> 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. The whole success of the scheme is due to the fact that equations of simple form do seem to work. The physicist is thus provided with a principle of simplicity, which he can use as an instrument of research.
> 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.
there is a quote from Pauli about Dirac : 'There is no God and Dirac is his prophet' Heisenberg and Dirac had different opinions on the existence of a god and Pauli, asked for his opinion, had the former to say
"I would like to put forward a suggestion as to how such a scheme might be realized. If we express the present epoch, 2 x 109 years, in terms of a unit of time defined by the atomic constants, we get a number of the order 1039, which characterizes the present in an absolute sense. Might it not be that all present events correspond to properties of this large number, and, more generally, that the whole history of the universe corresponds to properties of the whole sequence of natural numbers? At first sight it would seem that the universe is far too complex for such a correspondence to be possible. But I think this objection cannot be maintained, since a number of the order 1039 is excessively complicated, just because it is so enormous. We have a brief way of writing it down, but this should not blind us to the fact that it must have excessivly complicated properties."
this cannot be an early hint to the result made by chaitin; Chaitin’s Incompleteness Theorem?
Just to clarify; The term "2 x 109" is actually "2 x 10^9" or 2 billion years which was considered the present epoch i.e. accepted age of the universe at that time.
Correct (but now we know it should be somewhat higher than that), and 1039 should be 10^39, one of Dirac's Large Numbers, the size of the observable universe in atomic units, and also the ratio of electromagnetic to gravitational forces in the hydrogen atom.
Right. Added the wikipedia link to Dirac's LNH above.
Also found this; The fundamental constants and their variation: observational status and theoretical motivations - https://arxiv.org/abs/hep-ph/0205340
Yes, quite well in fact. This lecture was in 1939. In the 50's, Lie groups (previously just a pure, beautiful mathematics) were used in particle physics. Gell-Mann used SU(3) to predict some new particles before anyone observed them.
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.
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 ;)
>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
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.
I think this passage is more about the fact that mathematics is a form of symbolic computation people create and it is weirdly congruent to the physical measurements and models of reality. Not that mathematicians see certain models and objects used in physical models in a favorable way. What favor even means here?
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:
I don't see unreasonable effectiveness of mathematics. On one hand, it's a self fulfilling prophecy: if your math doesn't work, just make different math, and so on until it works (gravity was invented 6 times or so), so it's as effective as logic. On the other hand, it's fundamentally approximation limited by precision, so for any model we know when it breaks. And it's basically unworkable for chaotic or exponentially complex phenomena.
You're just claiming it's effective. We know it's effective. But there's no reason it should be effective. It's primarily because philosophy has not caught up and we still can't explain the foundations of mathematics and causality. These things seem to be baked into our very nature. It is possible to image an alien race with different conceptions of logic and causality that are just unable however hard they try to formulate either proper mathematics or proper physical laws from mathematics. (Take for example the Pythagoreans who could not believe irrational numbers existed, so the famous cube root of 2 was fake news to them. They could never build their little cubic altar to whomever.)
Dirac was talking about the fundamental laws. In favour of his point, and not known at the time he gave the talk, the standard model of particle physics is based on the symmetry groups U(1), SU(2), and SU(3), so Lie groups (which mathematicians find interesting) appear to have been chosen by nature.
I sort of have this suspicion that the kinds of phenomena that we can even apprehend are pretty closely related to what we can reason about mathematically, and the reason that math seems to work so well to model behavior that we understand says more about how our brain works than how the universe works.
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.
Well there are counting arguments to say that a higher level intelligence that somehow achieves say, 100000× brain efficiency of humans, still can't do that much more work than humans, if humans found the "best abstraction". Think about computability for example - that means you could feasibly solve problem instances of n+20 relative to what a human can solve.
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
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.
Perhaps symmetry is overrated and it is not the beauty that works, but it's economy. If something is symmetric depending on how symmetrical you can remove half or more of the required work during calculation. It is also useful to work in abstractions where it does not related to the system 1:1 but is some transformed representation (e.g. phase space). And it may be useful when you find a way to transform it to some easily solvable (symmetrical?) state, solve it there, and reverse the transform. Think of working in Cartesian and polar coordinates. Just a change in coordinate representations can improve your quality of life immediately on some problems with different symmetry. There's no intrinsic reason other than its much simpler on one than the other. Beauty doesn't have to come in. If it's round it's round. If it's square it's square.
