[PREV - EXCURSIONS_IN_THEORY] [TOP]
FOUND_MATH
September 1, 2020
The very name of the subject "the Foundations of Mathematics"
has an assumption built into it. It employs a metaphor,
suggesting that mathematics is like a large building, a
structure built-up on top of the fundamental elements which
are "the foundations".
At the outset, we don't *know* what those fundamental elements
are supposed to be, though the structure that's supposedly built
on them is familiar enough. It's as though we've found a lost
citadel out in the desert, and we're conducting investigations to
see what what it actually rests on.
This strikes me as a very 19th Century
understanding of the intellectual universe:
we explore to reveal pieces of the truth (A naive, limited
about a fixed tree of knowledge whose understanding of
structure we investigate and record. understanding?)
I think that's the kind of vision
Bertrand Russell started out chasing,
trying "to believe that there can be
such a thing as knowledge".
SUCH_A_THING_AS_KNOWLEDGE
THE_DOCTRINE_OF_POSTULATES
So, how do you choose your "starting points", the axioms or postulates
you're going to reason from to put all of math on a "solid foundation"?
The game is to try to come up with a small
number of things that seem obviously true, In which case it could be
which perhaps just means things that no one that our choice of
is likely to want to dispute. "foundations" is limited by
our intuition. Our perception
of what seems obvious
might turn out to just be a
quirk of cultural biases.
After all, how is it that
there can be *disagreement*
about the foundations of
mathematics if they're also
What we're after in our postulates: supposed to be self-evident?
(1) a small, minimal set
(2) of obviously true things
(3) that get us to I list these seperately
where we want to go because I think that they're
in tension with each other.
To start with point (3): myself, I think
that this blows the game, right here.
SIMON_ARTIFICIAL
In effect, these "postulates" we're
trying to contrive are not the
*real* foundations, the real
starting points are elsewhere
further up the structure: we know Russell and Whitehead-- following
something about that citidel Frege-- tried to take it back to
already, so the foundations we're the logical operations "or" and
speculating about have to be "not".
something the citidel can plausibly
rest upon. But in "The Philosophy of Logical
Atomism", Russell comments
approvingly on a system developed
by Sheffer & Nicod in which "and"
and "or" can both be derived from
a *single* operation, a test for
"incompatibility".
NOT_INCOMPATIBLE
Starting with just *one* thing is
"a good deal simpler" than
starting with *two*, see? That's
*waaay* better, right?
Damned if I can see why...
This brings us to point (1),
the drive to minimize the
number of postulates.
This alone strikes me as pretty nutty:
programmers call this "playing PLAYING_GOLF
golf", trying to achieve a task
using the smallest amount of code
possible.
I can see why, say, six postulates
would be better than 600, but for the MATHISM
life of me I can't fathom why you'd
prefer six to a dozen...
There's a drive toward a certain
conception of "elegance" here that EPISTEMS
strikes me as yet another kind of
madness.
And there's still point (2), the idea
that axioms should be things that just
seem true on the face of it---
Russell and Whitehead's attempt at rooting
everything in basic logical operations is a good
example: but there are problems with it that arise
very quickly-- notably Russell's paradox.
Russell tried various ways of dodging that
paradox over the years before just putting the
problem aside. His fixes were clunky and
unintuitive.
Currently, mathematician's have tended to place
their faith in ZFC, Zermelo-Franknel plus the
axiom of choice: the eight (or so-- I forget)
postulates of ZFC are hardly as transparently
obvious as just "and" and "not"... the need
to keep from running aground of paradoxes
requires tricking out the underlying theories
far too much to stick with point (2) very well.
It's very common for working
mathematicians to describe themselves You can make a case for Platonism--
as "Platonists": they don't feel like it's interesting that math invented as
they're inventing new intellectual a curiosity sometimes find practical
constructs, they feel like they're application much later-- but I suspect
discovering something that's *out Mathematicians of being a little
there* already. intellectually lazy on this point--
they take refuge in faith in Platonism
Like a lot of metaphysical to avoid confronting the "hole at the
issues there's a bottom of math".
pinheads-dancing-on-angels
quality to this one, and I MATH_HOLE
suspect it leaves most
sensible people going "but
who cares?"
I think a better way of looking at this is to ask
the question "is there only one way of doing math?"
Is it all a single, self-consistent logical edifice
or might there be an alternative conception of math
that might work as well-- or perhaps work even
better in some way, such as being useful for
particular purposes that conventional math doesn't
address.
There are directional metaphors we regularly
deploy that have problems buried in them: Heh: "buried".
Metaphors for dealing
The idea of higher realms built on lower with metaphors.
foundations is one, but there's also the idea
of "moving forward" from starting points to
destinations.
To me it seems clear that there's a certain equivalency in
different perspectives: at least some types of math can
be translated into each other, at least to some extent, e.g.:
o boolean logic
o set theory
o graph theory
o arithmetic
It could be that there really *is* one single True
Math, but instead of stacking up building blocks or
piecing together a hierarchy we should be thinking
about parallel implementations reflecting different
styles of thinking without worrying about which
one should be regarded as the "foundation".
But maybe that doesn't go far enough...
let me throw out an odd piece of wonky
speculation:
We've long since gotten used to a physics
with different domain dependent theories
(or "models") that have so far defied attempts
at incorporating them into one over-arching
understanding...
What if there's a similar situation lurking in
mathematics that we simply haven't noticed yet?
A mathematical "multiverse" undreamnt of by our
platonists...
--------
[NEXT - SUCH_A_THING_AS_KNOWLEDGE]