Talk:Where Mathematics Comes From

Terrible example
The following is actually a good example of the ignorance with which the book has been received in some quarters:

"It's difficult for me to conceive of a metaphor for a real number raised to a complex power, but if there is one, I'd sure like to see it." Joseph Auslander

Apparently Joseph Auslander did not read the book, because if he had he would have found a discussion of the conceptual metaphors involved in raising real numbers to an imaginary power at the very beginning of the section on Euler's equation. The whole point of that discussion is that raising a real number to a complex power is the metaphor.

I think this is a bad and misleading example of criticism of the book, and should be removed.

K.S. —Preceding unsigned comment added by 99.231.49.9 (talk) 01:25, 27 April 2009 (UTC)

A more charitable reading of Auslander would have him saying that he read the metaphor multiple times and carefully, and -- in his terms -- has difficulty conceiving of it. Metaphors, after all, aren't universal. Stacking them as high as WMCT does strains credulity of the resulting stack as a metaphor. Gene B. Chase 14:25, 15 March 2012 (UTC)

Random crap people couldn't bother to put a heading to themselves
''This page represents the latest discussion of Where Mathematics Comes From. For older discussion, ranging all over the map and including a discussion of quantum mechanics, see Talk:Where Mathematics Comes From/Archive.''
 * The archive should be mentioned at the top of this page. Otherwise the archive is difficult to find.  Tkuvho (talk) 16:14, 24 March 2010 (UTC)
 * Ok, I added an archive box at top. But I've never done this before and I'm not sure I understood all the instructions at Template:Archives. In particular, the archives were given a non-standard name, so the box cannot access them automatically. If someone knows how to do this better, please do. --seberle (talk) 22:01, 24 March 2010 (UTC)

I've renamed the article to the name of the Lakoff/Nunez book. The article "Cognitive science of mathematics" should be broader, covering all the findings of cognitive science related to math (and not just those identified in the book), and perhaps being less attached to these two authors' "embodied realism". --Ryguasu 06:11 Dec 27, 2002 (UTC)
 * all good moves

Now that structure of these articles is agreed on, and old talk gone, can we please discuss the book and the implications of the book and what can be said about it? If you look in the article history there was a great deal of material directly related to the book, including commentary on reviews etc.. This appears to have been deleted, contrary to wikipedia conventions, by people who evidently had not read the book nor understood its claims - perhaps bad writing was the issue - and perhaps some of that old text should be reviewed and re-incorporated by third parties? The book also has undergone some revisions and the authors have responded to criticisms. Does that response go here, or in cognitive science of mathematics ?


 * I think it would be highly appropriate to discuss criticisms about the book and responses thereto in this particular article. We can always move it somewhere else if that later turns out to be more appropriate. --Ryguasu

Am I missing something, or does the book not provide a way to conceptualize multiplication of a*b or a/b, when a and b are both non-integer, for any of the "4 Gs"? (The book provides ways to conceptualize many other simple operations, including these ones for integers.) If not, could somebody suggest a way to visualize such multiplications, or at least suggest why the standard procedure is reasonable? --Ryguasu 01:51 Feb 25, 2003 (UTC)

---

"This idea analysis is distinct from mathematics itself and cannot be performed by mathematicians not sufficiently trained in the cognitive sciences."


 * This is clearly a fantastic argument that beats all others: "my analysis is based on special techniques that you cannot understand, former experts in {insert subject here}".


 * Although there is certainly some room for the criticism that Lakoff, in particular, is too quick to presume himself one of the leading authorities on anything he's thought about, your paraphrase here is not particularly accurate. The argument is not that mathemeticians cannot understand the ideas in Where Mathematics Comes From; it's that a normal mathematical education doesn't teach anything like this. In particular, a normal mathematical education devotes approximately no time to considerations of A) what about the mind/brain allows it to do mathematics, or B) the philosophy of mathematics. These are precisely the topics that the book is about. Of course, the article could probably be clearer about all this. --Ryguasu 16:11 Mar 21, 2003 (UTC)

Well, your explanation sounds reasonable. How about replacing

"Idea analysis is distinct from mathematics and cannot be performed by mathematicians unless they are trained in cognitive science."

with

"A standard mathematical education does not develop such idea analysis techniques because it does not pursue considerations of A) what structures of the mind allow it to do mathematics, or B) the philosophy of mathematics."

