[FOM] Advertising an approach to the philosophy of mathematics(with some logical technical work)
Feng Ye
yefeng at phil.pku.edu.cn
Mon Sep 10 06:27:07 EDT 2007
Thank you all for the comments (including several private comments not
posted here). I like to clarify a few points and raise a question for
logicians.
1. This research does not assume or rely on the claim that the universe is
finite and discrete. The point is merely that our scientific theories
accurately describe things within some finite range only (from the Planck
scale up to the currently recognized cosmological scale), and infinity and
continuity in our mathematical models of nature are approximations to things
within that finite range.
2. I am not suggesting replacing classical mathematics by an alternative
mathematics. The goal is to offer a logical explanation of how exactly
infinite mathematics is applicable to physical things within that finite
range.
3. Regarding naturalism, my view is that naturalism, if taken literally and
seriously, does seem to imply anti-Platonism, because we cannot make sense
of how a physical brain 'grasps' abstract mathematical entities 'out of the
brain'.
A more interesting question for logicians here is: Can we identify a piece
of our scientific knowledge about real physical things within that strictly
finite range, such that reaching the knowledge must indispensably rely on
accepting mathematical axioms beyond strict finitism?
A false example: design a computer program simulating theorem-proving within
ZFC and predict that it will not output '0=1'. Now, from the naturalistic
point of view, human belief about the consistency of ZFC is actually an
inductive belief. That is, after practicing set theory for a long time and
based on reflecting upon our own mental activities, we come to believe that
we will not derive paradoxes in the future. This does not rely on 'accepting
the axioms of ZFC', whatever that means. The case is similar, if one designs
a program to verify Fermat's Last Theorem and predict that the computer will
not output 'false'. The advanced axioms for deriving Fermat's Last Theorem
can be replaced by the assertion about the consistency of those axioms
(because Fermat's Last Theorem is $\Pi _{1}^{0}$).
Another false example: design a computer program computing a fast growing
function provably recursive only in some strong system and predict that it
will output a value for each input. Unfortunately, the prediction is likely
to be literally false even for very small inputs, because the values will be
too large for a real computer in this universe to handle.
In searching for a piece of knowledge about real things in this universe
that indispensably depends on accepting advanced axioms, one must keep in
mind that this universe is ridiculously small from the logician's point of
view (the ratio of the cosmological scale to the Planck scale is less than
$\10^{100}$). Even the function $x^{x^{x}}$ obtained by a single iteration
of the power function never appears in natural contexts of scientific
applications.
I do not know if we can find other true examples. Logicians here may suggest
something?
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