A physicist, walking home at night, spots a mathematician colleague under a street lamp staring at the ground, "something wrong?" he asks; "I've dropped my keys" he replies, "whereabouts?" asks the physicist, keen to help. "Over there" says the mathematician pointing; "So why don't you look over there?" retorts the physicist, "the light is better here" says the mathematician.

Interviewer -- "Consider a situation where you are in your office and there is a fire outside in the hall. There is a fire escape outside your window but you can't reach it because the window is stuck. However, there is a hammer on the table. What do you do?"

Physicist -- "I use the hammer to break the window, allowing me to get out to the fire escape."

Interviewer -- "Now consider the same situation except that the hammer is on the floor. What do you do?"

Mathematician -- "I move the hammer from the floor to the table, thereby reducing it to the previously solved problem."

Software developer: It is more important to find out how the keys were dropped in the first place. And after I do that it will be more efficient to just generate new keys.

Physicist: but that doesn't get you any closer to a solution.

Mathematician: not yet, but if I wait here long enough someone will come by and drop their keys, which will then be retrieved, proving the possibility of retrieving lost keys when light conditions are optimal.

Non-software-developer humans often use things called "lamps" to illuminate spaces at night. Unfortunately, illumination inhibits effective nighttime coding.

Mathematicians like to develop new math right at the boundary of what is known. Physicists don’t have that luxury because they have to describe/model/build/etc things that correlate to what is actually going on in the world.

The mathematician of this joke would scan the edge of the light, finding nothing. Then he would keep lighting little lanterns at the perimeter to make the lighted area larger until finally his keys were within sight.

The physicist in this joke would presumably root around in the dark where she thinks her keys actually were. Upon finding them through brute force and luck, she might think “wow maybe one day this place will be illuminated so I can tell wtf I just did”

It is a common (but not universal[0]) practice in English writing to use either "he" or "she" as a pronoun for a person of unspecified gender to avoid the awkwardness of "they" or "one" or constructions like "he or she". No gender has been specified here, this is a neutral use of "she".

[0] Very few practices in English writing are universal.

I think your entire reasoning is in reverse. Using "she" where no gender needs to be specified is the awkward thing to do, and it's not neutral, it's explicitly designating a female gender. Using "they/them" is far less awkward, the reader doesn't have to have any kind of opinion about the text based on gender associations, making it less awkward.

Occasionally, they/them can get more confusing, especially in a story involving multiple people. It can become unclear if you're referring to just one of the people using singular they, or if you are in fact returning to more people using plural they.

The idea is that often a breakthrough in mathematics isn't achieved by tackling a problem directly, but converting it to a simpler problem, then solving that one.

> Hitchin agrees. “Mathematical research doesn’t operate in a vacuum,” he says. “You don’t sit down and invent a new theory for its own sake. You need to believe that there is something there to be investigated. New ideas have to condense around some notion of reality, or someone’s notion, maybe.”

This is kind of it I think. It's not just physics that drives interesting math, and it's not just recently that this relationship holds. Math is, IM humble O, the ultimate domain-specific language. It's a tool we use to model things, and then often it turns out that the model is interesting in its own right. Trying to model new things (ex. new concepts of reality) yields models that are interesting in new ways, or which recontextualize older models; and and so we need to reorganize, condense, generalize, etc; and so the field develops.

Unfortunately, he is unable to join the discussion. Deploying a ghost as an argument in and of itself is hardly useful.

Personally I tend to disagree with "I hope it isn't useful" or whatever the GHH quote is about maths being practicably applicable to the world/universe n that.

Why not tell us what you think? Mr Hardy's well documented positions on many things are well known but yours are not.

I don’t think his point and mine are incompatible, the language can have beauty in its own right and still be a model. Legos can be built into something which resemble a house, or something which resembles nothing real; their usage is a separate thing from the joy that comes from playing with them. Math is driven by utility and elaborated by enthusiasts.

I agree with you. What you say has even more support in the fact that the one of the most notable achievement of GH Hardy is in field of biological statistics (population genetics), and according to him, his discovery of Ramanujan.

That doesn't hold up to a reading of the history of maths. So much of it was invented by someone just noodling around with numbers and would find some use in physical science hundreds of years later.

I agree, I think. I would say it like so, that maths is a sort of highly technical, rigorous language, but like any language it will describe what you want it to. It is easy to think that it is describing the underlying terrain, but it is actually working on three (shared) and model which have of the terrain. So, as we consider different things, maths will follow.

One of my physics lecturers at university made the offhand observation that the distinction between physics and mathematics is a twentieth-century idea: it wasn't made during the nineteenth century or before, and it seems to be disappearing in the twenty-first.

That's because people were totally focused on physics, and math was just a useful tool sometimes. Doing physics was the true goal and observation the final arbiter of truth.

Nowadays, that distinction is blurred but for the opposite reason; people think that anything conceived by sound math must be true, and observation has taken a back seat.

To some extent, observation has taken a back seat because we're at the point in our physics journey where we pontificate about things that are too small or too dark and far away to see. We simply can't observe this stuff anymore.

I don’t agree with this. I think there are definitely people like Michio Kaku who have books to sell who spend their time pontificating. People just think that physics looks like pontificating because that’s what it looks like on TV.

But there are also active researchers doing real research. Physics postdocs aren’t just sitting around in a circle making up stories about what the universe is like.

The past decade is a difficult framing to ask the question in. Notable breakthrough results are usually understood in hindsight and a decade just isn't a lot of time for that context and understanding to develop. Science also doesn't necessarily develop in this way with consistent progress every X timespan. Usually you get lots and lots of breakthroughs all at once as an important paradigm is shattered and a new one is installed. Then observations with tiny differences slowly pile up and a very blurry/messy picture of the problems with the new paradigm takes shape. But none of those things feels like a breakthrough, especially to a layman.

That said: I'll submit the first detection of gravitational waves as two black holes merged together in 2020 as meeting the bar of "notable breakthrough in the last decade".

Absolutely, they're still handing out nobels after all.

Personally I think the ER=EPR conjecture and the complexity/action duality hypothesis are incredibly interesting. Technically ER=EPR was formulated in 2000s (maybe 90s?) and CA-duality is approaching if not just past 10 years old, but the thing about asking for breakthroughs is that they take a while to percolate. Ex Hawking radiation wasn't formulated until, like, 50-70 years after the "basis" (schwarzshild, Schrodinger) was formed.

Also JWST just keeps on giving, and gravitational waves were only confirmed in 2017. If you extend a bit further higgs was in the 2010s

So, in summary, in the late 10 years
- we've shown a break in our intuition of physics (nonloca-realness, that 2022 paper)
- proposed some novel yet elegant theories (CA-duality, and I'd hope you'd begrudge me er=EPR)
- confirmed some insane provings to the underlying reality (gravitational waves)

If those aren't noteworthy, I'd ask what you consider noteworthy any why you consider it noteworthy

The Higgs boson was predicted in 1964. Gravitational waves were predicted in 1916. Bell's theorem was published in 1964. Basically every recent discovery in physics has been observations confirming old predictions and refuting the endless zoo of poorly motivated, imaginary particles that seems to be standard practice these days.

There have been almost no truly significant, novel predictions that have a hope in hell of panning out in like, 40 years or more. The only mildly interesting, novel idea in physics has been quantum computing, and even that was first published in 1980.

> So, in summary, in the late 10 years - we've shown a break in our intuition of physics (nonloca-realness, that 2022 paper)

This paper showed no such thing, it has the same superdeterminism loophole as every other paper attempting to refute local realism.

Physics is stuck in a local QM-GR minimum, and some truly novel ideas are needed to kickstart things again. Oppenheim's postquantum gravity is the first truly novel idea I've seen in awhile.

I also agree that JWST is giving us great data, some of which has placed LCDM on the ropes, but astrophysicists are hard at work adding epicycles to keep it alive.