Giving 'er the good ol' college try, now. —Preceding unsigned comment added by 70.68.103.27 (talk) 00:23, 23 September 2010 (UTC)
 * Ok, hang on. We agree that Lakoff and Núñez are perhaps a bit too forceful in their argument that mathematicians cannot understand cognitive science and that your restatement is probably more palatable. However, the goal here is not to be more palatable. I believe that the original statement you deleted is a more accurate statement of what Lakoff and Núñez presented in WMCF. Please remember that this article is supposed to be presenting the argument from WMCF (unless clearly reporting criticism), not our own views. --seberle (talk) 13:31, 23 September 2010 (UTC)

Check me on this: "If one accepts logicism in its only coherent form, one must reject the outright denial of Lakoff, even if one accepts the findings of his research." Discarding the premise and looking only at the conclusion ("one must reject the outright denial of Lakoff, even if one accepts the findings of his research"), let me reword this to the semantically equivalent "if one accepts the findings of his (Lakodff's) research, one must reject the outright denial of Lakoff." Substituting "deny" for "reject" and "accept" for "deny the (outright) denial," we get "if one accepts the findings of his (Lakodff's) research, one must accept Lakoff." That's certainly true, but I do not understand why a tautology should be included. I'm writing a science fiction story, and I am no expert on logic or mathematics, but tautologies hurt my head. Can someone fix or explain?" -- Kencomer 8:06AM 23-Apr-2004 (UTC)

Kencomer, I've just rewritten that sentence to make what I take the original author's intentions to be a little more clear. See if the revision doesn't help you out. -Ryguasu 17:06, 24 Apr 2004 (UTC)~

Is it really necessary for every section to be headed with a question?? On a different note, this article needs a little NPOVing and input from math people. I've read parts of the book (esp. the infamous Euler identity part) and the authors are presumptuous about the way mathematicians do their work. In fact, many of the processes they accuse mathematicians of being unaware of or sufficiently informed about, are actually processes quite common in everyday mathematical thought and teaching. (For example, analogies and applications to the sciences and the "real world".) The way the Euler identity part of the book reads is virtually identical to the way I would explain it to someone learning it, and very similar to how it's presented in the classroom (not just textbooks). The authors may be talented cognitive scientists, but they are poor sociologists and observers of human behaviour. They should spend several months or a year following a mathematician around, just watching, observing, they would learn a lot. Revolver 01:52, 12 Jul 2004 (UTC)Revolver 01:50, 12 Jul 2004 (UTC)

Is this article about a book or about where math comes from? If it is about a book and is a faithful summary of this book, I think that it is a tragedy that such a book was ever written. If the article is indeed intended as an account of "where math comes from", then there is quite a bit lacking.

The entire article seems to mistake the subject of mathematics for the subject of applied mathematics, which I maintain are separate subjects. Applied mathematicians are not actually mathematicians at all, they are simply scientists in their field of application. A pure mathematician is one who concerns himself solely with the validity of mathematical statements. Whether they realize it or not, most pure mathematicians are formalists.

A formalist beleives that all of mathematics is just a game. A branch of mathematics is simply a set of deduction rules (a formal logic) and a set of axioms stated in this logic. Mathematicians concern themselves with the "game" of deciding what can and can't be deduced from these axioms using these rules. The question of their interpretation, their validity in the "real world", and their application is of no concern to a mathematician. That is the job of applied mathematicians who, I maintain, are not truly mathematicians at all. The doctrine of formalism is arguably the most successful answer to the question of where math comes from.

