They’re the same thing! Why doesn’t anyone ever say so?
Assume A and B are types; the continuation passing transform takes a function (here I’m using C++ notation)
B f(A a);
and produces a function
<X> CPT_f(<X> (*k)(B), A a) {
return k(f(a));
}
where X is any type. In CPT_f, instead of returning the value f(a) directly, [...]
There are several good introductions to category theory, each written for a different audience. However, I have never seen one aimed at someone trained as a programmer rather than as a computer scientist or as a mathematician. There are programming languages that have been designed with category theory in mind, such as Haskell, [...]
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 [...]
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); [...]
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, [...]
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 [...]
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 . [...]
We say a topological space has topological dimension if every covering of has a refinement in which every point of occurs in at most sets in , and is the smallest such.
Any point is clearly in at least one set; it’s the other sets that [...]
I was thinking about some stuff involving fractals and non-positive-real dimension. It’s still a very rough idea, though.
There’s the concept of topological dimension, which is necessarily an integer. It looks like it’s typically the floor of the Minkowski dimension.
One way of talking about iterated function systems is to consider patterns of digits in -ary [...]
David H. Bailey is an LDS mathematician, perhaps best known for his work on the “spigot algorithm” for pi. He has written rather extensively on the relationship between science and LDS theology.
