Our friend Rog needs 10 pages to opine that "the map is not the territory"? :-)That's right, the mathematics is the map that describes how to observe the physics, but it is not identical to the physics. His slogan would have made a great title.
I enjoyed reading your essay. It is really clearly written and thoroughly accessible, even to someone without a maths or physics background. You have set out your arguments very clearly and I might have been convinced had I not previously given this subject quite a bit of thought.In other words: Good argument, but my mind was already made up!
I'm somewhat in sympathy with Jonathan Burdick's pithy response, but of course in ten pages you do more that say that the map is not the territory. I take you also to say that the territory (reality) is not mathematical. ...I like his phrase "construction of a new systematization of experimental data." Yes, that is a laudable goal and mathematics is a terrific tool. I also "accept that Physics is the systematic description of reproducible experimental results."
What is left to do is very hard, in the usual story of all the low-hanging fruit having been picked, but we have made better tools than our forebears. It is also possible that there is some part of the territory that only ever happens once, so that it cannot be subject of Physics taken to be a repeatable experimental subject. ...
In any case, there has been a constant interplay between Mathematics and Physics, ...
My purpose is to better understand the limits to mathematical reasoning in physics. For example, consider the No-cloning theorem. If a physical state is perfectly representable by some numbers or other mathematical objects, then it is very hard to understand why a perfect copy cannot be made. Perfect cloning of mathematical objects is axiomatic. I say that the quantum state is great for systematizing experimental data, but when you take it too literally as being reality then paradoxes result. It is better to step back, and admit that our mathematical models may be necessarily imperfect.
My essay's public rating is currently a meager 4.4 out of 10. My essay goes against conventional wisdom, but I don't expect a high rating, but I hope that it is good enough to qualify for judging this fall. At least my essay answers the contest question:
Questioning the Foundations: Which of Our Basic Physical assumptions are Wrong?
What assumptions are ripe for rethinking? ...
What are the implicit assumptions we tend to forget we have postulated, or that have become so ingrained that they have become unquestioned dogma? ...
Note: Successful and interesting essays will not use this topic as an opportunity to trot out their pet theories simply because those theories reject assumptions of some other or established theory. Rather, the challenge here is to create new and insightful questions or analysis about basic, often tacit, assumptions that can be questioned but often are not.
Part of "the" problem: Maps cannot exist apart from cartographers and they must exist within territories. At a proper scale you can see that: the map at some resolution needs to map itself (recursivity). "Math maps" exists within the physical substrates of our minds, i.e. we're math-making machines. Not sure what good it does to wonder about the subject of "math independent of mathematicians" -- but I guess Plato managed to get some mileage out of it.ReplyDelete
I'm "replying" to my own comment, rather than posting a new one, but don't be deceived -- I can't seem to launch a new comment window and it's probably due to a Google bug. I discovered that even clicking on the "1 comment" link takes me to NNL (Never Never Land) unless I sign into Google first. Previously I could comment, then sign in to post it.Delete
Anyway, maybe I'm confused now. Which is the better analogy?
1. (my original) Physics as territory, math as the (a?) map
-- or --
2. Physics as map, math as set(s?) of cartography tools
Perhaps there's not much difference between the two. But I can get a sense of a difference if I think of them in terms of Roger's essay. You can think of his "abandoning the math substrate-of-physics" in (1) above as either "You need a completely different type of map" or "You need to be doing this without any map whatsoever". In terms of (2) above, it'd be something like "You need a different set of tools" or "You need to be doing this without a set of tools." Since some very basic qualitative physics can be discovered sans math ("apple falls from tree = 'gravity', and even if I can't truly numericize things, I can notice that the phenomenon seems pretty consistent), I take the view that throwing-out-math still leaves us with other tools.
But I'm still a bit muddled. Which is the better analogy: map-making, or navigation? Maybe no discernible difference.