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".
Given its strength in mathematics, it seems likely that AI should be of great help with discovering new physics. We just need to teach it to ask interesting questions.
Dirac comments that he thinks a potential path to the discovery of new physics is to start with a mathematical domain and work outwards from there, guided by mathematical beauty. This is what I took OP's comment to be about - of course setting aside questions on an LLM's ability to recognize beauty
Superstring theory has been based on the idea of mathematical beauty, but it's run into serious complications when it comes to empirical results. Science is empirical first and foremost, because we need to verify that theories actually predict the universe we live in by testing them.
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.
Actually there are people attempting to find new math via biological entities. Cells and such. The interesting thing here is it is easily falsifiable even though working at these scales is still insanely hard.
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.
There's this little pet project of mine, the Functional Universe [0]. It explores a direction suggested by Dirac: treating mathematical structure and transformation not merely as a language for describing physical reality, but as potentially constitutive of it. FU models physical reality in terms of functional state evolution, with an emphasis on composition, aggregation, and transitions. In that sense, I think there's an interesting point of contact with Dirac's emphasis on transformations: Dirac points toward transformations as fundamental mathematical structures from which physics might be derived; FU asks what happens if physical reality itself is formulated in terms of transitions and their composition.
Curious, in this model, is there any discussion on whether the mathematical structures underlying space-time are continuous or discrete? Usually the former is assumed, but it seems your "functional" approach may be able to accommodate the latter possibility.
universe is continuous / cant be proven otherwise but any measurement or simulation of it is discrete by nature same reason pi is computed infinitely and why calculus exists and why math exists at all to try and explain continuity of nature even if all of it is perceived discretely.
The question is what exactly is continuous. The quantum fields themselves? And we have no evidence the the quantum fields are discrete quantum states themselves, but the particles they produce appear otherwise.
It's always possible that continuous maths are a side effect of this field. Of course, the causality could be revered.
It is an interesting discussion you bring up but will take much more than a comment to talk about. There is a divide between mathematics and physics regarding numbers. Real numbers have supposedly infinite precision, and arbitrary size, which are not true in our physical universe. Indeed infinities have given physicists quite some headaches. It's a bit of a dark art as to where you can and cannot just use infinity willy nilly and many have expressed distaste to it. Also at some point you have to talk about computability, like are these infinite precision and magnitudes even real if nothing in the universe even gets close, what is omega to the omega, statements dream up by the utterly deranged?
I remember there being a talk or some article about this but can't find it anymore, but it is an interesting thing.
if you think of universe as minecraft lattice (which is arguably best way to think of it and explain all of physics currently in existence in intuitive level) then yea, it can be like eg a spin lattice discrete at some plank length from which field perturbations as particles emerge..look up spin lattice from https://arxiv.org/abs/hep-th/0507118 so basically bringing back idea of quantum ether cuz explaining empty space with waves without some underlying structure doesn't really make sense anyway since all other waves have mediums in nature.
I think Michael Levin is also doing work around platonic math being a thing that exists moreso than an actual concept. His work is related to biologies discovery of algorithms, the complexity levels of simple algorithms, and n-order effects that life is finding way to exploit for computational purposes.
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?
> 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. The whole success of the scheme is due to the fact that equations of simple form do seem to work. The physicist is thus provided with a principle of simplicity, which he can use as an instrument of research.
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.
there is a quote from Pauli about Dirac : 'There is no God and Dirac is his prophet' Heisenberg and Dirac had different opinions on the existence of a god and Pauli, asked for his opinion, had the former to say
There's a quote Atiyah has about spin geometry which he heard from his advisor Hodge, who had an office next to Dirac at Cambridge.
Only two people understand spinors: God, and Dirac. And Dirac's dead.
