An Aristotelian Realist Philosophy of Mathematics(newcriterion.com)
newcriterion.com
An Aristotelian Realist Philosophy of Mathematics
http://www.newcriterion.com/articles.cfm/The-sum-of-its-parts-7955
4 comments
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.
> 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.
Here's a maybe useful excerpt from the Stanford Encyclopedia of Philosophy, "In short, where platonism is an explicitly philosophical view, working realism is first and foremost a view within mathematics itself about the correct methodology of this discipline. Platonism and working realism are therefore distinct views."
I wonder if it's 'working realism,' as a methodological preference, that's really prevalent in mathematics. It seems like, if nothing else, the idea that some mathematical structure actually exists independently of one's thoughts on it would help to motivate a prolonged search. Maybe it's often transformed into a more full blown Platonism as a way of reconciling its methodological efficacy with one's other beliefs.
I wonder if it's 'working realism,' as a methodological preference, that's really prevalent in mathematics. It seems like, if nothing else, the idea that some mathematical structure actually exists independently of one's thoughts on it would help to motivate a prolonged search. Maybe it's often transformed into a more full blown Platonism as a way of reconciling its methodological efficacy with one's other beliefs.
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.
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.