Experimental validation of a major theory is major physics.

You can't rhetorically gloss over something as important as experimentally validating a 1964 prediction as though it doesn't matter or didn't happen.

If your contention is that a validation of something we already suspected to be true doesn't shatter/shift our paradigm, then how often would you expect that to happen? I would expect it a lot in small ways (so almost every person working in some niche area has probably had some "niche breakthrough" happen in their area that has really changed things) but not a lot in really fundamental overarching ways which for physics I think you could reasonably say has happened about 4 or 5 times in the last 400 years maybe idk: Newton, GR/SR/ quantum mechanics and then whichever ones you want to count out of Maxwell's equations and whatnot.

So to expect something like that every decade is not realistic.

You don't consider ER=EPR as novel? Or CA-duality? Agreed that Post-quantum gravity is really cool and "fresh"

Higgs/Bell/GW were experimental results, I was indeed trying to show that there's a huge lag between prediction and observation.

Imo the paradigm shift that we're slowly undergoing is thinking about physics from a information theoretic perspective instead of a kinematics one. I'd argue that's even more fundamental of a change than Newtonian physics to early GR & QM.

Yes, the Higgs boson is not a new prediction but that doesn’t mean discovering it wasn’t a major breakthrough. There are other theories besides the Standard Model which don’t contain Higgs particles and were ruled out when it was discovered.

Experimental validation of the Higgs Boson (2012 but close enough) [1]

Direct observation of gravitational waves (2015)[2]

exoplanets going from theoretically quite likely to being actually observable things that we find all over the place [3]

...would seem to be examples of very notable results during my lifetime. This is barely scratching the surface and I'm not a physicist but those seem very important to me and likely to stand the test of time and be thought of as important in the future.

Non-breakthroughs:

These guys who are responsible for the goddam blue leds that on every second device these days always keep me from getting a decent dark nights sleep when travelling until I hunt them all down in the hotel room I'm in or whatever and cover them up.[4]

This is an interesting take, the article touches on it too.

> “Physicists are much less concerned than mathematicians about rigorous proofs,” says Timothy Gowers, a mathematician at the Collège de France and a Fields Medal winner. Sometimes, he says, that “allows physicists to explore mathematical terrain more quickly than mathematicians.”

GP is right that the currently observed physical laws go far beyond our ability to observe them in reality because of the cost of observations. International effort over decades is required to create facilities capable of making helpful new observations: think of LHC, LIGO, James Webb, etc.

On the other hand, once the facilities are built and ground-breaking observations appear, we suddenly have a debt of theoretical and simulated exploration to understand all their implications. The low cost of computation greatly extends the value we can take from every truly new observation of reality.

In order to observe something new, we must be able differentiate it from something already understood. It seems like the physics and math communities are currently in a season of increasing our understanding of the existing models well enough to motivate trying to break them.

Furthermore, even if you can write down the math, it might not even be solvable (sometimes provably so) and simulations (or more accurately, numerical analysis and the finite element method) are our only option.

I’m struggling to see your point. My interpretation is that in each of those times, we used some math to talk about things we couldn’t observe until we were able to make experiments to observe them.

Are you saying that one day we will be able to devise experiments to observe these things?

Short answer: Yes, and new instruments to make those measurements possible. That's how physics has always progressed. Why would this point in time be suddenly different from the past 400 years? Because we can't see a clear path forward? That's always been the case. Insight and ingenuity of individuals is what gets us through it every time.

> Why would this point in time be suddenly different from the past 400 years?

Back in the late 1800s physicists thought they were done, other than adding a few more decimals to the values of fundamental constants. There were a few "small areas" where things didn't make sense, but they "would figure them out". One of those small areas turned out to be relativity, and the other quantum mechanics. There are some known areas where we still don't know what is going on, but a lot of physics is adding more decimals to constants (finding fundamental particles were we expect them for example)

The real question - that we cannot answer - are the things we don't understand small things we will figure out, or major things that will again turn our understanding of the universe upside down. Your guess is as good as mine.

Because the tools we need to measure differences between the proposed models are more than the current total economic output. We'll need several generations before its feasible.

> Are you saying that one day we will be able to devise experiments to observe these things?

The problem is that we might be on an exponential scale. So instead of decades it could be centuries assuming the humanity survives and keep develop new technologies and tools.

> pontificate about things that are too small or too dark and far away to see

I was bitten by this last week. I am enrolled in an aops physics course, titled Mechanics. So the last time I took any Mechanics was 40 years ago as a high school student in India. Most of the curriculum then was about stuff banging into each other aka collisions, & asking what happens to the result. Like some golf ball rolls down an inclined plane at some angle theta & hits a identical stationary ball & the objects stick together & we're supposed to compute where they end up. I was curious what American students learn, so I enrolled in this aops course.

Last week's assignment asks me - under which scenario will conservation of momentum make an accurate prediction. The 3 scenarios are - truck collides with car, eagle comes to rest at perch after flying from far away, and two galaxies colliding into each other to form a third megagalaxy.

I naturally picked truck & car - so aops knocks off 2 points for the wrong answer! Apparently if a truck collides with a car there will be so much thermal energy produced by the friction of the road, any prediction you make about the final velocity of the car assuming conservation of momentum will be bogus.

So then I pick eagle coming to rest - aops knocks off 2 more points for the wrong answer! Apparently when eagle comes to land, it will open its giant wings to create air resistance, so momentum won't be conserved.

Ok so that leaves the 2 galaxies. I pick that & get my correct answer, a pathetic score of 3/7. I'm left wondering how do we even know this is correct. Galaxies are too far away to observe. How is one supposed to compute the mass of 2 separate galaxies, & then find their moving velocities accurately, & then find the final velocity of the combined galaxy, & thus confirm momentum was conserved ? Seems very far fetched. I would rather go with the car & truck.

With the car & truck, and with the eagle, a lot of energy is lost into the environment. With two galaxies colliding, there is no environment to speak of. They're in space, and space is as close to a perfect vacuum as one might reasonably imagine. So the only real escape for energy is light, but I don't think there's any reason to believe colliding galaxies will produce absurd amounts of excess light (if it did we'd have observed it). Therefore the energy of the collision is preserved within the system, and so conservation of momentum will produce an accurate prediction.

I have zero issues with your logical reasoning - its the same reasoning aops gives. I'm simply asking - how do you know for sure that this actually happens in practice ? Like, have we observed such a thing ? I'm genuinely curious, no snark.

If all we have is reasoning, with zero observation, then its equivalent to what JFK says in Oliver Stone's picture - "Theoretical physics can prove that an elephant can hang from a cliff with its tail tied to a daisy."

I have no idea if we've observed it. Galaxy collisions take a very long time, I'm not sure how we'd ever be in a position to observe both the starting state and result.

That said, the point of the law of conservation of momentum is we don't need to observe it to know that it happens. Momentum is conserved, that's a fact. So the question becomes, what is it that makes this law not produce a good prediction about a scenario? And the answer is that when the scenario involves other factors that can take kinetic energy away from the system. In the car & truck scenario, we have friction with the road as a pretty huge factor, that removes a lot of energy. Even without a collision, friction is why a car needs to constantly burn fuel in order to keep moving at the same speed. So it should be no surprise that a car & truck collision will lose a ton of energy to friction. In the eagle landing scenario, that's not even a simple collision between the eagle and the branch, the eagle uses its wings to slow down, and the branch is fixed to a tree and absorbs the remaining energy. An eagle landing on a branch is an eagle that comes to rest on the branch, it doesn't just not conserve momentum it gets rid of all of it. But in the colliding galaxies scenario, there's no surface to have friction with, there's no wings beating against air, there's no tree anchoring one of the objects in place, nothing to absorb any energy, no environment to dissipate energy in. Without anything to get rid of kinetic energy, conservation of momentum will predict the results.

