Talk:Where Mathematics Comes From

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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)

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 w: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)

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.

Contents

[edit] 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...

[edit] 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.

[edit] 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.

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.

[edit] 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. It's positives were it's style and tone, which was 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)

[edit] 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.

[edit] Unclosed bracket

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)


[edit] 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)

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