What a well-written, approachable article. if I understand right, the summary is basically, there is a certain way, an abstraction from geometry, you can use to mathematically describe electromagnetic fields. You can also use this certain geometric abstraction to describe how different types of mathematical abstractions are entangled, like electricity and magnetism are entangled
Numberphile published an hour long video [1] about the Langland program, which dives a bit into this "grand unified theory of mathematics". I'm not a mathematician, but I find these topics fascinating.
I can also recommend Edward Frenkel's (the guest of this video) book "Love and Math" on the subject. He writes as well as he talks, the book is very approachable, not just for a mathematically literate audience, but also for those who "don't like math". Its about the Langlands Program, but also about his own personal history of becoming a mathematician in the Soviet Union and his experiences with antisemitism during that time. Fascinating Stuff, even though it only barely touches the surface of his research.
(Category Theory, of all things, being the best tool for formalizing dualities and all sorts of analogies between seemingly different objects of study.)
Coincidentally I just put down Edward Frenkel's "Love & Math" before opening HN to see this. In particular I was reading the chapter about the early 2000s meetings at IAS using DARPA funding to explore the links between the Langlands Program and physics. Highly recommend the book for those interested in an accessible overview of this area of math.
It is getting very hard to popularize modern research, especially when it is of the more abstract type (as in mathematics). The author does his best but I am not sure it is even a solvable problem.
The challenge is what level of familiarity with mathematics to assume of the reader. To transmit a sense of the actual accomplishment of the work, the actual concepts need to link to internalized concepts available to the readers (a sort of mathematical mapping that follows the pattern: A is to B what x is to y).
So its important to have a coherent set of x,y to work with (the assumed level of audience knowledge) and that set has to be expressive enough to convey the essence of the new development and its importance / limitations.
It feels still possible to achieve this using book-level amount of effort from author and audience (where entire early chapters lay the groundwork for the main course), but an article length might be prohibitevely short.
Something that could help, given the online format, is hyperlinks to further explanations (but they must be short and at a consistent level).
The title is clickbait - they found a connection between two mathematical objects by using a third mathematical object to translate between the two. The third mathematical object also happens to be useful in physics.
Not sure if it's clickbait, but there is good reason to believe in a deep connection between 4d gauge field theories (i.e. of the standard model) and number theory through representation theory. This involves other work than mentioned in this article though it's still inscrutable for most at this point (motivic cohomology/Hilbert–Pólya in RH, supersymmetry in YM mass gap, cosmic galois in renormalization).
From Peter Woit's blog:
> It’s worth noting that while there are many connections to the ideas originating with Langlands, this new work shows that the “Langlands program” has expanded into a striking vision relating different areas of mathematics, with a strong connection to deep ideas about quantization and quantum field theory. The way in which these ideas bring together number theory and quantum field theory provide new evidence for the deep unity of fundamental ideas about mathematics and physics.
[...] provide new evidence for the deep unity of fundamental ideas about mathematics and physics.
This makes no sense to me at all. Mathematics is a modeling language and when we use it to model fundamental aspects of the real world, then we call that physics. So there is obviously some relation between mathematics and physics but I do not understand why you would expect some kind of unity, mathematics is much richer than physics.
This makes no sense to you precisely because you think that mathematics is (only) a language. It's not. A language is invented; mathematics is discovered. In the US it's not called a science, but for all intents and purposes it indeed is.
A language is invented; mathematics is discovered.
I peg to differ. Mathematics is invented when you define the axioms of some structure, after that you discover the consequences of the axioms you picked. First you define the natural numbers and the operations on them, then you discover the primes. So actually invention and discovery are not mutually exclusive here, but invention is more fundamental as it defines what is there to be discovered.
You define, but the choice is in fact not quite arbitrary; rather, it is guided by something objective, i.e. by something that exists outside of your own mind.
What are the constraints? I can make up any set of axioms I like and see what they imply, there is no need that they are in any way related to the physical world. Sure, there is a lot of mathematics that was specifically invented in order to deal with the real world, but that is not a general requirement.
Why not? If I decide I want to study the properties of the space of total function from vectors of octonions with prime dimension to the surreal numbers, who is to say that this is not mathematics?
Yes, but that is a technical detail, it was just easier for me to come up with that than making up a set of axioms. How is an obscure combination of existing axioms still mathematics but not any set of axioms I make up? And how would we ever extend mathematics if coming up with new axioms is not mathematics?
But let us just take the integers with the common definitions for addition, subtraction and multiplication, but then redefine them so that every operation first performs the usual operation and then increments the result by one.
1 + 1 = 3
1 * 1 = 2
(1 + 2) * 3 = 13
(a + b) * c = a * c + b * c + c - 4
Still not quite what I had in mind, nothing completely new, but maybe at least different enough from the normal integers to have some weird properties. Not the numbers themselves, they are still just the integers, but the algebraic expressions involving the redefined operations.
The problem is that the word "any" stands for "random," or "arbitrary." It is not inconceivable that a scientist, say, would mix "random" substances together just to see what happens. But almost all such experiments would result in exactly nothing, nothing interesting anyway. Usually, a scientist first notices something, puts forward a concrete hypothesis, and then conduct an experiment. In mathematics, it is the same: for example, one notices something that certain things have in common and then tries to generalize this common into "axioms." Modern mathematics is highly evolved field full of such examples. Nobody starts with scribbling random doodles on a blank sheet of paper in the hope that something interesting would come out of it. While not impossible, it is extremely unlikely. In the end, one might say that what is mathematics and what is not is merely a matter of definition, but that, as well as calling the "language of doodles" mathematics, would make the word devoid of any useful meaning, IMHO.