The article discusses how Knot Theory was once conceived to explain different atoms and their properties. This physical explanation was abandoned after the electron was discovered (demonstrating the existence of sub-atomic structures). At that point, Knot Theory was correct math with no physical application.

I’m sure there are other examples but I’m not a mathematician.

This was the result of Philosophers ultimately winning, despite the fact that they are so annoyingly pedantic that we pretend not to care about their work, and also, by ignoring them we can invent iPhones which are really neat.

You can’t prove anything by observation. You can gather evidence through repeat experiment and become reasonably confident as your theory continues to not be incompatible with the observed universe. Then the problem of induction says, “well, it isn’t incompatible with the part of the universe… that you’ve observed, yet!” And then you say, “ok, but I want to use my theory to invent an iPhone, and I think there are enough people in the part of the universe that I’ve already observed. I looked very hard to find evidence against my theory, and I don’t think anyone will find evidence against it before I’ve sold enough iPhones to retire.”

Math, of course, is that stuff which can’t be invalidated by observations. But it is very hard to do enough math to retire off it.

> Math, of course, is that stuff which can’t be invalidated by observations.

This is a misunderstanding of what math is, I think. You can invent a perfectly valid mathematical theory that conflicts with observations. Math is just a sequence of "if this, then this" and if the conclusions follow from the premises, it's math. But if you demonstrate that in the universe we occupy, a premise isn't true, then the mathematical theory isn't any less valid, it's just not sound in our universe.

For example, there's a ton of completely valid mathematical work on the correspondence between anti-de Sitter spaces and Conformal Field Theory. However, much of this mathematics has no application in our universe, because our universe seems to be a de Sitter space (positive cosmological constant/expanding), not an anti-de Sitter space (negative cosmological constant/contracting). That doesn't make their math invalid, it just makes it not real.

You can also do a lot of math in Minkowski spaces, which are flat. But our universe isn't flat, it's curved. Doesn't mean it's not math, just that it's not real in our universe.

I think this is a disagreement about the word invalidate, and maybe it was a bad pick on my part. The observations don’t make the math wrong, maybe less useful.

But, I say the math can’t be invalidated by observations; and you describe some cases where the math is valid but might or might not be applicable to certain physical cases. So actually, I’m not clear as to what you are saying I’m misunderstanding.

No, proof by contradiction is a logical construction, has nothing to do with the real world. Proof by counter-example could or could not be observation, depending on whether your example comes from observation or from pure logic.

“Prove” could mean many things. For physics, prove what?

That a given model is the one and only model that accurately explains the known universe? Then I agree. Observation won’t get you there. Asymptotic at best.

But by “prove”, that it accurately or usefully makes predictions with respect to certain constraints (which may not be known)?

That’s a more modest use of
“prove”, where observation is certainly a key factor.

Maybe proof should be used for that more modest concept. In the sense of “bullet proof.” Similar to a piece of armor with a bullet proof; the ding from where it was shot for testing, a physics theory could be described as “microscope proof,” haha. We hit it with a bunch of microscopes and it didn’t fall apart.

Interesting linguistic aside: "the exception that proves the rule" makes use of this more old fashioned use of prove, to test. So it's the exceptional case that tests the rule.

With that in mind, "prove" is perfectly fine to use in the context of science.

Well also the idea of physics as the field we currently have didn't exist much before the 17th century. Movement of bodies, astronomy, fluid dynamics, electromagnetics, optics, etc. all kind of were their own thing (if they existed at all). Fundamental developments in calculus in the late 1600s enabled these subjects to be collected under one method of study/analysis which we now call physics. As much of modern math follows from the lineage of calculus the border between the things being modeled and the tools for modeling them is naturally kind of blurry, however the distinction did still exist quite strongly throughout this entire period. Look at ex. probability or algebra, although often researchers were pursuing both physics and math, they were aware that the subjects were distinct.

Math probably split off a bit because of the attempts at formalization. That was a useful tangent though, arguably giving us computer science via the lambda calculus, Turing machines, etc.

Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.

What does that mean? Physics is still empirical at the end of the day. Experiments decide what theories best explain the world. Math doesn't have such a requirement. It doesn't need to model natural phenomenon. Your physics lecturer sounds like a Platonist.

> "Your physics lecturer sounds like a Platonist."

I don't understand what this means, but it made me envision a McCarthy-esque witch hunt for "Platonist and Platonist sympathizers" lurking amongst the faculty

As fundamentally opposed to Mathematical Plantonism, I am fully of the opinion that I am nowhere near anything but a minority position among those that hold an opinion on the philosophy of mathematics. It would be kind of hard to have a witch hunt for something so common amongst working mathematicians:

The saying that “the typical working mathematician is a platonist during the week and becomes a formalist on Sunday” is becoming increasingly familiar. During working days, they are convinced that they are dealing with an objective mathematical reality that is independent of them, and when on Sunday they meet a philosopher who begins to question this reality, they claim that mathematics is in fact the juggling of formal symbols (see Davis et al., 2012, p. 359). The Platonist attitude of the working (rather than philosophizing) mathematician is so common that Monk (1976, p. 3) was tempted to make a subjective estimate to the effect that sixty-five percent of mathematicians are platonists, thirty percent formalists, and five percent intuitionists. [1]

[1] - A Metaphysical Foundation for Mathematical Philosophy (Wójtowicz,Skowron 2022)

Platonist meaning an assumption that mathematical objects have a real existence, and the universe is inherently mathematical, so we can just defer to mathematical reasoning instead of observation. I'm applying the term in a modern setting, not Plato or Aristotle debating the forms.

This debate played out in String Theory were some proponents claimed physics had progressed beyond the need for observation in favor of beautiful mathematical reasoning, that provided great explanatory power. But String Theory so far has failed to deliver a theory which describes our universe. Physics still needs to explain the actual world.

Just say witch hunt. McCarthy was entirely correct that there were a lot of communists and communist sympathizers, so many that many of the people he thought were helping him were themselves communists or communist sympathizers. Witches on the other hand are not real.

He missed most of the actual Soviet agents, though, right? Both seem to have mostly focused building up a lie to suppress some outside-the-norm element that the guy in charge wanted to hunt. That McCarthy also managed to have some real agents to miss is not a big difference.

I don't think OP is really wrong here. Wasn't there a debate in the late 19th century that basically asked if math had to have some mapping to the natural world, or should it work independently? I though there was some argument about this with Hilbert and Poincaré about this, and Hilbert more or less won.

Geometry? Lobachevsky actually proposed a test on measuring sum of angles of a celestial triangle to decide which geometry actually applied to the real world.

There are multiple geometries though, they don't have to describe the real world. In mathematics you are free to base your geometry on whatever axioms you want, whether they are realistic or not. I'd say the question of which geometry applies in the real world is more a question for physicists (or cosmologists, today).

No, it's trying to prove that Euclid's parallel postulate can be derived from other axioms is what goes back millennia. People were certain it's a) true, b) necessary consequence of other axioms. Gauss was probably the first to consider the possibility that it may be false; others at best tried reductio ad absurdum, arrived to some wildly unusual theorems, decided those were absurd enough to demonstrate the truth of the fifth postulate, and went back to trying to derive it.

It's either true as an axiom or true as derived from other axioms and/or theorems. In neither case does latching onto Euclid's other common notions/postulates/theorems as the selection it must be proved true from make sense as a 1.5kya long task.

I think there must have been a sense that it was true only as an axiom. Proving it from other axioms/theorems was then a goal to secure it's truth "further". But you'd only attempt that if you thought there was something questionable in the first place.

The real world determines which geometry applies. That's the crucial difference. Sometimes physicists use mathematical reasoning to figure things out that map onto the world and later observation validates their reasoning. But observation can also invalidate, and then physicists have to go back to the drawing board or devise new experiments.

