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Andler (2012) wrote:

What role does mathematics play in cognitive science today, what role should mathematics play in cognitive science tomorrow? The cautious short answers are: to the factual question, a rather modest role, except in peripheral areas to the normative question, a far greater role, as the peripherys place is reevaluated and as both cognitive science and mathematics grow. This paper aims at providing more detailed, perhaps more contentious answers.

References

  • Does Andler's (2012) article on 'Mathematics in Cognitive Science' provide an accurate picture of mathematics in cognitive science?
  • How do I prepare for the kind of advanced work described in the article?

Reference:

Andler, D. (2012). Mathematics in Cognitive Science. In Probabilities, Laws, and Structures (pp. 363-377). Springer Netherlands PDF

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can you please provide a link, url, DOI or some more details about the paper? ty –  user3543 Dec 6 '13 at 2:43
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edited question to what is presumably the intended article –  Jeromy Anglim Dec 6 '13 at 4:19
    
I apologize that I missed on sharing a link to the paper. This is the correct paper though. Thanks. :) –  Zeph Dec 6 '13 at 8:01
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There's another article on the future of mathematics in psychology - ncbi.nlm.nih.gov/pmc/articles/PMC2651093 if it is of interest. –  Zeph Dec 6 '13 at 8:30
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2 Answers

Thoughts on the paper

The paper appears to provide a high level overview of the role of mathematics in cognitive science. I'm not a sufficient expert in the overall field of cognitive science where I'd feel comfortable to truly judge the accuracy of the overall synthesis that Andler (2012) provides. That said, much of the paper is about providing examples of how mathematics integrates with cognitive science. And the examples seem reasonable. I could think of other examples that pertain to my work, but their absence does not really detract from the paper.

Andler also makes a number of distinctions about how mathematics can integrate with cognitive science. For example there is the kind of integration that statistics has with many experimental disciplines which is both fundamental and not very specific.

So in short, I think the paper provides a thought provoking big picture overview of the issues of interfacing maths with cognitive science.

Implications for you doing advanced work

This paper could potentially be motivating, but it might also be discouraging. It's so high level that it creates a vision, but the vision is so large that it might be overwhelming.

To do ground breaking research in cognitive science (or any area for that matter), you need to specialise. If you want to do research interfacing maths with cognitive science you would also specialise. The result is that you would only need a small subset of mathematics and cognitive science mentioned in the article. I use mathematics and statistics a lot in my research, and there's plenty of mathematics mentioned in that paper which I know little about.

If you are wanting to do research in this area, exposing yourself to a good undergraduate curriculum in mathematics, statistics, computing, and cognitive science would be a good start along with exposure to research. Then pursue a PhD with an appropriate advisor where you can hone your skills in a particular domain.

In terms of mathematics my own bias would favour learning calculus, linear algebra, probability, and statistics, but that's just my bias. It's also useful to learn how to code.

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This is great advice. Thanks! I really enjoy mathematics, but my research interests are in the domain of cognitive science. I have training in engineering, and I understand that I will have to gain wide exposure to mathematics to even progress little taking in Andler's high level vision. I want to be in a field where I can learn and be exposed to a variety of mathematics over a period of time. I might specialize in a sub-domain, but being able to learn about the interactions elsewhere would be worth looking forward to. :) –  Zeph Dec 6 '13 at 8:04
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There's another article on the future of mathematics in psychology - ncbi.nlm.nih.gov/pmc/articles/PMC2651093 if it is of interest. –  Zeph Dec 6 '13 at 8:29
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Thanks for sharing the article. I read the paper and what I take from it is a rather pessimistic view. He suggests that there is a crucial need for overarching proper mathematical modeling, but he makes it sound this is also a huge obsticle and we must wait (longer than a young persons academic career) to see the fruits of it.

I'm coming from a theoretical physics background, so I can't quite judge if the applications of math in the present he lists are exhausting. But as Jeromy Anglim said in his answer, it makes sense. Now if you ask "How do I prepare for the kind of advanced work described in the article?" the answer must depend on what your position in cognitive science actually is. You prepare yourself by leaning the math. You can't learn all the math. If you don't say what field interests you, learn the math that's most fascinating to you and try to interpret the cognitive science information your lean in these terms. This will also make you a happier person.

The article makes the crucial point that statistical methods for treating the data are to be distinguished from the math you need for the modeling of the subject matter, i.e. the math for the framework in which the data are interpreted themself. Obviously, if you analyse data, you need statistics and this is tied to calculus and linear algebra and you must master these - and for this you only need the right books, or good lectures. I can say that if you're not work with motorics, you'll probably not need differential geometry and Lie groups, although they are beautiful concepts. If you know you must code anyway, then try to look more into the fundamentals of computation - this also amounts to learn much of the logic which you can further apply to represent more adventurous concepts into which the field might lead you.

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