7 ms·

Mathematics is hard for mathematicians to understand too

134 points by mmaaz 10mo ago

geomark 10mo ago
I thought we were well past trying to understand mathematics. After all, John von Neumann long ago said "In mathematics we don't understand things. We just get used to them."
- ekjhgkejhgk 10mo ago
  Just because someone said it doesn't mean we all agree with it, fortunately. You know the meme with the normal distribution where the far right and the far left reach the same conclusion for different reasons, and the ones in the middle have a completely different opinion? So on the far right you have people on von Neumann who says "In mathematics we don't understand things". On the far left you have people like you who say "me no mats". Then in the middle you have people like me, who say "maths is interesting, let me do something I enjoy".
  - geomark 10mo ago
    Of course. I just find it hilarious that someone like von Neumann would say that.
    - ekjhgkejhgk 10mo ago
      von Neumann liked saying things that he knew would have an effect like "so deep" and "he's so smart". Like when asked how he knew the answer, claiming that he did the sum in his head when undoutedly he knew the closed-form expression.
      - srean 10mo ago
        I have tingling suspicion that you might have missed the joke. To date I have not met anyone who thought he summed the terms of the infinite series in geometric series term by term. That would take infinite time. Of course he used the expression for the sum of a geometric series. The joke is that he missed a clever solution that does not require setting up the series, recognising it's in geometric progression and then using the closed form. The clever solution just finds the time needed for the trains to collide, then multiply that with the birds speed. No series needed.
        ekjhgkejhgk 10mo ago
        Ah. I was going by memory, and I had those two as separate stories. I didn't remember that he said "I did the sum" on the trains problem.
- Davidzheng 10mo ago
  sorry but that is a dumb quote.
  - nyeah 10mo ago
    Yeah, I wonder how exactly he meant that. I doubt that Von Neumann believed in random plug-and-chug, which is what I'd probably mean if I said I had given up on understanding something. Possibly von N was being very careful and cautious about what "understanding" means. For example there's a story that von Neumann told Shannon to call his information metric entropy, telling S "nobody really understands entropy anyway." But if you've engaged with Shannon to the point of telling him that quantity seems to be the entropy, you really do understand something about entropy. So maybe v N's worry was about really undertanding math concepts fully and extremely clearly. Going way beyond the point where I'd say "oh I get it!"
- ekidd 10mo ago
  Many ideas in math are extremely simple at heart. Some very precise definitions, maybe a clever theorem. The hard part is often: Why is this result important? How does this result generalize things I already knew? What are some concrete examples of this idea? Why are the definitions they way they are, and not something slightly different? To use an example from functional programming, I could say: - "A monad is basically a generalization of a parameterized container type that supports flatMap and newFromSingleValue." - "A monad is a generalized list comprehension." - Or, famously, "A monad is just a monoid in the category of endofunctors, what's the problem?" The basic idea, once you get it, is trivial. But the context, the familiarity, the basic examples, and the relationships to other ideas take a while to sink in. And once they do, you ask "That's it?" So the process of understanding monads usually isn't some sudden flash of insight, because there's barely anything there. It's more a situation where you work with the idea long enough and you see it in a few contexts, and all the connections become familiar. (I have a long-term project to understand one of the basic things in category theory, "adjoint functors." I can read the definition just fine. But I need to find more examples that relate to things I already care about, and I need to learn why that particular abstraction is a particularly useful one. Someday, I presume I'll look at it and think, "Oh, yeah. That thing. It's why interesting things X, Y and Z are all the same thing under the hood." Everything else in category theory has been useful up until this point, so maybe this will be useful, too?)
- agumonkey 10mo ago
  It's probably a neurological artefact. When the brain just spent enough time looking at a pattern it can suddenly become obvious. You can go from blind to enlightened without the usual conscious logical effort. It's very odd.
borracciaBlu 10mo ago
