Existence proofs

Someone brought up the point that what we were doing by finding areas in the brain that respond to certain stimuli was providing 'existence proofs' that had the potential to provide constraints on future studies. I had to read up a bit to examine that claim:

http://www.mathpath.org/proof/proof.methods.htm
http://www.ams.sunysb.edu/~yildirim/handouts/proof-meth.pdf

Link 1 talks about what an existence statement is, and what the ways to prove it are. Link 2 talks about valid and invalid methods of proof in general.

An existence proof for P(x) is of the form 'there exists an x such that P(x) is true'. For example, we may want to prove 'if a is real, there exists real x that is a solution to 5x+3 = a'. A constructive proof would say, since a is real and (a-3)/5 is also real, there exists a real x such that P(x).

Two important things: the space is defined and is made known to you. If the space is undefined/not known to you, or worse, you assume a space, the solution becomes trivial because the problem is unconstrained - you can find a space such that P(x).

The brain presents a similar under-constrained problem. For example, you can ask 'show neurons such that these neurons process faces'. But you are imposing your framework by presupposing faces are represented at the level of single neurons, and you don't know the brain's framework.

The solution therefore cannot be constructive. The only other method of proof is non-constructive - to show that 'face cells' must exist as a consequence of previously known theorems. This necessarily requires a Theory of Vision - my personal bias is that it should be a mechanistic, bottom-up theory.

Proof by example is just plain wrong. For example, if I propose 'for any n, 3n is even', and show for n=4, 3n = 12, I am clearly wrong (for n=7, 3n=21 which is odd). Finding some face cells does not mean you've provided an 'existence proof' for face cells in a rigorous sense.

The point is, even an existence proof requires much more thought, and is ultimately much richer than a simple description. So in a sense, descriptions themselves are underconstrained, and may not be very useful as a prologue to more sophisticated questions. Of course some descriptions are really good and are probably true, but in a rigorous sense, we need a theory.

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