 14:34, 24 May 2006 (UTC)

JA: That message is an antique residue of the days when 32 Kb was the max file size. Articles on signficant subjects in WP need hardly worry about that, as most of them grow larger and larger and then undergo mitosis at the approriate stage of their protozoic life-cycles. Jon Awbrey 14:40, 24 May 2006 (UTC)

Fictionalism
Why does Fictionalism start its own heading? Shouldn't it be grouped with other schools? I will try to re-organize this. (Edit: I like it better now, though it's hardly ideal. JJL 14:39, 24 May 2006 (UTC)) JJL 14:34, 24 May 2006 (UTC)

On not getting in each other's hair
JA: JJL, please review the following general principles of WikiPolity:
 * 1) Five pillars
 * 2) Editing policy
 * 3) Simplified Ruleset

JA: I would like to call your particular attention to the following recommedations that I think are rather acutely pertinent here:

perfection isn't required
 * From Five pillars —

This policy in a nutshell: Improve pages wherever you can, and don't worry about leaving them imperfect. However, avoid deleting information wherever possible.
 * From Editing policy —

10. Particularly, don't revert good faith edits. Reverting is a little too powerful sometimes, hence the three-revert rule. Don't succumb to the temptation, unless you're reverting very obvious vandalism (like "LALALALAL*&*@#@THIS_SUX0RZ", or someone changing "6+5*2=16" to "6+5*2=17"). If you really can't stand something, revert once, with an edit summary something like "(rv) I disagree strongly, I'll explain why in talk." and immediately take it to talk.
 * From Simplified Ruleset —

JA: Thanks, Jon Awbrey 01:28, 24 May 2006 (UTC)

JA: Addressed to User:JJL. I am asking you one more time to cease beginning your editing sessions by reverting all of the edits that I have contributed in the intervening time. The charges of vandalism, invitations to "go troll elsewhere", aspersive metaphors of "broken clocks", and so on that you include in your edit lines are highly inappropriate and cross the line into constituting personal attacks. I refer you once again to the WP Policy of WP:No Personal Attacks. Thanks, Jon Awbrey 12:34, 24 May 2006 (UTC)

 * The charge of vandalism is proven; remember the "Big Muddy" link? I realize you're mounting a countertrolling defense here by charging me with everything I had previously accused you of (mass reverting of edits, using the site as your personal blog, misleading and insulting edit history lines, refusal to engage on talk, ignoring rules, etc.), and it may well end up being effective as no one will want to sort through the history to see how it all started, but whether or not you're a vandal isn't really up for discussion. I don't care to search each of your edits for more of your "humor" as you choose to call it. Keep citing rule after rule; I've seen it all before on USENET and web boards. Bottom line, you want to make this your own personal web page (and use it to demonstrate that you own many books, apparently), and are complaining that you're not being allowed to do so. I don't care to have to keep repeating this. JJL 13:43, 24 May 2006 (UTC)

JA: Addressed to User:JJL. I refer you once again to the general principles referenced above, with special reference to the elements of civil polity and collaboration facilitation that I copied out. Good faith edits are to be respected. An alleged offense of "flowerly language" does not justify blindly reverting many hours of another editor's bona fide work to improve the article. If there really is a problem with phrases like "secure territories" being too flowery, and so far we only have one person's opinion about that, then the phrases can be selectively deflowered by almost any editor, as some of us are quite used to this in all collaborative writing processes. An extreme lack of fit between the punishment and the alleged crime engenders suspicion that the real offense is simply in daring to contribute anything at all. And I know that you don't want to give that impression. So I ask you once again to respect the efforts of other participants in this joint venture. The words above are very good advice on how to do this. Thanks, Jon Awbrey 15:10, 24 May 2006 (UTC)

 * Alleged? So, the philosophy of mathematics may well be a major North American river, and I should respect those edits? Sounds like trolling to me, dude. JJL 15:17, 24 May 2006 (UTC)

JA: Addressed to User:JJL. I am formally requesting that you cease your practice of reflexively reverting the work of other editors. This is not respectful of the efforts that others make to improve the article. Please review the above principles of civil collaboration that I have directed you to on what is now a recurring basis.

