

An Aristotelian Realist Philosophy of Mathematics - rpenm
http://www.newcriterion.com/articles.cfm/The-sum-of-its-parts-7955

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onetimeusename
>The mathematician hunkered in a foxhole, earning his pay, finds it difficult
to set aside the prejudice that he is grappling with something real—to keep up
morale, if nothing else.

This is true of myself as much as I think Platonism leads to strange ideas
about things in other regards. There is also an idea known as logicism that I
think might explain a bit better what universal mathematical objects are.

I am not a mathematical philosopher myself, maybe some day, but when Franklin
says numbers can be relations to things, I think that the fact that there are
uncountable sets which means there is not a way to map the natural numbers in
any "relation" to that set seems like it undermines the Aristotelian idea of
linking mathematical objects with physical things.

~~~
NotAtWork
Disclaimer: my views on this are still being formed, and I don't necessarily
have good, concise explanations for some of the ideas the way I'd want.

> I think that the fact that there are uncountable sets which means there is
> not a way to map the natural numbers in any "relation" to that set seems
> like it undermines the Aristotelian idea of linking mathematical objects
> with physical things.

I try to think of infinite things as extending finite mathematics by replacing
a simple set with an equivalence class of pairs of (set, make_more_elms),
where set is a set and make_more_elms is a constructor to make set in to a set
with more of the "whole" set in it.

Then we can view acting on infinite sets (as long as we generate finite
results, which we by default always will) as interacting with these tuples
using finite math.

The fact that the naturals have a different cardinality than the reals can be
expressed by ({}, build_naturals) and ({}, build_reals) being in different
equivalence classes.

At no point do we have to deal with anything actually infinite to come to
these conclusions, we're just looking at objects that are both finite with a
pretend "infinity" constructed out of them.

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roywiggins
I'm not a philosopher, but this sounds awfully like embodied mathematics:
Mathematical ideas grow from embodied experience plus metaphor. Lakoff and
Nuñez's "Where Mathematics Comes From" is an extended effort at demonstrating
that this can be done in a convincing way.