The question which is the title of this article is never truly addressed in this article. The title should maybe be renamed "How can we jusify the application of mathematics?". I think that the article on the philosophy of math addresses the question of where math comes from much more thoroughly.

??
I don't understand how people can put forth things such as this in the Criticisms section: "WMCF does not explain where arithmetic comes from (if that is even possible or makes sense). Rather, it merely concluded that humans possess innate arithmetical ability." ?? I'd say it quite clearly follows, and should be obvious to any sensible person, that mathematics is but mathematical thought which directly maps to the physical world through the neural correlates of said thought--this is so trivial that I'm startled at the need to point it out! There is no contradiction in having mathematics both reflecting a structuring in the physical world and the subjective mental aspect, since mind is just an aspect of brain and part of that world. Thus saying "WMCF is entirely consistent with the Platonic philosophy which it rejects" is ludicrous since Platonism not only proposes mathematics is independent of mind, but that it is independent of the physical world, and that is a religious proposition and hardly has a place in a technical, rational discussion. ThVa (talk) 13:35, 21 July 2008 (UTC)

M.F.
Well, I know where this article comes from and someone should get a shovel. It reads like a bad review of the book or a personal essay, not an article in an encyclopedia. And who gives a fouc what the post-modernists have "developed" anyway?


 * Meanwhile, the postmodernists, most notably Michel Foucault, developed a deep critique of Western ethics, theology and philosophy, which focused on the absence of any model of the living and acting human body...

too much opinion
This article in general has too much opinion, e.g. René Descartes' "cogito ergo sum" seems to be under serious challenge. Wikipedia articles should be factual reports on what different sides in a controversy say.

The page is 'interesting', however not very encyclopic. As noted above, contains alot of opinion, and seems more like a book review.

This entry is seriously deficient
I own a copy of WMCF, and this entry does not do justice to the book at all. For starters, WMCF is a wonderful meditation on the cognitive origins of real analysis, complex numbers, the exponential function, and so on. There are also nice chapters on Boolean algebra, first order logic, and set theory, although the deepest passion of Lakoff and Nunez rests with analysis. Certain aspects of WMCF trace back to Lakoff's 1987 Women Fire and Dangerous Things. Nobody seems to notice that.

I think that Lakoff and Nunez have made a major contribution to our civilization's ongoing conversation about the philosophy of mathematics. The entry does not do justice to that conversation. For my part, I have long been suspicious of Platonism and the associated notion that mathematics is "discovered" rather than "invented." I agree with Lakoff when he writes "there is no way we can ever find out." No way, that is, until we interact with another technogically advanced civilization, which is unlikely to ever happen, if you agree with Barrow & Tipler, and Ward and Brownlee, that homo sapiens is the only technology manipulating species in our Galaxy.

I tell my students that mathematics is a vast "toolbox for the mind" and that mathematics, like all tools, was crafted by humans to serve human purposes. Euclidian geometry and number theory excepted, nearly all of our mathematics came into being after the start of the scientific revolution that began with Copernicus and Galileo, not before.

(I would except also plane and spherical trigonometry, probability, Gaussian elimination by Chinese to solve simultaneous linear equations, election theory, the mathematical logic of Ramon Llull, and so many other things that I can hardly count. In short, the Renaissance wasn't the beginning of an awful lot of things. -- Gene B. Chase 14:44, 15 March 2012 (UTC)) — Preceding unsigned comment added by GeneChase (talk • contribs)

It is very true that the education of all mathematicians does not prepare them at all to take on board claims like those of WMCF. And I can attest to the intense hostility nearly all mathematicians have for those sorts of claims. Nearly all working mathematicians are unwitting unreflective Platonists. Finally, it is incredibly true that nearly all mathematicians under 50 years of age take no interest in the philosophy of mathematics. No grants, no possible Fields medal, therefore worthless.

I believe that mathematics is the most successful human symbolic activity. This implies that understanding mathematics requires understanding the role of symbols in human communication, the subject matter of semiotics. This implies that the notation of mathematics is deserving of close scientific study. I doubt there is a single mathematician alive who thinks in this manner. A dead one who would have agreed was Charles Peirce.