JA: Perhaps you got the wrong impression from the title of "editor" that was granted to you on becoming a WP user. We are all editors here, and nobody has elected you Editor-in-Chief. We who are about to edit do not submit our performances to your Arena for the sole benefit of your imperious amusement and Thumb↑-∨-Thumb↓ approval. So please start trying to act like one of the proverbial Christians instead of like one of the proverbial Lions, so to speak. Gratia in futuro, Jon Awbrey 15:36, 26 May 2006 (UTC)

 * The last revert wasn't 'reflexive'. I've previously indicated an article size concern on this page and I indicated in the edit history that I thought the subject matter didn't fit and should be its own page. Perhaps you should have discussed it on this Talk page first. I note also that I have retained many of your edits, including ones with which I disagree.

 * As to complaints about simple reverts, first explain these reverts to me:, , , . Until then, the charge of hypocrisy sticks. You simply reverted my changes despite pleas that you work with me by editing my contributions in an iterative manner. I tried to engage on the Talk page, but you continued. I assumed good faith as best I could until you started vandalising the page. That's a very plain indication of bad faith. You fully lost the assumption of good faith with that; reverting edits that modified your contributions, mislabeling changes as 'corrections' or 'minor', etc., had started me on that path, but this was the final straw.

 * Now you continue to whine about how persecuted you are in the hopes that some admin will hear your plea, see that you have whined more than I have and take that as evidence of greater suffering, and return this page to your personal blog. Keep at it; I've seen such a tactic work before. Good luck. JJL 15:52, 26 May 2006 (UTC)

JA: Addressed to User:JJL. Getting to know new people in the columns of the WikiPersonals is always interesting, and occasionally challenging. I have commented on some of the trials and tribulations of the "Getting to Know You" phase, and I have already stipulated to the wrong-footed faux pas of my getting-offness in those early days of 1 and 2 weeks ago that you refer to above. As I also said above, I endeavor to maintain a 0-Revert-Rule in situations of hot dispute, and I think that I have maintained that rule since shortly after my last revert of 16:03 20 May 2006, the last that you reference above.

JA: Still, I will provide some additional explanation of what I was about in those dim days of our conjoint history:
 * 1) Item 1.  Edit line: "(revert -- not so)" -- Additional explanation: I reverted the insertion of a questionable statement, for which citations have repeatedly been requested since that time, and which remains unsourced to this day.
 * 2) Item 2.  Edit line: "(reverting -- this is a wiki, remember? -- things become clearer over time)"  -- Additional explanation: I had bona fide (Eng: good faith) reasons for placing that text where I did.  The fact that its point was not immediately clear to you is not sufficient reason for you to relocate it somewhere else.  It occurred to me that perhaps it did not occur to you that I had good reasons for putting that text where I did, so I was informing you of that fact.
 * 3) Item 3.  Edit line: "(reverting mass deletions -- please proceed more slowly & go to talk page if you wish discuss particular statemnts)" -- Additional explanation:  Here I simply reverted one of your many mass deletions of intervening additions, and informed you of the standard WP practice of discussing edits of mass destructure on the talk page.

JA: I hope that will put to rest the events of 1 and 2 weeks ago. On to the present, and, who knows, the future. Jon Awbrey 17:34, 26 May 2006 (UTC)

Mention & Use in WikioPolis
JA: I can't say that I'm fond of the usage, but italics are standard for the mention of phrases in WP, so we have to be more sparing than usual in using them for emphasis. Jon Awbrey 15:01, 25 May 2006 (UTC)

On Being Neither God Nor His Prophet
JA: Will WP ever need a page for G.H.R. Parkinson, who translated, edited, and wrote an introduction to a volume of Leibniz's logical papers? Being Neither God Nor His Prophet, I really cannot say. But I am fallibly, finitely, and humanely guessing that infidels who have yet to imbue themselves with the accumulated wisdom of Five pillars are not the ones to speak for what WP needs. Jon Awbrey 15:02, 26 May 2006 (UTC)

