> it wasn't made during the nineteenth century

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.

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.

Ok, I see what you mean now.

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

> I say the math can’t be invalidated by observations

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

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.

I suppose even in pure math, if you postulate a set of axioms, then proceed to to prove some theorems, you're still at risk of someone providing something like a counter-example showing that your axioms are not consistent, and that not all of them can be true at the same time.

Meaning that even if the inductive logic is 100% correct, the theory can be incorrect due to using mutually exclusive (in some non-obvious way) axioms.

“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.

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.

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.

In Physics, you just remove all possible theories that are inconsistent with observations.

For instance, you can prove that Newton's laws are not 100% accurate.

> You can’t prove anything by observation.

Nice observation.

Epistemology is a harsh mistress.

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.

Furthermore, we might not be able to observe it but we can observe simulations of it.

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

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.

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.

Like the atom was in the 18th century? Electrons and protons in the 19th? Quarks in the 20th?

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.

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.

please forgive my ignorance, has there been notable progress in the field of physics in the last decade? noteworthy breakthroughs i mean

i ask out of layman curiosity

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".

>first detection of gravitational waves as two black holes merged together in 2020 a

2015. (Your point is otherwise taken).

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

No worries. I’m (mostly) a layman too, I just happened to have watched this video recently:

https://youtu.be/d_o4k0eLoMI

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

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.

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

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.

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.

As Frank Yang predicted in 1980 , "the party's over".

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.

Eh, I suppose that was more the Field era

They might be novel mathematical constructs but seemingly have no bearing on our universe. ER=EPR doesn't really solve anything because GR remains fundamentally incompatible with QM's linearity. That's the problem that needs to be solved. The core idea of ER=EPR wasn't even particularly novel, as Hadley effectively did something very similar back in 1997 [1].

CA-duality is again mathematically interesting, but physically dubious because it's based on anti-de Sitter space, which does not describe our universe.

Information theoretical formulations of QM are mildly interesting, but I don't think they will be revolutionary, and I don't think they are tackling the core problem, which is QM's linearity where we classically observe a non-linear universe.

[1] https://arxiv.org/abs/quant-ph/9706018

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

I'm not, I'm pointing out that theoretical progress has stagnated. Experimentalists are doing great.

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

I'm not expecting it every decade, but we've had 4 decades of recycling the same ideas using the same failed approaches to try to patch gaps in existing theory using bogus arguments, which ends up funding poorly motivated experiments that then find nothing. I think Sabine Hossenfelder elaborated the problems here in excellent detail (see "Lost in Math").

Smart theoretical physicists of today should bring their Lie-group algebras and help build better neural nets, and then use those nets to make new discoveries in Physics.

I think those qualify more as interesting suggestions and experimental confirmations than as breakthroughs.

I suspect "breakthrough" is supposed to mean "huge definitive paradigm shift." We haven't had many of those in all of history, and we certainly haven't had one in the last decade.

Everyone is still very, very confused about quantum fundamentals. Non-local realness is really a Bohmian idea, and that's certainly not new. Universe-as-information is new but there's a huge gap between that and the Standard Model.

And so on. None of these problems are settled in the way that GR and QM settled various issues.

You may say that's too high a bar and things are moving. But there's been more of a history of missteps (string theory, supersymmetry, so far at least) that were sold as potential breakthroughs than genuinely transformative insights.

The confirmation of the Higgs boson is pretty huge...

I should also add, this page has a section for previously unsolved physics problems that were solved in the past 30 years:

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

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]

[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/

> 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.

That is, until gravitation waves carry away the energy :)

It doesn't matter much if they carry away energy, though the waves also carry away some momentum, I suppose, unless they're symmetric on both sides.

The question is testing if your reasoning is in line with what you were taught. It is not questioning your ability to validate your reasoning experimentally.

I came to the same conclusion as the replier almost immediately as I was reading the question. It was obvious to me. But I do agree it’s a bad question as it relies on a lot of assumptions. Maybe the question is just testing if your ability to make reasonable estimates assumptions.

