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"
I had a lecturer (for a fairly advanced set theory course) who said he told his young son, when teaching him to count, that he, as a set the theorist, ‘doesn’t do finite’. I guess when his son got to school 1+2 might have been ‘the successor of 2’, but anything else would have been ‘less than omega’.
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".
Buddhist would say - There is just changing patterns, sensations and thoughts that create an illusion that there is a sheep outside and seperate you watching it. In reality there is just emptiness.
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”
Non-software-developer humans often use things called "lamps" to illuminate spaces at night. Unfortunately, illumination inhibits effective nighttime coding.
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.
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.
> It's not just physics that drives interesting math, and it's not just recently that this relationship holds
When you say
> So much of it was invented by someone just noodling around with numbers
I think you're ignoring where the numbers being played with came from. Very rarely does someone just invent a fresh problem de novo and start messing with it AFAIK; the 'playful mathematics' approach is still reusing tools which were developed in application, just (typically) a long period of time after that original application. Euclid's geometry doesn't exist without the invention of a compass and rule for drafting. Yes he's playing with the concepts freely, but it's not just some arbitrary toy ideas, they're rooted in a practical reality (albeit deeply).
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.
Its grammar is limited in ways we don't understand, because the grammar is based on abstractions of our experiences of the world and the relationships between them.
If we can't imagine entire classes of relationship - likely because we do not have limitless intelligence - then the model will always be a partial analogy, not a full and complete abstraction.
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.
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.
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
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.
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.
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.
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?
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'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.
Readit News
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."
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"
Disclosure, I'm a software developer
[1] https://en.wikipedia.org/wiki/Nasreddin#cite_ref-32
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!
"Don't use statistics like how a drunkard uses a lamp-post, for support rather than for illumination".
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".
Disclosure: I’m a software developer
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”
Disclosure: I'm a software developer.
https://xkcd.com/435
Deleted Comment
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.
> It's not just physics that drives interesting math, and it's not just recently that this relationship holds
When you say
> So much of it was invented by someone just noodling around with numbers
I think you're ignoring where the numbers being played with came from. Very rarely does someone just invent a fresh problem de novo and start messing with it AFAIK; the 'playful mathematics' approach is still reusing tools which were developed in application, just (typically) a long period of time after that original application. Euclid's geometry doesn't exist without the invention of a compass and rule for drafting. Yes he's playing with the concepts freely, but it's not just some arbitrary toy ideas, they're rooted in a practical reality (albeit deeply).
If we can't imagine entire classes of relationship - likely because we do not have limitless intelligence - then the model will always be a partial analogy, not a full and complete abstraction.
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.
I’m sure there are other examples but I’m not a mathematician.
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.
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
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.
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.
https://archive.org/details/ourmathematicalu0000tegm
Deleted Comment
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.
— V.I. Arnold: "On teaching mathematics" (1997)
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.
Deleted Comment
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
(I was once a reasonably successful Physicist, so I might be biased :D)
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.