It's a very weak loose statement, but I think the idea is that leading scientific and mathematical thinkers (think Newton as a quintessential example) were "natural philosophers" who studied whatever caught their interest and took it wherever it went. Astronomers invented lenses and ground them and studied the starts and developed algebra and calculus to model the observations.

Some people were more narrowly focused, like Gauss who did mostly math (but an amazing breadth of math!)

There was a lot of hesitancy about math that couldn't be empirically illustrated by building out of atoms, like irrational numbers and then transcendental numbers and imaginary numbers and then infinite structures.

That's metaphysics not science. I'm not against metaphysics, but philosophy is distinct from physics. And it's Tegmark's particular metaphysics. Interesting but hardly a consensus among physicists or philosophers.

I judge the beauty of math and code by the how concisely and simply they can model the object in question - if the math exists abstractly and isn't trying to model any actual thing or concept it isn't particularly beautiful to me because it isn't an accomplishment of expressiveness - it's instead just a coincidence that if you put some symbols together you only need those symbols.

There are, of course, times that concise expressions aren't possible and multiple strange and arbitrary values come into play (coefficients of friction or earth's gravity at sea level aren't particular nice numbers or expressions) and that just tends to highlight how beautiful things are when those icky real-world numbers can be canceled out and you're left with a clean expression.

This is really interesting to me. It could imply that an ugly theorem is less valuable. What if a theorem is ugly but useful vs a beautiful but esoteric one?

Then you should study Algorithms, where "the perfect answer is impossible, without simply trying all permutations, but this process gets you a close answer 99.9% of the time"

> Might there be certain laws of physics that are also “necessary” in the same way? In his paper, Molinini argues that the principle of conservation may be one such law. In physics, some properties of a system, such as energy or momentum, can’t change. A bicyclist freewheeling down a hill, for example, is converting her gravitational potential energy into movement energy, but the total amount of energy she and her bike have stays the same.

Arithmetic itself is a consequence of physical conservation: if you have a collection of four acorns, another collection of three acorns, then combine them without dropping an acorn, then you must have a collection of seven acorns. It is our deep physical understanding of space and causality which leads to simple arithmetic being intuitively true to most (if not all) vertebrates. (If the squirrel only got six acorns after combining then there must be a causal explanation for the quantitative discrepancy; another squirrel stole an acorn from the older stash, or maybe it fell in a hole.)

Do we know if it’s better at creating new math than other fields? For example, computers sure created a lot of new math. Statistics was entirely driven by external pressure from medicine, social sciences, and business. Finance and economics created a lot of math around modeling and probability. And so on.

Could this be a case of physics being more "tangible", thus leading to more obvious paths? Like, if you only study pure maths and you stay in your field, can you really point to a concrete direction for where to look for new stuff? In physics, your job is literally to study how the universe already behaves, so you have a frame of reference to take inspiration from and the efforts are a bit more concentrated. In fact, since models don't describe reality perfectly, you can always observe where it fails to know in which direction to attack. In maths, on the other hand, everything that is proven is correct forever. The model is reality. So it seems to be more difficult to find criticalities to look for. It's more of a "I wonder if this property does or doesn't hold" ordeal, which seems much more vague. Just a question.

I understand that one huge reason for Ed Witten's optimism about strong theory is this very fact. That, in his terms, the process of building out string theory has led to the uncovering of so much "buried treasure" in the form of novel developments of maths.

Of course it's not anything like a proof but something that bolsters an intuition.

Witten has revolutionized mathematical physics, and his development of topological quantum field theory was nothing short of monumental.

Much of Witten's own point above is that advancements in string theory have cashed out in revolutionary new mathematical approaches that would be of lasting value even string theory itself never receives any experimental confirmation.

I think article highlights something very beautiful about how physics, including string theory, have lead to the creation of new math, and how that is suggestive of an unmet promise. To ignore that just to come in and repeat for the 1000th time the world's most repeated thing about string theory, and take a completely unnecessary cheap shot at Ed Witten is the perfect embodiment of why comment sections can too often be a depressing waste of time.

The only cheap thing here is the amount of actual physics that came out of all of this ptolemaic endeavor.

And writing ptolemaic is probably too charitable because the Almagest at least predicted movements quite well at the time (apparently it now deviates too much).

Is your preferred remedy that quantum gravity be entirely defunded, or instead that more funding be redirected to any of the other programs to study quantum gravity? If the latter, which ones in your opinion are more likely to be productive than string theory?

Your suggestion implicitly asserts that string theory was productive, which is exactly the claim that seems to be in contention.

I don't think it's too wild to suggest that, without the constraints of string theory imposed by advisors, lots of novel approaches would have been tried. We have no idea what could have been produced.

As for quantum gravity specifically, arguably not much progress will be made without more data, and we now have some proposed experiments that can be conducted here on Earth to test them.

There are in fact exceptionally strong incentives to discover alternatives to quantum gravity which could be tested in experiments. These are the same incentives that always drive the scientific process, and new theories cost next to zero to produce. The reason string theory is popular is not because string theorists somehow prevent funding of other directions. It is because string theory has given us tools like AdS/CFT that are useful in other contexts to understand real physics—-and the alternatives have not (yet). There are many physicists who spend their lives studying alternatives to string theory with 100% of their time. I hope for their sake that there is a similar pot of gold at the end of their rainbows. It has not yet materialized.

Oh Ads, you mean that space that emphatically does NOT describe our universe? Ads/CFT is overblown. It's just an interesting mathematical result that hasn't borne much meaningful fruit for actual physics.

I'm sorry, but string theorists absolutely do prevent funding other research because funding is finite, grad students have to research something their advisors think is worthy, and their advisors have their heads full of "beautiful math" so that's what they tell their students to work on if they want their PhDs, and that's what they hire their post grads to work on if they want a job.

Only now as the strong theory haze has started dissipating are we starting to see novel approaches, like Oppenheim's post quantum gravity theory.

No, this is a shallow understanding of AdS/CFT. If you want to study quantum gravity when it is weakly coupled to matter, you can use AdS/CFT regardless of whether the background space is asymptotically AdS by embedding a brane near the boundary and working in a perturbative expansion. If you want to study the physics of quantum de Sitter space with a field theory dual, you can study any of the recent work on TTbar deformations. And anyway, surely you aren’t arguing that conformal field theories are irrelevant for physics? Because that would obviously be an untenable position, and the whole point is that quantum gravity AdS (basically) is CFT (it’s an equality! It goes both ways), just in different variables. You can actually study non-gravitational physics with it, using a gravitational language. That’s awesome stuff! Please don’t dismiss this fascinating field too quickly.

By the way, I know Oppenheim personally. He gets funding from string grants. Nobody is angry about that. Anybody can do this. I don’t think his theory is going to pass any experimental validation (it requires a really severe violation of a physical principle we have tested over and over) but the entire community has always supported and listened. He gives talks at major universities. He’s not an outcast or renegade or something.

Something I've speculated about is that we just don't know if and when the next answer will arrive. The past few centuries have certainly stoked our expectations for rapid progress, but it's not guaranteed. There's no success-or-failure metric for the search, until an answer is found.

It took humanity something like 1000 years between when the ancient Greeks could formulate general quadratic equations, and someone came along who solved them. And another several centuries before the solution was extended to included complex numbers.

The problem that they tried to solve with string theory: Will it be solved in a decade, a century, a millennium, or ever. Is it possible that bending the rules of scientific methodology, even temporarily, will help us find an answer?

I've long thought physics is a subsection of maths and reality is a mathematical object that exists the same way that prime numbers or the Mandelbrot set do. Hence the unreasonable effectiveness of one in the other.

This is the well known litany from string theorists to try and justify the inordinate amount of money threw at them to get back nothing of physical value: no falsifiable prediction.