I was writing a small article about [Set, Set Builder Notation, and Set Comprehension](https://adropincalm.com/blog/set-set-builder-natatio-set-comprehension-explained/ https://adropincalm.com/blog/set-set-builder-natatio-set-com...) and while i was investigating it surprised me how many different ways are to describe the same thing. Eg: see all the notation of a Set or a Tuple. One last rant point is that you don't have "the manual" of math in the very same way you would go on your programming language man page and so there is no single source of truth. Everybody assumes...
- BlackFingolfin 10mo ago
  I find it strange to compare "math" with one programming language. Mathematics is a huge and diverse field, with many subcommunities and hence also differing notation. Your rant would be akin to this if the sides are reversed: "It's surprising how many different ways there are to describe the same thing. Eg: see all the notations for dictionaries (hash tables? associative arrays? maps?) or lists (vectors? arrays?). You don't have "the manual" of programming languages. "
  - segfaultex 10mo ago
    Not the original commenter, but I 100% agree that it's weird we have so many ways to describe dictionaries/hash tables/maps/etc. and lists.
  - worthless-trash 10mo ago
    > You don't have "the manual" of programming languages. " Well, we kinda do when you can say "this python program" the problem with a lot of math is that you can't even tell which manual to look up.
    - nkrisc 10mo ago
      Someone not educated in programming would not know that a given text is Python source code.
johngossman 10mo ago
Mathematics is such an old field, older than anything except arguably philosophy, that it's too broad and deep for anyone to really understand everything. Even in graduate school I often took classes in things discovered by Gauss or Euler centuries before. A lot of the mathematical topics the HN crowd seems to like--things like the Collatz conjecture or Busy Beavers--are 60, 80 years old. So, you end up having to spend years specializing and then struggle to find other with the same background. All of which is compounded by the desire to provide minimal "proofs from the book" and leave out the intuitions behind them.
- Davidzheng 10mo ago
  actually a lot of minimal proof expose more intuition than older proofs people find at first. I find it usually not extremely enlightening reading the first proofs of results, counterintuitively.
- ekjhgkejhgk 10mo ago
  > A lot of the mathematical topics the HN crowd seems to like--things like the Collatz conjecture or Busy Beavers--are 60, 80 years old. Do you know the reason for that? The reason is that those problems are open and easy to understand. For the rest of open problems, you need an expert to even understand the problem statement.
- scotty79 10mo ago
  > Mathematics is such an old field, older than anything except arguably philosophy If we are already venturing outside of scientific realm with philosophy, I'm sure fields of literature or politics are older. Especially since philosophy is just a subset of literature.
  - saithound 10mo ago
    > I'm sure fields of literature or politics are older. As far as anybody can tell, mathematics is way older than literature. The oldest known proper accounting tokens are from 7000ish BCE, and show proper understanding of addition and multiplication. The people who made the Ishango bone 25k years ago were probably aware of at least rudimentary addition. The earliest writings are from the 3000s BCE, and are purely administrative. Literature, by definition, appeared later than writing.
ikyr9999 10mo ago
Just the other day I was listening to EconTalk on this: https://www.econtalk.org/a-mind-blowing-way-of-looking-at-math-with-david-bessis/ https://www.econtalk.org/a-mind-blowing-way-of-looking-at-ma...
- MrDrDr 10mo ago
  Thank you for posting! - I was not aware of this.
MrDrDr 10mo ago
I think this would be extremely valuable: “We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics—with consequently less energy on the most recent results.” I’ve long thought that more of us could devout time to serious maths problems if they were written in a language we all understood. A little off topic perhaps, but out of curiosity - how many of us here have an interest in recreational mathematics? [https://en.wikipedia.org/wiki/Recreational_mathematics https://en.wikipedia.org/wiki/Recreational_mathematics]
- segfaultex 10mo ago
  Yeah, I don't want to be uncharitable, but I've noticed that a lot of stem fields make heavy use of esoteric language and syntax, and I suspect they do so as a means of gatekeeping. I understand that some degree of formalism is required to enable the sharing of knowledge amongst people across a variety of languages, but sometimes I'll read a white paper and think "wow, this could be written a LOT more simply". Statistics is a major culprit of this.
  - bell-cot 10mo ago