"I would like to put forward a suggestion as to how such a scheme might be realized. If we express the present epoch, 2 x 109 years, in terms of a unit of time defined by the atomic constants, we get a number of the order 1039, which characterizes the present in an absolute sense. Might it not be that all present events correspond to properties of this large number, and, more generally, that the whole history of the universe corresponds to properties of the whole sequence of natural numbers? At first sight it would seem that the universe is far too complex for such a correspondence to be possible. But I think this objection cannot be maintained, since a number of the order 1039 is excessively complicated, just because it is so enormous. We have a brief way of writing it down, but this should not blind us to the fact that it must have excessivly complicated properties."
this cannot be an early hint to the result made by chaitin; Chaitin’s Incompleteness Theorem?
Related;
From Heisenberg to Goedel via Chaitin - https://arxiv.org/abs/quant-ph/0402197
Just to clarify; The term "2 x 109" is actually "2 x 10^9" or 2 billion years which was considered the present epoch i.e. accepted age of the universe at that time.
Dirac large numbers hypothesis - https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis
Correct (but now we know it should be somewhat higher than that), and 1039 should be 10^39, one of Dirac's Large Numbers, the size of the observable universe in atomic units, and also the ratio of electromagnetic to gravitational forces in the hydrogen atom.
Right. Added the wikipedia link to Dirac's LNH above.
Also found this; The fundamental constants and their variation: observational status and theoretical motivations - https://arxiv.org/abs/hep-ph/0205340
Did Dirac’s prediction that studying “beautiful” mathematics would lead to breakthroughs in the understanding of Physics pan out?
Yes, quite well in fact. This lecture was in 1939. In the 50's, Lie groups (previously just a pure, beautiful mathematics) were used in particle physics. Gell-Mann used SU(3) to predict some new particles before anyone observed them.
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.
The standard model is completely accidental, no? Also hierarchy problem.
as an example of this, pretend probability is a velocity: https://www.mathpages.com/home/kmath216/kmath216.htm
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 ;)
Also see Relationship between Mathematics and Physics - https://en.wikipedia.org/wiki/Relationship_between_mathemati...
>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
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.
I think this passage is more about the fact that mathematics is a form of symbolic computation people create and it is weirdly congruent to the physical measurements and models of reality. Not that mathematicians see certain models and objects used in physical models in a favorable way. What favor even means here?
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...
I don't see unreasonable effectiveness of mathematics. On one hand, it's a self fulfilling prophecy: if your math doesn't work, just make different math, and so on until it works (gravity was invented 6 times or so), so it's as effective as logic. On the other hand, it's fundamentally approximation limited by precision, so for any model we know when it breaks. And it's basically unworkable for chaotic or exponentially complex phenomena.
You're just claiming it's effective. We know it's effective. But there's no reason it should be effective. It's primarily because philosophy has not caught up and we still can't explain the foundations of mathematics and causality. These things seem to be baked into our very nature. It is possible to image an alien race with different conceptions of logic and causality that are just unable however hard they try to formulate either proper mathematics or proper physical laws from mathematics. (Take for example the Pythagoreans who could not believe irrational numbers existed, so the famous cube root of 2 was fake news to them. They could never build their little cubic altar to whomever.)
Dirac was talking about the fundamental laws. In favour of his point, and not known at the time he gave the talk, the standard model of particle physics is based on the symmetry groups U(1), SU(2), and SU(3), so Lie groups (which mathematicians find interesting) appear to have been chosen by nature.
I sort of have this suspicion that the kinds of phenomena that we can even apprehend are pretty closely related to what we can reason about mathematically, and the reason that math seems to work so well to model behavior that we understand says more about how our brain works than how the universe works.
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.
Well there are counting arguments to say that a higher level intelligence that somehow achieves say, 100000× brain efficiency of humans, still can't do that much more work than humans, if humans found the "best abstraction". Think about computability for example - that means you could feasibly solve problem instances of n+20 relative to what a human can solve.
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
In case you haven't read it; see "Science and Method by Henri Poincare" - https://archive.org/details/sciencemethod00poinuoft
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.
Perhaps symmetry is overrated and it is not the beauty that works, but it's economy. If something is symmetric depending on how symmetrical you can remove half or more of the required work during calculation. It is also useful to work in abstractions where it does not related to the system 1:1 but is some transformed representation (e.g. phase space). And it may be useful when you find a way to transform it to some easily solvable (symmetrical?) state, solve it there, and reverse the transform. Think of working in Cartesian and polar coordinates. Just a change in coordinate representations can improve your quality of life immediately on some problems with different symmetry. There's no intrinsic reason other than its much simpler on one than the other. Beauty doesn't have to come in. If it's round it's round. If it's square it's square.
