Mathematical Platonism and its Opposites
Barry Mazur
January 11, 2008
Comment 1: In this essay, Professor Barry Mazur brings a very romantic sensibility to what is definitely a courageous and an honest attempt at introspection. The courage and honesty that he shows are much more than can be reasonably expected from any modern scholar. However, mathematical Platonism deserves an even better defense than this article provides.
We had the sky up there, all speckled with stars, and we used to lay on our backs and look up at them, and discuss about whether they was made or only just happened–Jim he allowed they was made, but I allowed they happened; I judged it would have took too long to make so many.
mused Huckleberry Finn. The analogous query that mathematicians continually find themselves confronted with when discussing their art with people who are not mathematicians is:
Is mathematics discovered or invented?
Comment 2: The relationship between the Huckleberry Finn quotation and The Question, mentioned below, is far from a natural analogy. The Huckleberry Finn quotation would be more fitting to describe the scientific controversy called Intelligent Design that has gained a lot of attention in recent times. For example, this controversy would ask whether the stars were created by an Intelligent Supreme Being, or they just came to be from more elementary particles through procedures governed by scientific laws.
Perhaps what the author means is that just like that the Intelligent Design question has taken center-stage today among enthusiasts of science, the question that occupies an analogous position among enthusiasts of mathematics (both professional and amateurs) is The Question. If so, see my last few comments on this article.
I will refer to this as The Question, acknowledging that this five-word sentence, ending in a question-mark
Comment 3: In formulating this question so carefully and precisely, is the author willing to put his reputation behind its resolution? Or at least accept that he, as a professional mathematician, would seriously like to know more about The Question.
—and phrased in far less contemplative language than that used by Huck and Jim—may open conversations, but is hardly more than a token, standing for puzzlement regarding the status of mathematics.
Comment 4: Here, the author seems to be referring to mathematical amateurs whom an established mathematician often meets in social occasions and who try to show off their knowledge in mathematics by asking about ‘hot’ topics like The Question which they have read about in some magazine or expository article on mathematics.
In acknowledging that the question is ‘hardly more than a token’, there is a conflict of interest, in that, the standing of the author, as a famous mathematician, makes him suggest that many people think there are far deeper things to modern mathematics than The Question, and hence the ‘puzzlement regarding the status of mathematics’. On the other hand, the author would like to propose The Question as the intellectual equivalent of the question on Intelligent Design, which has thrown the whole of modern science into a state of confusion, and has sent many great scientists on personal journeys of soul-searching. Come on, Professor Barry Mazur, why don't you just accept that this is all just a storm in a tea-cup, relatively speaking.
One thing is—I believe—incontestable: if you engage in mathematics long enough, you bump into The Question, it won’t just go away1
Footnote 1: Garrison Keillor, a wonderful radio raconteur has in his repertoire a fictional character, Guy Noir, who tangles indefatigably with “life’s persistent questions.” This is all to the good. We should pay particular honor to the category of persistent questions even though—or, especially because—those are the chestnuts that we’ll never crack.
Comment 5: With reference to footnote 1, what is the author’s philosophical disposition towards The Question? Does he think this is a serious philosophical issue or just performing arts? The ‘persistent questions’ in life are those one memorizes and remembers well enough to throw around off-hand during parties and social conversations? Woody Allen meets Sophocles?
If we wish to pay homage to the passionate felt experience that makes it so wonderful to think mathematics, we had better pay attention to it.
Comment 6: Rather than frame it as ‘life’s persistent questions, could the author explain the reasons why The Question is important to mathematicians, as I have done in my correspondence with Professor Dana Scott?
Some intellectual disciplines are marked, even scarred, by analogous concerns. Anthropology, for example has a vast, and dolefully introspective, literature dealing with the conundrum of whether we can ever avoid—wittingly or unwittingly—clamping the templates of our own culture onto whatever it is we think we are studying: how much are we discovering, how much inventing?
Comment 7: Note that the words discovery and invention are used here with a different meaning than in The Question. Discovery here means using an objective set of guidelines to make observation about ‘whatever it is we think we are studying’. Invention here means that ‘we’ are making up the observation much influenced by our own cultural prejudices.
On the other hand, there is another interpretation to the use of discovery and invention here. Perhaps earlier civilizations had already known a lot about 'whatever it is we think we are studying'. And our lop-sided views of history and science, make us think we are really inventing (for the first time), when we are only discovering for ourselves what earlier civilizations had actually invented.