    Gatekeeping, or self-promotion? You don't get investors/patents/promotions/tenure by making your knowledge or results sound simple and understandable.
    - segfaultex 10mo ago
      Why not both? And that's a good point, there are a LOT of incentives to make things arbitrarily complex in a variety of fields.
    - master-lincoln 10mo ago
      Is that really the case or are you just assuming so? Seems counter-intuitive to me.
  - bncndn0956 10mo ago
    3blue1brown proves your point. The saying, "What one fool can do, another can," is a motto from Silvanus P. Thompson's book Calculus Made Easy. It suggests that a task someone without great intelligence can accomplish must be relatively simple, implying that anyone can learn to do it if they put in the effort. The phrase is often used to encourage someone, demystify a complex subject, and downplay the difficulty of a task.
isolli 10mo ago
I recently came to realize the same things about physics. Even physicists find it hard to develop an intuitive mental picture of how space-time folds or what a photon is.
- abraxas 10mo ago
  Well, that's just the esoterical nature of physics, no? I mean the old adage that "if you think you understand quantum physics you do not understand quantum physics" is a reflection of this.
pathikrit 10mo ago
I love math but the symbology and notations get in my way. 2 ideas: 1. Can we reinvent notation and symbology? No superscripts or subscripts or greek letters and weird symbols? Just functions with input and output? Verifiable by type systems AND human readable 2. Also, make the symbology hyperlinked i.e. if it uses a theorem or axiom that's not on the paper - hyperlink to its proof and so on..
- zwnow 10mo ago
  I'd love getting rid of all the weird symbols in favor of clear text functions or whatever. As someone who never learnt all the weird symbols its really preventing me from getting into math again... It is just not intuitive.
  - Jensson 10mo ago
    Those are used since it makes things easier, if you write everything out basically nobody would manage to learn math, that is how it used to be and then everything got shortened and suddenly average people could learn calculus.
    - zwnow 10mo ago
      Yea because hieroglyphs are more understandable than the name of a function
      - Jensson 10mo ago
        That is exactly it, a long text is much harder to understand than a one liner, we see that time and time again in problem solving if you write the same problem as a long text many fewer students manage to solve it than if you write it as a one liner.
      - deleted 10mo ago
        [deleted]
      - SetTheorist 10mo ago
fithisux 10mo ago
Mathematics is hard when there is not much time invested in processing the core idea. For example, Dvoretzky-Rogers theorem in isolation is hard to understand. While more applications of it appear While more generalizations of it appear While more alternative proofs of it appear it gets more clear. So, it takes time for something to become digestible, but the effort spent gives the real insights. Last but not least is the presentation of this theorem. Some authors are cryptic, others refactor the proof in discrete steps or find similarities with other proofs. Yes it is hard but part of the work of the mathematician is to make it easier for the others. Exactly like in code. There is a lower bound in hardness, but this is not an excuse to keep it harder than that.
zerofor_conduct 10mo ago
"The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet finally it surrounds the resistant substance." A. Grothendieck Understanding mathematical ideas often requires simply getting used to them
voidhorse 10mo ago
As someone who has always struggled with mathematics at the calculational level, but who really enjoys theorems and proofs (abstract mathematics), here are some things that help me. 1. Study predicate logic, then study it again, and again, and again. The better and more ingrained predicate logic becomes in your brain the easier mathematics becomes. 2. Once you become comfortable with predicate logic, look into set theory and model theory and understand both of these well. Understand the precise definition of "theory" wrt to model theory. If you do this, you'll have learned the rules that unify nearly all of mathematics and you'll also understand how to "plug" models into theories to try and better understand them. 3. Close reading. If you've ever played magic the gathering, mathematics is the same thing--words are defined and used in the same way in which they are in games. You need to suspend all the temptation to read in meanings that aren't there. You need to read slowly. I've often only come upon a key insight about a particular object and an accurate understanding only after rereading a passage like 50 times. If the author didn't make a certain statement, they didn't make that statement, even if it seems "obvious" you need to follow the logical chain of reasoning to