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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, \mathbb{Z}, the integers. Any ring \mathbb{Z}/n \mathbb{Z} is somehow more ‘confused’, a condition has been applied to identify elements.

Now, consider the spaces RP(n), the set of straight lines through the origin in \mathbb{R}^{(n + 1)}, or projective real n-space. Each is covered by its corresponding n-sphere. The fibre over a point has two points — forming a \mathbb{Z}_2 bundle. But none is quite as simple as it could be. We need to embed each RP(n) in the next higher space and take a suitable limit to form RP(\infty). The bundle over it with total space the infinite dimensional sphere is ‘universal’ among all \mathbb{Z}_2-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.

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1. Charles - February 18, 2008

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.

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2. James - February 20, 2008

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.

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