Hi Justin,
Nice meeting you. This question about mathematics also
interests me.
JUSTIN:
-As far as mathematics, Conceptualism states that we
invent math, not discover it- it is our own modelling
of reality in a way that we can intellectually
understand. Math seems to fit nature so well (as per a
Neo-Platonist claim) because we have designed it to do
so.
I've read about a school of thought called Formalism that
sounds similar to Conceptualism. The Formalists believed
that arithmetic could be built up from logic and set
theory into a formal system. Their goal was to prove this
system was consistent and complete. They also believed
mathematics was a kind of intellectual game, a pure
invention that didn't exist before humans hit the scene and
doesn't exist without us.
The opposing view was held by the Platonists, who said that
theorems of mathematics are truths that exist independently
of humans and are being discovered by us.
Now it's clear that mathematics was invented. We have systems
of geometry that start out with fundamental axioms that seem
self-evident to us based on our concepts of space. Axioms of
arithmetic are also self-evident but the concepts here
concern numbers, counting, and induction. The invention really
stops at this point. What's involved henceforth is deriving
theorems from the axioms, and new theorems from existing ones,
and while this certainly involves ingenuity of the highest
calibre, this is in retrospect merely following the rules of
the game.
Up to this point it doesn't sound like the Platonists have a
leg to stand on. But the game gets very interesting, in fact so
interesting that it defies all expectations you had at the start.
You could never have guessed, given Peano's 5 rather boring
axioms, that irrational numbers exist, that pi appears again and
again in the sum of so many different infinite series, or that
there's an infinitude of primes. But all of these can be proved
from this simple starting point.
It would still be considered no more than a fascinating game
except now we see that nature is following rules, and they are
mathematical. So when you say "Math seems to fit nature so well
because we have designed it to do so.", the answer is yes, but
no one could have anticipated the spectacular degree to which
this is so. For example, the ideas that guided the foundations
of math could never have predicted that gravity behaves according
to a mathematical law (an inverse square law), and further, no
mathematical inventions need be conjured up on the spur of the
moment to explain it. So I partly disagree with your premise. Math
is indeed an invention, but it is such a superb invention that its
results seem like, or may very well be, truths that have been
waiting for an eternity to be discovered.
Your analog to this statement is "Quality seems to fit nature so
well because Pirsig has designed it to do so." You are suggesting
that quality, like math, may be just an invention that works at
explaining reality, without being real itself. Well, Pythagoras
was so enchanted by numbers that he thought they were the primal
stuff of the universe. If you feel the same way about quality,
you could say the same of it. Pirsig himself recently argued that
quality is reality because of the harmony it produces. Presumably
he means by this that it solves or dissolves the SOM platypuses and
provides guidelines for solving practical moral problems. I'm not
convinced, but you have to be the judge of this yourself.
Glenn
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