> […] the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.
There's a quote about the relationship between physicist, mathematicians/engineers that I'm blanking on and have been trying to find/remember -- it goes something like this: Physicists are concerned with discovering and understanding the fundamental laws underpinning the structure of things while mathematicians/engineers are interested in discovering all the possible worlds that could follow from the fundamental laws physicists discover. Does anyone here remember how that goes?
Reminds me of how the AlphaGo imstance that trained against human experts was equalled or bested by the version that just bootstrapped by playing against itself. (With the math side being analagous to the self-trained instance)
Let us start with Geometry. The sum of angles of a triangle is >, = or < 180 degrees. Is it possible maths describe the world. No. Maths covers all ground. Unless the universe is illogical, even if the world is multi-verse still only one rule.
The maths provide what human can understand logically. Universe can be a total mystery, inconsistent or just one of the possible maths world.
Do note the issue godel and hence there might be hiles even in maths world.
Whilst we have faith do nite the issue of dark energy and matter etc. No one guarantee everyone is ok.
>The dominating idea in this application of mathematics to physics is that the equations representing the laws of motion should be of a simple form.
But are they the rules of nature and not a set of relationships in nature accurately communicated by the language of maths?
For example, famously, Newtonian gravity turned out not to be fundamental rules by which nature followed, but further inquisition found a deeper, suprisingly more complex relationship of rules.
Feynman talked about this, saying "nature may be for the foreseeable future like an onion, where we just keep pulling away the layers and there are more and more layers."
I'm not an expert in physics nor math, but it seems we still have little-to-no fundamental understanding of what the fundamental rules or causes of the game of reality are. The rules we are finding (that physics seems to have stalled out on discovering, but that's a tangent) are not really fundamental truths or rules, but perhaps even more accurately mental models on describing relationships in the universe. Einstein himself actually said math was an invention of the human mind, and I believe the jury is still not settled on that.
Perhaps someone here has a better understanding and can help me here?
The view that Dirac is putting forth, that the fundamental equations need to be simple, is certainly not the modern one endorsed by theorists today. Since the 70s or 80s, the point of view that has come to dominate is that our most accurate theories (quantum field theories) are what are known as "effective field theories", where the adjective 'effective' refers to the idea that they are only only low-energy approximations to some other more complete theory. In this point of view the complete theory would have more complicated, even non-renormalizable terms that violate conservation laws, but these additional terms are suppressed by powers of something like the plank scale and so are virtually undetectable at low energies. Thus the low energy approximations (the effective field theories) have the simple, renormalizable form that we know. There is a good accessible discussion of this in Weinberg's popular science book 'Dreams of a Final Theory'.
Based on my understanding, I don't think there is any particularly compelling evidence that there is an entirely new layer beneath what we know. But it's always a possibility. For example, it wouldn't be too surprising if modern physics breaks down beneath the Planck scale (equivalently, at very high energies) [1], or inside black holes.
In other words, I think both the possibility that we are nearing a complete understanding of the rules and the possibility that we are nowhere close are both quite plausible.
I doubt many people of the time thought that the photovoltaic effect was compelling evidence that hinted at a whole new layer of physics beneath the classical theories. Only hindsight is 20/20. There a lot of loose threads in modern physics that seem unrelated or even inconsequential, but that might ultimately turn out to be revolutionary. So like you said, us being close or very far from a total understanding are both plausible.
I am by no means an expert on this, but intuitionism [1], one of the major foundational philosophies of mathematics, does view mathematics as an invention of the human mind, and as the "one true language" we use to communicate with each other. Physics then, is also "just", a mental creation.
Fair! I implicitly assumed that physics and mathematics are isomorphic (as speculated by Dirac in the essay). And so from this assumption, and my first sentence, the second follows.
Think of it as a linguistic problem instead. What is the minimum number of symbols needed to express the complexity of nature together with the complexity of the interpreter needed to parse them? Simple forms are just those that compress notions well.
> There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with reasonable success.
Might be semantics, but there is a lot of logical reason why mathematics should work to describe nature, because mathematical proofs are made based on logics. There is no empirical reason, though.
> I would suggest, as a more hopeful-looking idea for getting an improved quantum theory, that one take as basis the theory of functions of a complex variable.
Quantum physics is based on complex Hilbert spaces. This is quite different to the theory of functions of a complex variable. I think Dirac was just making s!!t up here.
But anyway, by now modern theoretical physics has hoovered up quite a lot of 19th century pure mathematics, including this theory mentioned by Dirac. Specifically I'm thinking of things like conformal field theory.