 * Why don't we link G.H.R. Parkinson to the Nile River, then? In the meantime, feel free to add pages for these people. I just did Paul Weiss, though much more needs to be done there. Otherwise, filling a page with redlinks isn't helpful. JJL 15:08, 26 May 2006 (UTC)

Perennial questions
First, there was nothing 'improper' about the deletion. However...focusing on one question in phil. of math. seems wrong for this article, and it also seems an awkward place to put it. At the least, I would place it after Historical overview. I do concur with Article_size that "[r]eaders may tire of reading a page much longer than about 6,000 to 10,000 words, which roughly corresponds to 30 to 50 KB of readable prose. Thus the 32 KB recommendation is considered to have stylistic value in many cases; if an article is significantly longer than that, then it probably should be summarized with detail moved to other articles" and so I am leery of seeing someone starting a major new section. The stub tag suggests it may indeed grow large. I propose that, like the foundations article, this be moved to its own article. It could be something like Source/Origin of Math. It'd be nice to have discussions of all the major questions and schools of thought--but not on the main phil. of math. page. JJL 19:44, 26 May 2006 (UTC)

JA: By "improper" I mean that it is not recommended practice under the applicable guidelines, policies, and just plain good advice — for example, Five pillars, Editing policy, Simplified Ruleset, to name just a few — for editors to delete or revert good faith edits by others. I cite once again the following bits of WP policy, recommendation, and just plain good advice:
 * 1) Perfection isn't required.
 * 2) This policy in a nutshell: Improve pages wherever you can, and don't worry about leaving them imperfect.  However, avoid deleting information wherever possible.
 * 3) Particularly, don't revert good faith edits. Reverting is a little too powerful sometimes, hence the three-revert rule. Don't succumb to the temptation, unless you're reverting very obvious vandalism (like "LALALALAL*&*@#@THIS_SUX0RZ", or someone changing "6+5*2=16" to "6+5*2=17").  If you really can't stand something, revert once, with an edit summary something like "(rv) I disagree strongly, I'll explain why in talk." and immediately take it to talk.

JA: Concerns about article size are no excuse for deleting or reverting good faith edits. These are incidental issues that can be discussed in the Committee of the Whole, not unilaterally dictated by a single editor. Very few articles in WP that take up signficant subjects, as this one clearly is, can be expected to sum it up in 32 kb, and a reader who opens up an article on the Philosophy of Mathematics may be expected to expect that. At any rate, these are pragmatic issues that can be discussed by all interested parties when the artcle reaches critical mass, which it is nowhere near yet. In the meantime, deleting good faith text is not a productive way to develop either the one or the prospectively many articles. Jon Awbrey 20:22, 26 May 2006 (UTC)

 * That's great. Any response to my attempt to discuss the new section you're starting? I wrote: "focusing on one question in phil. of math. seems wrong for this article, and it also seems an awkward place to put it. At the least, I would place it after Historical overview" and "I propose that, like the foundations article, this be moved to its own article. It could be something like Source/Origin of Math. It'd be nice to have discussions of all the major questions and schools of thought--but not on the main phil. of math. page" in an attempt to initiate dialogue on the point at hand. If there are other interested parties, they can speak up right here. JJL 20:44, 26 May 2006 (UTC)

JA: Addendum. JJL, perhaps your worries about file size can be assuaged by reading the next paragraph of WP:SIZE: No need for haste Do not take precipitous action the very instant an article exceeds 32 KB. There is no need for haste. Discuss the overall topic structure with other editors. Determine whether the topic should be treated as several shorter articles and, if so, how best to organize them. Sometimes an article simply needs to be big to give the subject adequate coverage; certainly, size is no reason to remove valid and useful information. JA: Hope that helps a little. Jon Awbrey 05:40, 27 May 2006 (UTC)

What are the Perennial questions being discussed here? I still don't see the point of this section, other than to drop names and quotes. For example, "Aristotle routinely derives his initial philosophical impulses from the parables of his predecessors, especially Plato, but his natural attraction to earthly topics just as dependably brings him back to empirical grounds." goes nowhere. The stub tag is gone...and I still don't see the point, now how it is to fit into the rest of the article. I'd like to hear someone else's comments on this. JJL 20:16, 30 May 2006 (UTC)