 * Though this may be true, just in the same way that you praise the book, changing the article reflects your personal opinion about mathematics, mathematicians and the book, which, just as the article, may do no justice to mathematicians or mathematics. I suppose the best option is that if you feel strongly about this, you find sources that both agree and disagree with the book, as this would be the purpose of an encyclopedia. At the same time, someone could interpret your thoughts on mathematicians to be very opinionated such as "unwitting" and "unreflective". Any group can make such a claim about any other group and I think we can agree that it rarely holds. So even if understand your dismay towards this article, its problem is precisely that it reflects opinions whereas it should reflect a general consensus.GabKBel (talk) 23:20, 14 May 2010 (UTC)

Mathematics and Politics?
Does this article need the "Mathematics and Politics" section? The authors of the book make conscientious attempts to disassociate themselves and their theses from the excesses of postmodernism, and I think this article should preserve that sentiment.

In addition, I think the argument that "the failure of Principia Mathematica to ground arithmetic in set theory and formal logic" was a "failure" needs a link to Godel's incompleteness theorem or better citations. Was this stated or implied goal of Principia? Has Godel "plagued" philosophers of mathematics? On the contrary I think it took about 50+ years to digest Godel but mathematical philosophy is alive and kicking, "thanks" to Godel. (Witness Chaitin, Wolfram ...)

As someone with a deep (but not professional) interest in the history and philosophy of mathematics and someone who generally would describe himself as a Platonist, I found this book to be frustratingly thought-provoking and refreshing, and I respected its strongly worded and generally well-constructed arguments, who's points could be refuted or conceded line-by-line. Its positives were its style and tone, which were smart enough NOT to attach mathematics to politics or the Baghavad Ghita. Unfortunately it seems as if the leftist intellectuals have co-opted Lakhoff/ Nunez's bold yet constrained theses and hence somehow we get this Wikipedia article.--209.128.81.201 00:45, 22 March 2006 (UTC)


 * I like the substance and thrust of nearly everything you write above. It seems, however, that much of the material to which you (and I) objected strongly has been excised, to be replaced by -- nothing.
 * Principia is historically very important, but a substantive failure. Its notation is wallpaper, and its proofs turgid and pedantic, a long winded horror. A much better approach is that of Norm Megill's Metamath, although even his approach is long winded. One day someone will write a taught little book showing how to ground analysis, algebra, and geometry in first order logic (a logic which is simply than the way it is typically taught) and a minimum of set theory.


 * Thanks anon202. (I did the revert, and I'm 209 above.)  I agree that the knife was widely cut.  The article could be a little longer, with some mention on the critique of mathemetician's emphasis on the concept of closure (which I appreciated), the critique of mathematical induction, the overall importance of metaphor...  Can you think of a way to expand it?
 * I agree with you about Principia. But I didn't read WMCF as a critique of formalism or Hilbert's Program in general.  Godel & Turing & recently as mentioned above Chaitin & Wolfram & Robins & ... did it better.  WMCF seemed more a critique of "Platonism," or what Nunez and Lakhoff have described as the "Romance of Mathematics."  Thoughts? --M a s 23:33, 9 May 2006 (UTC)

Expand tag
The article is long enough as it is. I'm going to remove the tag unless someone can explain why I shouldn't. Gene Ward Smith 19:32, 12 May 2006 (UTC)


 * Go for it. Could you then desection some?  (It might look strange to have a section with only 2 sentences.)  Thanks!  --M a s 20:04, 12 May 2006 (UTC)

I did that and a lot more; the section on the response of the mathematical community was incorrect and seriously failed the NPOV test, and I've completely rewritten and expanded that section so it's not just an ad for the book any more. Gene Ward Smith 19:07, 13 May 2006 (UTC)


 * Good edits Mr. Smith. Thanks! --M a s 01:08, 15 May 2006 (UTC)


 * Now the article has turned back to crap. No wonder I'm giving up on Wikipedia.