make sure. 4. Translate into natural english. A lot of math books will have whole sections of proofs and /or exercises with little to no corresponding natural language "explainer" of the symbolic statements. One thing that helps me tremendously is to try and frame any proof or theorem or collection of these in terms of the linguistic names for various definitions etc. and to try and summarize a body of proofs into helpful statements. For example "groups are all about inverses and how they allow us to "reverse" compositions of (associative) operations--this is the essence of "solvability"". This summary statement about groups helps set up a framing for me whenever I go and read a proof involving groups. The framing helps tremendously because it can serve as a foil too—i.e. if some surprising theorem contravene's the summary "oh, maybe groups aren't just about inversions" that allows for an intellectual development and expansion that I find more intuitive. I sometimes think of myself as a scientist examining a world of abstract creatures (the various models (individuals) of a particular theory (species)) 5. Contextualize. Nearly all of mathematics grew out of certain lines of investigation, and often out of concrete technical needs. Understanding this history is a surprisingly effective way to make many initially mysterious aspects of a theory more obvious, more concrete, and more related to other bits of knowledge about the world, which really helps bolster understanding.
youoy 10mo ago
> As Venkatesh concludes in his lecture about the future of mathematics in a world of increasingly capable AI, “We have to ask why are we proving things at all?” Thurston puts it like this: there will be a “continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true.” This type of resoning becomes void if instead of "AI" we used something like "AGA" or "Artificial General Automation" which is a closer description of what we actually have (natural language as a programming language). Increasingly capable AGA will do things that mathematitians do not like doing. Who wants to compute logarithmic tables by hand? This got solved by calculators. Who wants to compute chaotic dynamical systems by hand? Computer simulations solved that. Who wants to improve by 2% a real analysis bound over an integral to get closer to the optimal bound? AGA is very capable at doing that. We just want to do it if it actually helps us understand why, and surfaces some structure. If not, who cares it its you who does it or a machine that knows all of the olympiad type tricks.
karmakurtisaani 10mo ago
A lot of people here suggesting they'd be great mathematicians if only it wasn't for the pesky notation. What they are missing is that the notation is the easy part..
- nh23423fefe 10mo ago
  Indeed, confused people say things that don't make sense.
- matheme 10mo ago
  > What they are missing is that the notation is the easy part. This is so wrong it can only come from a place of inexperience and ignorance. Mathematics is flush with inconsistent, abbreviated, and overloaded notation. Show a child a matrix numerically and they can understand it, show them Ax+s=b, and watch the confusion.
  - gjulianm 10mo ago
    Well, obviously they will be confused because you jumped from a square of numbers to a bunch of operations. They’d be equally confused if you presented those operations numerically. I am not sure what it is you want to prove with that example. I am also not sure that a child can actually understand what a matrix is if you just show them some numbers (i.e., will they actually understand that a matrix is a linear transformer of vectors and the properties it has just by showing them some numbers?)
    - matheme 10mo ago
      > a bunch of operations. Sorry, the notation is bit confusing. The 'A' here is a matrix.
      - gjulianm 10mo ago
        I know it is a matrix, the notation is not confusing at all. I am saying that the concept of a matrix as a set of numbers arranged in a rectangles and the concept of operations on a matrix are very different things, the confusion will not come from notation.
zkmon 10mo ago
The views quoted are just as cryptic as modern mathematics. Did mathematicians lose the ability to convey stuff tin plain simple ways? Probably they are trying to romanticize something that may not sound good if told plainly. Face it. Mathematics is one of fields strongly affected by AI, just like programming. You need to be more straight forward about it rather than beating around the bush. To simply put, it appears to be a struggle for redefining new road map, survival and adoption in AI era.
matheme 10mo ago