JA: The intent is to cover whatever questions we end up with in the lead, but for my part I'm taking them as they occur to me, and not worrying if the $$\cup$$ runneth over a little, as it has always seemed like a feasible plan to integrate historical glosses with the text to some extent. My edit line for the Aristotle bit said "begin expo …" and there's more coming. We can put the stub tag back if it makes anybody feel better. Time for dinner here, further explanation later. Jon Awbrey 21:10, 30 May 2006 (UTC)

Historical overview
The historical overview looks like a good start... But the order is a little off. The Pythagoreans (if they were) were the ones who threw out sqrt(2) into the sea. (Socrates) Plato, then Euclid, demanded straight-edge and compass. Archimedes and Appolonius didn't seem to care. --M a s 21:56, 26 May 2006 (UTC)

 * Thanks. I'm working from memory until next month when my library and I are reunited. Right now it's more "at some time someone believed this" unfortunately. I'll try to straighten it out and hope to add more. JJL 16:40, 27 May 2006 (UTC)

Regarding the worldview issue, I agree...yet, isn't this page inherently (Western) Philosophy of Mathematics? There's been no attempt yet to draw in views other than those that originate with the Greeks, it seems to me. JJL 20:55, 19 June 2006 (UTC)

 * Thanks. What do other philo of ... pages have?  And I guess just as importantly other encyclopedias?  I don't know myself.  I put the tag on with the concern that it would invite spurious claims of propriety.  Is there a weaker worldview tag that draws attention but allows for exceptions?  Thanks, --M a s 01:02, 23 June 2006 (UTC)

 * Looking at phil. of logic, language, science, physics, I saw only science having any mention of non-Western thought, and that was quite brief. Looking at Eastern_philosophy, I'm not sure what to say. But, comparing to other entries, I'd leave the tag off. It ain't right, but it's common. JJL 01:29, 23 June 2006 (UTC)

Reorganization : Questions & Schools
JA: I am going to make an effort to reorganize the subject matter a bit, making a short list of major questions that arise in the philosophy of mathematics, highlighting these questions briefly before diving into the phil-political crossfires of the various 20th century schools of thought. In connection with this alternative topology for the material, I think that it would also serve understanding to trace the questions back in time, to the roots of the 20th century quakes and revolts. Jon Awbrey 14:42, 23 May 2006 (UTC)

 * Good with me. A lot of this article looks as if it's setting up a straw-man of most early 20th century philosophies, and then introducing late 20th / early 21st century postmodern stuff, and then implying that all the new stuff solves the problems of the old.  Early 20thC schools of course had problems, but they were developed to solve problems of late 19thC (e.g. Cantor), and those were developed to solve problems of mid-19thC (e.g. Fourier), etc.

 * PhiloMath has a history going back as you rightfully state, 2000 years. Unfortunately whatever I can add will only be Eurocentric.  --M a s 16:52, 23 May 2006 (UTC)

 * The article has a strong bias toward the 20th century, which I think is unfortunate. We need more on ancient Greek ideas ("1" was a unit, not a number, etc.) and so on, and perhaps less on the current schools of thought (each of which may well merit its own page). While the aftereffects of set theory and Godel are fascinating, there's so much more. I'd also like to see a bit more about the phil. implications of interactions with physics. I'm traveling and away from my references now, unfortunately. Right now a lot of deck chair rearranging is going on. Hopefully next month when I'm back home I can add some stuff on ancient ideas, either here or as a linked article. JJL 16:58, 23 May 2006 (UTC)

JA: What the article has a strong bias toward is the personal opinions of JJL. Jon Awbrey 02:56, 24 May 2006 (UTC)

JA: With regard to this query from JJL:

