(t + xli + ylj + zlk) * (t - xli - ylj - zlk)
= t^2 - x^2 - y^2 - z^2
This is something that gets repeated again and again but I don't find this puzzling at all. Mathematics is an extremely general tool for modeling all kinds of things, so if something follows some sort of rules, you most certainly can use mathematics to describe, model, and analyze it. While I am not aware of any specific results or hypotheses, I would still guess that mathematics - or even some small subset like set theory - is in some sense universal for modeling things in a way similar to universal models of computation like Turing machines or lambda calculus.
I can not even imagine a universe with laws of physics that could not be tackled with mathematics. Even if the universe was pure chaos, everything happening totally randomly without any recognizable rules, we still could try throwing statistics at it, at least ignoring the fact that such an universe would probably not be a place for life to flourish. Finally mathematics does not exist in a vacuum, we invented large parts of it specifically to deal with real world phenomenons. And then of course »some« more that does not describe anything existing in the physical world.
Somewhere deep down below there are certainly many puzzling issues, from why there is a universe at all and why it is the way it is to human consciousness and why and how we are able to make sense of the universe, but that mathematics can describe the universe does not seem to be one of those, at least to me.
I think there's some hidden complexity at play here also. Yes, the number of symbols required is small. But those symbols are only meaningful to someone with the education to understand them. Namely, usually, a human being with a very complex brain, both in terms of requiring millions of years of evolution and years of training time. Actually, those alone aren't enough, since these people are building on top of hundreds of years of previous work in the field.
Yeah, except, maybe the number of symbols isn't so small. The number of symbols needed to model very simple relationships is small, like the relationship between two gravitational bodies.
Now add a third body, and yes, in some sense this system is still described by the simultaneous relationship between pairs of bodies, but it's not really usable in that form. You've got to explode it, and you've got to do a lot more calculation.
I'm not totally confident here. I do have some intuition that the Navier–Stokes equations, for example, fully describe the motion of a very complex phenomenon in a very small number of symbols. But then if they require so much computation to actually apply, is my intuition wrong that this really counts as a full description of the system?
Is there a distinction between running the equations and running a simulation? And what exactly is the distinction between running a simulation and setting up the system and just seeing what happens?
It's definitely different, but I'm not so sure it's this amazing compression that it's made out to be.
The third body makes the system nonlinear and non-analytic, so the equations can only be investigated via numerical simulation. But the number of symbols is still small.
It may be the case that most of the universe is nonlinear and we only pay attention to the linear parts because they're easy. It might also be the case that a lot of the universe requires a lot of symbols to describe but we only pay attention to the other part.
Well, the key word being relatively. Relative to what? Mathematics of QFT is not simple by any measure (and neither is that of fluid dynamics, for example), nor is algebraic geometry that is being pushed as a new tool of theoretical physics.
Then there's always geometric algebra!
Simply put, "the first quantization." (I am amazed how difficult popularizers of science often make it to understand what the hell they are talking about.)
Nobody does that for Quantum Mechanics.
The case of Quantum Mechanics is strange, there is no clear name to choose. I guess Heisenberg an Schrödinger are the best candidates, but their names are associated in popular culture with some specific effect instead of the whole theory.
Good reference books?
But first and foremost any recommended primers into algebraic topology?
>One reason is that quantum physics is based on algebra, while general relativity involves a lot of topology. But that suggests an avenue of attack: If we can figure out how to reduce topology to algebra, it might help us formulate a theory of quantum gravity.
huh? everything boils down to big matrices and hard differential equations in the end
>My physics colleagues will let out a howl here, and complain that I am oversimplifying. Yes, I’m oversimplifying:
no, you're not making any sense. i stopped reading here
> He is known for his work on spin foams in loop quantum gravity. For some time, his research had focused on applications of higher categories to physics and other things.
So he probably has _some_ understanding of physics, no?