> > What is your evidence for expecting inconsistencies in our mathematical
> > systems?
>
> My only evidence is that I've been persuaded by the logicians that we have
> no way of telling whether or not our systems (provided they're
> sufficiently interesting) are consistent, unless we actually *find* an
> inconsistency (e.g. a certain algorithm fails, but a certain proof says it
> won't).
>
> All I'm saying is that a certain type of person might feel, because
> of the incompleteness theorems, that proofs are not very reassuring. One
> is reassured by proofs only if they secretly assume that mathematics is
> consistent, i.e. that you won't get contradictory statements as theorems.
>
> So, in my opinion, the question of weather a proof is more reassuring than
> a large number of successful cases depends upon where you come from and
> what you happen to find reassuring. From an objective standpoint, there's
> no good reason that either one should convince anyone of anything.
>
I disagree, though I'm admittedly not an expert on this.
(I seem to have an opinion on everything today.) We don't have
any perfect assurances in the situations you've mentioned, but
I think that, when we understand well enough what is going on,
then some assurances are STRONGER than others. Neither dollars
nor rubles are 100% reliable, but this year I'd rather have dollars.
Let me first put aside the question of mistakes. I admit, we could make
mistakes, and we sometimes do, so nothing is absolutely certain.
Moreover, complicated arguments may be more prone to error than simple
arguments, unless the complicated arguments are carried out by very
careful mathematicians and checked by very careful referees. But modern
mathematical arguments have the advantage that they can be (and
sometimes are) formulated very very precisely, so that they *can* be
checked for errors. In contrast, arguments in most other disciplines
cannot be formulated quite so precisely. Pure math has the advantage
that it is essentially a finite, closed system -- a mathematical theorem
has no hypotheses that we do not know about, no ingredients that were
too small for us to detect.
Now, I will admit that Godel's Incompleteness Theorems were
discouraging news for those of us who long for certainty.
For instance, we will never be able to prove the consistency
of ZF set theory. However, we still have the benefits of
relative consistency arguments. For instance, Godel showed that
if ZF is consistent, then ZF+AC is consistent,
and Cohen showed that
if ZF is consistent, then ZF + not-AC is consistent.
There are many other results like these in the literature -- i.e.,
"relative consistency" results. I think they're quite reassuring,
for these reasons:
ZF itself has stood the test of time. For nearly a century, many
brilliant mathematicians have studied ZF, fully aware that if they could
find an inconsistency, they would instantly become famous (at least,
among mathematicians). Thus, there is now strong *empirical evidence*
to believe that ZF is consistent. It's not a certainty, but in my mind
it is pretty close. And then any other consequences (such as the
consistency of ZF+AC) inherit the same degree of empirical consistency.
To make an analogy: We can't be certain that the laws of physics will
always work the way that they have worked so far. Perhaps God has built
the universe so that the laws of physics will suddenly change on January
1, 2000, and gravity will no longer work right, and we'll all float
away. But that seems unlikely (to me), and so we have strong reason to
believe that we will all stay anchored on the earth in the next
millenium in much the same fashion as now. Now, imagine that we have
some sort of device that functions properly if gravity is working
normally, and imagine (this part is a little far-fetched) that we have
somehow done a "proof" in mechanical engineering, so that we are somehow
ABSOLUTELY CERTAIN that the device will function properly as long as
gravity works normally. Then our certainty about the device inherits
the near-100% certainty we have about the reliability of gravity.
Of course, you can't really do those absolute certainty proofs in
mechanical engineering; there's always the possibility that we've
overlooked some particle that was so small that we couldn't detect it
without a superaccelerator that we're still building. But you can do
absolute certainty proofs in math.
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Eric Schechter *** http://www.math.vanderbilt.edu/~schectex/
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