Instead of reasoning on the worth of the effort spent in this direction to investigate nature (a very tangible companion) they try to steer the discourse toward this nonsense. We spent >50 years listening to these tales and the time has long passed since we are required to stop playing with these smoke and mirrors.

> justify the inordinate amount of money threw at them

They're theorists, you're paying for pencils and paper. String theory may not have produced a theory of quantum gravity yet, but neither has any other line of inquiry.

You are right, the amount of money spent in string theory proposals has been staggering if you take into account the size of the field. For decades competing (or even just non-aligned) research lines have been starved to feed this behemoth.

Particle accelerators have been built since way before any string theory was formulated.

The biggest and most powerful existing accelerator (LHC) has been built to fulfill the high energy/high luminosity requirements to explore the Highs boson energy regime (that has been found) and at most the lightest supersymmetric particles (not found as of today).

The Higgs boson is a cornerstone of the Standard Model. Supersymmetry is an extension to it that does not involve strings.

And right now we have no reason to think either would produce more useful math. Math at least is honest about studying math for the sake of math (or truth and beauty). Nothing wrong with throwing money at math, but physics is supposed to be about understanding the universe so if we only get math we get a bad return on investment no matter how nice the math is. Even if the math turns out to be useful elsewhere we didn't get what we wanted out of the investment.

They produced a family of theories with an infinite amount of underlying possible geometries, and we still don't know if these reduce in the low energy limit to the standard model. I humbly suggest this reading:

I'm not a physics or math whiz but isn't the relationship more of a virtuous cycle?

I think I read that the 20th century was a revolution because of the marriage between physics and math. Quarternions are key to relativity. Discrete math is littered all over quantum mechanics and the Standard Model. Like U(1) describes electromagnetism, SU(2) describes the weak force and SU(3) describes the strong nuclear force. In particular the mass of the 3 bosons that mediate the weak force is what led directly to the Higgs mechanism being theorized (and ultimately shown experimentally).

One of the great advances of the 20th century was that we (provably) found every finite group. And those groups keep showing up in physics.

The article mentions how string theory has led to new mathematics. This is really interesting. I'm skeptical of string theory just because there's no experimental evidence for "compact dimensions". It seems like a fudge. But interestingly there have been useful results in both physics and maths based on if string theory was correct.

Physics is also great for machine learning, though the approaches can be rather unintuitive. For example message passing and belief propagation in trees/graphs (Bayesian networks, Markov random fields etc.) for modeling latent variables are usually taught using the window/rainy weather marginal probability analogy and involves splitting out a bayesian/statistical equation into subcomponents via the marginalization chain rule. For physicists however, they tend to teach it using Ising models and magnetic spin, which is a totally different analogy.

A lot of the newer generative ML models are also using differential equations/Boltzmann distribution based approaches (state space models, "energy based" models) where the statistical formulations are cribbed wholesale from statistical physics/mechanics and then plugged into a neural network and autodiff system.

The best example is probably the Metropolis-Hastings algorithm which was invented by nuke people.

One that many people may be familiar with is Stable Diffusion, which is used in many AI image generators today. There is an analogy between random noise -> image and a random distribution of gas particles -> concentrated volume of particles.

The common method for choosing the next output token for an LLM is sampling from a Boltzmann distribution. If you have seen the term "temperature" in the context of language models, that is a direct link to the statistical gas mechanics.

i don't find the connection between softmax and boltzmann really all that deep tbh (compared to say, the connection between field theory/ising models and EBM)

Discoveries are made and measurements are taken with the tools available.

The measurements, theories, and currently understood or applicable math may not match up with observations.

People ponder and discover, then attempt to explain the observations and measurements with a new theory. If the theory pans out, a deeper explanation of that theory is necessary and that's where the new math's at.

It's not that physics is good at creating math. Physics is good at describing our observations /with/ math. That's kind of its whole job.

Next time you look at raindrops in a puddle, try to imagine how you would describe those movements scientifically. One needs math for that.

Sometimes the available tools and math are sufficient for a thorough explanation, and sometimes one needs to invent a universe of math to describe a tiny fluctuation.

The subject of study of physics is "physical quantity" which is defined as a number with a unit. Physical quantity doesn't have to be a "physical" quantity. So physics does not study exclusively physical objects. I think this is how mathematics and physics are related, mathematics does not deal with units (except unity).

Physics research gets funded because of applications, existing or promising, for curiosity, fundamental science, the economy, medicine, and national security, etc. Since math can help physics research, that research is funded and motivated to make applications of math, old or new. Math alone is less involved with applications.

I’m not sure how it could be otherwise. On some level mathematics is a description of reality that we can use to compute things in reality.

For example, pi is the ratio of a circle’s circumference to its diameter. It’s just what a circle is in two dimensions. The value of pi isn’t any more mysterious or connected to physics than the existence of this thing called a circle. If you have some other Euclidean shapes you’ll have other ratios and values that have other relationships to other things in physical reality.

And if reality was different, hence the physical laws were different then the math would be different.. and the beings in that world might wonder why their math and physics were so interconnected.

> On some level mathematics is a description of reality that we can use to compute things in reality.

This is contested by nominalists. They'd say you have it backwards. Mathematics is just an abstraction/language that can be used as a tool. The reason we're able to understand the world through mathematics says more about the power of mathematics than it does about the world. If the physical world were different, math would still work.

Well if the fundamental constants were different, math would still work the same way. But if a universe acts in a completely different way, say like a dream world, where they is no logical physical causaulity, our universe's math would be irrelevant. You could still think about it but it would be irrelevant to that universe, and not applicable. And so you would also not be able to derive that math from within this universe, at least not the same math as ours. So we have to admit that our mathematics is tied to the universe we inhabit.

Math is informed by physics, but not constrained by it.

Very loosely speaking, pi can take a different values on non-euclidean planes. This ends up becoming relevant on the surface of the earth or, say, the saddle of a horse. I'm not sure if the motivation was from looking at curved surfaces, but it just as easily could've come from the rejection of Euclid's parallel postulate and seeing what results. Similarly, I think imaginary numbers were motivated by the math well before they found applications in reality.

There are also plenty of other mathematical constructions that are informed by reality (since that's what our brains are constrained to,) but I'm pretty sure are far from actually describing reality. Transfinite cardinals/ordinals, fast growing hierarchies, Turing degrees, Goldbach's conjecture, how the hypervolume of a hypersphere eventually decreases as dimension increases...

You can even reject the standard axioms and construct math that can not be compatible with reality. Or argue that the standard axioms permit too much wiggle room to create concepts that have no relation to reality. (But maybe you shouldn't; that sounds like philosophy.)

Not sure that I agree as does a mathematical circle actually exist? We can produce things that approach the concept of a circle and similarly, we can measure circumference and diameter to a level of precision to approach the value of pi, but we never have a perfect circle or the exact value of pi.

I tend towards maths being distinct from physics as some areas of maths deal with concepts that can only have a passing resemblance to reality - the Banach-Tarski paradox is an example. (Similarly, pretty much any treatment of infinities ends up to have little relation to reality such as with Hilbert's Hotel).

If this was true then every bit of math must have a real-world application. But math doesn’t need to have real-world applications in order to be valid.

“Physicists are much less concerned than mathematicians about rigorous proofs.
...
That allows physicists to explore mathematical terrain more quickly than mathematicians.”

Disclosure, I'm a mathematician.

Interviewer -- "Consider a situation where you are in your office and there is a fire outside in the hall. There is a fire escape outside your window but you can't reach it because the window is stuck. However, there is a hammer on the table. What do you do?"

Physicist -- "I use the hammer to break the window, allowing me to get out to the fire escape."

Interviewer -- "Now consider the same situation except that the hammer is on the floor. What do you do?"

Mathematician -- "I move the hammer from the floor to the table, thereby reducing it to the previously solved problem."

Disclosure, I'm a software developer

Mathematician: not yet, but if I wait here long enough someone will come by and drop their keys, which will then be retrieved, proving the possibility of retrieving lost keys when light conditions are optimal.

