From: Scott Roberts (jse885@cox.net)
Date: Fri Jul 29 2005 - 00:24:04 BST
Sam,
> Scott said:
> Well, I don't think you have made that point about mathematics.
<snip>
Sam said:
My point is that the perception of the truth of, eg 2+2 = 4 is not
independent of human character, eg honesty. I think we are too familiar with
2+2=4 and the like for the possibility of an alternative to be readily
understandable. But consider Fermat's last theorem - there were a number of
'false starts' where people thought it had been solved, before Wiles
actually did so (and even now people think that there must be a 'better' (ie
simpler and more aesthetically pleasing) proof to be found). I think what
seems to be 'independence' is simply a deeply embedded social level of
agreement.
Scott:
Still not convinced. Because they are fallible, mathematicians need peer
review, that I don't deny. Further, there can be fashions in mathematics, in
the sense of people working in one area rather than another, but that does
not invalidate the unfashionable areas.
In the case of 2+2=4, there is no alternative, if one accepts the Peano
axioms of arithmetic. Compare it to moving a knight in chess. One can move a
knight, say, four squares diagonally, but if you do so, you are no longer
playing chess. Similarly, if you say 2+2=5 you are no longer playing
arithmetic, which does not necessarily imply that you aren't playing
something else interesting. The case of Fermat's Last Theorem is the same,
except there is a likelier possibility that Wiles' proof contains an error.
Suppose 50 years from now, someone discovers such an error. What that means
is that anything built on FLT has the same unproven status as the FLT. It
does not mean that the validity of the FLT depends on a social level of
agreement. Consider the Axiom of Choice (that from a set of non-empty sets
one can assume the existence of a set containing one element from each of
the sets, without specifying the elements.) This cannot be proven from the
usual axioms of set theory, but it turns out that a lot of interesting
mathematics can be built up by assuming the Axiom of Choice as well as the
other axioms. So there is a lot of mathematics of the form "if the Axiom of
Choice, then ...". Since showing an error in the proof of FLT does not imply
that the FLT is false (just unproven), all that stuff built on the FLT in
the next fifty years just becomes "if FLT, then ...", and could still be
interesting mathematics.
I also want to comment on your emphasis on honesty. As I see it, in this
context, honesty is just the lack of dishonesty. By this I mean that
intellect is hurt by dishonesty (hidden social and biological agendas,
self-deceptions, unrecognized assumptions, and so on). Hence increasing
honesty is a matter of decreasing dishonesty, which is done through
intellect.
- Scott
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