Sure, randomly generating axioms would probably not be very productive, most of the time you will continue from known territory, generalize something, add additional constraints, whatever. And if that is what you mean with discovery, identifying things that might be interesting to change in one way or another, fine, that is not an unreasonable description of the process. But you are still inventing a new structure when you decide what to change and write down the new rules. If you have the naturals, you might wonder what would happen if there was an inverse of addition or of multiplication for each natural, if you pick the former you will invent the integers, if you pick the later the non-negative rationals. You can also call that a discovery if you like, but it is not a discovery in the sense of say discovering the electron, you did not discover something that has always been there, you actually brought it into existence by the choices you made.
You will discover them, not invent. Albeit subtle, there is a difference between these notions, and you will indeed discover the integers in (almost) exactly the same sense as electron was discovered - what you do is, you put forward an idea, play with it, test it, correct mistakes, and in the end you find what you have been looking for. What you can invent in mathematics, is a proof; but even in this case the path may lead to a series of discoveries. Whether it does, depends on the attitude towards it (Grothendieck's analogy between building a proof and opening a nut).
I would say it is exactly the opposite - you invent the integers, you discover a proof. Discovery for me means finding something that already exists, inventing means bringing something into existence. You discover the electron, you invent the transistor.
Before you write down the axioms for integers, they do not exist, you invent them. That there are infinitely many primes is a consequence of the axioms, you just have to notice those special numbers and give them a name. Here I find it actually quite tempting to say you invent the primes, or maybe better the idea of primes, but I still think it is more correct to say you discover them, given the axioms they were always there even if you have not yet noticed, named and described them. The same for the proof that there are infinitely many of them, given the axioms and the laws of logic, the truth of that statement and the possible proofs for it are fixed, you just have to find one.
Maybe back to the integers. Yes, you may play around with different ideas, tweak definitions and so on. But what you are actually doing is inventing a whole bunch of similar structures and then you pick the one that you like the most, that works the best or whatever. Some of your attempts might be inconsistent, some might not do what you want, but you all invented them. And among them you might discover one that works just like wanted it to work.
But unlike the transistor, and like the electron, the natural integers do exist in nature (i.e. have "natural existence"). To quote my other comment here, [number 3] exists as the thing that is common between three apples and the three horses you might want to feed these three apples to; and the fact of the (almost) physical existence of that common (i.e., the number) can be proven by you being bitten by the third horse if you only happen to have two apples instead of three. This common exists, objectively (no sane person can say that it does not), so does the number 3 (because that's what it is called). Now, you can say that we invented the infinity, but that's like saying "we invented the idea that jumping off a cliff is dangerous" - because in both cases the "invention" is unavoidable, as it is dictated by the objective reality which tends to harshly punish those who fail to "invent" the facts.
What then about mathematical structures that do not exist in the real world? Do you discover some and invent others? Or do you think all mathematical structures are realized?
You say that no sane person will deny the existence of three, I am actually willing to go even further, I am willing to deny the natual existence of your apples and horses. The idea of cutting the universe into pieces is arguably an invention. It is of course an extremely useful idea for comprehending the universe, but if you think about it, you are somewhat arbitrarily drawing borders around collections of atoms.
There is one forest. And a hundred trees. And millions of cells. Is the water in a tree part of the tree? What about the water inside of the cells? When does it become part of the tree? What about a water molecule just evaporating from a leave, when does it become part of the atmosphere?
But even if there are things that can be counted, or if quarks and gluons behave in a way that can be described using SU(3), does that really imply the existence of the mathematical structure used to describe those things? Where do you get multiplication from? Or tetration? Are they invented while the basic structure of the integers is discovered? What exactly does existence even imply or require in this case?
And another thing just came to my mind. In which way do the apples and horses tell us anything about three? I can see that you can probably demonstrate that you can pair your apples and your horses, but that is quite far from telling us anything about three as you can do the same with any number of objects.
You also say the common thing is three, but that does not really help to pick out what three is, therevare many common things. Also note that you introduced the idea of a set, where does that come from?
Three is not something that applies to one of the apples or horses, it applies to the sets of them. Does the set of three horses exist or is that not something you made up, this, this and that, this are my three horses?
And now that we are talking about sets, because that is where three comes into play, why did we talk about apples and horses to begin with, it totally doesn't matter what you put into your sets, just how many things. So did we just lose the relationship to the real world or at least conclude that it never mattered?
I would at least say that it is certainly not as easy as you want it to be, there is much more nuance to this than three apples, three horses, therefore natural numbers. It actually sound circular. Let me pick three apples and three horse, now look, here are two sets of three things, so three exists.
Maybe it would be a nice challenge to tell me how to find three in nature. You obviously can not say take three horses. It seems tempting to say start without a horse, then add one, then another and yet another. But did you then not just invent that, did you not just come up with the axioms of the natural numbers, just expressed with horses instead of symbols?
A blind person would definitely disagree, having “invented” things he keeps bumping into, one after another… See, knowledge is never invented, it’s based on discovery; and mathematics is a form of knowledge.
> anything about three
Well, the discussion wasn’t about any particular number or concept, but if you want to define 3, you should look no further than the special property of a three-legged stool, the smallest polygon, or, if you look close enough, the proton. The general notion of the number, then, may come from a comparison (trying to see if there is a one-to-one correspondence) between the legs of a stool that is safe to sit on and one you risk breaking your neck if you try. If that’s full of “nuance,” I don’t know what isn’t.
A blind person would definitely disagree, having “invented” things he keeps bumping into, one after another… See, knowledge is never invented, it’s based on discovery; and mathematics is a form of knowledge.