Physicist drops keys.

Mathematician: Eureka!

on edit: can I stay at your house for a bit while waiting for publication?

[1] https://en.wikipedia.org/wiki/Nasreddin#cite_ref-32

"Don't use statistics like how a drunkard uses a lamp-post, for support rather than for illumination".

Disclosure: I’m a software developer

I call those "monitors."

The mathematician of this joke would scan the edge of the light, finding nothing. Then he would keep lighting little lanterns at the perimeter to make the lighted area larger until finally his keys were within sight.

The physicist in this joke would presumably root around in the dark where she thinks her keys actually were. Upon finding them through brute force and luck, she might think “wow maybe one day this place will be illuminated so I can tell wtf I just did”

[0] Very few practices in English writing are universal.

it's explicitly designating a female gender. Using "they/them" is far less awkward, the reader doesn't have to have any kind of opinion about the text based on gender associations,making it less awkward.And annoying certain kinds of pedant is a bonus more than anything.

Disclosure: I'm a software developer.

https://xkcd.com/435

This is kind of it I think. It's not just physics that drives interesting math, and it's not just recently that this relationship holds. Math is, IM humble O, the ultimate domain-specific language. It's a tool we use to model things, and then often it turns out that the model is interesting in its own right. Trying to model new things (ex. new concepts of reality) yields models that are interesting in new ways, or which recontextualize older models; and and so we need to reorganize, condense, generalize, etc; and so the field develops.

https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology

Personally I tend to disagree with "I hope it isn't useful" or whatever the GHH quote is about maths being practicably applicable to the world/universe n that.

Why not tell us what you think? Mr Hardy's well documented positions on many things are well known but yours are not.

twentieth-century idea: it wasn't made during the nineteenth century or before, and it seems to be disappearing in the twenty-first.That's because people were totally focused on physics, and math was just a useful tool sometimes. Doing physics was the true goal and observation the final arbiter of truth.

Nowadays, that distinction is blurred but for the opposite reason; people think that anything conceived by sound math must be true, and observation has taken a back seat.

But there are also active researchers doing real research. Physics postdocs aren’t just sitting around in a circle making up stories about what the universe is like.

i ask out of layman curiosity

That said: I'll submit the first detection of gravitational waves as two black holes merged together in 2020 as meeting the bar of "notable breakthrough in the last decade".

2015. (Your point is otherwise taken).

https://en.wikipedia.org/wiki/First_observation_of_gravitati...

Personally I think the ER=EPR conjecture and the complexity/action duality hypothesis are incredibly interesting. Technically ER=EPR was formulated in 2000s (maybe 90s?) and CA-duality is approaching if not just past 10 years old, but the thing about asking for breakthroughs is that they take a while to percolate. Ex Hawking radiation wasn't formulated until, like, 50-70 years after the "basis" (schwarzshild, Schrodinger) was formed.

There's also been a ton of productive research integrsting computer science and physics lately ( on hn last week: https://arxiv.org/abs/2403.16850 and 2022 novel prize; https://www.scientificamerican.com/article/the-universe-is-n...)

Also JWST just keeps on giving, and gravitational waves were only confirmed in 2017. If you extend a bit further higgs was in the 2010s

So, in summary, in the late 10 years - we've shown a break in our intuition of physics (nonloca-realness, that 2022 paper) - proposed some novel yet elegant theories (CA-duality, and I'd hope you'd begrudge me er=EPR) - confirmed some insane provings to the underlying reality (gravitational waves)

If those aren't noteworthy, I'd ask what you consider noteworthy any why you consider it noteworthy

There have been almost no truly significant, novel predictions that have a hope in hell of panning out in like, 40 years or more. The only mildly interesting, novel idea in physics has been quantum computing, and even that was first published in 1980.

> So, in summary, in the late 10 years - we've shown a break in our intuition of physics (nonloca-realness, that 2022 paper)

This paper showed no such thing, it has the same superdeterminism loophole as every other paper attempting to refute local realism.

Physics is stuck in a local QM-GR minimum, and some truly novel ideas are needed to kickstart things again. Oppenheim's postquantum gravity is the first truly novel idea I've seen in awhile.

I also agree that JWST is giving us great data, some of which has placed LCDM on the ropes, but astrophysicists are hard at work adding epicycles to keep it alive.

You can't rhetorically gloss over something as important as experimentally validating a 1964 prediction as though it doesn't matter or didn't happen.

If your contention is that a validation of something we already suspected to be true doesn't shatter/shift our paradigm, then how often would you expect that to happen? I would expect it a lot in small ways (so almost every person working in some niche area has probably had some "niche breakthrough" happen in their area that has really changed things) but not a lot in really fundamental overarching ways which for physics I think you could reasonably say has happened about 4 or 5 times in the last 400 years maybe idk: Newton, GR/SR/ quantum mechanics and then whichever ones you want to count out of Maxwell's equations and whatnot.

So to expect something like that every decade is not realistic.

Higgs/Bell/GW were experimental results, I was indeed trying to show that there's a huge lag between prediction and observation.

Imo the paradigm shift that we're slowly undergoing is thinking about physics from a information theoretic perspective instead of a kinematics one. I'd argue that's even more fundamental of a change than Newtonian physics to early GR & QM.

https://youtu.be/d_o4k0eLoMI

Maybe a more direct answer to your question would be the discovery of the tetraneutron.

Direct observation of gravitational waves (2015)[2]

exoplanets going from theoretically quite likely to being actually observable things that we find all over the place [3]

...would seem to be examples of very notable results during my lifetime. This is barely scratching the surface and I'm not a physicist but those seem very important to me and likely to stand the test of time and be thought of as important in the future.

Non-breakthroughs:

These guys who are responsible for the goddam blue leds that on every second device these days always keep me from getting a decent dark nights sleep when travelling until I hunt them all down in the hotel room I'm in or whatever and cover them up.[4]

[1] https://www.nobelprize.org/prizes/physics/2013/summary/

[2] https://www.nobelprize.org/prizes/physics/2017/summary/

[3] https://www.nobelprize.org/prizes/physics/2019/summary/

[4] https://www.nobelprize.org/prizes/physics/2014/summary/

https://en.m.wikipedia.org/wiki/List_of_unsolved_problems_in...

If it was not possible to simulate, I think we'd be less invested in the math and physics of it.

> “Physicists are much less concerned than mathematicians about rigorous proofs,” says Timothy Gowers, a mathematician at the Collège de France and a Fields Medal winner. Sometimes, he says, that “allows physicists to explore mathematical terrain more quickly than mathematicians.”

GP is right that the currently observed physical laws go far beyond our ability to observe them in reality because of the cost of observations. International effort over decades is required to create facilities capable of making helpful new observations: think of LHC, LIGO, James Webb, etc.

On the other hand, once the facilities are built and ground-breaking observations appear, we suddenly have a debt of theoretical and simulated exploration to understand all their implications. The low cost of computation greatly extends the value we can take from every truly new observation of reality.

In order to observe something new, we must be able differentiate it from something already understood. It seems like the physics and math communities are currently in a season of increasing our understanding of the existing models well enough to motivate trying to break them.

Are you saying that one day we will be able to devise experiments to observe these things?

Back in the late 1800s physicists thought they were done, other than adding a few more decimals to the values of fundamental constants. There were a few "small areas" where things didn't make sense, but they "would figure them out". One of those small areas turned out to be relativity, and the other quantum mechanics. There are some known areas where we still don't know what is going on, but a lot of physics is adding more decimals to constants (finding fundamental particles were we expect them for example)

The real question - that we cannot answer - are the things we don't understand small things we will figure out, or major things that will again turn our understanding of the universe upside down. Your guess is as good as mine.

The problem is that we might be on an exponential scale. So instead of decades it could be centuries assuming the humanity survives and keep develop new technologies and tools.