That's great. Any response to my attempt to discuss the new section you're starting? I wrote: "focusing on one question in phil. of math. seems wrong for this article, and it also seems an awkward place to put it. At the least, I would place it after Historical overview" and "I propose that, like the foundations article, this be moved to its own article. It could be something like Source/Origin of Math.  It'd be nice to have discussions of all the major questions and schools of thought -- but not on the main phil. of math. page" in an attempt to initiate dialogue on the point at hand. If there are other interested parties, they can speak up right here. JJL 20:44, 26 May 2006 (UTC)

JA: I discussed my reasons for trying a new outline in advance of beginning the trial. There is certainly no intention of focusing on one question, but it is necessary to begin somewhere or not at all. I began with a question that is certainly of primary importance. Not only that, but the sourced material that I selected to add was chosen for the fact that it touches on issues of central importance, lying at the intersection ($$\cap$$) of several of the identified questions. For instance, the question of relating mathematics to physics and the other empirical sciences is motivated in a quite natural way.

JA: Yes, proper placement is an eventual consideration, but one that is best addressed after the material has developed a little further. Not knowing the "perfect" place to locate a bit of useful material is still no excuse for snipping it in the bud. There are many discussions in WP as to where to place the historical sections. I can remember several people citing a guideline somewhere that says to put the history last, but that may have been advice specific to technical articles in the sciences, and it does not seem to befit the current case. Still, I anticipate that the history sections are likely to expand over time, and it did not seem apt to put off a second pass at the major questions of the field until the history was done. Again, there is no rush about deciding these things in advance of having some concrete text to work with.

JA: The idea that the Perennial Questions are more important than divvying up the turf between the Scholastic Departments of the 1900s is hardly a new idea. It is every bit as well motivated as the current breakdown by Schools. Jon Awbrey 03:16, 27 May 2006 (UTC)

 * A case can be made either way for Big Questions vs. Big Schools of Thought. The info. on both should be somewhere in WP but not necessarily all in this article. Sorting that out can indeed wait. I would like to simplify some of the philosophical language in the Perennial Q's section. I think the main phil. of math. article should be more non-phil.-friendly. JJL 18:54, 27 May 2006 (UTC)

JA: Of course, folks I know in sociology would have already introduced a "matrix" table with Questions as column heads and Schools as row leaders, or vice versa, depending on their orientation, but this is only a second pass at the Questions and it's not possible or necessary to say everything at once. It is also possible to distribute little bits of historical floss and gloss throughout the discussion of Questions, for instance, mention the relation between what G. de B.R. said and Aristotle's distinction between "what is closer to nature" and "what is closer to us", or mention the relation between what Putnam said and Kant's ideas about the synthetic a priori. All in good time, all in good space. Jon Awbrey 19:18, 27 May 2006 (UTC)

What constitutes a valid mathematical proof?
That sound too much like the detailed mathematical question. This should specify how the mathematical answer is interpreded by phylosophy.

JA: I have in the meantime generalized the notion of proof to include all forms of inquiry that play a role in mathematical practice, for example, analogy, conjecture, and others. Jon Awbrey 06:48, 30 May 2006 (UTC)

Bedeviled Chicken, Bedeviled Egg
JA: I've been putting off the "relation between logic and math" question, as this is always the big fly in the ointment for foundations of maths. So it's likely to be a bit rough at the outset. Be gentle, be patient, etc. Jon Awbrey 20:16, 1 June 2006 (UTC)

Russell, IMP, 1919
JA: Storing for future reference and abstracting a few long excerpts from Russell's Introduction to Mathematical Philosophy (1919). Jon Awbrey 03:45, 2 June 2006 (UTC)