TIL AoPS has physics classes now

i am essentially a rayiner on steroids, in that i believe america has absolutely nothing meaningful to offer to the world at large other than aops and rsm. its such an extraordinary institution. large number of international signups - so tons of indian and chinese kids burning the midnight oil solving woot alongside their american compatriots. just an amazing, spectacular creation all around. if the nobel committee had any brains, rusczyk would have gotten atleast a peace prize by now.

"I am essentially a rayiner on steroids" has to be the most unhinged thing I have read on this site today lmao

> conservation of momentum make an accurate prediction.

Accurate prediction of what?

If it's the velocity and location in the end state, the best answer is the eagle (once it comes to rest). The momentum of Earth is so huge that the eagle will end up precisely on the perch (if we disregard the inability of a living bird to be completely still)

For the cars, the exact end state depends on many other factors, as you say.

Finally, for the colliding galaxies, there may be some uncertainty about how dark matter and dark energy (or their absence) affect galaxy collisions. This may be beyond the understanding of whoever wrote the test, though.

You still have to do experiments and verify that what you did has ground.

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.

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.

It—that is, science at the time—all fell under the umbrella term of natural philosophy.

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

"In this house we side with Aristotle."

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.

So you claim it was a total coincidence he went after Soviet agents? The witch analogy here is that there are real witches but they are so powerful/well hidden witch hunters cannot find them and are diverted to patsies. This wouldn't mean witch hunting is bad, just that we needed better witch hunters.

But Platonists, ironically on many levels, are also real. Sometimes complex. But never fully imaginary.

There are no Platonists, only imitations of the one true heavenly Platonist.

Genuinely chuckling.

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.

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)

You should read up, in detail, how the founders of modern physics (Paul Dirac in particular) made their discoveries.

The most famous case of this is the discovery of antimatter, but it goes so much farther than this.

It's truly astounding how, given shockingly few predicates, you can reach the conclusion that electromagnetism as we know it has no choice but to exist by way of pure math.

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.

> Math doesn't have such a requirement

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).

This is a very late-XIX century development. Geometry used to be an integral part of physics, vide Newton's Principia.

Wondering if Euclid’s parallel postulate was independent goes back millennia though. And that’s like the key ingredient for multiple geometries.

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.

> But you'd only attempt that if you thought there was something questionable in the first place.

No, that's not the only reason. Come on, people actually wrote why they tried to prove it in their commentaries on Euclid's Elements.

For example, some people found that this postulate, compared with the first four, is not really that self-evident and also has a sudden jump in the complexity of its formulation. That's why some courses on geometry replaced it with something different (but equivalent), like "the sum of angles of any triangle is 180 degrees", or "for a line and a point not on it, there is exactly one line parallel to it that passes through that point", or "there are triangles with arbitrary large areas", etc.

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.

Add negative numbers to your list as well. Mathematicians used to rearrange terms specifically to avoid them (e.g. before the concept existed).

See Our mathematical universe : my quest for the ultimate nature of reality by Max Tegmark

https://archive.org/details/ourmathematicalu0000tegm

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.

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.

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

Physics is the subset of mathematics with physical application. However mathematics need not have physical applications in order to be valid math.

Pure Calculus is not Physics even though it has applications in Physics.

Just like English isn't Physics even if you can have describe aspects of Physics using English.

For Math and English to be Physics, it needs to specifically define what physical meaning the variables and parameters involved have.

Typically, you recognize it as Physics when it comes with units.

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.

I would argue that in the twenty-first century the distinction is even greater thanks to computer science.

Haha in my country half the computer science degrees are titled "Applied Mathematics".

Such a title makes me feel like I might come off smarter than I actually am -- I like it. Especially when considering my mathematically understanding and abilities.

I think you don't understand enough physics, maths or cs if you think there's a way to draw a neat line between any pair of them.

Saying that there's a distinction between them is different than drawing a neat line. Surely you aren't arguing that they're the same discipline instead?

> One of my physics lecturers at university made the offhand observation that the distinction between physics and mathematics is a twentieth-century idea:

It's actually a 19th century idea. The discovery or acceptance of non-euclidean geometry in the 19th century untethered math from physics or physics from math.

> and it seems to be disappearing in the twenty-first.

It can't disappear because math is no longer tied to the physical world. Math is simply theorem generation regardless of whether the axioms and theorems apply to the physical world.

The math used in physics is only a tiny subset of possible math.