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When the infinite is simpler *February 18, 2008*

*Posted by dcorfield in Uncategorized.*

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I’d like to open a discussion of cases in mathematics where one resorts to the infinite just because things work out simpler there. A classic case of this is the free ring, , the integers. Any ring is somehow more ‘confused’, a condition has been applied to identify elements.

Now, consider the spaces , the set of straight lines through the origin in , or projective real -space. Each is covered by its corresponding -sphere. The fibre over a point has two points — forming a bundle. But none is quite as simple as it could be. We need to embed each in the next higher space and take a suitable limit to form . The bundle over it with total space the infinite dimensional sphere is ‘universal’ among all -bundles.

So, often the infinite shows up in some universal object. John Baez uses the term walking such and suches to cover many of these.

Some cases I’ve come across in algebraic geometry:

First and foremost, it’s MUCH easier to work over an algebraically closed field (which must be infinite) than a finite field, though this might not be so much in the vein of what you’re talking about.

Probably a bit more on topic, there are quite a few proofs in algebraic geometry that require a field of “sufficiently high transcendence degree” which often means not just infinite, but in fact uncountably many transcendental elements. For instance, a couple of quick proofs of Hilbert’s Nullstellensatz either require that you have infinite transcendence degree over the prime field or that the field itself is in fact uncountable.

Indeed, I would say that the whole point of infinity is to simplify things. I think that the most interesting mathematical problems are fundamentally finite. The use of infinite constructions is to “mod out by” complications that we deem inessential.

[…] January 12, 2009 Posted by dcorfield in Uncategorized. trackback Continuing our discussion of free entities, from Fiore and Leinster’s A simple description of Thompson’s group F we […]