I started working on what I suppose you could call Foundations of Mathematics when I was in Sweden back in 2019, but I’m reluctant to call it that, because I didn’t produce any meaningful axiomatic systems, or results, though it looks like I did solve the Continuum Hypothesis (but I didn’t realize it at the time). Because I was producing an incredible amount of useful work in Physics and Artificial Intelligence, I gave up my work on Foundations of Mathematics, picking it up here and there.

One problem that’s always bothered me is the affect Mathematics has on the observable world, that is plainly beyond Physics. Specifically, all combinatorial properties of a system (e.g., counting or connective properties) are always true, with certainty, for a given system, at all moments in time. For example, if a system has 5 components, then there are 5 choose 2 = 10 ways of selecting any pair of components from the system. This is never subject to serious doubt, and what’s even more strange, is that there’s no process that causes the Theorems of Combinatorics to hold. In more formal terms, the belief in the veracity of the Theorems of Combinatorics is not empirical, and its machinations (for lack of a better word), appear to be atemporal, and eternal, in the most real sense, that they do not begin or end, and instead simply apply at all times, at all points in space.

Observations of this sort in my opinion take what was previously Philosophy, and turn it into something that is close to Mathematics and Science, and I think I have my first real result in what could be an entirely new way of thinking about reality and Mathematics. Specifically, I believe I’ve constructed a credible argument that (i) Mathematics exists in 3-space, i.e., what we would call the observable world, and (ii) Mathematics does not have a location.

The argument is as follows:

Assumption 1: If a system does not exist, then that system cannot have an effect on the observable world.

Now assume that Mathematics does not exist. We just showed above that Mathematics certainly has an admittedly peculiar, but immutable effect on the observable world, and in fact, on all systems in the observable world. Therefore, mathematics exists.

I don’t want to over formalize prong (ii), but I think it’s fair to say there is no known cosmic ledger that contains all the true of Theorems of Mathematics. If you want to convince yourself that’s an empirical result, that’s fine by me, but in my life, I’m going to treat that as a given, and therefore, Mathematics exists in the observable world, but does not have a location in the observable world, and instead only has effects on the observable world.

Tonight, I derived what I think is a fascinating corollary, by analyzing and abstracting the structure of the argument, which is in essence a forbidden property. Specifically, if a system does not exist, then it cannot have property P, which in Assumption 1, is an observable effect. Now ask whether there is some other property P*, specifically:

Query 1: Is there some other forbidden property P*, such that if a system S does not exist, then S cannot have property P*?

Assume that P* does exist, S does not exist, and that S has property P*. How would we know that S has property P*, except for the effects of property P* on the observable world? Note that any property imputed by deduction or inference generally must have began with an observation. Therefore, the set of properties that contradict Assumption 1, are the set of measurable properties. In essence, there is only one forbidden property, perhaps abstracted as measurability. In summary, if a system does not exist, then the unique forbidden property of that non-existent system is measurability.