I was bitten by this last week. I am enrolled in an aops physics course, titled Mechanics. So the last time I took any Mechanics was 40 years ago as a high school student in India. Most of the curriculum then was about stuff banging into each other aka collisions, & asking what happens to the result. Like some golf ball rolls down an inclined plane at some angle theta & hits a identical stationary ball & the objects stick together & we're supposed to compute where they end up. I was curious what American students learn, so I enrolled in this aops course.

Last week's assignment asks me - under which scenario will conservation of momentum make an accurate prediction. The 3 scenarios are - truck collides with car, eagle comes to rest at perch after flying from far away, and two galaxies colliding into each other to form a third megagalaxy.

I naturally picked truck & car - so aops knocks off 2 points for the wrong answer! Apparently if a truck collides with a car there will be so much thermal energy produced by the friction of the road, any prediction you make about the final velocity of the car assuming conservation of momentum will be bogus.

So then I pick eagle coming to rest - aops knocks off 2 more points for the wrong answer! Apparently when eagle comes to land, it will open its giant wings to create air resistance, so momentum won't be conserved.

Ok so that leaves the 2 galaxies. I pick that & get my correct answer, a pathetic score of 3/7. I'm left wondering how do we even know this is correct. Galaxies are too far away to observe. How is one supposed to compute the mass of 2 separate galaxies, & then find their moving velocities accurately, & then find the final velocity of the combined galaxy, & thus confirm momentum was conserved ? Seems very far fetched. I would rather go with the car & truck.

If all we have is reasoning, with zero observation, then its equivalent to what JFK says in Oliver Stone's picture - "Theoretical physics can prove that an elephant can hang from a cliff with its tail tied to a daisy."

That said, the point of the law of conservation of momentum is we don't need to observe it to know that it happens. Momentum

isconserved, that's a fact. So the question becomes, what is it that makes this law not produce a good prediction about a scenario? And the answer is that when the scenario involves other factors that can take kinetic energy away from the system. In the car & truck scenario, we have friction with the road as a pretty huge factor, that removes a lot of energy. Even without a collision, friction is why a car needs to constantly burn fuel in order to keep moving at the same speed. So it should be no surprise that a car & truck collision will lose a ton of energy to friction. In the eagle landing scenario, that's not even a simple collision between the eagle and the branch, the eagle uses its wings to slow down, and the branch is fixed to a tree and absorbs the remaining energy. An eagle landing on a branch is an eagle that comes to rest on the branch, it doesn't just not conserve momentum it gets rid of all of it. But in the colliding galaxies scenario, there's no surface to have friction with, there's no wings beating against air, there's no tree anchoring one of the objects in place, nothing to absorb any energy, no environment to dissipate energy in. Without anything to get rid of kinetic energy, conservation of momentum will predict the results.I’m sure there are other examples but I’m not a mathematician.

You can’t

proveanything by observation. You can gather evidence through repeat experiment and become reasonably confident as your theory continues to not be incompatible with the observed universe. Then the problem of induction says, “well, it isn’t incompatible with the part of the universe… that you’ve observed, yet!” And then you say, “ok, but I want to use my theory to invent an iPhone, and I think there are enough people in the part of the universe that I’ve already observed. I looked very hard to find evidence against my theory, and I don’t think anyone will find evidence against it before I’ve sold enough iPhones to retire.”Math, of course, is that stuff which can’t be invalidated by observations. But it is very hard to do enough math to retire off it.

This is a misunderstanding of what math is, I think. You can invent a perfectly valid mathematical theory that conflicts with observations. Math is just a sequence of "if this, then this" and if the conclusions follow from the premises, it's math. But if you demonstrate that in the universe we occupy, a premise isn't

true, then the mathematical theory isn't any less valid, it's just not sound in our universe.For example, there's a

tonof completely valid mathematical work on the correspondence between anti-de Sitter spaces and Conformal Field Theory. However, much of this mathematics has no application in our universe, because our universe seems to be a de Sitter space (positive cosmological constant/expanding), not an anti-de Sitter space (negative cosmological constant/contracting). That doesn't make their math invalid, it just makes it notreal.You can also do a lot of math in Minkowski spaces, which are flat. But our universe isn't flat, it's curved. Doesn't mean it's not math, just that it's not real in our universe.

But, I say the math can’t be invalidated by observations; and you describe some cases where the math is valid but might or might not be applicable to certain physical cases. So actually, I’m not clear as to what you are saying I’m misunderstanding.

I took your word "invalidate" to mean "be proven false" whereas I meant "valid" as in "logically coherent, even if not true."

I meant, isn't a counter-example or proof by contradiction an invalidation by a type of observation?

That a given model is the one and only model that accurately explains the known universe? Then I agree. Observation won’t get you there. Asymptotic at best.

But by “prove”, that it accurately or usefully makes predictions with respect to certain constraints (which may not be known)?

That’s a more modest use of “prove”, where observation is certainly a key factor.

With that in mind, "prove" is perfectly fine to use in the context of science.

Nice observation.

Epistemology is a harsh mistress.

— V.I. Arnold: "On teaching mathematics" (1997)

I don't understand what this means, but it made me envision a McCarthy-esque witch hunt for "Platonist and Platonist sympathizers" lurking amongst the faculty

The saying that “the typical working mathematician is a platonist during the week and becomes a formalist on Sunday” is becoming increasingly familiar. During working days, they are convinced that they are dealing with an objective mathematical reality that is independent of them, and when on Sunday they meet a philosopher who begins to question this reality, they claim that mathematics is in fact the juggling of formal symbols (see Davis et al., 2012, p. 359). The Platonist attitude of the working (rather than philosophizing) mathematician is so common that Monk (1976, p. 3) was tempted to make a subjective estimate to the effect that sixty-five percent of mathematicians are platonists, thirty percent formalists, and five percent intuitionists. [1]

[1] - A Metaphysical Foundation for Mathematical Philosophy (Wójtowicz,Skowron 2022)

This debate played out in String Theory were some proponents claimed physics had progressed beyond the need for observation in favor of beautiful mathematical reasoning, that provided great explanatory power. But String Theory so far has failed to deliver a theory which describes our universe. Physics still needs to explain the actual world.

hadto have some mapping to the natural world, or should it work independently? I though there was some argument about this with Hilbert and Poincaré about this, and Hilbert more or less won.Geometry? Lobachevsky actually proposed a test on measuring sum of angles of a celestial triangle to decide which geometry actually applied to the real world.

Principia.reductio ad absurdum, arrived to some wildly unusual theorems, decided those were absurd enough to demonstrate the truth of the fifth postulate, and went back to trying to derive it.theselection it must be proved truefrommake sense as a 1.5kya long task.I think there must have been a sense that it was true only as an axiom. Proving it from other axioms/theorems was then a goal to secure it's truth "further". But you'd only attempt that if you thought there was something questionable in the first place.

Some people were more narrowly focused, like Gauss who did mostly math (but an amazing breadth of math!)

There was a lot of hesitancy about math that couldn't be empirically illustrated by building out of atoms, like irrational numbers and then transcendental numbers and imaginary numbers and then infinite structures.

Our mathematical universe : my quest for the ultimate nature of realityby Max Tegmarkhttps://archive.org/details/ourmathematicalu0000tegm

There are, of course, times that concise expressions aren't possible and multiple strange and arbitrary values come into play (coefficients of friction or earth's gravity at sea level aren't particular nice numbers or expressions) and that just tends to highlight how beautiful things are when those icky real-world numbers can be canceled out and you're left with a clean expression.

It’s far more difficult to come up with novel mathematics without some external inspiration.

