The Relation Between Mathematics and Physics by Paul Dirac (1939)
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https://arxiv.org/abs/math-ph/0009007
> Occam's Razor as a Formal Basis for a Physical Theory
> We introduce the principle of Occam's Razor in a form which can be used as a basis for economical formulations of physics. This allows us to explain the general structure of the Lagrangian for a composite physical system, as well as some other artificial postulates behind the variational formulations of physical laws. As an example, we derive Hamilton's principle of stationary action together with the Lagrangians for the cases of Newtonian mechanics, relativistic mechanics and a relativistic particle in an external gravitational field.
Only two people understand spinors: God, and Dirac. And Dirac's dead.
this cannot be an early hint to the result made by chaitin; Chaitin’s Incompleteness Theorem?
From Heisenberg to Goedel via Chaitin - https://arxiv.org/abs/quant-ph/0402197
Dirac large numbers hypothesis - https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis
Also found this; The fundamental constants and their variation: observational status and theoretical motivations - https://arxiv.org/abs/hep-ph/0205340
going beyond that page, think about beta distributions (and dirichlets). they are probability distributions about probabilities. try recentering the beta distribution between -1 and 1 first (representing c).
going even further, recall the bernoulli-binomial-beta connection. treat the recentered beta distribution's parameters as bernoulli trials. try placing them in a light cone. let me know what interesting things you find ;)
And then similar stuff happened later in quantum mechanics, with gauge theory, but I understand that only at a handwavy level. I think overall the "standard model" is a perfect example of what Dirac predicted.
Whether this still holds up in the past 40 years is another question. I don't have a great example from my lifetime.
1) Dirac actually developed big parts of Lie Theory and its application in relativistic particle physics, without using the mathematical formalisms at all (until later on).
His work was subsumed/beautified by mathematicians after the fact (30s), an 'optimization' really which helped to extend and understand it further, not to discover it. The Gell-Mann prediction mirrors Dirac's own positron (non)prediction closely but was done without Lie formalisms.
The applications of Lie theory in relativistic fields theories began even earlier, with the Noether/Klein/Hilbert/Einstein collaboration in 1910s.
2) Continuous symmetries are a convenience and not essential to particle physics IMHO.
e.g. It's obvious we can swap positive and negative charge and get the same physics. That's a discrete symmetry. But in gauge theories the symmetry is not +ve <-> -ve, but rather the continuous rotation of the unit circle in a complex plane. electron becomes positron via complex numbers. It doesn't mean there's a complex electron with +-ie charge, but it does work better with relativistic Lagrangian fields theories where phase changes are continuous and frame dependent. Noether's methodology of using an invariant action integral leverages the Lie theory very effectively, but there are other ways to skin that cat though.
3) Beauty is in the eye of the beholder, but I doubt the standard model and the perturbative methods (renormalization, etc) underpinning it would be to Dirac's liking.
This is quantifiable, right? Should be possible to measure how many mathematicians favor ideas like turbulence, macroscopic quantum physics, anthropic principle, analog geometry, finitism. I think, mathematics doesn't really try to match physics yet.
It is quantifiable: several mathematical objects and categories people invented to do symbolic computation in letters and published papers are somehow very useful to model and understand data from physical measurements of reality.
I highly recommend reading the "Unreasonable effectiveness of mathematics" essay by Wigner:
https://www.hep.upenn.edu/~johnda/Papers/wignerUnreasonableE...
You ought to learn some advanced mathematics to avoid making such absurd claims as "mathematics is too simple to model complex physics"
Pray tell what parts of physics are too complicated for mathematics
If a phenomenon isn’t model-able by relatively simple math it doesn’t even look like a phenomenon to us, it looks like a chaotic mess.
Which is to say that unless something has mathematical coherence it is more or less invisible to our experience.
Of course, I think putting numbers to wishy washy meta-quantities like "how efficiently does a certain conceptual scheme help you" are super loaded and hard to properly talk about (incommensurability). I've been toying with trying to make a repository of all the possible "moves" one can make in this kind of abstract analysis - constrain the problem statement, argue something like "the system is what it does", dissolving, etc. but even that seems hard
Here is a summary - https://thetelos.org/science-and-method-science-et-methode-h...
Here is a video discussion - https://www.youtube.com/watch?v=sQ-t8-igDZo
The thesis is that Science is made for efficiency and hence does not include everything (which is infinite). It only looks for a hierarchy of interesting facts and focuses on simple foundational recurring phenomena from nature. It then uses the language of Mathematics to economize and impose order on the complexity to make it tractable.
Finally, our subconscious prefers aesthetic attributes and hence we often find/create harmony/symmetry/beauty in our mathematical products.
So I think symmetry might just be one of the tricks that keep working and keeps on giving back and so we love it and call it "beautiful".
I don't know how LLMs would help sorting through the 10^500 possible universes when it comes to superstrings. You need to be able to run experiments or obtain observations.
Take the golden rule for example. It's a simple algorithm that shows up everywhere in nature. It represents the least amount of energy needed to assemble all kinds of structures. The gist here is nature finds these algorithms via evolution over billions of years and quadrillions of individual life experiments. We see life finding simple algorithms and platonic maths, how complex of algorithms has it found?
You no longer have to do an impossible number of experiments, instead you have to tease apart gene expressions to turn them on and off. Now, that is still a monumental task, but it's still doable in a reasonable amount of time.
Added: https://en.wikipedia.org/wiki/Xenobot
Individual cells like this can perform all kind of higher order operations that would appear to be intelligent actors.
It's always possible that continuous maths are a side effect of this field. Of course, the causality could be revered.
I remember there being a talk or some article about this but can't find it anymore, but it is an interesting thing.
While not being deeply familiar with the work but I think the short of it is that life has found algorithms at many different orders that we've not discovered them yet, and by 'running' these different algorithms it turns many P problems into NP problems.
The interesting thing about it is you can make falsifiable tests around this and do actual science around it.
In particular, students of Hindu Philosophical Schools will find lots of parallels here. Bohm was heavily influenced by Jiddu Krishnamurti which inspired his take on quantum theory.
Bohm also wrote a great book, "Quantum Theory" based on the Copenhagen Interpretation which contains separate parts on "Physical Formulation" and "Mathematical Formulation" of quantum theory.
Kaffee: Corporal, would you turn to the page in this book that says where the mess hall is, please?
Barnes: Lieutenant Kaffee, that’s not in the book, sir.
Kaffee: You mean to say in all your time at Gitmo you’ve never had a meal?