> Venkatesh argued that the record on this is terrible, lamenting that “for a typical paper or talk, very few of us understand it.” > "few of us" You see, if you plebs are unable to understand our genius its solely due to your inadequacies as a person and as an intellect, but if we are unable to understand our genius, well, that's a lamentable crisis. To make Mathematics "understandable" simply requires the inclusion of numerical examples. A suggestion 'the mathematics community' is hostile to. If you are unable to express numerically then I'd argue you are unable to understand.
- xigoi 10mo ago
  A lot of math is not about numbers, so not everything can have a numerical example.
assemblyman 10mo ago
I find software engineers spend too much time focused on notation. Maybe they are right to do so and notation definitely can be helpful or a hindrance, but the goal of any mathematical field is understanding. It's not even to prove theorems. Proving theorems is useful (a) because it identifies what is true and under what circumstances, and (b) the act of proving forces one to build a deep understanding of the phenomenon under study. This requires looking at examples, making a hypothesis more specific or sometimes more general, using formal arguments, geometrical arguments, studying algebraic structures, basically anything that leads to better understanding. Ideally, one understands a subject so well that notation basically doesn't matter. In a sense, the really key ingredient are the definitions because the objects are chosen carefully to be interesting but workable. If the idea is that the right notation will make getting insights easier, that's a futile path to go down on. What really helps is looking at objects and their relationships from multiple viewpoints. This is really what one does both in mathematics and physics. Someone quoted von Neumann about getting used to mathematics. My interpretation always was that once is immersed in a topic, slowly it becomes natural enough that one can think about it without getting thrown off by relatively superficial strangeness. As a very simple example, someone might get thrown off the first time they learn about point-set topology. It might feel very abstract coming from analysis but after a standard semester course, almost everyone gets comfortable enough with the basic notions of topological spaces and homeomorphisms. One thing mathematics education is really bad at is motivating the definitions. This is often done because progress is meandering and chaotic and exposing the full lineage of ideas would just take way too long. Physics education is generally far better at this. I don't know of a general solution except to pick up appropriate books that go over history (e.g. https://www.amazon.com/Genesis-Abstract-Group-Concept-Contribution/dp/0486458687 https://www.amazon.com/Genesis-Abstract-Group-Concept-Contri...)
- matheme 10mo ago
  > If the idea is that the right notation will make getting insights easier, that's a futile path to go down on. I agree whole heartedly. What I want to see is mathematicians employ the same rigor of journalists using abbreviations: define (numerically) your notation, or terminology, the first time you use it, then feel free to use it as notation or jargon for the remainder of the paper.
carlCarlCarlCar 10mo ago
Applied math is little more than semantics compression. This fundamental truth is embedded in the common symbols of arithmetic... + ... one line combined with another ...linear...line wee - ...opposite of + one line removed x ...eXponential addition, combining groups •/• ... exponential breaking into groups ...also hints at inherent ratio From there it's symbols that describe different objects and how to apply the fundamental arithmetic operations; like playing over a chord in music The interesting work is in physical science not the notation. Math is used to capture physics that would be too verbose to describe in English or some other "human" language. Which IMO should be reserved for capturing emotional context anyway as that's where they originate from. Programming languages have senselessly obscured the simple and elegant reality of computation, which is really just a subset of math; the term computer originated to describe humans that manually computed. Typescript, Python, etc don't exist[1]. They are leaky abstractions that waste a lot of resources to run some electromagnetic geometry state changes. Whether it's politics, religion or engineering, "blue" language, humans seem obsessed with notation fetishes. Imo it's all rather prosaic and boring [1] at best they exist as ethno objects of momentary social value to those who discuss them
moi2388 10mo ago
I think notation and motivation matters greatly, in the context of learning. Awesome that for mathematicians notation does not matter, an every solved problem is trivial.. But for a student this is not the case yet. Take the simple pi vs tau debate. Of course it doesn’t matter which you use once you understand them. But if you don’t understand it yet, and learn about it for the first time, tau makes everything a lot more intuitive.

