Mathematics and the mind reflect each other. 🙂
The question: How much will the study of mind mimic the foundations of math and find use of math?
The fields of computational neuroscience and cognitive science both use quite a bit of mathematics.
The answer hangs on what is meant by the word “mimic” in the question. One way to interpret the question is this: Will ideas about cognition mirror ideas about abstract mathematical concepts? From one perspective you could say this is necessarily the case: since human minds produce mathematics, the structures of mathematics must conform to the constraints of the structures of human thought.
Going in the other direction, mathematics is a key tool for understanding the natural world, and so if we study the mind/brain using methodologies derived from physics, chemistry and engineering, we will no doubt find ourselves using mathematical terminology, analogies, and concepts to describe mental processes.
This blog post I wrote gives an overview of the theoretical and mathematical approaches to the mind/brain: The Pentagon of Neuroscience — An Infographic/Listicle for Understanding the Neuroculture
And this book may be of interest in exploring the psychological roots of mathematical concepts: Where Mathematics Comes From
Based on an answer I wrote on Quora.
Image: Drawing Hands by MC Escher.
Neurologism 
Wouldn’t mathematics be a subset of human cognition? We still don’t know what consciousness is and it might take us centuries more to understand it. And it might require a new kind of science to describe consciousness and this new science might be beyond anything we have right now – including mathematics. I feel as we understand the human mind/brain more, we will probably use the mind/brain concepts to model the physical world rather than the other way around. As you said, mathematics is a product of the human mind but we don’t know what the mind itself is. So once we know what that is, why use a lower form of science to model the world when we can model the world using the concepts of the mind?!
Not sure exactly what you mean. Science uses mathematics as its language, so the two are hard to separate. And I don’t see why there should be “lower” and “higher” forms of science. There is a great chain of being, and there are different lenses with which you can look at each link in the chain.
In any case I suspect we can never escape a metaphysical debate: if you choose to take mental experience as “ontologically prior” to physical existence then math and science become subsets of psychology. If you go with the traditional western materialism, matter and space are ontologically prior to subjective experience, so psychology becomes a subset of physical sciences.
But whichever ontology you choose, you will find mathematics useful to express ideas in other domains.
The General Theory of Relativity is a better science than Newton’s theory of gravitation. So in the future we might/will come up with a science of the mind which is better than the one we have right now. Thats what I mean by lower and higher forms of science. Some sciences are so primitive that we no longer consider them as science in the general sense because they deviate so much from our current sciences – http://en.wikipedia.org/wiki/List_of_creation_myths
To support my second claim, here is an excerpt from the book – Incognito: The Secret Lives of the Brain
So the brain has its own way to model the world. So we might be able to better model the world by using the brain’s logic to represent the world thereby improving the current sciences we have.
Strictly speaking mathematics is not a science though. Only a subset of mathematical concepts deal with natural processes.
Are you trying to say that an understanding of human intuition will help us formalize more concepts?
The analogy with Newton and Einstein only goes so far. Both Newton and Einstein were studying the same topic: gravity. Einstein’s physics was not a different form of science, but an improvement within a particular from of science — physics. They operate in essentially the same domain.
But mathematics and psychology study things that are not really in the same domain, unless your ontological position is that mental experience is “fundamental”. In that case everything from geology to economics is a branch of psychology or phenomenology. But I don’t see how this would help make progress in these fields.
Think of particle physicists. They are ever-closer to a “theory of everything”. But this theory has almost nothing to say about human-scale phenomena in biology, psychology and sociology. I would imagine that the same would hold if we had a better theory of mind. We would know more about “fundamental” mental entities, but that might not help us make progress in other areas.
Saying that one type of science is more fundamental, or “higher” than another is a form of reductionism. There are some powerful arguments against reductionism made by physicists:
PW Anderson’s classic paper “More is Different” talks this: http://robotics.cs.tamu.edu/dshell/cs689/papers/anderson72more_is_different.pdf
Nobel Laureate Robert Laughlin has a very readable pop sci book that reflects the condensed matter physicists’ approach: A Different Universe: Reinventing Physics from the Bottom Down. He has way of looking at cosmology from a solid state physics perspective. http://www.physicscentral.com/explore/writers/laughlin.cfm
I also recommend his PNAS paper with David Pines called The Theory of Everything. http://www.pnas.org/content/97/1/28.full
Are you trying to say that an understanding of human intuition will help us formalize more concepts?
Yes, not only that but I am saying that they will eventually supplant the current theories because they are not good enough. Here is another example from the above book,
Now if everybody on this planet had the above condition, then would our laws of motion be similar to what we have today? I trying to say that our brains impose a limit and we can overcome that limit by decoding the brains and applying that knowledge to other sciences. For example, quantum mechanics are now being used in chemistry and computer science concepts are also being used in computational physics and chemistry.
Think about it, we have the theory of relativity now and we still use Newtonian laws because even though they are less accurate we can do with less accuracy for most of our day to day work. But if we had enough data and computational power then I am guessing that we would eventually use relativity instead of Newtonian laws in all case (though this is debatable).
I am not qualified in quantum mechanics but here is a video that talks about “visible quantum objects” – http://www.ted.com/talks/aaron_o_connell_making_sense_of_a_visible_quantum_object
So if we discover tomorrow that there is a Theory of Everything, then why wouldn’t we use that to explain everything? 🙂 I mean, why use Newtonian physics when we have the Theory of Everything?
And I will definitely read the material you have provided in the above comment. I love this stuff! 🙂