I was not talking about inventing the things themselves but logically dividing them up. For the blind person it makes not difference whether he bumps into a tree because you decided to divide the forest into several individual trees or whether he bumps into the entire forest. But if you want to count things, then it of course makes a huge difference whether there is just the forest or whether there is a collection of trees. This subdivision of the universe - or the forest - into several individual object - or trees - is arguably invented, not the universe or the forest itself.
Knowledge is a thing you have, your awareness of some fact. Mathematics is not that, it is some form of fact you can be aware of. You can have knowledge about mathematics but it is not knowledge itself. If I name my dog Beethoven, that establishes the fact that the name of my dog is Beethoven, that is a kind of invention. If I tell you about this, you gain knowledge about the name of my dog. No discovery involved, neither when establishing the fact nor when you learn about it.
[...] look no further than the special property of a three-legged stool, the smallest polygon [...]
Okay, I found a stool that does not wobble on uneven ground and a triangular rock. Nothing about them on their own is related to three. You can of course explain to me what a leg is or the corner of a triangle and then ask me to form the sets of legs and corners and point out to me that the sets have the same number of elements, but there is lot of stuff going on here. You made me find things related to three, and I never doubted that there are such things, the number of dimensions of space would be another good candidate. But you mostly gloss over the hard part of actually extracting the natural numbers in general or three in particular.
It is of course trivial in everyday language as we learn about pairing and counting things relatively early in our life and humanity has made use of those ideas for a long time. Look, the number of legs equals the number of corners. And there are as many of them as I have horses and apples. But notice that those sentences are full of ideas and words related to numbers - number of legs, equals, as many.
But just as with the horses and apples, nothing about the non-wobbly stool or the smallest polygon - actually you mean the polygon with the fewest number of sides and note that there the idea of numbers already sneaked in again - is intrinsically related to three. It is the set of legs and the set of sides that are related to three, it is the carnality of the sets that behaves like the naturals. The process of forming those sets does a lot of heavy lifting to get you towards finding numbers in nature. And I do not think you can just brush that under the rug, you will have to justify that forming sets and looking at their cardinality is not something that humans invented.
> it makes not difference whether he bumps into a tree because you decided
Exactly. It makes no difference for him that you want to see a proof of individual trees' objective existence - because he already knows this for a fact! That's what that pesky objective reality does, sometimes forcing knowledge about itself upon us, whether we like it or not, or whether we would prefer some other "proof." The proof is in the pudding, as they say.
> Mathematics is not that.
Sure it is. One who knows about numbers knows more about the objective reality than those who don't. One who knows about Lie groups knows even more.
> you gain knowledge about the name of my dog. No discovery involved
That's not quite true: I discover that you have a dog (as long as you did not "invent" it).
> Nothing about them on their own is related to three.
It does, if you look at it from the right angle. There's a different thing at play here. While the number of legs could be easily matched with the corresponding number of apples, by itself this correspondence does not necessarily make any particular number stand out (although in some cases, like, say, in the case of a non-wobbly stool, a triangle, or the number of eyes and hands, it would - simply because there are many pairs of eyes, etc.); what's also at play here is different ways to look at numbers, which includes seeing them not only as "cardinals" (which is what you are still limiting yourself to) but also as "ordinals": the number 1 "stands out" as the smallest ordinal (greater than "nothing"), the number 2 is what follows it, etc. It is all these aspects combined that form the true content of the notion of the number.
It makes no difference for him that you want to see a proof of individual trees' objective existence - because he already knows this for a fact!
You are missing my point here. The blind guys knows he bumped into something, so something exists. He could just say he bumped into a part of the universe, he is not forced to say he bumped into a forest or a tree. He could even consider himself part of the universe and say one part of the universe bumped into another part of the universe, just as one part of you bumps into another part of you when you clap your hands. The consequence of that is that there are no distinct objects to count, it is just one really complex object, the universe, interacting with itself.
Sure it is. One who knows about numbers knows more about the objective reality than those who don't. One who knows about Lie groups knows even more.
No, that is a kind of map territory thing. You can have knowledge of mathematics but mathematics is not knowledge. You can have knowledge of my dos's name but my dog's name is not knowledge.
It does, if you look at it from the right angle.
Switching from the cardinals to the ordinals will not really make a difference, you are glossing over a lot of heavy lifting. I have to repeat myself, the non-wobbly stool is not related to three, it is the set of its legs that is related to three. On to get there, you have to single out parts of the stool and combine the parts into a set and the take about the cardinality of it. There are a lot of steps and concepts on that way which makes it at least very non-obvious how the number three was always there and is not just the result of that process.
I am open to an example how I would find the ordinals in nature, I am not sure it will be any easier than with the cardinals. Non of the trees in the forest is the first one, you will have to impose an order an them. Maybe something with time, sunrises or days, they are at least already ordered.
That probably depends heavily on the field, there is certainly a huge difference whether you are trying to better understand established physics or whether you are trying to come up with a new theory in an area where we are lacking experimental data.
Well, what I mean is that (to give you perhaps an extreme example) classical mechanics, while not "lacking experimental data," has for the last 200 years been effectively considered part of (applied) mathematics (known as "analytical mechanics"). Thermodynamics, electromagnetism, quantum field theory, general relativity - they all seem to be purely mathematical now.
Then I probably misunderstood you, I thought you were talking about how closely the math is tied to the actual world, say all the reformulations of classical mechanics as opposed to some supersymmetric theory.
In the end physics is applied mathematics, once you have a theory that describes your observations, you can just study the mathematical structure and look for interesting things, only occasionally going back to the lab to verify your predictions, improving confidence that you picked the correct model.
There might well be a connection between the two. However, there are much more obvious connections between electromagnetism, or all of physics for that matter, and numbers [1].
Even though I prefer to interpret things differently, the paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" [2] might be a good avenue to think about these things.