Such a discovered/invented perplexity may or may not be a burning issue for other intellectual pursuits, but it burns exceedingly bright for mathematics, and with a strangeness that isn’t quite matched when it pops up in other fields.
Comment 8: The author gives two reasons below for why The Question is so uniquely a burning issue for mathematicians: (i) mathematician works with layer after layer of ideas built over the more concrete objects like circles, triangles, and numbers, (ii) mathematical investigation is distinctly different from other artistic activities like writing or painting. In its core, the mind does not think in terms of pictures or words, and it is hard to give a location for the mathematical 'hunting grounds'. But, there is a third reason why The Question is a burning issue -- plain old ambition. This reason is stated eloquently by G H Hardy in his 'A Mathematician's Apology': "Immortality may be a silly word, but probably, a mathematician has the best chance of whatever it may mean".
For example, if you were to say—as Thomas Kuhn once did—“Priestley discovered oxygen but Lavoisier invented it” I think I know roughly what you mean by that utterance, without our having to synchronize our private vocabularies terribly much. But to intelligently comprehend each other’s possibly different attitudes towards circles, triangles, and numbers, we would also have to come to some—albeit ever-so-sketchy—understanding of how we view each view, and talk about, a lot more than mathematics.2
Footnote 2: For a start: you and I turn adjectives into nouns (red cows |-> red; five cows |-> five) with only the barest flick of a thought. What is that flick? Understanding the differences in our sense of what is happening here may tell us lots about our differences regarding matters that can only be discussed with much more mathematical vocabulary.
Comment 9: The author has made a deep insight here. An high school science major would have a fairly concrete (and complete) model of oxygen in his/her mind (periodic table properties, atmospheric composition, required for burning, etc). In contrast, a circle can be thought of pictorially, or as the solution set in Euclidean space of a polynomial equation, or could be defined using an ideal in a polynomial ring. Likewise, a triangle could be a picture, or the intersection of subspaces in a linear space, and its geometric properties can depend on what the ambient space is -- whether it is Euclidean, Spherical or Hyperbolic. Likewise, the concept of a number could be a simple counting device, or it could be understood to have deep similarities with geometric shapes through modern developments in algebraic geometry. Layer upon layer of ideas are built on the basic physical (geometric) concepts that are familiar to the faculties of all humans. Thus more advanced studies are required in math for getting these additional perspectives on our basic cognitive faculties. So, by the time one gets to the point of finding new results in math, there is a lot of effort involved. In other words, one is working from an advanced standpoint on the same basic concepts of circles, triangles and numbers. And would what results the mathematician establishes be discoveries or inventions? Discovery would have the somewhat unfair connotation that what the mathematician has established is not that far advanced from the knowledge of the non-expert.
Moreover, if the reader thought that it is just a temporary accident that, in math, there are layers of advanced concepts that one requires to learn as background, then it is useful to think of G H Hardy’s words: ‘A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas’. By the nature of his/her work, a mathematician needs to work with much more abstract concepts than the natural scientist, and this require years to get familiar with.
For me, at least, the anchor of any conversation about these matters is the experience of doing mathematics, and of groping for mathematical ideas. When I read literature that is ostensibly about The Question, I ask myself whether or not it connects in any way with my felt experience, and even better, whether it reveals something about it. I’m often—perhaps always—disappointed.
Comment 10: On the other hand, there are significant restrictions to the nature and extent of the romantic ‘felt experience’ of the working mathematician. For example, if a mathematician claims that he/she has come to realize that the Riemann Hypothesis is true because of a metaphysical experience that he/she has been through, it would not be accepted that he/she has established the RH.
The felt experience, as the author suggests later, may be the sole property of the working mathematician but he/she does not have arbitrary freedom to define what it is! The common man can still hold the working mathematician accountable on many counts.
The bizarre aspect of the mathematical experience—and this what gives such fierce energy to The Question—is that one feels (I feel) that mathematical ideas can be hunted down, and in a way that is essentially different from, say, the way I am currently hunting the next word to write to finish this sentence. One can be a hunter and gatherer of mathematical concepts, but one has no ready words for the location of the hunting grounds.