134 points by mmaaz 10mo ago

I thought we were well past trying to understand mathematics. After all, John von Neumann long ago said "In mathematics we don't understand things. We just get used to them."

ekjhgkejhgk 10mo ago
Just because someone said it doesn't mean we all agree with it, fortunately. You know the meme with the normal distribution where the far right and the far left reach the same conclusion for different reasons, and the ones in the middle have a completely different opinion? So on the far right you have people on von Neumann who says "In mathematics we don't understand things". On the far left you have people like you who say "me no mats". Then in the middle you have people like me, who say "maths is interesting, let me do something I enjoy".
- geomark 10mo ago
  Of course. I just find it hilarious that someone like von Neumann would say that.
  - ekjhgkejhgk 10mo ago
    von Neumann liked saying things that he knew would have an effect like "so deep" and "he's so smart". Like when asked how he knew the answer, claiming that he did the sum in his head when undoutedly he knew the closed-form expression.
    - srean 10mo ago
      I have tingling suspicion that you might have missed the joke. To date I have not met anyone who thought he summed the terms of the infinite series in geometric series term by term. That would take infinite time. Of course he used the expression for the sum of a geometric series. The joke is that he missed a clever solution that does not require setting up the series, recognising it's in geometric progression and then using the closed form. The clever solution just finds the time needed for the trains to collide, then multiply that with the birds speed. No series needed.
      - ekjhgkejhgk 10mo ago
        Ah. I was going by memory, and I had those two as separate stories. I didn't remember that he said "I did the sum" on the trains problem.
Davidzheng 10mo ago
sorry but that is a dumb quote.
- nyeah 10mo ago
  Yeah, I wonder how exactly he meant that. I doubt that Von Neumann believed in random plug-and-chug, which is what I'd probably mean if I said I had given up on understanding something. Possibly von N was being very careful and cautious about what "understanding" means. For example there's a story that von Neumann told Shannon to call his information metric entropy, telling S "nobody really understands entropy anyway." But if you've engaged with Shannon to the point of telling him that quantity seems to be the entropy, you really do understand something about entropy. So maybe v N's worry was about really undertanding math concepts fully and extremely clearly. Going way beyond the point where I'd say "oh I get it!"
ekidd 10mo ago
Many ideas in math are extremely simple at heart. Some very precise definitions, maybe a clever theorem. The hard part is often: Why is this result important? How does this result generalize things I already knew? What are some concrete examples of this idea? Why are the definitions they way they are, and not something slightly different? To use an example from functional programming, I could say: - "A monad is basically a generalization of a parameterized container type that supports flatMap and newFromSingleValue." - "A monad is a generalized list comprehension." - Or, famously, "A monad is just a monoid in the category of endofunctors, what's the problem?" The basic idea, once you get it, is trivial. But the context, the familiarity, the basic examples, and the relationships to other ideas take a while to sink in. And once they do, you ask "That's it?" So the process of understanding monads usually isn't some sudden flash of insight, because there's barely anything there. It's more a situation where you work with the idea long enough and you see it in a few contexts, and all the connections become familiar. (I have a long-term project to understand one of the basic things in category theory, "adjoint functors." I can read the definition just fine. But I need to find more examples that relate to things I already care about, and I need to learn why that particular abstraction is a particularly useful one. Someday, I presume I'll look at it and think, "Oh, yeah. That thing. It's why interesting things X, Y and Z are all the same thing under the hood." Everything else in category theory has been useful up until this point, so maybe this will be useful, too?)
agumonkey 10mo ago
It's probably a neurological artefact. When the brain just spent enough time looking at a pattern it can suddenly become obvious. You can go from blind to enlightened without the usual conscious logical effort. It's very odd.

borracciaBlu 10mo ago

I was writing a small article about [Set, Set Builder Notation, and Set Comprehension](https://adropincalm.com/blog/set-set-builder-natatio-set-comprehension-explained/ https://adropincalm.com/blog/set-set-builder-natatio-set-com...) and while i was investigating it surprised me how many different ways are to describe the same thing. Eg: see all the notation of a Set or a Tuple. One last rant point is that you don't have "the manual" of math in the very same way you would go on your programming language man page and so there is no single source of truth. Everybody assumes...

