Well, I don't see what I do as education. I would call it more like exploration. On the one hand, we do want a pure mathematics, and you might argue that closure is a form of purity, but on another wavelength we'd like a math which actually generates reality. This then will be a pure theoretical basis. This blue sky figure yields spacetime and the atoms and molecules in it; the photons, and even the mesons, which I barely have any awareness of.
I get caught in this morass in the empirical view as attacking the theoretical view. Then flipping it around, and so forth: it may be that my arguments here are confused, but at least they are reaching. I see the natural value argument that I make, while I still do perceive that natural value just as I was trained; as the simplest form of number. Well: unity is ultimately the simplest form of number, and unification may as well be renormalized to unitification, and possibly all would be well. That the universe is avoided in our awareness, and yet as we confess that every practical count will require a partition of that universe be cleanly secured, the very stability of our situation to arrive with the natural value is fortunate. Dragons do not fly out of cans of coffee when you open them up in the morning, do they? I guess it's bags of coffee for most people these days, but I buy the cheap stuff.
I suppose at some level, and this involves stepping out away from the morass, we can witness that we are involved in a progression; that this progression is not necessarily straight-forward, and that our involvement in the progression is to find our contribution to it. I think all who bother to post here hope to do just this. Quite a few of us wind up finding the need to tear quite a lot down in the process. Facing the accumulation I find this entirely acceptable, but as well our constructions have to stand after the fact, or else we've merely burned the thing down.
That's really going too far, and while it may seem that the thing burns too well, possibly another approach is to disentangle the parts. It is bizarre to me that functional analysis takes a seat in the basis of values and operators, as is the formal language today. I say that without values and operators what can a function even portray? As to which is elemental, and which is built upon those elements: this inversion at least deserves scrutiny. This is a compiler level error which Peano committed, and later the abstract algebraist, too. As to who is being destructive to mathematics: I could readily point my finger to them rather than at myself. Have I failed to appreciate the mathematician's function? Am I too entrained on my computer's version? The fact that we can substantiate such a complete structural inversion within human thought is very troubling to me.
Now, taking the number and its hardware form more seriously, we witness several formal syntax usages which deserve structural scrutiny. One is the sign, and the other is the decimal point. Sure enough our hardware versions do bother to keep track of these augmentations to the natural value. Meanwhile the mathematician views them as evolutionary set theory, somewhat through the closure argument that I've opened with here. Staying in the digital view, which is consistent with modern computing hardware, we see that we'd certainly like closure to hold up, but it does not universally. In this moment I am brought to a new name; should we augment the notion of the digit and allow that these elemental conglomerations as if they have achieved a singular larger digit? A mudigit, for instance? I don't feel settled on that, but upon tying those binary bits together in hardware such that they do indeed spit out another in the same form, then we do in fact validate this molecular form. In a sense this brings us around to taking the original digit more seriously as well, which is anathema to Peano's way.
As to finding order in the universe: pretty clearly our immediate form is troubled at our scale of existence. All that we can do is concern ourselves with some partition of it, as for instance a bean pod containing five fertile beans introduces some semblance of order, whereas the quantity and varieties of beans in the universe is unknown. This value likely goes gray as refinements occur. The very definition of bean will need to be reformed; refined. At the human scale we will not achieve elemental status this way; the bean is not fundamental. Even the atom is just a stop along the way. Enter the muon. A return to the classical exposes so many contradictory constructions along the way that the very mathematics that are in use along the way become open.
I am going to stick to my claim that the trifurcation to physics, mathematics, and philosophy is invalid. Heck, these days your king is a queen, right?
Whether philosophy is just a jack, or rather, an ace: this has to be determined. It depends upon the philosophy, you see? My hope is that a semi-classical approach will work; that we can make sense of reality. The alternative is; well; it will go dark before our eyes. I don't really want to say it, but the analysis speaks for itself practically: quantum physicists are nihilists. I suppose that should be another topic of conversation, and I don't mean to place it too seriously, but from the semi-classical approach there are plenty of quotes direct from their mouths to prove it.