Comment 11: Professor Mazur definitely has a valid point here. At its inner core, the mind does not think in terms of pictures or words. In fact, in recent decades, cognitive scientists have proposed the notion of ‘conceptual metaphors’ as the medium of thinking for the human brain. In any case, it is definitely acceptable that the experience of doing mathematics is not the same as using language to write prose or poetry. Having said that, perhaps it is not inappropriate to ask the author to explain how come several Fields Medalists (Gowers, Tao, Borcherds) have taken to writing pages and pages of blogs on their websites. Such blog-writing must take at least 2 to 3 hours of their daily activity. When exactly are they pursuing intellectual activities that are different from ‘hunting for the next word to write'? I am not trying to just be sarcastic here. The issue is not so simple. Conventional wisdom among mathematicians highly valued results produced with prolonged, isolated, individual efforts for their originality. However, in the past twenty years, the whole world has taken to communicating through e-mails, and on websites and blogs, and a lot of information and knowledge is being exchanged vigorously. There is definitely a great dilemma for a serious mathematician here. The funding agencies and the mathematicians at universities don't seem to appreciate the urgency and the seriousness of this issue.
Of course we humans are beset with illusions, and the feeling just described could be yet another. There may be no location.
Comment 12: Try a historical location, like the Renaissance Enlightenment, as I have done in my comments to Professor Dana Scott.
There are at least two standard ways of—if not exactly answering, at least—fielding The Question by offering a vocabulary of location. The colloquial tags for these locations are In Here and Out There (which seems to cover the field).
Comment 13: All this seems to be an artificial set-up to project the importance of The Question. Only historical context would provide a credible background for The Question. (ref: my comments to Professor Dana Scott).
The first of these standard attitudes, the one with the logo In Here—which is sometimes called the Kantian (poor Kant!)—would place the source of mathematics squarely within our faculties of understanding. Of course faculties (Vermogen) and understanding (Verstand) are loaded eighteenth century words and it would be good—in this discussion at least—to disburden ourselves of their baggage as much as possible. But if this camp had to choose between discovery and invention, those two too-brittle words, it would opt for invention.
Comment 14: Although Immanuel Kant taught mathematics for many years, he is too weak a choice, as a mathematician, to represent the anti-Platonist position. In his Critique of Pure Reason, Kant proposed that Cartesian (Euclidean) geometry was a matter of innate knowledge for humans, and moreover the shape of physical space was Euclidean because the innate knowledge of humans was directly related to the nature of external reality. (... stars above me, and the moral law within me). Obviously, this grand scheme of things did not hold up, since Einstein's relativity theory explained convincingly several decades later that physical space was curved. In any case, Professor Mazur portrays the anti-Platonist from several different perspectives in the last section of this article. So, why not use several different famous personalities (e.g., Aristotle, Descartes, Kant), to represent the anti-Platonist.
The “Out There” stance regarding the discovery/invention question whose heraldic symbol is Plato (poor Plato!) is to make the claim, starkly, that mathematics is the account we give of the timeless architecture of the cosmos. The essential mission, then, of mathematics is the accurate description, and exfoliation, of this architecture. This approach to the question would surely pick discovery over invention.
Strange things tend to happen when you think hard about either of these preferences.
For example, if we adopt what I labeled the Kantian position we should keep an eye on the stealth word “our” in the description of it that I gave, hidden as it is among behemoths of vocabulary (Vermogen, Verstand). Exactly whose faculties are being described? Who is the we? Is the we meant to be each and every one of us, given our separate and perhaps differing and often faulty faculties? If you feel this to be the case, then you are committed to viewing the mathematical enterprise to be as variable as humankind. Or are you envisioning some sort of distillate of all actual faculties, a more transcendental faculty, possessed by a kind of universal or ideal we, in which case the Kantian view would seem to merge with the Platonic.3
Footnote 3: A more general lurking question is exactly how are we to view the various ghosts in the machine of Kantian idealism—for example, who exactly is that little-described player haunting the elegant concept of universally subjective judgments and going under a variety of aliases: the sensus communis or the allgemeine Stimme?
Comment 15: Work of Aristotle, Rene Descartes, Charles Darwin, Sigmund Freud, Noam Chomsky are also important here, along with Immanuel Kant’s work.
If we adopt the Platonic view that mathematics is discovered, we are suddenly in surprising territory, for this is a full-fledged theistic position. Not that it necessarily posits a god, but rather that its stance is such that the only way one can adequately express one’s faith in it, the only way one can hope to persuade others of its truth, is by abandoning the arsenal of rationality, and relying on the resources of the prophets.
Comment 16: Very good observation! Does the author have references here, other than prophets and poets? How about the Bhagavad Gita? It would hardly do justice to the Gita if one portrayed it merely as a poem or as a prophet’s inspired utterances. Yet, the Gita has been of great interest to mathematicians, particularly Andre Weil and Kurt Godel.
