Mathematical constructs are not inventions, but discoveries. This is why a mathematical construct can not be patented. You can't patent a method for finding the circumference of a circle because the relationships are not something you construct. It just is. You can discover it, but you can't invent it.
My understanding of the "running on a computer" type of argument is that building a machine to do something is an invention -- even if that machine relies on mathematical facts. In fact, all machines rely on physics, and chemistry, etc that can be modelled with these mathematical facts.
Building a machine that relies on mathematical facts and executes an algorithm, does not in itself mean that the machine is not an invention. It does not in itself mean that the invention is not of the same class of things that patents were meant to work with.
My personal favourite example to think about is PKI type applications. The fact that one way hashes are hard to factor is a discovery, not an invention. The actual algorithms we use in PKI, no matter how much creativity was used in discovering them, are still simply discoveries. We don't create math, we explore it. However, is a machine that uses this math to create a signature that is hard to fake an invention? Is it any less an invention than a machine that uses the laws of physics (which we also don't invent) to fly?
Please note that I don't think software patents should exist. However, there are many people who disagree. The first step in getting them to change their mind is to understand why they think the way they do. It's easy to oversimplify the situation and to just imagine that it is bad will and greed that drives your opposition. However, if you don't deal with their reasonable positions, you won't make any progress. The bad guy/good guy analogy feels good, but the world is unfortunately a lot more nuanced than that.
With software, the machine is not merely "relying" on physics which in turn "rely" on math. The functionality is purely mathematics. This is very very different from a patent on a machine.