Unclosed bracket
Can someone add the missing bracket to the quote in the first section? Should it enclose "and human communities"? &mdash;Viriditas | Talk 01:41, 17 October 2006 (UTC)

Mathematics and nature stimuli
Most of mathematics is analogious to easy navigation or to moving objects in a seen landscape. The pictorial form of thinking is humans' most efficient way of handling information. The capacity of humans who live in a nature environment is enermous: compare the number of technical kind of details in a nature landscape (lines, curves, shapes, structures, etc) to the number of them in a city landscape. My memory for mathematical things used to be much better when I had wandered in nature. Has there been any research on this?Htervola 10:30, 8 December 2006 (UTC)

The romance of mathematics
I think the following can be deleted:


 * It can be argued that only the first of the above quotes really describes Platonism. The second is rather vague, the third is obviously true (logic is a subset of mathematics), while the fourth is not necessarily associated with Platonism. For example, the Platonists Kurt Godel and Roger Penrose deny that artificial intelligence is possible.

"It can be argued" is classic weasel wording - who argues this? Secondly, the section above does not appear to be direct critique of platonism anyway, but rather a critique of several common romantic ideas about mathematics. Thirdly, the third statement is not "obviously true" to me, nor to the authors of the book who apparently "dismiss it as an intellectual myth"! Reasoning can be non-logical (eg. linguistic or emotional reasoning).

Any objections to removing this small paragraph? ntennis 03:36, 4 May 2007 (UTC)

P.S. The same anonymous editor made some |other additions to the article at the same time, which begin with "it has been pointed out that..." and "another criticism is..." but what follows looks like an original critique.

Further, there's a large section at the bottom of the article in the "summing up" section that looks like it needs to go:


 * Mathematics has grown into an extremely powerful toolbox for the mind, one whose potential applications extend well beyond those traditional bastions of mathematical application, science and technology. For example, logic and abstract algebra have much to offer to the social sciences and humanities. But communicating these riches to the wider community of nonmathematicians has proved difficult, and the problem is worsening. Even formal systems as basic as first order logic and axiomatic set theory are nowadays learned only by the more technical philosophy majors, and by a small fraction of mathematics students. Hence only a few specialists learn any mathematics beyond calculus, applied statistics, differential equations, and a bit of linear algebra. Just how many persons with a university education know what an equivalence class, partial order, or morphism are? What it means for a collection of axioms to have a model? If the cognitive approach to mathematics suggests improvements to the basic mathematical toolbox and better ways to communicate that toolbox to nonspecialists, it will move humanity closer to the fulfillment of Leibniz's great dream of a universal symbolistic.

Again, any objections? ntennis 03:59, 4 May 2007 (UTC)


 * OK, a week has passed so I removed the passsages. They are retained here on the talk page should anyone want to argue for their reinclusion. ntennis 01:20, 11 May 2007 (UTC)

material removed from Critical response
I have removed the text below, which is variously irrelevant, OR, unencyclopedic in style, ungrammatical, or doesn't reflect a coherent understanding of the text. Please feel free to hash this out in sandbox or talk, and be sure that it fits the book before posting in the article. And cite sources.


 * But this critique can also be challenged under the light of theory of multiple intellegences provided by Howard Gardner. So a person might posses good intellegence of certain types but not necessarily of other types. One such example is Munshi Premchand - the famous story writer and novelist in Hindi literature - who was weak in mathematics. Further, by the same logic, the claim - " ... it would be required to show that intelligence and mathematical ability are separable, and this has not been done" can also be said to be untenable, since here the author seems to presume that intellegence is of one universal nature.


