## Rational numbers

In a previous post, I discussed the basic concept of number, beginning with natural numbers, or counting numbers. I am pursuing a notion here that certain kinds of numbers go together with certain kinds of operation. In the case of the natural numbers, the fundamental operation is counting. We begin with no thing and add one thing to it. The addition is equivalent to a count of one. Intellectually, if we have a set of, say, three items in front of us, and we are going to count them, we begin with nothing (“0”), then count “1”, then “2”, then...

Read More## Maxwell’s Demon and counting

I have already written about Maxwell’s demon in a previous post some time back. Before going on with this post, unless you know the demon well, you should read that previous post. He is a most interesting wee beasty. In the early days of the demon, the notion was that he could decide which molecules had greater or lesser energy, and with this knowledge, decrease the entropy of the system by increasing the temperature of one side of the container and decreasing that of the other. By doing this without any expenditure of work, he violates the second law of thermodynamics. This operation...

Read More## Numbers: sets or successors?

When I was in high school, I played football. My position was defensive tackle, it’s a rough game; and eventually, I wound up with torn cartilage in my left knee. I had surgery for that when I was sixteen, and recovered quickly. When I got to university a year later, I was a philosophy major. I was reasonably good at math & science, but I didn’t find it very challenging; so philosophy for me. Between my first and second year in that, I picked up a summer job washing cars. All the up & down movement, bending over, and et cetera, re-injured my knee. So, off to hospital for...

Read More## Big Sinhy Spheres in Space

OK, my title is really corny. Call it geek humour of the worst kind, if you want. But you’ll see where I’m coming from soon enough. In my last post, I was going over the three kinds of space that could be broadly consistent with a homogeneous distribution of matter; this leading to a criterion of uniform curvature. To review briefly, this yields either a universe of positive energy and negative curvature in the form of a 3-hyperbola or a universe of zero energy and zero curvature in the form of a Euclidean 3-space or a universe of negative total energy and positive curvature in...

Read More## Three kinds of space

Assume once more that space is homogeneous, at least at large scales; that is, for distances greater than (where , the Hubble distance is roughly 14 billion light-years). We know that below these scales, the distribution of ordinary matter appears to have a correlation dimension of around 1.77±0.04, implying a fractal structure similar to that of diffusion limited aggregation (DLA). But if the universe has a homogeneous distribution of matter at large scales, then we might think about it like, for example, the paved surface of a asphalt road. Stand back, and the surface is uniform; look...

Read More## What’s the matter?

In a previous post, I derived a simple form of the Friedmann equations using an argument based upon the total energy of a galaxy in a cosmological field of matter of uniform density. In that derivation, I made the working assumption that the total energy was . That was a restrictive assumption that I want to lift now. Let me write that equation for the energy down again: where I am continuing with the same notation that I set up in that previous post: is the mass of some test galaxy under consideration, is the total mass within a sphere of radius from the origin of coordinates, is...

Read More