Of course, professional philosophers are in the business of formulating anti-metaphysical or meta-physical positions, decorticating them, defending them, and refuting them4. Mathematicians, though, may have another—at least a prior—duty in dealing with The Question. That is, to be meticulous participant/observers, faithful to the one aspect of The Question to which they have sole proprietary rights: their own imaginative experience. What, precisely, describes our inner experience when we (and here the we is you and me) grope for mathematical ideas? We should ask this question open-eyed, allowing for the possibility that whatever it is we experience may delude us into fabricating ideas about some larger framework, ideas that have no basis5.
Footnote 4: A very useful—and to my mind, fine—text that does exactly this type of lepidoptery is Mark Balaguer’s Platonism and Anti-Platonism in Mathematics, Oxford Univ. Press (1998).
Footnote 5: When I’m working I sometimes have the sense—possibly the illusion—of gazing on the bare platonic beauty of structure or of mathematical objects, and at other times I’m a happy Kantian, marveling at the generative power of the intuitions for setting what an Aristotelian might call the formal conditions of an object. And sometimes I seem to straddle these camps (and this represents no contradiction to me). I feel that the intensity of this experience, the vertiginous imaginings, the leaps of intuition, the breathlessness that results from “seeing” but where the sights are of entities abiding in some realm of ideas, and the passion of it all, is what makes mathematics so supremely important for me. Of course, the realm might be illusion. But the experience?
Comment 17: If mathematicians are going through truly poetic experiences in producing their discoveries, how come a majority of them are solely dependent on the university for employment? In contrast, many poets and writers had to endure poverty, social exclusion and uncertainty of employment. How different are the experiences of mathematical discoveries and poetry writing? Perhaps poetic sensibility only accounts for a part of the mathematical experience. The other part is simply ‘discovery as in any other science’ (ref: G H Hardy’s A Mahematician’s Apology).
This attitude has the curious effect of reducing some of the urgency of that staple of mathematical life: rigorous proof. Some mathematicians think of mathematical proof as the certificate guaranteeing trustworthiness of, and formulating the nature of, the building-blocks of the edifices that comprise our constructions. Without proof: no building-blocks, no edifice. Our step-by-step articulated arguments are the devices that some mathematicians feel are responsible for bringing into being the theories we work in.
Comment 18: Grothendieck, in his letter to Gerd Faltings, has written eloquently about the nature of proof and certainty. Also, Thurston’s Proof and Progress is relevant here. Perhaps the author is referring to the controversy surrounding the computer-based proof of the Four Color Theorem when he says ‘certificate guaranteeing trustworthiness’.
This can’t quite be so for the ardent Platonist, or at least it can’t be so in the same way that it might be for the non-Platonist. Mathematicians often wonder about—sometimes lament—the laxity of proof in the physics literature. But I believe this kind of lamentation is based on a misconception, namely the misunderstanding of the fundamental function of proof in physics. Proof has principally (as it should have, in physics) a rhetorical role: to convince others that your description holds together, that your model is a faithful re-production, and possibly to persuade yourself of that as well. It seems to me that, in the hands of a mathematician who is a determined Platonist, proof could very well serve primarily this kind of rhetorical function—making sure that the description is on track—and not (or at least: not necessarily) have the rigorous theory-building function it is often conceived as fulfilling.
Comment 19: Thurston, in his Proof and Progress article, explains how too much emphasis on a rigorous proof can often behave as a hindrance to progress in a given area of mathematics. Another point to note is that different areas in mathematics have required different standard of rigor in proofs.
My feeling, when I read a Platonist’s account of his or her view of mathematics, is that unless such issues regarding the nature of proof are addressed and conscientiously examined, I am getting a superficial account of the philosophical position, and I lose interest in what I am reading.
Comment 20: The author has correctly identified the nature of proof as one of the central issues of mathematical Platonism. Perhaps he is influenced by the fact that traditionally algebraic number theory has had higher standards of rigors for proofs than other areas. In any case, the author suggests that a conscientious examination of this issue, by the community of modern mathematicians, would provide an authentic account of the role of proof in mathematical discoveries. One feels that the author would be surprised to learn about a different aspect of the role of proofs. He should know that no account of the nature of proof would be complete without considering Brahminical scholarship. After all, Srinivasan Ramanujan, the mathematician rated by G H Hardy even above David Hilbert, came from the Brahminical tradition.
