[FOM] What produces certainty in mathematical proofs?
Feng Ye
yefeng at phil.pku.edu.cn
Fri Sep 14 05:30:10 EDT 2007
Perhaps we should first make explicit about certainty for what. Three
aspects seem most prominent:
(1) Certainty of mathematical theorems as literal truths about
mind-independent abstract mathematical entities;
(2) Certainty regarding the consistency of an axiomatic system;
(3) Certainty of the scientific assertions about physical things drawn with
the help of mathematics.
(1) is obviously very tricky. On what ground can we decide certainty, or
evaluate a judgment regarding certainty? What proofs or evidences can one
offer for a judgment regarding certainty?
On the other side, our belief about the consistency of a formal system can
be interpreted as a very realistic belief about the outcome of our own
mental activities in doing mathematical proofs. It is then a belief about
concrete things in this universe. Different opinions among mathematicians
regarding the consistency of a system are similar to chess plays' different
opinions regarding the possible outcomes of a chess game. Intuitive
evidences supporting or against such a belief can be drawn from experiences
in constructing proofs or entertaining relevant mathematical concepts. In
the end, time will tell who is right, or time will silence the skeptics
about consistency. (Time won't silence anti-Platonists, for their point is
that Platonism is conceptually meaningless.)
As for the certainty in (3), it seems that a very critical point is
frequently ignored. The scope of this physical universe to which we really
apply mathematics is strictly finite, from the Planck scale (10^{-35}m) up
to the currently recognized cosmological scale. Infinity and continuity in
our mathematical models of nature are only for glossing over details that we
have no idea about or that we want to ignore by the models. It means that
realistic applications can never be logically completely rigorous, because
they always involve such approximations, and then absolute certainty of pure
mathematics in itself does not imply certainty of the results of
applications. On the other side, this observation also leads to the thought
that perhaps a strictly finitistic axiomatic system is already in principle
sufficient for the applications to physical things within that finite range,
from the Planck scale up to the cosmological scale (which is ridiculously
small if one considers those fast growing functions provably recursive only
in strong systems). Some technical work supports this thought. (It develops
some advanced applied mathematics in a fragment of the quantifier-free PRA.
See my message 'Advertising an approach to the philosophy of mathematics' a
week ago for more details.) If that is correct, then practical certainty in
(3) for our ordinary applications of mathematics in sciences is actually not
affected even if someday ZFC (or even PA) is discovered to be inconsistent.
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