Talk:Cognitive science of mathematics

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Please see Talk:Numerical cognition.

Others

since this page is now clearly about the research program and not about Lakoff and Nunez specifically, it's worth asking who else has done work on this? a few obvious suggestions:

w:Eugene Wigner, in his 'unreasonable effectiveness', raised questions that go beyond issues of w:cognitive bias and get directly into human-species-specific perspective. He asked the role of other species in our discovery of limits of our cognition. Note that his position on this is very different from that of Lakoff who says 'we cannot possibly know' how real math is to other species, which seems to be contrary to other cognitive science experimental programs, which tend to be quite heavy on ape/primate experiments. Primates are learning math now. What is relevance of this to the research program?

w:Rene Descartes had a specific cognitive view that is often confused with his philosophy of mathematics. Interestingly, he commented on the degree to which animal emotion could be said to be like human: not at all, "the screaming of an animal in pain is like the chiming of a clock". In denying the relevance of non-human non-linguistic experience, he's a bit like Lakoff. But like Lakoff he also felt it necessary to define metas (like 'w:cogito ergo sum') in order to justify his disinterest in pre-linguistic experience.

w:J. J. Gibson, the perceptual psychologist, thought ecology was the most basic science, since human senses were evolved to perceive it and interactions within it, and were thus only imperfectly applied to other problems, including inter-human social problems (which only make a difference to the individual once the ecology is mastered). This has parallels to the Lakoff/Nunez position, but inverts it to suggest that the most realistic perspective is that of the ecology acting on the evolving being and its cognition/perception.

There's a lot of work on w:mathematical practice in some schools of w:philosophy of mathematics. Also on w:quasi-empirical methods. To what degree is this going to intersect with research based on cognitive science methods? Is it totally disjoint 'social science'? Or is the rigorous cooperation of mathematicians a kind of w:collective intelligence?

These are badly stated, I guess, but let's talk about it, to establish what is the consensus on the actual scope of the field.

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Interesting questions are raised above. My strongest conviction about this page is that it should not become just a collection of replies to Lakoff and/or Nunez. What "cognitive science" means here should not be defined as "similar to Lakoff and Nunez's book", but more along the lines of "scientific results about mathematics". I think "cognitive science of mathematics" applies chiefly to empirical results/claims about mathematics, coming in particular from psychology, neuroscience, neural modelling, and cultural anthropology. (Example: psychologists have arguments that infants have an "innate mathematics", capable of dealing with numbers about three and under.) The more philosophical claims made by some cognitive scientists (Lakoff in particular) should probably find their main discussion in Philosophy of mathematics.

The stuff mentioned about Eugene Wigner, Rene Descartes, and J. J. Gibson should almost certainly not enter into this article, unless they it concerns mathematics in particular. This does not mean, however, that the material is inappropriate for inclusion in, say, George Lakoff or Where Mathematics Comes From.

--Ryguasu 01:58 Jan 11, 2003 (UTC)

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Okay, there's an ambiguity problem that may keep this article from ever working out. "Cognitive science of mathematics" can arguably refer to at least four different things:

1. The techniques of cognitive science, applied to mathematics. This applies not only to the techniques of Lakoff's specialty (cognitive linguistics), but also those of cognitive psychology, neuroscience, etc..
2. The scientific parts of the Lakoff/Nuñez book (as opposed to the philosophical stuff)
3. Anything related to the Lakoff/Nuñez book
4. Any attempt to establish some kind of "embodied" position in the philosophy of mathematics, through the use of science, as exemplified by the Lakoff/Nuñez book.

I prefer the first definition, but I think the cat is already out of the bag and that people will continue to use the other three at least as long as the Lakoff/Nuñez book is popular.

Perhaps the best thing is to make this page a disambiguation page. The first definition could have an article called, say, "scientific results pertaining to mathematics". The second and third can probably be covered adequately in Where Mathematics Comes From. I thought philosophy of mathematics would be an adequate place to discuss the fourth definition, but other people seem to disagree. I'm not sure what a #4 article might be called, however. "Embodied mathematics" is one option, but I think that name is ambiguous -- it could reasonably be about just science, or about science and philosophy together.