But the main task of the Platonist who wishes to persuade non-believers is to learn the trade, from prophets and lyrical poets, of how to communicate an experience that transcends the language available to describe it. If all you are going to do is to chant credos synonymous with “the mathematical forms are out there,”—which some proud essays about mathematical Platonism content themselves to do—well, that will not persuade.
Comment 21: Perhaps the traditional affinity of mathematicians for music is also relevant here.
- For the Anti-Platonists. Here there are many pitfalls. A common claim, which is meant to undermine Platonic leanings, is to introduce into the discussion the theme of mathematics as a human, and culturally dependent pursuit and to think that one is actually conversing about the topic at hand. Consider this, though: If the pursuit were writing a description of the Grand Canyon and if a Navajo, an Irishman, and a Zoroastrian were each to set about writing their descriptions, you can bet that these descriptions will be culturally-dependent, and even dependent upon the moods and education and the language of the three describers. But my having just recited all this relativism regarding the three descriptions does not undermine our firm faith in the existence of the Grand Canyon, their common focus. Similarly, one can be the most ethno-mathematically conscious mathematician on the globe, claiming that all our mathematical scribing is as contingent on ephemeral circumstances as this morning’s rain, and still one can be the most devout of mathematical Platonists.
Comment 22: How about claims based on anthropology that substantiate Platonism? (ref: Hindu mathematics).
Now this pitfall that I have just described is harmless. If I ever encounter this type of mathematics is a human activity argument when I read an essay purporting to defuse, or dispirit, mathematical Platonism I think to myself: human activity! what else could it be? I take this part of the essay as being irrelevant to The Question.
Comment 23: Here, is the author referring to social constructivism?
A second theme that seems to have captured the imagination of some anti-Platonists is recent neurophysiological work—a study of blood flow into specific sections of the brain—as if this gives an insider’s view of things6. Well, who knows? Neuro-anatomy and chemistry have been helpful in some discussions, and useless in others. To show this theme to be relevant would require a precisely argued explanation of exactly how blood flow patterns can refute, or substantiate, a Platonist—or any—disposition. A satisfying argument of that sort would be quite a marvel! But just slapping the words blood flow—as if it were a poker-hand—onto a page doesn’t really work.
Footnote 6: Like the old Woody Allen movie Everything you wanted to know about sex but were afraid to ask
Comment 24: Here, the author seems to be seriously underestimating the relevance of 20th century developments in biology, cognition and psychology to mathematics
Sometimes the mathematical anti-Platonist believes that headway is made by showing Platonism to be unsupportable by rational means, and that it is an incoherent position to take when formulated in a propositional vocabulary.
Comment 25: Here, the author answers to the Logicist’s perspective, famously pioneered by Bertrand Russell.
Sometimes the mathematical anti-Platonist believes that headway is made by showing Platonism to be unsupportable by rational means, and that it is an incoherent position to take when formulated in a propositional vocabulary.
Comment 26: In his rebuttal to the Logicist’s perspective, the author neglects the impact of computers in mathematics (not the usual Four Color Theorem stuff), but the role of computers in reformulating our whole scientific worldview (ref: the movie Matrix), and also the use of computers in Genomics and other biological sciences. In fact, Discovery Channel gives ample evidence that the biological sciences and geological sciences have been able to harness the power of the computer in extending the knowledge in these areas. In contrast, the mathematicians have not been able to do it (other than the P = NP problem).
So, when is there harm? It is when the essayist becomes a leveller. Often this happens when the author writes extremely well, super coherently, slowly withering away the Platonist position by—well—the brilliant subterfuge of making the whole discussion boring, until I, the reader, becomes convinced—albeit momentarily, within the framework of my reading the essay—that there is no “big deal” here: the mathematical enterprise is precisely like any other cultural construct, and there is a fallacy lurking in any claim that it is otherwise.
Comment 27: Perhaps the author is referring here to the book, “Where Mathematics Comes From: How the Embodied Mind brings Mathematics into Being”. In this extremely well-written book, the authors attempt to explain all of mathematics from a cognitive perspective. Then they go on to refer to mathematical Platonism as the ‘Romance of Mathematics’, which, in their opinion, is quite an unnecessary disposition, since they have shown that mathematics is a construction of the human mind. They also fault mathematical Platonism for its eliticism and social exclusiveness. One point I want to make here is that the ‘rational embodied’ viewpoint has many adherents among natural scientists and engineers since it is crucial for their world view and the progress they have made in recent centuries.
The Question is a non-question.
