I keep on going back and forth on the question whether the beauty of mathematics is something surprising and metaphysically significant. I find myself going between two views.
Deflation: Mathematical beauty is just a matter of selection. There are many beautiful theorems. But there are many, many more ugly theorems. It's just that the ugly theorems don't get published, unless they are of practical importance or are appropriately connected with beautiful mathematics. Imagine that we got a book of all the theorems of arithmetic. There would be many beautiful things in the book. But intuitively a large part of the book (if that makes sense to say: it's an infinite book after all!) will just be boring theorems like "18883 x 77891 = 1470815753" or "The equation x2+9873773873+8383883=0 has no solutions."
Theology: Mathematics seems to be have more in the way of surprising beauty than we would expect from the selection hypothesis. It happens not infrequently that as a working mathematician one writes down some obvious necessary conditions for something to happen, and then one proves—often in a highly nontrivial fashion—that these necessary conditions are also sufficient. Or maybe there is just a little bit to add, and then they become sufficient. Of course, often no such thing happens—we're just stuck with necessary conditions. But the number of times that the necessary conditions are also sufficient is surprisingly large, large enough to call out for an explanation.
And that need for explanation pulls me in one of two theological directions. First, there is Augustine's idea that mathematical objects are in the mind of God, and so we would expect to find beauty in them, since God is supremely beautiful. Second, one might have the thought that we are divinely designed, among many other things, for the kind of reasoning found in mathematics. Of course, one might also offer a naturalistic evolutionary explanation. But I am not sure that will be satisfactory: finding utterly exceptionless necessary and sufficient conditions is just not something that happens much in the practical life that our evolutionary development is driven by.
Hello Alexender, I love mathematics and work on the computer simulation of chemistry.
ReplyDeleteLike you I believe that mathematic belongs to a platonic realm of idea.
The problem is that it could also exist without God, and I don't see what the creation of the laws of non-contradiction could mean.
Friendly greetings from Europe.
Lothars Sohn – Lothar’s son
http://lotharlorraine.wordpress.com
That was a beautiful post.
ReplyDeleteLothar, what do you think of this:
1. Platonic Forms are ideas.
2. Ideas never exist apart from a mind.
3. Platonic Forms are necessary (obtain in all logically possible worlds).
4. Therefore, all logically possible worlds include at least one mind.
5. There are logically possible worlds in which no contingent mind exists.
6. Therefore, there must be at least one logically possible world in which one necessary mind exists.
7. If there is at least one logically possible world in which a necessary mind exists then there is at least one logically possible world in which a necessary being exists.
8. Therefore, there is at least one logically possible world in which a necessary being exists.
9. If there is at least one logically possible world in which a necessary being exists then a necessary being exists in all logically possible worlds.
10. Therefore, a necessary being exists in all logically possible worlds.
11. If a necessary being exists in all logically possible worlds, then a necessary being exists.
12. Therefore, a necessary being exists.
Lothar:
ReplyDeleteI certainly don't think God creates the laws of mathematics. Rather, they depend on his essential nature, just as all metaphysical necessity does. And as the other commenter said, I am inclined to think that if there is a Platonic realm, it is in the mind of God.
I have a thought, which I am not sure how to express, that the beautiful theorems (and scientific laws) are not scattered randomly among the ugly ones. Rather, part of what is surprising is that things start, at the foundation, with something very simple and elegant. There are a lot of ugly theorems but I conceive of them as so to speak side-effects of the beautiful parts of mathematics. That the trunk or core is beautiful is surprising.
ReplyDeleteBut I am not sure I can really make sense of those intuitions.
Heath:
ReplyDeleteVery helpful!
See my post on mathematical structures today.