### Afternoon Fishing

#### Posted by David Corfield

Fishing about at the Café for material to extract for $n$Lab, I was reminded of a question I had never got around to asking. I was also led to the third of the talks listed here (note caveat there concerning change of sentiments, and here concerning the second half), where Minhyong Kim writes:

For years, I’ve felt the need to deny the popular conception of mathematics that equates it with the study of numbers. It is only recently that I’m returning to a suspicion that mathematics is perhaps about numbers after all. It is said that in ancient Greece the comparison of large quantities was regarded as a very difficult problem. So it was debated by the best thinkers of the era whether there were more grains of sand on the beach or more leaves on the trees of the forest. Equipped now with systematic notation and fluency in the arithmetic of large integers, it is a straightforward (albeit tedious) matter for even a schoolchild to give an intelligent answer to such a question. At present, our understanding of the complex numbers is about as primitive as the understanding of large integers was in ancient Greece.

Earlier in the essay he writes

…in contrast to the continuum picture of the complex plane, a number theorist is more likely to perceive of each individual number or groups of numbers in a discrete fashion, and in nested hierarchies reflecting various complexities, and even attach a symmetry group to each individual number. It is not that number theorists avoid the plane model, since it is also an important tool in much of number theory. It is just that the plane has a much more grainy and elaborate shape, with many levels of microscopic detail and structure.

This neatly leads to the question I wanted to ask.

James shared with us his image of $Spec (\mathbb{Q})$:

I like to picture $Spec (\mathbb{Q})$ as something like a 2-manifold which has had all its points deleted. The extra complication is that what we think of as the points are actually very small circles. So it’s really a three manifold with all of the loops inside it deleted…Maybe some should be seen as bigger than others, corresponding to the fact that there are prime numbers of different magnitudes.

Now, when we were discussing that the fundamental group of $Spec(\mathbb{Q})$ is $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$, the issue of base points arose, which allowed Minhyong to tell us that

…for $\mathbb{Q}$, a choice $\mathbb{Q} \embedsin \bar{\mathbb{Q}}$ of an algebraic closure will correspond to a map $Spec(\bar{\mathbb{Q}}) \to Spec(\mathbb{Q})$ that can then be considered a base-point.

This was in response to Jim Dolan’s remark

…the fundamental group

oidof $Spec(\mathbb{Q})$ is (at least morally) the groupoid of algebraic closures of the field $\mathbb{Q}$.

So how should I square this image with James’ deleted loops?

Perhaps I don’t have long to wait for some useful insight since:

I’ve been helping James Dolan develop some new ways of thinking about algebraic geometry. We’ll release a preliminary version of a paper on that any day now.

Wait a minute, I thought the spectrum of a ring was already a groupoid for you guys. Could Jim have dropped ‘the fundamental group*oid* of ’ from his comment?

## Re: Afternoon Fishing

Cool stuff! We almost need a whole blog on three dimensional thinking about numbers - oh wait, Lieven Le Bruyn already has one!