I wouldn't be surprised if numbers and physics go hand in hand, in the sense that we can only observe these, and not the more irregular patterns that make up the total chaos that we might exist in.
Maxwell's equations are not the simplest formulation for electromagnetic theory, and they're not a foundation, but merely a formulation of the physics.
The more elegant mathematical foundation for understand electromagnetism is Noether's theorem.
The more elegant formulation of the physics is just:
∇F = J
Which uses natural units and the geometric algebra of spacetime to encode everything contained in Maxwell's equations, but in a way that is coordinate-free and relativistic.
In all fairness, it should be noted that, for better or worse, GA is not part of the mainstream; modern treatments use differential forms instead. For an accessible account, see, for example, https://www.jpier.org/ac_api/download.php?id=14063009.
Physics is littered with incomplete implementations of GA using a mish-mash of random bits of mathematics because of a perverse insistence on refusing to use the appropriate algebra.
> I wouldn't be surprised if numbers and physics go hand in hand, in the sense that we can only observe these, and not the more irregular patterns that make up the total chaos that we might exist in.
I am curious to see how far we can get with the logic and mathematics that we understand (as humans). Perhaps there is a perfectly "logical", consistent and complete description of why the universe exists and what it looks like that is beyond our comprehension. Maybe we wouldn't even recognize the description as such if it were shown to us (comparable to an abstract painting with two lines and one dot that is supposed to depict a woman).
This kind of reasoning is slightly problematic. What does it mean if properties of "logic" are "logical"? Does that lead to tautology, recursion, symmetry, or worse?
I notice that you put "logical" in quotes, so you seem to be aware of this problem. Any idea how to get around it?
For those too lazy to look it up: ˘ is called a "breve", and it is rounded, not even pointy as the typical "hat" is in physics and mathematics literature.
Obviously, a rounded hat like that will stay on top of a human head only for a brief amount of time. Not sure if that is the actual reason (people in this thread are fairly pedantic about causality), but it may help to remember that a breve is used to signify short vowels.
“Echoes” suggest a causal relationship, which would be a fascinating to see occur emanating from physical properties and reflecting in mathematical abstractions.
What this actually sounds like is “Reflecting on work by physicists helps mathematicians find analogous relationships in their own field” — which is cool and makes for a good story, but is very much not the same thing.
While this is a valid nitpick, I think people sometimes overdo it.
Mathematical objects can be both abstractions and systems in and of themselves. They are abstract "enough" to be useful as models, but anyone who infers causality has a fundamental misunderstanding of what mathematics is.
I think this comes from our priority in the educational system of teaching mathematical techniques applicable to specific subjects rather than focusing on mathematics as the art of using and manipulating mathematical objects.
On the other hand, I had an interesting chat with sommeone who had earned a pH.D in mathematics and physics (and one other discipline which I forget) and he regretted wasting his time learning about mathematical systems that were not applicable to physics research.
I didn't retort, but if I had I would have said simply that one never knows how useful links might be found and that there have been a lot of surprises.
Well I think part of the problem is that our education system results in many people thinking mathematics is anything other than a language. Most often people think that math is something we are uncovering about the universe. The OP's critique makes perfect sense from the perspective that physicists (and other sciences) utilize mathematical language. The "causal" relationship is as interesting as "English linguists discover new insights about colors by learning Russian." Similarly I would say physics is a language, though much more grounded in the observable world (since this is explicitly what physicists are trying to do: model our world/universe/reality).
> and he regretted wasting his time learning about mathematical systems that were not applicable to physics research.
Which shows the above claim about language. Though I'm a bit surprised since these ideas often connect due to the abstractions. 3 PhDs is a bit insane. 2 is even odd. Why go back to school when you already know how to research?
> many people thinking mathematics is anything other than a language.
I disagree with the claim that mathematics is only language, if by language you mean terms and definitions.
The EASY part of mathematics is the language.
There are deep truths, too, and they are not just language.
For a (kind of) popular example, the question of "Where do the zeros of the Riemann zeta function lie?" is way way more than just vocabulary. There is a 20 minute video by 3Blue1Brown on the subject. [1]
This is where I highly disagree. Doing the math is doing the language. I'm not sure which part is "easy". Even setting up problems is often incredibly difficult. This is where many physics students often struggle, despite having strong math skills there's a challenge to describe the problem to be solved. Even strong mathematicians often struggle to translate between mathematics and their native spoken language. Though 3B1B is quite good at doing this, though there is quite a lot of simplification to goes on due to the target audience. Like I said to the sibling comment, do not mistake how powerful and impressive language is. It created our modern world and there's a reason it is more important to us than even our phones. But all languages, including math, are never complete. They cannot describe everything, unfortunately. But I thought my name would have indicated that I was well aware of this notion.
> "Where do the zeros of the Riemann zeta function lie?" is way way more than just vocabulary
And a language isn't just vocabulary. That's like saying a programming language (which remember has a direct correspondence. They're equivalent) is just tokens. Idk if you want to look into CH Correspondence, Lambda Calculus, Abstract Algebra, Category Theory, or if I should just quote Poincare about mathematics being the study of relationships.
But since mine is easier, I'll just lay some stuff out. Which is real? Finite ZF? ZF? ZFC? Are you calling one of these math and the others not? What distinguishes them? Are infinities real? Are they not? No matter your choice we're going to lead to a decision where math isn't simply due to our ability to have mathematical systems in either form. Similarly I can mathematically describe geometries that cannot exist in our universe or create algebras which are not in fact useful. I lay out a lot more in the other comments. But I really don't appreciate the "trust me bro" responses. They are intellectually lazy and I have no reason to actually trust you.
Trust has little if anything to do with knowledge. I know calculus not because I trust my professor. Trust no one, only yourself - after you have read the book, of course.
