When feasible, I still use my CPOC (Constituent, Process, Observable, Causality) method for designing and constructing simulations. My claim is that any given system can (usually) be equivalently described in any one of those four “languages”. When I first make that claim, many in the (whatever) audience balk. But I side-step most objections by a) hand-waving about possible isomorphisms between various formalisms and b) submitting a caveat that I really just want them to model agnostically, with as little preloaded bias as possible … so it doesn’t really matter if the claim is rigorously true in every (or any) case.
Last weekend, a friend told me about the recent kerfuffle over Numberphile’s proof that the sum of the natural numbers comes to -1/12. (I personally like this discussion the best.) I didn’t believe it, then, but hadn’t run across it, and wanted to keep an open mind. In any case, my friend and I then launched into a discussion about Platonists vs. the constructivists. I suggested that none of these results are “real” in any sense. They are all just artifacts of the way we’ve formulated the questions we’re asking. My friend lands on the other side, that these results have some (ontological) reality, existence, and we discover them … at least some of them. (Of course, I knew he felt that way, which I why I chose to be a constructivist for this conversation. To be honest, I’m agnostic and try to think either way depending on the circumstances.)
The superstitious tend to assert that things come in 3′s. So, to validate that, I also noticed these 2 blurbs this morning:
- How the language you speak changes your view of the world
- “Is There Something Mysterious About Math?”
Of course, this flows right along the lines of my previous post about Lee Rudolph’s comments on Hardy’s “astonishingly beautiful complex”. So, the superstition is really just a cognitive bias.
Anyhoo… Near the end of Scott’s post, one of my old saws was evoked: the extent to which the content of our minds/brains is separated from the environment in which we’re embedded. This topic also came up with my friend over the weekend. Our discussion was largely about the very natural language of humans, that of things, objects, “nouns”, as contrasted with the language of behaviors, processes, or “verbs”. W.r.t. the equivalency of models written in different languages, I can confidently assert that a system can be equivalently described in terms of states (objects) versus state transitions (processes). In that discussion, one of the examples I used was a human (or any organism). We think if it as an object. But given that our skin is semi-permeable, our cells die off and are replaced, we eat, defecate, etc., where I end and you begin is really not very well defined. A human is, I argued, more accurately described as a large cluster of processes. It’s not an object at all.
But that was all simply background for what I want to point out in this log entry, related to Scott’s post and all the rest. I tend to think that math is only mysterious where everything else is mysterious, at that border between what’s inside versus what’s outside our selves. Hence, when we talk about a proof or disproof of something that’s intuitive, we’re really talking about a) our own internal, isolated, ways of thinking and b) the socially constructed “complex” providing the medium for our inter-individual communications. Often, when something is counter-intuitive to one person but intuitive to another, perhaps it’s because one person’s internal “complex” (or “landscape”) is more like the socially agreed upon “complex” that is the body of math as a whole.
Finally, one of my ongoing (though still agnostic) assertions is that our brains/minds are really rather flat. What’s inside us is directly defined by what comes in and what goes out. You are mostly a naively traceable, albeit stigmergic, product of the stimulus you received from your environment. (Now, I have an alternative ongoing assertion: that we are all little isolated universes of thought, which is tragic, actually, because that means when someone dies, a universe of knowledge dies with them. Hence, though it may be counter-intuitive, the idea that we’re predetermined bags of meat, defined completely by our stimulus, is more optimistic than that alternative. If the alternative is true, just think of the knowledge we lose when a species goes extinct. Ugh. That perspective is profoundly depressing.)