| Mathematics and logic, historically speaking, have been entirely | distinct studies. Mathematics has been connected with science, | logic with Greek. But both have developed in modern times: | logic has become more mathematical and mathematics has | become more logical. The consequence is that it has | now become wholly impossible to draw a line between | the two; in fact, the two are one. They differ as | boy and man: logic is the youth of mathematics and | mathematics is the manhood of logic. This view is | resented by logicians who, having spent their time | in the study of classical texts, are incapable of | following a piece of symbolic reasoning, and by | mathematicians who have learnt a technique | without troubling to inquire into its | meaning or justification. Both types | are now fortunately growing rarer. | So much of modern mathematical work | is obviously on the border-line of | logic, so much of modern logic is | symbolic and formal, that the very | close relationship of logic and | mathematics has become obvious | to every instructed student. | The proof of their identity is, | of course, a matter of detail: | starting with premisses which | would be universally admitted | to belong to logic, and arriving | by deduction at results which as | obviously belong to mathematics, | we find that there is no point | at which a sharp line can be drawn, | with logic to the left and mathematics | to the right. If there are still those | who do not admit the identity of logic and | mathematics, we may challenge them to indicate | at what point, in the successive definitions and | deductions of 'Principia Mathematica', they consider | that logic ends and mathematics begins. It will then | be obvious that any answer must be quite arbitrary. | | Russell, IMP, pp. 194-195. | | Bertrand Russell, Introduction to Mathematical Philosophy, 1919. | Reprinted, Routledge, London, UK, 1993.

Realism
JA: The use of the term "realism" is extremely problematic in this article, in the absence of any sourced or standard definitions flip-flopping back and forth between technical senses and non-technical uses at will with no notice of the severe equivocation involved in doing this. I will add some sourced and standard definitions and leave it to others to justify the use of its "street" meaning in this context. Jon Awbrey 02:24, 8 June 2006 (UTC)

Deleted Material: Historical Overview
JA: Citations requested for this material have gone unsupplied. Moving it here pending substantiation. Jon Awbrey 02:01, 10 June 2006 (UTC)

There are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Western philosophizing about mathematics has a history that goes at least as far back as Plato, who considered the ontological status of mathematical objects, and Aristotle, who considered logic and issues related to infinity (actual versus potential). Greek views of quantity strongly influenced their views of other areas of mathematics. At one time, the Greeks held the opinion that 1 (one) was not a number, but rather a unit of arbitrary length (so that 3, for example, represented 3 such units and truly was a number). At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair. Of course, this was well before 0 was considered a number. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: The first line drawn had unit length, and numbers represented multiples of it. Greek ideas of number were upended by the discovery of the irrationality of the square root of two, showing that the diagonal of a unit square was incommensurable with its (unit-length) edge: There was no number that represented how much longer the diagonal was than an edge. This caused a significant re-evaluation of Greek philosophy of mathematics, as non-Euclidean geometry would do to European philosophy of mathematics two millenia later. Beginning with Leibniz, the focus shifted strongly to the relationship between mathematics and logic. This view dominated the philosophy of mathematics through the time of Frege and of Russell, but was brought into question by developments in the late 19th and early 20th century.

 * Is any of that controversial?142.177.110.74

Deleted Paragraph: Mathematical Philosophy
JA: The statement that "the terms philosophy of mathematics and mathematical philosophy are often taken to be synonymous" continues to go unsupported by evidence after repeated requests for citations. The rest of the paragraph actually contradicts the initial statement, and the overall result is a wholly irrelevent distraction. Moving it here until somebody thinks of a way to (1) support it (2) explain its relevance. Jon Awbrey 02:34, 10 June 2006 (UTC)

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonymous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

 * OK, I've tried giving you time, working with you, etc. Moving one section seems to have led to a series of 'retaliatory' redactions of things I've written (like the hist. overview, which I said here that I'd cite in another week's time when I return to my books) and this section, which you and I had previously reached a compromise on. No discussion here first--just elimination of good faith edits while you rewrite the page to your stilted view of how it should be. We'll go back to doing this the hard way, since you seem unable to work constructively with others and unable to accept views other than your own. JJL 05:38, 12 June 2006 (UTC)

JA: Unsourced statements can be deleted any time. It is of course a nice idea to request citations and allow a decent interval for sources to be supplied, but some of these claims have gone begging for months now. Jon Awbrey 05:50, 12 June 2006 (UTC)

