2007 September 19 – 2:36 pm
How many one-element sets are there? Well, given any set we can construct the one-element set so the collection of one-element sets has to be a proper class, a mindbogglingly enormous collection far larger than any mere set could be. However, they’re all the same from the point of view [...]
2007 September 15 – 3:44 pm
The last example in the previous post said that the collection of all algebraic gadgets of a given kind and structure-preserving maps between them forms a category. The example given was the category of rings. It’s also true that a category itself is an algebraic gadget with structure (the ability to compose morphisms); [...]
2007 September 12 – 1:35 pm
The last example in the previous post was the monoid consisting of all functions from a set to itself under composition. We could multiply the elements (i.e. compose them) in any order because the source and the target were the same, the set .
For arbitrary sets, we still know how to compose, [...]
2007 September 12 – 9:19 am
A set has no structure. It’s just a collection of things, all of them equally unimportant.
Figure 1. A set.
Figure 2. Another set, the one-element set we’ll call “1.”
A function, or map, “goes between” sets. It has a source set (also called the domain) and a target set (also called the range). To [...]
2007 September 6 – 2:09 pm
I was trying to understand Wick rotation by applying it in the case of a finite-dimensional Hilbert space, and came up with something strange. The way I’ve worked it out, it seems to map classical observables to quantum states! I’ve never heard anything like that before.
Say we have an -qubit Hilbert space . [...]