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".
Given its strength in mathematics, it seems likely that AI should be of great help with discovering new physics. We just need to teach it to ask interesting questions.
How does this comment relate to the article?
Dirac comments that he thinks a potential path to the discovery of new physics is to start with a mathematical domain and work outwards from there, guided by mathematical beauty. This is what I took OP's comment to be about - of course setting aside questions on an LLM's ability to recognize beauty
Superstring theory has been based on the idea of mathematical beauty, but it's run into serious complications when it comes to empirical results. Science is empirical first and foremost, because we need to verify that theories actually predict the universe we live in by testing them.
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.
Actually there are people attempting to find new math via biological entities. Cells and such. The interesting thing here is it is easily falsifiable even though working at these scales is still insanely hard.
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.
There's this little pet project of mine, the Functional Universe [0]. It explores a direction suggested by Dirac: treating mathematical structure and transformation not merely as a language for describing physical reality, but as potentially constitutive of it. FU models physical reality in terms of functional state evolution, with an emphasis on composition, aggregation, and transitions. In that sense, I think there's an interesting point of contact with Dirac's emphasis on transformations: Dirac points toward transformations as fundamental mathematical structures from which physics might be derived; FU asks what happens if physical reality itself is formulated in terms of transitions and their composition.
[0] https://voxleone.github.io/FunctionalUniverse/
Curious, in this model, is there any discussion on whether the mathematical structures underlying space-time are continuous or discrete? Usually the former is assumed, but it seems your "functional" approach may be able to accommodate the latter possibility.
universe is continuous / cant be proven otherwise but any measurement or simulation of it is discrete by nature same reason pi is computed infinitely and why calculus exists and why math exists at all to try and explain continuity of nature even if all of it is perceived discretely.
The question is what exactly is continuous. The quantum fields themselves? And we have no evidence the the quantum fields are discrete quantum states themselves, but the particles they produce appear otherwise.
It's always possible that continuous maths are a side effect of this field. Of course, the causality could be revered.
It is an interesting discussion you bring up but will take much more than a comment to talk about. There is a divide between mathematics and physics regarding numbers. Real numbers have supposedly infinite precision, and arbitrary size, which are not true in our physical universe. Indeed infinities have given physicists quite some headaches. It's a bit of a dark art as to where you can and cannot just use infinity willy nilly and many have expressed distaste to it. Also at some point you have to talk about computability, like are these infinite precision and magnitudes even real if nothing in the universe even gets close, what is omega to the omega, statements dream up by the utterly deranged?
I remember there being a talk or some article about this but can't find it anymore, but it is an interesting thing.
if you think of universe as minecraft lattice (which is arguably best way to think of it and explain all of physics currently in existence in intuitive level) then yea, it can be like eg a spin lattice discrete at some plank length from which field perturbations as particles emerge..look up spin lattice from https://arxiv.org/abs/hep-th/0507118 so basically bringing back idea of quantum ether cuz explaining empty space with waves without some underlying structure doesn't really make sense anyway since all other waves have mediums in nature.
I think Michael Levin is also doing work around platonic math being a thing that exists moreso than an actual concept. His work is related to biologies discovery of algorithms, the complexity levels of simple algorithms, and n-order effects that life is finding way to exploit for computational purposes.
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.
Btw it's unredable on mobile because the table of contents box blocks the actual contents
Now dive down into Math with Penrose and David Bohm with the implicate order.
Quite independently of the Mathematics, Bohm's ideas of "Implicate and Explicate Order" are very interesting and comprehensible by the layman at a high-level - https://en.wikipedia.org/wiki/Implicate_and_explicate_order
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.
Great answer.
Shouldn't this have a "(1939)" suffix on the title according to HN rules? ;)
Should it though? I don't see such a rule.
https://news.ycombinator.com/newsguidelines.html
unwritten rules
Why should anyone know or care about it then? This is not a static platform with a static userbase.
Your quote should give proper attribution according to the rules.
If an article is a few years old then it could be out-of-date. But if an article is almost 100 years old then its classic.