 * OK. Most of what you wrote for Perennial Questions takes a sourced quote and bases your personal POV of how things should be around it. All that needs sourcing. I note from this edit that you have declared yourself an expert on the Truth and are now deciding who does and does not have access to the true Truth. People like you, who believe that their way is the only way, are the problem here. You're using WP's popularity to espouse your personal views and sense of 'humor'. Get a blog. You'll always have the Usenet Salute. JJL 06:31, 12 June 2006 (UTC)

Deleted Paragraph: Badiou
JA: Badiou is not a major figure in the philosophy of math, and the following account of his position strikes me as sheer gibberish, whose fault that is I cannot say, so I'll place it here until somebody can justify it on the grounds of (1) notability (2) sensibility. Jon Awbrey 05:25, 10 June 2006 (UTC)

It should be noted that this reading of "Platonism" is rejected by modern philosopher Alain Badiou, who considers the "empiricist" relationship between object and subject (where objects external to one's mind act, through the senses, on an internal subjective realm) utterly foreign to Platonic thought, according to which this location of mathematical entities is irrelevant to their ontological status. Badiou, in fact, identifies mathematics with ontology, considering mathematical discovery to be the scientific investigation of Being qua Being.

OOPS forgot no continental philosophy on wikipedia my bad lol

Intuitionism & Constructivism
JA: Though they share many tenets, I-ism and C-ism are really distinct movements, and require separate sections. It will take a day or so to look up some primary sources and sort out which bits of text belong to which sections. Jon Awbrey 05:32, 10 June 2006 (UTC)

 * On the one hand, this is correct and I'm glad to see them be split up given the current structure. On the other hand, I'm of the opinion that most schools of though can be grouped broadly under the headings of Platonism (or Realism), Formalism, and Constructivism, where those terms are construed rather more broadly than they are in the article. Given the article's length, I would prefer to see fewer separate entries and more Wikilinks to detailed discussions of these theories. Right now the list of different schools of thought is too forbidding for those who would benefit from this article. JJL 15:15, 12 June 2006 (UTC)

JJL and Jon Awbrey - 3RR
User:JJL and User:Jon Awbrey, please stop revert warring. You're already both in violation of the 3 revert rule, and if either of you reverts again, I'll block both of you, until you each agree to chill out and pursue a more productive strategy. If you can't sort it out between you, please seek outside opinion. Please do not revert war. -GTBacchus(talk) 06:58, 12 June 2006 (UTC)

 * OK. JJL 15:02, 12 June 2006 (UTC)

Synonomy of Phil. of Math. and Math. Phil.
I previously cited the example of Mathew Mount, "Classical Greek Mathematical Philosophy", on this page. I've also referenced on the article page the book by Maziarz and Greenwood, "Greek Mathematical Philosophy", which discusses the changing nature of the Greek phil. of math. There is also Cassius Keyser, "Mathematical Philosophy", among many others. Of course, there are also books more specific to one working mathematician's views on math., but books like Carolyn Eisele, "Studies in the Scientific and Mathematical Philosophy of Charles S. Pierce" (M. Martin, ed.), are discussing the subject of a philosopher's phil. of math. From a review of Korner's book: "a thorough survey of the issues in modern mathematical philosophy, see Morris Kline's Mathematics: The Loss of Certainty." At Questia.com, both math. phil. and phil. of math. go to the same page (phil. of math. [www.questia.com/library/philosophy/ philosophy-of-mathematics.jsp ]). Here's a St. Lawrence University course : "Platonic mathematical philosophy". At Wright State University, look at PHL424 Mathematical Philosophy : "Investigation of philosophical theories concerning the nature of mathematics, the ground of mathematical knowledge, the necessity of mathematical truth, the empirical relevance of mathematics, and the relationships between mathematical philosophy and general philosophy". At Oberlin College, someone is researching the origins of what has become the "Unreasonable Effectiveness" issue : "Numbers and Things: Nominalism and Constructivism in 17th Century Mathematical Philosophy". This looks to me like phil. of math., as he discusses natural phil. being mathematized, not general phil. Look here for a College Math. Journal article: "New Thoughts on the Mathematical Philosophy of the Pythagoreans". From here, we have: "The study of the principles and the nature of mathematics is known as the philosophy of mathematics, or sometimes mathematical philosophy." The author is no expert, but the point is that they are used synonomously, not that they should be. From here : "the Tarskian classical correspondence theory of truth in modern mathematical philosophy" (emphasis in original). From this Word file : "Logicism – This is a mathematical philosophy " (emphasis in original HTML version ). They describe intuitionism and formalism similarly. From a University of Windsor seminar series : "A Whirlwind of Ideas in Mathematical Philosophy", which appears to discuss standard phil. of math. topics. From here : "Thought Experiments in Galileo and Newton’s Mathematical Philosophy".