What we are talking about here is a rather deep (and obviously confusing) philosophical topic, and HN is not the place to be discussing it in full, but the gist of it is this. Mathematical reality is somewhat different from the physical reality ("our universe"). The focus of mathematics is on abstractions and generalization. But abstractions are direct reflections of the objective (i.e. what is outside our mind).
You are asking about ZF, ZFC, etc. Try asking yourself about XYZQJJX (it's language, after all, which you are free to invent, right?) and you won't be getting anywhere. You better trust me on this one.
> Mathematical reality is somewhat different from the physical reality ("our universe").
I think in general when we're referring to "reality" we're being synonymous with physical reality. Include the reality beyond observable (like String Theory predicts) if you want. With your usage I honestly have no concept of what you mean by "reality" that I can differentiate from any other abstract concept. If the apple in my head is part of reality (excluding the literal physical state of my brain analogous to bit state of computer memory) then I feel this discussion is rather moot and we can go ahead and conclude that nothing is invented and everything is discovered. But then why not discuss with the definition of reality I'm using (and what I'm assuming is fairly status quo)?
And at what point is math "real?" Is any axiomatic system real? If so we can create arbitrary reality and go back to the above point. If not, then not all math is real. I don't see how we can have something that is self consistent here. Unless you're saying that "math" is whatever rules the universe operates on and not these other inaccurate set theories/systems that we use to approximate this thing." But then again the latter is what we call math and I guess we need to come up with a new definition for many things that exist within recreational mathematics. Sorry happy numbers, idk what you are but I guess not math?
> You are asking about ZF, ZFC, etc. Try asking yourself about XYZQJJX (it's language, after all, which you are free to invent, right?) and you won't be getting anywhere.
Sorry, you're going to have to define it if you want me to determine if it goes anywhere or not.
I'm referring to Zermelo–Fraenkel set theory and Zermelo–Fraenkel Choice set theory respectively[0]. (I figured it wasn't hard to Google, but you're right, I should clarify) These are for all intents and purposes different "maths." They have different axioms. Different rules. Different conclusions.
I'm unconvinced that utility has anything to do with something being real or not. Your usage suggests you aren't considering language real but it's one of the most useful things animals have invented (especially spoken and written language). Nor is the an arbitrary computer image "real" in the same sense. There's a lot of things that belong to this same categorical set.
And while we're at it, just give me anything, 1 or more, that wouldn't be considered real. I'll represent them with a set and I can do math with them... After all, math is the study of relationships between arbitrary objects. See Poincare or ... Category Theory
To be clear, we can create arbitrary set theories that will go "nowhere" and not have utility in physics or other common things. So if you don't want to call this math (I'm not sure who would say that. It looks like math, acts like math, and even mathematicians call it math) what would you call it?
Here, I'll provide an example:
∀ u(u ⊂ X≡u ⊂ Y)⇒X≠Y. : (Axiom of non-Extensionality) When x and y have the same members they are not the same set
Idk we can have the rest of ZF if we want here but it isn't really going to make sense because we destroyed uniqueness. But we can still do math from here. This is a more extreme version of what I was suggesting with Finite ZF vs ZF vs others. They have different axioms and they lead to different things. ZFC is in contention specifically because the axiom of choice leads to nonsensical things (again, see Banach-Taraki paradox). Are these both math? Back to paragraph 1.
The apple (or a "mathematical" whole number) in your head is not part of any sort of reality that I meant. The difference between any given apple and, say, the number 3, is that, while both do actually exist in nature, the number 3 exists in a subtly less direct way, it exists as the thing that is common between three apples and the three horses you might want to feed these three apples to; and the fact of the (almost) physical existence of that common (i.e., the number) can be proven by you being bitten by the third horse if you only happen to have two apples instead of three.
(I'll need more time in order to address the rest of your points.)
More time is fine. I am genuinely interested, just not understanding. Because even still what you are describing to me is impossible for me to distinguish from a social construct. Like in the sense the borders are real. But I wouldn't call borders "real" other than "a real agreement" and even then there's lots of arguments. Definitely borders are not inherent to the world/reality other than... our social constructs. So I think where the main point of contention is coming from is me not understanding your definition of "real." It appears like you understand the one I'm using, and is it fair to say that this is the standard one?
I know. What I meant was, these axioms reflect something real, like for example a news report reflects objective events; whereas you allow XYZQJJX to be something that is invented, i.e. to be quite arbitrary, like, to continue with the analogy, a "science fiction" piece (at best).
> you aren't considering language real
What I am saying is that mathematics is more than a language, and using it as if it were just a language, ignoring its positive content, would lead nowhere. I'd say, the content is its most important part, although the language, of course, is nice to have. Moreover, the language (including mathematical notation) is often abused, for various reasons - personal and otherwise. My other point was that even what we consider abstractions does reflect properties of the objective reality - just as do the notions these abstractions generalize. But the more you treat mathematics as merely a language and start playing with words (or symbols, or even concepts), changing their order and "inventing" new combinations of them, the chances that you get anywhere become small (to put it mildly). Yes, you can play with axioms, but notice that even then, as a sane person, you will be testing an idea rather than throwing stuff around or making wild sounds.
> Why go back to school when you already know how to research?
There's a couple of reasons I can think of. One is that if you want to do fundamental research in a particular field then you probably need other people to work with and that's mostly going to be in academia and PhD is the pathway. The other is that if you take a break from academia after a PhD it's really hard to come back. Three PhDs does seem extreme though!
I'm sort of in the latter position. I did a PhD in math, have spent 15 years in govt and industry but a part of me wishes I could go back to pure math research. I think the only path to do that would probably be to do a PhD in a related field such as theoretical physics.