BlackFingolfin 10mo ago
I find it strange to compare "math" with one programming language. Mathematics is a huge and diverse field, with many subcommunities and hence also differing notation. Your rant would be akin to this if the sides are reversed: "It's surprising how many different ways there are to describe the same thing. Eg: see all the notations for dictionaries (hash tables? associative arrays? maps?) or lists (vectors? arrays?). You don't have "the manual" of programming languages. "
- segfaultex 10mo ago
  Not the original commenter, but I 100% agree that it's weird we have so many ways to describe dictionaries/hash tables/maps/etc. and lists.
- worthless-trash 10mo ago
  > You don't have "the manual" of programming languages. " Well, we kinda do when you can say "this python program" the problem with a lot of math is that you can't even tell which manual to look up.
  - nkrisc 10mo ago
    Someone not educated in programming would not know that a given text is Python source code.

johngossman 10mo ago

Mathematics is such an old field, older than anything except arguably philosophy, that it's too broad and deep for anyone to really understand everything. Even in graduate school I often took classes in things discovered by Gauss or Euler centuries before. A lot of the mathematical topics the HN crowd seems to like--things like the Collatz conjecture or Busy Beavers--are 60, 80 years old. So, you end up having to spend years specializing and then struggle to find other with the same background. All of which is compounded by the desire to provide minimal "proofs from the book" and leave out the intuitions behind them.

Davidzheng 10mo ago
actually a lot of minimal proof expose more intuition than older proofs people find at first. I find it usually not extremely enlightening reading the first proofs of results, counterintuitively.
ekjhgkejhgk 10mo ago
> A lot of the mathematical topics the HN crowd seems to like--things like the Collatz conjecture or Busy Beavers--are 60, 80 years old. Do you know the reason for that? The reason is that those problems are open and easy to understand. For the rest of open problems, you need an expert to even understand the problem statement.
scotty79 10mo ago
> Mathematics is such an old field, older than anything except arguably philosophy If we are already venturing outside of scientific realm with philosophy, I'm sure fields of literature or politics are older. Especially since philosophy is just a subset of literature.
- saithound 10mo ago
  > I'm sure fields of literature or politics are older. As far as anybody can tell, mathematics is way older than literature. The oldest known proper accounting tokens are from 7000ish BCE, and show proper understanding of addition and multiplication. The people who made the Ishango bone 25k years ago were probably aware of at least rudimentary addition. The earliest writings are from the 3000s BCE, and are purely administrative. Literature, by definition, appeared later than writing.

ikyr9999 10mo ago

Just the other day I was listening to EconTalk on this: https://www.econtalk.org/a-mind-blowing-way-of-looking-at-math-with-david-bessis/ https://www.econtalk.org/a-mind-blowing-way-of-looking-at-ma...

MrDrDr 10mo ago
Thank you for posting! - I was not aware of this.

MrDrDr 10mo ago

I think this would be extremely valuable: “We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics—with consequently less energy on the most recent results.” I’ve long thought that more of us could devout time to serious maths problems if they were written in a language we all understood. A little off topic perhaps, but out of curiosity - how many of us here have an interest in recreational mathematics? [https://en.wikipedia.org/wiki/Recreational_mathematics https://en.wikipedia.org/wiki/Recreational_mathematics]

segfaultex 10mo ago
Yeah, I don't want to be uncharitable, but I've noticed that a lot of stem fields make heavy use of esoteric language and syntax, and I suspect they do so as a means of gatekeeping. I understand that some degree of formalism is required to enable the sharing of knowledge amongst people across a variety of languages, but sometimes I'll read a white paper and think "wow, this could be written a LOT more simply". Statistics is a major culprit of this.
- bell-cot 10mo ago
  Gatekeeping, or self-promotion? You don't get investors/patents/promotions/tenure by making your knowledge or results sound simple and understandable.
  - segfaultex 10mo ago
    Why not both? And that's a good point, there are a LOT of incentives to make things arbitrarily complex in a variety of fields.
  - master-lincoln 10mo ago
    Is that really the case or are you just assuming so? Seems counter-intuitive to me.
- bncndn0956 10mo ago
  3blue1brown proves your point. The saying, "What one fool can do, another can," is a motto from Silvanus P. Thompson's book Calculus Made Easy. It suggests that a task someone without great intelligence can accomplish must be relatively simple, implying that anyone can learn to do it if they put in the effort. The phrase is often used to encourage someone, demystify a complex subject, and downplay the difficulty of a task.

