FOM: Mathematical Intuition

[Joe Shipman]

Pruss:

<<The strong species of mathematician's intuition, however, makes
stronger assertions, such as: "It's now easy to show that p follows from

q", which mathematicians (I submit, based on personal experience, which
I
think is not idiosyncratic)  make without actually having gone through
the
proof that q entails p.  Obviously, outside of logic, almost no math
paper
includes every step of the proof.  The standard when refereeing is that
the reader should be capable of filling in the missing steps.  But I
suspect that in practice in many published papers there are steps that
neither the referee nor the author has filled in.  Nonetheless, both the

referee and the author have a strong mathematician's intuition that
these
steps can be filled in. >>

A very good observation.  There's a lot going on in mathematicians'
brains which may not be so easy to reduce to a completely rigorous proof
in written form, but which is in its own way reliable.  This is probably
because the mathematician has formal structures in his brain which he
does not explicitly understand but which represent reliable intuition
because of the way they have developed over the course of the
mathematician's life of studying proofs -- they arose via a gradual
pushing of mathematical reasoning-steps below the level of consciousness
as more mathematics was learned and the same types of arguments were
seen over and over.

In the same way all sorts of explicit information about sequencing of
neural events for skills like piano-playing or driving or dancing is
"compiled" by the brain when the same steps are practiced repeatedly, so
that the conscious mind (all of which is cerebral activity) only deals
with higher-level chunks representing groups of mental events.  The
individual low-level events are executed by the unconscious mind, much
of which is activity of the cerebellum rather than the cerebrum (people
with cerebellar damage have trouble with patterned sequential activities
like walking because they must make each step consciously--I wonder if
they also have trouble following sketchy proofs!).

-- Joe Shipman