 * From this point of view, whatever one's views on Platonism (right, wrong, meaningless), the 'invention' of mathematical concepts such as number would be impossible since they are hard-wired into our brains from the moment we are born. Also, the word "invention" insinuates that things could somehow be different, so that we could have invented a number theory where 1+1=3, or prime decomposition is false. However, any such number theory immediately falls apart in the face of simple reasoning. In reality, 1+1=3 is obviously false, and Euclid and the ancient Indians stumbled upon prime decomposition rather than inventing it. [But - It can be said that these critics have not enough grounding in the ideas of Lakoff to raise the relevant, valid criticism. Lakoff, Nunez can be interpreted that the experiences of the bodies we have with the external world has resulted in the emergence of unique 'affordance' ( J.J. Gibson's term) characteristic of the type of bodies we have. This has laid down the basis of emergence of mathematics we have. So it is merely regarding the origin of basis of 'ours type' of mathematics which the work of Lakoff, Nunez is refering.]


 * Another criticism is that WMCF does not explain where arithmetic comes from (if that is even possible or makes sense).[ Again, they have but simply the critic don't seem to have understood that]. Rather, it merely concluded that humans possess innate arithmetical ability. Some argue that WMCF is entirely consistent with the Platonic philosophy which it rejects. [ No way it seems tenable. Rather the central attack of Lakoff is to undermine Platonic view of mathematics]

"alyosha" (talk) 05:51, 6 April 2009 (UTC)

original research
I agree with Seberle's criticism that this article contains a considerable amount of original research and synthesis. Much of it is less than entirely sympathetic to the book under discussion. Tkuvho (talk) 15:04, 21 March 2010 (UTC)
 * It should not be too difficult to clean this article up, but I am unfamiliar with the criticism of this book beyond the single working link offered in the criticism section. Here is what needs to be done by someone knowledgeable of the criticism of the book:


 * There are two criticism sections that need to be combined. The first one appears to be entirely original research. (If it is, it can simply be deleted.)
 * Any criticism which is original research should be removed.
 * Any criticism which is explaining what critics have said needs to be referenced.
 * The two quotes need to have their links fixed. If there are no longer valid links, simple footnoted references would suffice.
 * There are a few good sentences which can be kept, such as the criticism of math error and the authors' response, which is properly linked. This sentence could be expanded to include some of the other responses included in that link, such as the authors' insistence that the embodiment aspect of their claims is being overlooked. It might also be good to reference the original Bonnie Gold review which the authors are responding to and mention the criticisms contained in that review.
 * As Tkuvho pointed out, everything should be stated as a neutral report on what critics have said and the authors have responded. Some of the current sentences are POV in that they are directly criticizing the authors rather than stating what critics have said. This is both POV and original research. We don't want to know Wikipedia editors' thoughts on this book; we want to know what the public reaction of experts has been. --seberle (talk) 16:24, 22 March 2010 (UTC)
 * I would like to raise the following additional issue. Jbottoms76 reports above that the link to the book site is, regrettably, a prohibited one.  Who prohibited it and why?  The link should certainly be included here.  Is it correct to assume that the site was prohibited by an overzealous wikipedian who happens to disagree with the thesis contained in the book?  Tkuvho (talk) 17:00, 22 March 2010 (UTC)
 * I am not sure what Jbottoms76 is referring to. The link actually no longer works. You can access an archived copy of the link from June 20, 2008, at http://web.archive.org/web/20071101135617/perso.unifr.ch/rafael.nunez/2ndprint.html. But I don't think it is proper to link to an archived copy. (Is that what he meant by "prohibited"?) I have not yet been able to locate the new site. Perhaps there is not one? In that case, this link should simply be removed. --seberle (talk) 20:06, 22 March 2010 (UTC)

removing POV tag with no active discussion per Template:POV
I've removed an old neutrality tag from this page that appears to have no active discussion per the instructions at Template:POV:
 * This template is not meant to be a permanent resident on any article. Remove this template whenever:
 * There is consensus on the talkpage or the NPOV Noticeboard that the issue has been resolved
 * It is not clear what the neutrality issue is, and no satisfactory explanation has been given
 * In the absence of any discussion, or if the discussion has become dormant.

