At 12:07 PM 9/17/2014, Frederik wrote:
I think it follows from these observations [that MRI scans require
mathematics] that it is a preposterous claim to say that mathematics
is the study of neurological structures - or that mathematics could,
in any way, be reduced to neuropsychology.
HP: I agree that there are too many MRI papers, and that MRI images
by themselves explain nothing. But also logic by itself explains
nothing. I'll repeat my point that, like other primitive concepts,
understanding mathematics requires complementary models. What is
preposterous is to claim that any one view of the
<http://en.wikipedia.org/wiki/Foundations_of_mathematics>Foundations
of Mathematics is the only non-preposterous view. Preposterous views
in physics are common, but accepted only if supported by experimental
evidence. The evidence I know is that only human brains (and their
artifacts) actually do mathematics. I have no evidence that inanimate
nature does even simple mathematics.
Human math began with geometry and numbers. No one has discovered a
point or a triangle or a number, the infinite or the infinitesimal,
in Nature. From Zeno and Aristotle ("That which moves does not move
by counting") to Cantor and Dedekind, discreteness and continuity
have been impossible to combine in one logical model. Peirce spent
many years trying to describe precisely how points could form a
continuous line. Like Aristotle, he concluded that they couldn't.
We now believe(from MRI images) that concepts of discrete objects,
like points, and concepts of continuous motions and structures, like
lines, are formed in different regions of the brain, or at least by
different neural codes. It all began 400 million years ago when the
earliest sensorimotor controls based on vision required brains to
distinguish discrete objects from their continuous motions.
Howard
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