The nature of number

[Angrboda]

I'm getting lost because my maths is weak and my brain is old and gray. Still, I enjoy trying. So please continue.


I have this.... let's call it a notion, as the word "idea" is perhaps too strong. My idea is that if you look at the set of all possible universes, varying on whatever variables are possible for a universe, some subset of those universes will have regularities in them; in our universe, we call such regularities the laws of physics. And inside that subset, arrangements of "stuff" will occur that are able to capitalize on the inherent predictability and order implicit in such regularities. Call that life. Now I so lack the necessary math and physics to prosecute this statement, but my hunch is that in the phase space of all possible universes, these subsets span a significant portion of the entire space. In my view, it's possible that life and intelligent life aren't rare chance occurrences, but likely built into the relationship between order and the possible. I rather suspect that the potential universes without some forms of regularity and order may be a minority. (This dips into questions of the meaning of causality, questions which I'm going to side step for the moment.)

Now, given this understanding, one can view life forms, and especially intelligent life forms, as having evolved mechanisms for extracting the utilitarian value of the predictatability of order by mental mechanisms (evolution itself being an algorithmic process which capitalizes on the ordered regularities and "builds them into" machines that behave in ways that are useful if the regularities are assumed to hold true [they have predictive value]). (Just a side example, a fish doesn't view the minnow it wants to eat as existing or moving in more or less than 3 dimensions, nor as if it might disappear here and appear there. It doesn't "know" these things in the sense of having thought them, but the regularities are built into how its brain thinks about the possible, so that what it thinks is often quite probable, and following the probable is profitable — that profitability being the ratchet that drives evolution.)

Now I'm going to make a gross simplification for the sake of example. When we look at brains, and human brains specifically, they appear to be composed of neural networks arranged in a connectionist model sort of way, with a bunch of nodes at the input or perception side, a bunch of nodes at the output side, and a bunch of nodes in between that are doing the work. (See ANN, artificial neural networks.) Now this isn't an invariant property of such networks, but many can be characterized as having the nodes at the input side dealing with the entire input in all its particularity, there are no generalizations or inferences made at this level. However, as you go farther back into the network, you encounter things which can be described as abstractions: expressions of properties and aspects of the particulars but which aren't themselves particulars. Example.... humma. We don't see a circle, what we see are a bunch of points that cause one of those higher order abstraction levels to gravitate both to shape and the particular shape, but neither shape nor that particular shape "exist" in the real world, nor in our particulars.

Now to connect the dots. As life forms that capitalize on the order and regularity in our environment, some of that order and regularity can be expressed as abstractions. Math probably being the important one. If my view is correct, then math and number are just pure forms of what is happening in our neural networks; they express awareness of the output of layers that output abstractions of the real as real things themselves (e.g. square, triangle, circle are not real, but thinking about them in relation to perception or idea is real, because our thoughts are the categories which our brains create for containing what we think; one layer, then, gravitates towards idealizations of shape as idealizations — as idealizations qua idealizations; these aren't truly abstractions or generalizations in the pure sense, they are more "creations"). However, given what gave rise to these mental forms, they have an intimate connection to the real that can never be completely severed.

It is in this latter sense I would call to mind Heidegger's notion that "there is sameness in difference and difference in sameness". In that same sense, the particulars of real world circles in some sense "infect" our mental, platonic circles, and yet they are also quite distinct; the circle inside us is not the circle outside us, it is just a useful way for brains to model regularities outside, regularities that the circle inside shares with them.


Anyway, I hope this hasn't been an inchoate mess. For whatever it's worth, which is surprisingly little in today's economy.