Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

tchow_at_lsa.umich.edu
Date: 01/14/05 

Date: 14 Jan 2005 03:59:01 GMT

In article <cs6ep3$lso$1@ra.nrl.navy.mil>,
Ralph Hartley <hartley@aic.nrl.navy.mil> wrote:
>But you are presumably a mathematician, and when mathematicians make
>unqualified statements, with no other context, they usually *mean* "In ZFC
>...".

As a matter of sociological fact, this is definitely false. For a trivial
counterexample, consider:

  (*) "sqrt(2) is irrational" is provable in ZFC.

When mathematicians assert (*), they mean (*). They don't mean

  (**) `"sqrt(2) is irrational" is provable in ZFC' is provable in ZFC.

They couldn't, obviously, because this would instantly lead to an infinite
regress. Even you seem comfortable with asserting (*) flat out; that is,
it isn't a white lie or a shorthand for something like (**)---it is
meaningful on its own. So we have at least one class of examples of
mathematical statements that are meaningful on their own. Mathematicians
typically include "sqrt(2) is irrational" and "every differentiable function
is continuous" and so forth among the mathematical statements that are
meaningful on their own, and that are *true* in an absolute sense, just
as (*) is true in an absolute sense.

Now maybe they are wrong to do so, but that's how they are.

There's another way to see that ZFC doesn't, in practice, have the status
that you assign it. Suppose that someone were to find a contradiction
in ZFC. Would this make any difference to mathematics? It would
depend on the specific contradiction, but in general, it wouldn't make
any difference. Logicians would just pick some other foundation for
mathematics with no known contradiction. ZFC is way too strong for most
of ordinary mathematics anyway. All the theorems in the books would
remain intact, except for the few that were affected by the specific
contradiction. What would you say then? That in this new situation,
"sqrt(2) is irrational" no longer means "`sqrt(2) is irrational' is
provable in ZFC" but now means "`sqrt(2) is irrational' is provable in X,"
where X is the new foundation? Given that the usual proof of sqrt(2)'s
irrationality is left unchanged by the discovered contradiction in ZFC,
it's a little bizarre to think that its meaning has changed. It is surely
more plausible that "sqrt(2) is irrational" meant, and still would mean,
"sqrt(2) is irrational" and not "`sqrt(2) is irrational' is provable
in something-or-other." And that's the way most mathematicians view it.
The familiar proof that sqrt(2) is irrational consists of a sequence of
meaningful statements that we can read and understand and that leads us
to accept the truth of "sqrt(2) is irrational"; mimicking this proof
formally in this or that formal system does not yield the "true meaning"
of the statements in question. In fact, it's almost the other way around;
we only accept (as a candidate for foundations) formal systems that
faithfully mimic what we *already* recognize to be correct reasoning.
Where do you think ZFC came from in the first place?

This is not to say that your point of view, which is roughly some kind of
finitism or formalism, is untenable. But it involves a whole host of
assumptions, many of which don't agree with how mathematicians actually
work with and view mathematical statements. 

-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences

Relevant Pages

  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... >> But you are presumably a mathematician, and when mathematicians make ... The most obvious exception is most statements about ZFC. ... Given that the usual proof of sqrt's irrationality is ... > bizarre to think that its meaning has changed. ...
    (comp.theory)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... >> But you are presumably a mathematician, and when mathematicians make ... The most obvious exception is most statements about ZFC. ... Given that the usual proof of sqrt's irrationality is ... > bizarre to think that its meaning has changed. ...
    (sci.math)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... >> But you are presumably a mathematician, and when mathematicians make ... The most obvious exception is most statements about ZFC. ... Given that the usual proof of sqrt's irrationality is ... > bizarre to think that its meaning has changed. ...
    (sci.logic)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... When mathematicians assert, they mean. ... There's another way to see that ZFC doesn't, in practice, have the status ... Suppose that someone were to find a contradiction ... irrationality is left unchanged by the discovered contradiction in ZFC, ...
    (sci.logic)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... When mathematicians assert, they mean. ... There's another way to see that ZFC doesn't, in practice, have the status ... Suppose that someone were to find a contradiction ... irrationality is left unchanged by the discovered contradiction in ZFC, ...
    (sci.math)