(Yes, I do think there is some arbitrariness in separating science and philosophy, but it may nonetheless make some sense to organize along these lines, since most everybody thinks in terms of them at least somewhat.)

--Ryguasu 01:49 Jan 24, 2003 (UTC)

Are there any works other than Lakoff/Nunez in #1? Do the people in #4 other than Lakoff/Nunez really use the term "Cognitive science of mathematics"? If the answer to both is no, then this article could simply redirect to the book, where the term was used for the first time as far as I know. AxelBoldt 23:06 Jan 24, 2003 (UTC)

As for your first question, I don't know if there are are popular works, but there's certainly research into what infants know about number, how people who never formally study mathematics develop some mathematical knowledge, if and how we can teach neural networks to do arithmetic, etc.. (The sort of research Lakoff/Nuñez describe towards the beginning of their book, and then some.) If you were feeling broad about the meaning of "cognitive science", you could also include theories about how much genes influence math. Thus if, as a native speaker of English, you think #1 sounds like a reasonable definition (I'd argue that before the Lakoff/Nuñez book it was the only reasonable one), there's certainly non-Lakovian stuff to place in an article with the current title.

As for the second, I'm not sure if there actually are any others. Supposing there are, however, I doubt they currently use the same term, although I bet copycats (especially, I imagine, sundry postmodernists) will soon be raising the "cognitive science of mathematics" (or perhaps "embodied mathematics") banner at every possible opportunity. --Ryguasu 23:31 Jan 24, 2003 (UTC)

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If y'all are interested, I linked to the sociology of knowledge and Michel Foucault articles because sociology of knowledge is all about what a cognitive view of mathematics represents: a shift in the way we view the world. Foucault wrote about this quite a bit, especially in Archaeology of Knowledge and The Order of Things. Of course, sociology of knowledge is not the same thing as a cognitive science of mathematics, but they are closely related. I included Platonism because, with respect to numbers, it represents a distinctly opposing view - numbers, mathematical relationships, and so on, actually exist in some sense. Especially with such a bare article on such a complex subject, defining things by linking to articles that are related (though of course not identical!) seems to me to be a good idea.

Seth Mahoney 19:02, 31 Oct 2003 (UTC)

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The comment about Euler's Identity is too strongly worded I think. WMCF uses Euler's Identity in a case study to emphasize the application of metaphor to the practice of mathematics, and it in itself is not central to the "argument that justifies." I'll try and reword. Thanks. --209.128.81.201 00:40, 13 April 2006 (UTC)

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Re Question number 1: I have independently started an entry, Numerical_Cognition which reviews/outlines (perhaps too pedagogically?) some of the work on the techniques of cogntive science applied to mathematics. The current version focuses more on numbers/numerosity, but this provides the foundations for mathematics, and I intend to expand this into higher-level mathematics. The numerical cognition entry draws on my current post-doctoral research with Stanislas Dehaene who summarized, and help direct the field with his 1997 book, The Number Sense (TNS)

Re Quesiton number 2: I agree with Ryguasu that there is really nothing that has been done on questions like the BMI aside from the Lakoff and Nunez's WMCF. Note, I took a 10-week PhD seminar with Rafael Nunez at UCSD, and we started with Dehaene's book. Nunez also had a list of non-WMCF readings, but most were just background articles. I'll look for them.

Re Question number 4: "numerical cognition" is the standard term for this field in the active research community, which is what I had searched for first, and only found this entry after having started my own. However, I feel like WMCF is part of the broader numerical cognition field (and as I note in my numcog entry, numerical cognition is a sub-discipline of cognitive science).

Perhaps we can combine these two entries, or in some way make them more clearly complimentary. My opinion is that WMCF starts where TNS leaves off, although Dehaene and Nunez do not agree on everything. Perhaps by situating WMCF in the broader research program of numcog, some of the debate over this particular book can be eliminated Edhubbard 23:15, 5 August 2006 (UTC)