Comment 28: The Question is a non-question for a very different reason, as I have explained in my correspondence with Professor Dana Scott. The Question was the central question in mathematics a hundred years ago, in view of the pre-occupation of mathematicians and physicists with the shape of physical space. Today there are more important questions for mathematicians to ponder. For example, did mathematics progress or decline in the 20th century? Nearly every working mathematician would claim that mathematics progressed, highlighting the insularity, ignorance and the irrelevance of the modern mathematician.
But someone who is not in love won’t manage to definitely convince someone in love of the nonexistence of eros; so this mood never overtakes me long. Happily I soon snap out of it, and remember again the remarkable sense of independence—autonomy even—of mathematical concepts, and the transcendental quality, the uniqueness—and the passion—of doing mathematics. I resolve then that (Plato or anti-Plato) whatever I come to believe about The Question, my belief must thoroughly respect and not ignore all this.
Comment 29: This is a very romantic finish to what is definitely a very brave and honest attempt at self-examination. However, mathematical Platonism deserves an even better defense.
Today, mathematics remains one of the last bastions of Platonism. The working mathematician however would rather portray himself/herself as different personalities at different times—Platonist, Logicist, Formalist, Intuitionist, Social-constructivist, etc. Thus it is a cautious move on the author’s part to diplomatically take an equidistant position from the Platonist and the anti-Platonist. How about taking a unified perspective? Would the author know how to go about it?
Notes
I. Professor Barry Mazur's article appeared in the June 2008 issue of The Newsletter of the European Mathematical Society.
4 comments:
Damn, that was..well....er...now I need a drink. ;) - Tub.
Happy birthday BTW. - Tub again.
Thanks Tub.
Response from Professor Barry Mazur
Date: February 23, 2008
Dear Dr. Selvakumaran,
I want to thank you for your detailed commentary, and to assure you that I do not think that the comments are impertinent!(Impertinent in neither the first nor the second sense of the word as given in Webster's Dictionary.) My aim, in the little four and a half page article I wrote, is to gently nudge authors---writing either with a platonic bent or with any other bent, about these issues --to write with more focus. An editor of a journal had passed on to me a few manuscripts (about this) that he was contemplating publishing and asked me to write him my reaction. My little article was my reaction.
I do have a single comment that is a response (explicitly) to one of your comments, but also, more generically, it hints at a response to many of your comments. To explain it, let me begin by listing three activities--and these are all significantly different.
1. Doing Mathematics.
2. Thinking about the nature of mathematics.
3. Thinking about one's own felt experience of doing mathematics.
Now I do believe that when a mathematician engages in activity (2), he or she might do well to make use of his or her own felt experience (in doing mathematics) or reflections on this felt experience, i.e., in activity (3). BUT: (1), (2), and (3) are entirely differently activities. One's thoughts about (2) may in some subtle way affect some shade of one's way of doing activity (1) but that's a very subtle connection between the two activities.
So, when you write that there are
"significant restrictions to the nature and extent of the romantic 'felt experience' of the working mathematician. For example, if a mathematician claims that he/she has come to realize that the Riemann Hypothesis is true because of a metaphysical experience that he/she has been through, it would not be accepted that he/she has established the RH."
you are---as I'm sure you know---stating the obvious. Namely, that (1) and (3) are different, as indeed they are. Different, and---each of them---interesting, activities. But, and this one of my points, each are interesting for their own purposes.
Also, a tiny comment about your comment about Kant. You wrote:
"Although Immanuel Kant taught mathematics for many years, he is too weak a choice, as a mathematician, to represent the anti-Platonist position. In his Critique of Pure Reason, Kant proposed that Cartesian (Euclidean) geometry was a matter of innate knowledge for humans, and moreover the shape of physical space was Euclidean because the innate knowledge of humans was directly related to the nature of external reality."
First, I think that you vastly underestimate Kant's mathematical sensibility. Anyone who writes so cogently about orientation (in spherical geometry, no less) as he does in the Prolegomena---this type of thing was NOT in the literature of mathematics at his time---has a real gift for mathematics, and the antinomies in the first critique are extraordinary. Second, whether or not he was a good mathematician is irrelevant. He says things that teach me (and, I believe, everyone else) things. Third, if you think hard about the First Critique you will realize that Kant's position need not be construed as very pointedly anti-platonist---it mainly has a different emphasis, that's all. (Also your second sentence is in very funny vocabulary, if it is to reflect faithfully what goes on in the Critique of Pure Reason.) Do take another look at the Critique of Pure Reason and see if you don't agree.
May I convey, again, my thanks to you for reading with my article with such energy, and for your comments!
Sincerely,
Barry Mazur
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