Others maintain a distinction: See the question about Russell's views here or the review. Most of the exceptions I find are just those addressed in the article here: Those who take Russell's view or one like it, and those who use math. phil. when referring to a single mathematician's (or specific group of mathematician's) views. But the Greek math. phil. books above are about the views of philosophers, not mathematicians, over the whole classical period. (Granted, it was harder to distinguish a mathematician from a philsopher at the time.) There are many uses of math. phil. as a synonym for phil. of math., by authors and academics. I think the section should stay, and that it is supportable. I don't think a simple matter of definition such as this needs an in-article citation. JJL 17:28, 13 June 2006 (UTC)

JA: Good. Now was that so hard? Inconvenient, perhaps, to be sure. But now that we have your long-anticipated compliance with a routine citation request, we can begin to examine the issue in a proper scholarly fashion as to just how often, and who, and in what sense, some may identify the two disciplines in question. Jon Awbrey 22:12, 14 June 2006 (UTC)

 * Could these sources be added inline into the article?????? --Vesal 12:46, 7 October 2006 (UTC)

The 'the'
This has at least been mentioned in the edit history, but I'm not sure it's been discussed on Talk: The leading 'the' isn't used in the Phil. of Sci., Phil. of Lang., etc., entries, so for consistency I think it should be omitted here too. JJL 13:59, 15 June 2006 (UTC)

JA: Consistency being a 2-way street, one might as well argue that the others should be changed to be consistent with this. But it's a very weak argument, and a very minor hobgoblin, either way. Jon Awbrey 14:22, 15 June 2006 (UTC)

Aesthetics 2
Shouldn't this section be merged into Mathematical beauty? Ben Finn 20:21, 12 July 2006 (UTC)

 * That's a reasonable suggestion. JJL 20:27, 12 July 2006 (UTC)

 * You could leave a short section on aesthetics, on symmetry, etc., here, and put some of the material over there. But it must be mentioned. 142.177.110.74

Delisted GA
Im sorry, but with so many "un-verifiable" tags and that cleanup tag, I can't tell whether the article is or is not, in fact, well-referenced or well-written, despite the really very extensive book list at the bottom you've got. Therefore, I don't think readers would probably figure it out either. Furthermore, it seems nobody actually gave this article a real review in the first place, and we've been having a problem with people tagging articles as GA's without reviewing them at all and comprimising the integrity of the GA system, I recommend that if the verifiability tags have no merit due to the books at the bottom you remove all those tags and re-nominate this article, so hopefully somebody more familiar with this sort of math-stuff can give the article a real review. Right now however, this does not look like a Good Article. Homestarmy 16:55, 8 August 2006 (UTC)

JA: The sad fact is that certain editors on this article simply fail to supply sources that are adequate to verify the statements that they insist on keeping in the article. Until that changes, nothing can be done about the quality of the article. Jon Awbrey 17:32, 8 August 2006 (UTC)

 * The sad fact is that people have not read all the archived discussions and addressed the questions they raise. This article needs some kind of subpage discussing issues, like Talk:Philosophy_of_mathematics/issue so we can keep this all sorted out instead of threading it all out like this.  The JA/JJL troll war seems to be what got this article kicked out of GA.142.177.110.74

 * He's been banned for trolling, vandalism, etc. See the discussion here. He was littering the article with cite/verify tags as a form of tactic. Hopefully things can now proceed in a reasoned manner. Feel free to contribute to the article itself, not just here. JJL 22:18, 9 October 2006 (UTC)

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