## When the infinite is simpler February 18, 2008

Posted by dcorfield in Uncategorized.

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.

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.

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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