Interesting. I totally get the desire to have collaborators but at the same time I'm particularly burnt out on academia. And most of my PhD has been teaching myself things so tbh I'm not sure what I need the institution for. Not that there aren't useful things that they can provide, but there's also problems that come with it too.
> Well I think part of the problem is that our education system results in many people thinking mathematics is anything other than a language. Most often people think that math is something we are uncovering about the universe.
And those people would be right, as the mathematical universe is part of our shared universe/reality. It's not just language and games. A Turing Machine either halts, or it doesn't. But if you want to solve a concrete problem (in physics, for example), it is best to develop/find/learn the mathematics you need to solve your problem, and not the other way around.
I'm not sure what your argument proves. No one said it is just games either. I think you're turning what I said into something else. Languages are incredibly impressive and powerful things. I feel like you're making languages appear simple and of low utility. But language has been one of the, if not the most, powerful tool humans have ever created, especially when you consider written language. The latter allowing us to pass along information asynchronously through time and space. Do not miss the power of this tool just because it is something we use so frequently. There's a reason we use it so frequently.
The reason we use the language of math is because it is so precise. But you just described a turing machine without using math. Keep in mind that physics is also a language. Physics is an ever evolving model, a description of our reality. But it is not more than that, a model. But all models being wrong does not equate to models being useless. Physicses (yes, there are multiple ones), are clearly quite powerful and useful models that have enabled us to do quite amazing things as well. But I'm also not going to start programming in Brainfuck or with Magic cards. But I do have a deep passion for mathematics and it is something I use every day and spend quite a lot of time learning even outside my work.
Your argument demonstrates the usefulness of mathematics, but does not demonstrate that it isn't a language. I also would like to remind you that you can create pretty useless algebras. Maybe also checkout Wolfram's section on recreational mathematics. Maybe also dig into my name sake's multiple theorems. Most people misunderstand the results, but I don't think it diminishes their worth, they are quite important and amazing. And the incompleteness theorem is deeply related to turing completeness as well as superposition and a famous logic paradox.
I am having trouble following your opinion through this thread.
> Your argument demonstrates the usefulness of mathematics, but does not demonstrate that it isn't a language.
What is “a language” to you? What is an “isn’t a language” to you?
I can grok you referring to axioms and lemmas as a “mathematical language”, but I see such as just the way we communicate something more essential and wholly independent of any need to have been communicated.
A lot of contemporary research mathematics is layered and wrought of “useful” complexities for its desired domain, but how do you dismiss the essential and seemingly unrealness of its abstraction from our perceived reality?
Counting is an example.
Subjective boundaries illuminate the essentialism of distinctness. 2 apples describe the same abstract phenomena as 2 atoms, or 2 galaxies, or 2 orientations of stereoisomers.
What is the “language” here? The word/symbol 2? The subjective boundary that separates something more continuous into discrete forms?
Transcendentals and irrationals alight my meditation on what the hell all this is that we’re experiencing.
You have a triangle with edges that terminate at each vertex, but if two of those edges have equal length than you can interpret their length as unit 1 where the third edge then has a length of (sqrt 2) which is a number without a finite decimal expansion.
What language can be used to defend an infinitesimal equating to a finite value?
This points at an essentialism to me.
Any amount of “language” is incapable of both explaining this completely or explaining it away.
Similar with pi and its relation to a circle which has a well defined circumference that somehow expresses itself with a number that is itself incapable of being expressed or defined.
As you brought up the incompleteness theorems, they too have a similar “infinite in finite” quality.
I am unsure how you can understand godel but argue against the essentialism of the sur-real abstractions he brings attention to.
> "Infinitesimal" is just an idea, as far as I know. Nothing real is infinitesimal.
The unreal (re: abstracted) aspect is what places it outside the confines of “language” for me.
Are black holes real? Do they have singularities? If yes, that can be an example of your “real” infinitesimal.
My opinion is that infinitesimals are more than real they are essential. They are the building blocks of all that is “real”.
Ultimately, what we’re talking about is a philosophical debate that would require one to step “outside” reality to confirm or deny outright so we are just providing our opinions on an unknowable concept.
What is “real” in this context?
Is pi “real”? Is the plank constant? The former was my path to the essentialism of infinitesimals. The latter my path to the essentialism of discrete counting.
The math leads us there but I don't think anyone is particularly happy about it.
> Is pi “real”?
¯\_(ツ)_/¯
> Is the plank constant?
¯\_(ツ)_/¯
Fuck man, I can't tell you if a quark is real. I'm also not aware of anyone who can. The best we got is our interpretation that the model being indistinguishable from the real thing might as well be the real thing. Metaphysics and metamathematics are mind bending areas that require a deep understanding of the non-meta concepts first.
But given all you've said, I highly suggest looking into the various set theories I mentioned previously. Specifically start with Finite ZF set theory and Peano Arithmetic, where you'll find you can indeed operate on such concepts as pi without infinities.
> What is “a language” to you? What is an “isn’t a language” to you?
A language is an abstract concept that describes a method of communication. It need not be spoken (such as English), written (such as what we're doing now). We frequently use body language to communicate, and so do many animals. We have braille, smoke signals, maritime flags, we communicate with knots on a string, and so many more things. You're right that language is quite a broad and vague thing. But recognize that all these things are also not of the universe, but of us humans (or similar of other animals). Something like English is something we may better refer to as a social construct, as it is a collective agreement, though body language may be a bit more ingrained but I still do not think you would refer to it as something other than language or something of the universe (distinct from us being of the universe in the trivial sense).
> What is the “language” here? The word/symbol 2? The subjective boundary that separates something more continuous into discrete forms?