Since there's no evidence of ongoing discussion, I'm removing the tag for now. If discussion is continuing and I've failed to see it, however, please feel free to restore the template and continue to address the issues. Thanks to everybody working on this one! -- Khazar2 (talk) 00:02, 30 June 2013 (UTC)

External links modified
Hello fellow Wikipedians,

I have just added archive links to 1 one external link on Where Mathematics Comes From. Please take a moment to review my edit. If necessary, add after the link to keep me from modifying it. Alternatively, you can add to keep me off the page altogether. I made the following changes:
 * Added archive https://web.archive.org/20060718161407/http://www.unifr.ch:80/perso/nunezr/warning.html to http://www.unifr.ch/perso/nunezr/warning.html

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Removed criticism
Well, it had been tagged as possibly (and is hard to see it otherwise) OR for about 7 years, so I went ahead and removed it. Here's what I cut out: In set theories such as Zermelo–Fraenkel one can indeed have {1,2} = (0,1), as these are two different symbols denoting the same object. The claim that there is an anomaly because these are "fully distinct concepts" is on the one hand not a clear scientific statement, and on the other hand, is on par with such statements as: ""The positive real solution of $$x^2=2$$" and "$$\sqrt{2}$$" cannot be equal because they are fully distinct concepts.".

The apparent anomaly stems from the fact that Lakoff and Núñez identify mathematical objects with their various particular realizations. There are several equivalent definitions of ordered pair, and most mathematicians do not identify the ordered pair with just one of these definitions (since this would be an arbitrary and artificial choice), but view the definitions as equivalent models or realizations of the same underlying object. The existence of several different but equivalent constructions of certain mathematical objects supports the platonistic view that the mathematical objects exist beyond their various linguistical, symbolical, or conceptual representations.

As an example, many mathematicians would favour a definition of ordered pair in terms of category theory where the object in question is defined in terms of a characteristic universal property and then shown to be unique up to isomorphism (this was recently mentioned in an article on mathematical platonism by David Mumford).

The above discussion is meant to explain that the most natural and fruitful approach in mathematics is to view a mathematical object as having potentially several different but equivalent realizations. On the other hand, the object is not identified with just one of these realizations. This suggests that the intuitionistic idea that mathematical objects exist only as specific mental constructions, or the idea of Lakoff and Núñez that mathematical objects exist only as particular instances of concepts/metaphors in our embodied brains, is an inadequate philosophical basis to account for the experience and de facto research methods of working mathematicians. Perhaps this is a reason why these ideas have been met with comparatively little interest by the mathematical community.

Non-existent Mac Lane
In the article it is written: "Lakoff and Núñez cite Saunders Mac Lane (the inventor, with Samuel Eilenberg, of category theory) in support of their position. Mathematics, Form and Function (1986), an overview of mathematics intended for philosophers, proposes that mathematical concepts are ultimately grounded in ordinary human activities, mostly interactions with the physical world."

I have searched the book by Lakoff and Nunez for this fact. However, Mac Lane is never mentioned (except of listing his book in a general bibliography, without any comment).

Zbisem (talk) 14:31, 7 November 2018 (UTC)

Criticism deletion
The last few paragraphs under the "Human cognition and mathematics" section were recently deleted. User:Metalmikebot explained the deletion as follows:
 * I deleted the last several paragraphs, which were attached as criticisms without relevant citations; these would be better suited to a separate section under "criticisms", in which criticisms were **cited** rather than written into the entry like an op-ed. They appeared to use original research and did not accurately represent the book.)

I do not have sufficient expertise on this book and its criticism to judge the quality of the deleted paragraphs. However, in apparent contradiction to the stated reasons for the deletion, the deleted paragraphs did have several citations. Also, there does exist a separate "Critical response" section where the paragraphs could have been moved to, in contradiction to the implication that such a section needs creating. I am restoring the deleted paragraphs until a better justification can be given for their removal. Perhaps there are genuine reasons for believing the references were not valid? Perhaps these paragraphs should be moved to the "Critical response" section? --seberle (talk) 15:28, 8 January 2022 (UTC)