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."
Physics faculty wants to buy a new expensive research machine. University rector is furious at all this spending and tries to talk some sense into them: "Why aren't you more like the mathematicians, they just need a paper a pencil and an eraser. Or like the philosophers, they just need a paper and pencil"
Or one more related to this article: Mathematicians waste time designing the topology of coats for people with 3 arms. Physicists find people like that.
Oh and my favorite: Mathematician's son goes to school for the first time. The teacher asks: "Who knows how much is 1+2?", the son stands up and says "I don't know how much it is but I do know that it's the same as 2+1 as addition is commutative in the monoid of natural numbers"
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.
An engineer, a physicist, and a mathematician are on a train from London to Edinburg. It will be the first time any of them have been to Scotland.
In Scotland the train passes a field and there is a single sheep standing in that field. The sheep is black.
The engineer says, "Look! The sheep in Scotland are black!".
The physicist sighs, shakes his head, and says, "No...at least one sheep in Scotland is black".
The mathematician sighs, shakes his head, and rolls his eyes, and says, "No...at least one sheep in Scotland is black on at least one side at least some of the time".
> 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.
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.
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.
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.
> 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.
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.
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.
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.
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.
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.
> 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.)
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.
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.
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.
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).
I think it’s the other way: math is unreasonable good at describing physics!
Imagine a universe where the laws are best described in iambic hexameters under the condition that the last letters of the stanzas form specific words.
The ancients held some believes like that: kabala, astrology and the like. How wonderfully absurd it must have felt to them that the answer was something even more removed from reality.
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).
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.
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.
[+] [-] jjgreen|1 year ago|reply
Disclosure, I'm a mathematician.
[+] [-] wrp|1 year ago|reply
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."
[+] [-] yread|1 year ago|reply
Or one more related to this article: Mathematicians waste time designing the topology of coats for people with 3 arms. Physicists find people like that.
Oh and my favorite: Mathematician's son goes to school for the first time. The teacher asks: "Who knows how much is 1+2?", the son stands up and says "I don't know how much it is but I do know that it's the same as 2+1 as addition is commutative in the monoid of natural numbers"
[+] [-] thruway516|1 year ago|reply
Disclosure, I'm a software developer
[+] [-] seanhunter|1 year ago|reply
[1] https://en.wikipedia.org/wiki/Nasreddin#cite_ref-32
[+] [-] bryanrasmussen|1 year ago|reply
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!
[+] [-] billfruit|1 year ago|reply
"Don't use statistics like how a drunkard uses a lamp-post, for support rather than for illumination".
[+] [-] tzs|1 year ago|reply
In Scotland the train passes a field and there is a single sheep standing in that field. The sheep is black.
The engineer says, "Look! The sheep in Scotland are black!".
The physicist sighs, shakes his head, and says, "No...at least one sheep in Scotland is black".
The mathematician sighs, shakes his head, and rolls his eyes, and says, "No...at least one sheep in Scotland is black on at least one side at least some of the time".
[+] [-] mettamage|1 year ago|reply
Disclosure: I’m a software developer
[+] [-] tijtij|1 year ago|reply
https://xkcd.com/435
[+] [-] tech_ken|1 year ago|reply
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.
[+] [-] groos|1 year ago|reply
https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology
[+] [-] namaria|1 year ago|reply
[+] [-] verisimi|1 year ago|reply
[+] [-] cjs_ac|1 year ago|reply
[+] [-] tines|1 year ago|reply
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.
[+] [-] goatlover|1 year ago|reply
[+] [-] tech_ken|1 year ago|reply
[+] [-] speedchess|1 year ago|reply
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.
[+] [-] waterhouse|1 year ago|reply
— V.I. Arnold: "On teaching mathematics" (1997)
[+] [-] naasking|1 year ago|reply
[+] [-] SkiFire13|1 year ago|reply
[+] [-] marcosdumay|1 year ago|reply
[+] [-] slashdave|1 year ago|reply
[+] [-] Onavo|1 year ago|reply
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...
[+] [-] janalsncm|1 year ago|reply
https://arxiv.org/abs/1503.03585
[+] [-] scarmig|1 year ago|reply
[+] [-] whimsicalism|1 year ago|reply
[+] [-] Anon84|1 year ago|reply
(I was once a reasonably successful Physicist, so I might be biased :D)
[+] [-] jmyeet|1 year ago|reply
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.
[+] [-] zeroonetwothree|1 year ago|reply
[+] [-] aithrowaway1987|1 year ago|reply
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.)
[+] [-] throwawaymaths|1 year ago|reply
[+] [-] pyb|1 year ago|reply
[+] [-] 77pt77|1 year ago|reply
[+] [-] Almondsetat|1 year ago|reply
[+] [-] ezst|1 year ago|reply
[+] [-] imchillyb|1 year ago|reply
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.
[+] [-] crazydoggers|1 year ago|reply
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.
[+] [-] vehemenz|1 year ago|reply
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.
[+] [-] NegativeK|1 year ago|reply
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.)
[+] [-] ndsipa_pomu|1 year ago|reply
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).
[+] [-] niemandhier|1 year ago|reply
Imagine a universe where the laws are best described in iambic hexameters under the condition that the last letters of the stanzas form specific words.
The ancients held some believes like that: kabala, astrology and the like. How wonderfully absurd it must have felt to them that the answer was something even more removed from reality.
[+] [-] verzali|1 year ago|reply
cf. string theory
[+] [-] nyc111|1 year ago|reply
[+] [-] glenstein|1 year ago|reply
Of course it's not anything like a proof but something that bolsters an intuition.
[+] [-] 77pt77|1 year ago|reply
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.
[+] [-] throw0101b|1 year ago|reply
* https://web.archive.org/web/20210212111540/http://www.dartmo...
* https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...
[+] [-] jordanpg|1 year ago|reply
I've always loved this line and paraphrase it often. It's an eminently reasonable and yet accidentally profound thing to say.
[+] [-] graycat|1 year ago|reply