On 2/7/2017 2:16 AM, Telmo Menezes wrote:
Yes. The relation of mathematics to facts in the world is one of
description. That a dx/dt = -x has a decaying exponential as a solution is
not a fact about the world. As any engineer will tell you, it means that if
the differential equation is a good description of something about the world
then the decaying exponential will be a good description of something about
the world. The analogy with Godel's theorem is that if we create an AI
system to prove theorems, no matter how fast or long it runs it will not be
able to prove all true theorems.
And, importantly, it will not be able to prove its own consistency. If
we are machines ourselves, this also applies to us, right?
One cannot prove his own consistency as a formal axiomatic inference
machines - but very little of our reasoning is in that form. Even that
which is in that form may not rely on axioms of infinity. Finite
mathematics can certainly be consistent - every computer implements it.
And nothing precludes someone else from proving your consistency.
Brent
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