The top set is 0.95, the bottom set is 1.05, and 1 is the flat set in the middle. I’ve inserted a small sphere to mark the origin.

It’s apparent here that the sets for converge rapidly on as they approach the origin. As in the previous images, I’m projecting down to 3 dimensions by dropping the imaginary component of .

]]>So, approaches 1 as we approach the origin along any such line.

The only way to get values other than 1 is to make exponentially smaller than , and approach the origin along a curve such as:

We then have:

So the level set for any complex number comes arbitrarily close to the origin in , but all these sets for approach the surface very rapidly as they approach the origin.

]]>It’s also true that the empty set is an n-manifold for each n if we say “an n-manifold is a topological space such that each point has a neighborhood homeomorphic to ”

There’s also a category of manifolds where different components are allowed to have different dimensions, and this is a kind of rig category, with finite products and coproducts. In this category, the question “what is the dimension of this manifold?” doesn’t make sense; the right question is “what is the dimension of this component of a manifold?”, and the empty manifold has no components.

]]>So the point is that it’s really just a matter of being mindful of the surrounding context — I don’t think there’s any controversy about that point for practicing mathematicians.

]]>Whereas you use the latter to define what 0^0 must be. So I find that point in your derivation unconvincing, a sleight of hand.

Let me try a different approach to the same point.

There are all kinds of ways that we *could* define , but there is only one definition of it that doesn’t call for special exceptions to be added to the general catalog of statements of mathematical facts.

Anders gave a good list of these facts, in the first comment to this post.

One of the clearest there is this: how many functions are there from a set of size A to a set of size B? Clearly, it is .

In the article, I pointed towards another such fact: the multiplicative inverse of is .

Now, if is defined as anything other than 1, we would have to modify the statements above, to say: the number of functions is , except when A and B are empty, then it is 1. And the multiplicative inverse of is , except when a and x are both zero.

So, if nothing else other than sheer laziness, why don’t we go with the most “globally parsimonious” definition?

I also think that these are hints about what the gods of mathematics prefer :)

]]>The determinant of a square n x n matrix is obtained as the sum, with appropriate signs, of n factorial products, each of which is composed of n factors. If we let n be zero, this is zero factorial terms each composed of zero factors – that is to say one term composed of no factors and therefore possessing the value one. Hence the sum must be one.

]]>SNARKs can be generically constructed from probabilistically checkable proofs and Merkle trees, but I’ve heard that early versions of the PCP theorem had O(Bekenstein entropy of the galaxy) constant factors. Having trouble finding a citation for that right now though.

]]>