What are proofs?
Most of my undergraduate math students seem to be intimidated
by the word "proof." I try to give them a reassuring and
informal explanation; I tell them that it just means
"try to convince someone that something is true."
That may also be enough when we mathematicians talk
to a non-mathematically-inclined scientist or to a
layperson. But of course, when we mathematicians talk
to one another, we want more.
If you contrast the Babylonian math with
Euclid, the Euclidean math attempts to put things into
more abstract and general terminology, so that it is
made *explicit* just what cases the method can be applied to.
The Babylonian approach is more along the lines of "here is
a method that works in a lot of examples, and it will
probably work in a lot more examples too, but we're not
going to specify in advance just precisely *which* examples
it applies to, since we don't have sufficiently abstract
language to make such a specification."
I think that what is new in Euclid is not the depth of insight
or the reliability of the reasoning, but the explicitness
with which the language handles generality. That explicitness
may become more important in some more advanced topics,
where one can no longer rely on pictures and other
intuitively obvious notions; one may need the
abstract language itself for one's reasoning.
The Axiom of Choice is not really true or false in any
concrete sense; it is just an artifact of our abstract language
-- a figment of our imagination, if you will.
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Eric Schechter http://www.math.vanderbilt.edu/~schectex/
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