(This is HN, so I'm going to assume you're familiar with programming languages.) If I give you these 14 characters (p, t, k, s, m, n, l, j, w, a, e, i, o, u) are we able to communicate? Maybe after some trial and error, but certainly not something we could throw into a translation machine. It'd be hard to call these even tokens since we have not distinguished consonants from vowels or if that even is a thing here, so we can't really lex. We need words, phrases, and context before you can even from syntax. Then we need to build our syntax, which is equally non trivial despite looking so (build a PL, it is a great exercise for any computer scientist or mathematician. For the latter, build your own group, ring, field, ideal, and algebra. You'd do this in an abstract algebra course). We need all this to really start making a real means to communicate. These are things we take for granted but are far more complex when we actually have to do them from scratch, forcing our hands.
Do you have a problem calling a programming language a language? I'd assume not because we collectively do so tautologically. Great, you agree that math is a language. Thank you lambda calculus. We can have an isomorphic relationship between programming languages and various mathematical systems. I'll point out here that there are different algebras and calculus with different rules and forms, though many that are not deep in mathematics may not be exactly familiar with these. I think this is often where the confusion arises, since we most often are using our descriptions that are most useful, just like how no one programs in brainfuck and just how most drawings are communicative visualizations rather than abstract art. I again remind you of Poincare who says that mathematics is not the study of numbers, but the study of relationships. He does not specify numbers in the latter part, on purpose. Category theory may be something you wish to take up in this case, as it takes the abstraction to the extreme. Speaking of which
> but how do you dismiss the essential and seemingly unrealness of its abstraction from our perceived reality?
I could ask you the same about English. Why is this any different? Is that because you are aware of other languages that people speak? Or is it because you recognize that these languages are a schema of encoding and decoding mechanisms which result in a lossy communication of information between different entities?
You discuss counting, but are not recognizing that you can not place an apple into text, nor atoms, galaxies, or stereoisomers. It is because mathematics is the map, the language, not the thing itself. We can duplicate these at will or modify them in any way. Math is not bound to physical laws like an apple is. Its bound is the same of the apple that exists in my mind, not in my hand. (If you want to make this argument in the future, a stronger one might revolve around discussion of primes and their invariances)
> What language can be used to defend an infinitesimal equating to a finite value?
If this is the essential part, I think this is probably the best point to focus on. Specifically because infinities are not real. Nor are they even numbers. If you disagree then you disagree with physics. Rather infinities are a conceptual tool that is extremely useful. But if we were able to count and use infinities then we'd have the capacity for magic via the Banach-Tarski Pardaox, and completely violate the no cloning theorem. But our universe does not appear to actually have arbitrary precision, rather our tool does due to its semantics. Maybe finite ZF is a better choice than ZF or ZFC set theory. Why not NBG which has a finite number of axioms or why not MK which isn't?
Infinities, singularities, and such are not things in our universe. You may point to a black hole but this would represent a misunderstanding of our understandings of them. We cannot peer in beyond the event horizon, which certainly is not a singularity and has real measurable and finite volume. It is what is inside that is the singularity. But can you say that this is not in fact just an error in the math? It wouldn't be the first time such a thing has happened. Maybe it is at the limits of our math and so thus is a result of the inconsistency of axiomatic systems? There are many people working on this problem, and I do not want to undermine their hard work, and neither should you.
You're biased because you're looking at how we use the tool rather than what the tool is itself. We use mathematics as the main descriptive language for science because of its precision. But we've also had to do a lot of work to ensure its consistency and make it more precise along the way. But I think you may have not been exposed to the levels of abstraction that math has, as this is not seen by most people until well beyond a calculus class.
> I am unsure how you can understand godel but argue against the essentialism of the sur-real abstractions he brings attention to.
And I cannot see the reverse. Are you saying that the universe is incomplete? Are you saying that the universe is not consistent? This sounds like a better argument for the idea that the universe is a simulation (as in we are being simulated, not as in you can represent and draw parallels between the universe and simulation. The former begs the question "on what" and we get turtles all the way down). Rather, as the old saying goes, I do not believe that the map is the territory. Just like how our brain creates an incomplete model of the world we live in, mathematics too is used to create an incomplete model to help describe not only what we can see but what we don't. But do not trivialize or diminish the notion of a model, as I certainly would not claim our brains and senses are useless. Models are quite powerful things, there is a reason we use them. But a model is not the thing itself.
Mathematics certainly serves as a language. It is also full of nice games. But it is not just that. Mathematics is real. As I said, a Turing Machine is real, and it either halts, or it doesn't. And what turns a Turing Machine into something real is not that you and I agree on what it is, communicating with each other, but that it is a mathematical construct I know the exact definition of, and the mathematical object behind that definition wouldn't change even if you would not exist.
I've seen people having trouble with my use of "real" before. When I say real, I mean that these mathematical objects exist in our reality. There is only one reality, and we share it. That's why you and I can use these mathematical objects to communicate with each other, and that's why mathematics also serves as a language. These "pretty useless algebras" exist nevertheless as themselves, you don't need to find some other, physical objects in the world that correspond to them.
> Turing Machine is real, and it either halts, or it doesn't.
*for computable functions. There's plenty of non-computable mathematics. Ideally everything would be computable but that's just not the case (pun specifically intended).
> but that it is a mathematical construct I know the exact definition of, and the mathematical object behind that definition wouldn't change even if you would not exist.
I don't think this is true. Wouldn't the definition change were we to consider finite ZF set theory? Since you cannot have an infinitely long tape?
There are algorithms where it is not possible to determine if they halt or not (Godel says "True v False is not accurate, it's True v False v Indeterminate"). There are also algorithms that halt in exactly infinite time. We can make that countable steps on uncountable. (Finite ZF is going to definitely change things here) But then there's hypercomputation and we run into the same problem.
