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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)
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.
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)
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 w:cognitive science of mathematics ?
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."
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)
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)
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?
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.
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.
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)
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)
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)
Can someone add the missing bracket to the quote in the first section? Should it enclose "and human communities"? — Viriditas | Talk 01:41, 17 October 2006 (UTC)
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)
I think the following can be deleted:
"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 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:
Again, any objections? ntennis 03:59, 4 May 2007 (UTC)
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.
"alyosha" (talk) 05:51, 6 April 2009 (UTC)
Jbottoms76 ( talk) 15:22, 1 September 2009 (UTC)
It appears that the link now points to an active website. Closed Jbottoms76 ( talk) 23:01, 14 September 2012 (UTC)
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)
I've removed an old neutrality tag from this page that appears to have no active discussion per the instructions at Template:POV:
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)
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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 " and "" 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 citation needed.
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 citation needed).
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.
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)
The last few paragraphs under the "Human cognition and mathematics" section were recently deleted. User:Metalmikebot explained the deletion as follows:
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)
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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)
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.
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)
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 w:cognitive science of mathematics ?
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."
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)
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)
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?
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.
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.
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)
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)
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)
Can someone add the missing bracket to the quote in the first section? Should it enclose "and human communities"? — Viriditas | Talk 01:41, 17 October 2006 (UTC)
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)
I think the following can be deleted:
"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 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:
Again, any objections? ntennis 03:59, 4 May 2007 (UTC)
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.
"alyosha" (talk) 05:51, 6 April 2009 (UTC)
Jbottoms76 ( talk) 15:22, 1 September 2009 (UTC)
It appears that the link now points to an active website. Closed Jbottoms76 ( talk) 23:01, 14 September 2012 (UTC)
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)
I've removed an old neutrality tag from this page that appears to have no active discussion per the instructions at Template:POV:
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)
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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 " and "" 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 citation needed.
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 citation needed).
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.
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)
The last few paragraphs under the "Human cognition and mathematics" section were recently deleted. User:Metalmikebot explained the deletion as follows:
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)