Hey Patrick,
PATRICK: A friend of mine has argued that mathematics is one big
tautology. You seem to be saying the same. It's a very relevant point, I
think. But a tautology normally exists of two assumptions explaining one
another. In mathematics there are whole 'buildings' of assumptions and
theorems that all relate in some exact way to each other. You can
calcute Pi with different algorithms to the infinite, but the
assumptions of both algorithms lead to different theorems. Ah. I can't
in just a few words explain to you what is floating intuitively in my
teacup. In my opinion the tautological notion of mathematics has to do
with Goedel's Argument, but it's just an intuition. Maybe someday I'll
try to formulate this intuition in an essay for the MoQ site, if anyone
is interested (and if I decide to stick around here, which right now I
do plan to!) Just one thing still I'd like to mention is, if mathematics
is just saying A=A, why then does it take so long to develop new
mathematics? And another point, why does mathematics work so well in
describing reality?
Yes, that's what I'm saying. Math is, IMHO, basically, tautological.
You've really hit my 'math comprehension wall' here, so I'm not sure how
much more I can contribute. But as for your last two questions, I'll hazard
some guesses... (1) It takes a long time to develop new math because, while
in the end it can all break down to 'A=A', what you usually wind up with is
a situation where each"A" is a highly complex equation (if it were really
all as easy as 'A=A', then I've got some explaining to do about my undergrad
math grades). It's easy to spot 'A=A' when it's written that plainly, but
when it 'disguised' as a long equation or the like, it will naturally take
longer. (2) Sure Math works well in describing reality... arguably, it is
the most accurate description of reality we have... but if you'll follow
this thread back, we can say that math offers us no guarantee that it will
always be a high quality description of reality. Perhaps someday a higher-Q
description will emerge with an accuracy and descriptive power that is so
much better, the Math of old will relegated to the status of phrenology....
(not likely, but logically possible, and that's enough for the point we
started on here).
thanks for your contribution,
rick
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