Arithmetic itself is a consequence of physical conservation: if you have a collection of four acorns, another collection of three acorns, then combine them without dropping an acorn, then you must have a collection of seven acorns. It is our deep physical understanding of space and causality which leads to simple arithmetic being intuitively true to most (if not all) vertebrates. (If the squirrel only got six acorns after combining then there must be a

causalexplanation for the quantitative discrepancy; another squirrel stole an acorn from the older stash, or maybe it fell in a hole.)isreality. So it seems to be more difficult to find criticalities to look for. It's more of a "I wonder if this property does or doesn't hold" ordeal, which seems much more vague. Just a question.https://youtu.be/obCjODeoLVw?si=2akBzyo-fC2j90OH

Entertaining viewpoint

cf. string theory

Of course it's not anything like a proof but something that bolsters an intuition.

Even Witten's achievements objectively reduce to an alternate proof of the positivity energy theorem in GR.

This is an abject failure by all metrics.

Much of Witten's own point above is that advancements in string theory have cashed out in revolutionary new mathematical approaches that would be of lasting value even string theory itself never receives any experimental confirmation.

I think article highlights something very beautiful about how physics, including string theory, have lead to the creation of new math, and how that is suggestive of an unmet promise. To ignore that just to come in and repeat for the 1000th time the world's most repeated thing about string theory, and take a completely unnecessary cheap shot at Ed Witten is the perfect embodiment of why comment sections can too often be a depressing waste of time.

And writing ptolemaic is probably too charitable because the Almagest at least predicted movements quite well at the time (apparently it now deviates too much).

I don't think it's too wild to suggest that, without the constraints of string theory imposed by advisors, lots of novel approaches would have been tried. We have no idea what could have been produced.

As for quantum gravity specifically, arguably not much progress will be made without more data, and we now have some proposed experiments that can be conducted here on Earth to test them.

I'm sorry, but string theorists absolutely do prevent funding other research because funding is finite, grad students have to research something their advisors think is worthy, and their advisors have their heads full of "beautiful math" so that's what they tell their students to work on if they want their PhDs, and that's what they hire their post grads to work on if they want a job.

Only now as the strong theory haze has started dissipating are we starting to see novel approaches, like Oppenheim's post quantum gravity theory.

By the way, I know Oppenheim personally. He gets funding from string grants. Nobody is angry about that. Anybody can do this. I don’t think his theory is going to pass any experimental validation (it requires a really severe violation of a physical principle we have tested over and over) but the entire community has always supported and listened. He gives talks at major universities. He’s not an outcast or renegade or something.

But I refuse to say "thank you very much" when sand has been thrown at my eyes for decades.

It took humanity something like 1000 years between when the ancient Greeks could formulate general quadratic equations, and someone came along who solved them. And another several centuries before the solution was extended to included complex numbers.

The problem that they tried to solve with string theory: Will it be solved in a decade, a century, a millennium, or ever. Is it possible that bending the rules of scientific methodology, even temporarily, will help us find an answer?

And that amounted to 60 years (and counting) of absolutely nothing in terms of how the physical world works.this is in part because it's outside the scope of existing technology

Newly devised math also seems to be

unreasonablygood at creating new particles.Not out of thin air, nothing like that, truly out of stuff thinner than air!

Instead of reasoning on the worth of the effort spent in this direction to investigate nature (a very tangible companion) they try to steer the discourse toward this nonsense. We spent >50 years listening to these tales and the time has long passed since we are required to stop playing with these smoke and mirrors.

They're theorists, you're paying for pencils and paper. String theory may not have produced a theory of quantum gravity yet, but neither has any other line of inquiry.

Do string theorists not use models on supercomputers?

Particle accelerators have been built since way before any string theory was formulated.

The biggest and most powerful existing accelerator (LHC) has been built to fulfill the high energy/high luminosity requirements to explore the Highs boson energy regime (that has been found) and at most the lightest supersymmetric particles (not found as of today).

The Higgs boson is a cornerstone of the Standard Model. Supersymmetry is an extension to it that does not involve strings.

https://home.web.cern.ch/science/physics/supersymmetry

theoretical physics(the thing they told are trying to work on).You forgot to add "with our current technology ability to probe the required energy scale".

https://www.americanscientist.org/article/is-string-theory-e...

If not realistically "non falsifiable" this sounds to me at least "not scientifically relevant" in the context of physics.

I think I read that the 20th century was a revolution because of the marriage between physics and math. Quarternions are key to relativity. Discrete math is littered all over quantum mechanics and the Standard Model. Like U(1) describes electromagnetism, SU(2) describes the weak force and SU(3) describes the strong nuclear force. In particular the mass of the 3 bosons that mediate the weak force is what led directly to the Higgs mechanism being theorized (and ultimately shown experimentally).

One of the great advances of the 20th century was that we (provably) found every finite group. And those groups keep showing up in physics.

The article mentions how string theory has led to new mathematics. This is really interesting. I'm skeptical of string theory just because there's no experimental evidence for "compact dimensions". It seems like a fudge. But interestingly there have been useful results in both physics and maths based on if string theory was correct.

A lot of the newer generative ML models are also using differential equations/Boltzmann distribution based approaches (state space models, "energy based" models) where the statistical formulations are cribbed wholesale from statistical physics/mechanics and then plugged into a neural network and autodiff system.

The best example is probably the Metropolis-Hastings algorithm which was invented by nuke people.

https://web.archive.org/web/20150603234436/http://flynnmicha...

https://arxiv.org/abs/1503.03585

The measurements, theories, and currently understood or applicable math may not match up with observations.

People ponder and discover, then attempt to explain the observations and measurements with a new theory. If the theory pans out, a deeper explanation of that theory is necessary and that's where the new math's at.

It's not that physics is good at creating math. Physics is good at describing our observations /with/ math. That's kind of its whole job.

Next time you look at raindrops in a puddle, try to imagine how you would describe those movements scientifically. One needs math for that.

Sometimes the available tools and math are sufficient for a thorough explanation, and sometimes one needs to invent a universe of math to describe a tiny fluctuation.

For example, pi is the ratio of a circle’s circumference to its diameter. It’s just what a circle is in two dimensions. The value of pi isn’t any more mysterious or connected to physics than the existence of this thing called a circle. If you have some other Euclidean shapes you’ll have other ratios and values that have other relationships to other things in physical reality.

And if reality was different, hence the physical laws were different then the math would be different.. and the beings in that world might wonder why their math and physics were so interconnected.

This is contested by nominalists. They'd say you have it backwards. Mathematics is just an abstraction/language that can be used as a tool. The reason we're able to understand the world through mathematics says more about the power of mathematics than it does about the world. If the physical world were different, math would still work.

Very loosely speaking, pi can take a different values on non-euclidean planes. This ends up becoming relevant on the surface of the earth or, say, the saddle of a horse. I'm not sure if the motivation was from looking at curved surfaces, but it just as easily could've come from the rejection of Euclid's parallel postulate and seeing what results. Similarly, I think imaginary numbers were motivated by the math well before they found applications in reality.

There are also plenty of other mathematical constructions that are informed by reality (since that's what our brains are constrained to,) but I'm pretty sure are far from actually describing reality. Transfinite cardinals/ordinals, fast growing hierarchies, Turing degrees, Goldbach's conjecture, how the hypervolume of a hypersphere eventually decreases as dimension increases...

You can even reject the standard axioms and construct math that can not be compatible with reality. Or argue that the standard axioms permit too much wiggle room to create concepts that have no relation to reality. (But maybe you shouldn't; that sounds like philosophy.)

I tend towards maths being distinct from physics as some areas of maths deal with concepts that can only have a passing resemblance to reality - the Banach-Tarski paradox is an example. (Similarly, pretty much any treatment of infinities ends up to have little relation to reality such as with Hilbert's Hotel).

* https://web.archive.org/web/20210212111540/http://www.dartmo...

* https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

I've always loved this line and paraphrase it often. It's an eminently reasonable and yet accidentally profound thing to say.

The end.

There's no magic here.