But maybe we're discussing this the wrong way. Let's do this differently. Certainly we can operate a "Turing Machine" purely through spoken language, right? No mathematical symbols or equations required. Now I understand you might just say "yes, but that's still math." Which is a valid point. But here we have the conundrum of distinguishing any arbitrary communication from math.
And while we're at it, let's figure out this problem. What is the relationship between {math} and {language}. Is math a superset of language? Subset? Are they completely disjoint? If they intersect then what is in the set of math that is not in the set of language? And what is in the intersection? Fuck, we're doing math right now!
> These "pretty useless algebras" exist nevertheless as themselves, you don't need to find some other, physical objects in the world that correspond to them.
Sure, but now we're at circular logic. Go back to how you started. You said math is real (claim, no proof). Then you follow by giving an example of Turing Machines stating (incorrectly) that they either halt or don't (damn tricky indeterminate option!) and then state that you know the precise definition. Your evidence here is one of utility. I'm also not entirely convinced you know the __exact__ (overly pedantic) definition as we'd need to describe that starting with our choice of preferred axioms/set theory. Yes, you can wave your hand and say that there is an exact definition in ZF, ZFC, FZF, NBG, MK, KFC, or Godelsk's-Fucked-up-ZF Set Theory, or whatever. But this arbitrariness just makes a more compelling case for math being something humans made up. We invented math to be very precise and highly consistent description of relationships between things but I'm not even sure what you're saying math "is". I don't even know what you mean by "just". As if there's something disgraceful about language that tarnishes the purity of math. Which really just means you haven't looked under the covers because it's pretty fucked up under there.
Why would math be any lesser were it a made up thing? I want to ask, what would be __wrong__ with it? If it is made up what would it change? Would it really be anything other than our perspective of it? Honestly, I think there is something wrong if the quality of being "invented" tarnishes something, with that specific quality in isolation (being precise here). So if we can't address the above can we at least address this? What would be wrong with math being invented? Honestly I think that just would speak more to the ingenuity of humans.
Side note: I'm also highly convinced that were we to meet an alien species we'd find a lot of similarities between our maths. But that would not convince me that this is real, but rather that there's just a convergent solution space. The whole process, when applied to describing our world, is in fact to make the least amount of assumptions and to make the assumptions as uncontestable as possible (but we can never escape assumptions). It would make quite a lot of sense that there'd be a lot of similarities in that and very good reasons to create such things. Just like I would expect every alien civilization to invent knives, scissors, wheels, and other such things. I mean these are rather consequences of interacting in our universe and dealing with the least energy rather than the wheel existing as a thing that is to be discovered. But if so, we come back to language, and I again cannot differentiate how the wheel or math can be existing in reality (by this abstract definition of minima in solution spaces) but language is not.
Morning! I thought you would answer along these lines ;-)
> for computable functions.
No, I believe Busy Beaver numbers are real, too. Because TMs either halt, or they don't, independently of what you can prove in some axiom system about them.
> I don't think this is true. Wouldn't the definition change were we to consider finite ZF set theory? Since you cannot have an infinitely long tape?
No, the definition would not change: Yes, we can have an infinitely long tape, as a mathematical object, that is part of the overall mathematical object that is the TM. Axiom systems describe mathematical objects, so your choice of axioms to describe a TM is not arbitrary, but must make a TM a model of these axioms. So if you are describing a TM with an infinitely long tape with axioms that demand finiteness, your description is wrong.
> You said math is real (claim, no proof)
Yes, I have no proof for this, and there might never be, because, as Gödel showed, you cannot prove everything that is true (note that Gödel also thought that math is real). But it is the only thing that makes sense, and actually allows you to move forward doing proper mathematics, instead of saying things such as "there is no infinitely long tape", although you know perfectly well how such an infinitely long tape looks like.
> Why would math be any lesser were it a made up thing?
It is not a moral judgement on my part. I don't think language is bad, I really like language. And you can make up descriptions in math of course, using language and axioms, I do it all the time. And some of these descriptions will describe real mathematical objects that exist, and some of your descriptions will be inconsistent, and so the things you described won't exist. In general, it is not possible to prove which of your descriptions are consistent, you will need faith here, although you can make relative judgements. I have faith in the natural numbers, for example.
> I'm also highly convinced that were we to meet an alien species we'd find a lot of similarities between our maths. But that would not convince me that this is real, but rather that there's just a convergent solution space.
I cannot prove you wrong, so you can keep your opinion. But I find it more convincing simply to accept that math is real, than that there is a "convergent solution space", whatever that is (and is it real?).
This would be a good thing to provide a citation for.
> you will need faith here,
Sorry, I don't do that.
> than that there is a "convergent solution space", whatever that is (and is it real?).
Well yes. Physics does optimize for the least energy. For example, if you intend to build a device that you hold in your hand and can smack things you're going to probably come up with a hammer. You're probably also going to come up with a hammer who's head is a cylinder too. But that's because we live in a universe and that the thing we're doing is bound by its rules and influences.
Idk man, it seems like your argument is just consistent with "everything is real." I've asked and you can't define to me what is real and what isn't real, with you clearly using a nonstandard definition. That's not a proof, that's just definition.
There is no proof for this kind of thing, just faith. And there is a proof that faith must be enough. So I know if you don't even have faith in your foundational position, you have actually nothing at all. You can call it "conviction", if you want. If you want to know more about Gödel, there is a thing called the Internet. Look it up [1].
Well, I think it is just a clear sign of someone who is not interested in a proper discussion, and it is also somewhat insulting: To ask for a reference, when you can just look it up yourself, in about the same amount of time it takes to ask for a reference. Especially if you name your account after Gödel (it's Gödel, not Godel, by the way), then it just shows that you have no clue what you are actually talking about.