isolli 10mo ago

I recently came to realize the same things about physics. Even physicists find it hard to develop an intuitive mental picture of how space-time folds or what a photon is.

abraxas 10mo ago
Well, that's just the esoterical nature of physics, no? I mean the old adage that "if you think you understand quantum physics you do not understand quantum physics" is a reflection of this.

pathikrit 10mo ago

I love math but the symbology and notations get in my way. 2 ideas: 1. Can we reinvent notation and symbology? No superscripts or subscripts or greek letters and weird symbols? Just functions with input and output? Verifiable by type systems AND human readable 2. Also, make the symbology hyperlinked i.e. if it uses a theorem or axiom that's not on the paper - hyperlink to its proof and so on..

zwnow 10mo ago
I'd love getting rid of all the weird symbols in favor of clear text functions or whatever. As someone who never learnt all the weird symbols its really preventing me from getting into math again... It is just not intuitive.
- Jensson 10mo ago
  Those are used since it makes things easier, if you write everything out basically nobody would manage to learn math, that is how it used to be and then everything got shortened and suddenly average people could learn calculus.
  - zwnow 10mo ago
    Yea because hieroglyphs are more understandable than the name of a function
    - Jensson 10mo ago
      That is exactly it, a long text is much harder to understand than a one liner, we see that time and time again in problem solving if you write the same problem as a long text many fewer students manage to solve it than if you write it as a one liner.
    - deleted 10mo ago
      [deleted]
    - SetTheorist 10mo ago

fithisux 10mo ago

Mathematics is hard when there is not much time invested in processing the core idea. For example, Dvoretzky-Rogers theorem in isolation is hard to understand. While more applications of it appear While more generalizations of it appear While more alternative proofs of it appear it gets more clear. So, it takes time for something to become digestible, but the effort spent gives the real insights. Last but not least is the presentation of this theorem. Some authors are cryptic, others refactor the proof in discrete steps or find similarities with other proofs. Yes it is hard but part of the work of the mathematician is to make it easier for the others. Exactly like in code. There is a lower bound in hardness, but this is not an excuse to keep it harder than that.

zerofor_conduct 10mo ago

"The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet finally it surrounds the resistant substance." A. Grothendieck Understanding mathematical ideas often requires simply getting used to them

voidhorse 10mo ago

As someone who has always struggled with mathematics at the calculational level, but who really enjoys theorems and proofs (abstract mathematics), here are some things that help me. 1. Study predicate logic, then study it again, and again, and again. The better and more ingrained predicate logic becomes in your brain the easier mathematics becomes. 2. Once you become comfortable with predicate logic, look into set theory and model theory and understand both of these well. Understand the precise definition of "theory" wrt to model theory. If you do this, you'll have learned the rules that unify nearly all of mathematics and you'll also understand how to "plug" models into theories to try and better understand them. 3. Close reading. If you've ever played magic the gathering, mathematics is the same thing--words are defined and used in the same way in which they are in games. You need to suspend all the temptation to read in meanings that aren't there. You need to read slowly. I've often only come upon a key insight about a particular object and an accurate understanding only after rereading a passage like 50 times. If the author didn't make a certain statement, they didn't make that statement, even if it seems "obvious" you need to follow the logical chain of reasoning to make sure. 4. Translate into natural english. A lot of math books will have whole sections of proofs and /or exercises with little to no corresponding natural language "explainer" of the symbolic statements. One thing that helps me tremendously is to try and frame any proof or theorem or collection of these in terms of the linguistic names for various definitions etc. and to try and summarize a body of proofs into helpful statements. For example "groups are all about inverses and how they allow us to "reverse" compositions of (associative) operations--this is the essence of "solvability"". This summary statement about groups helps set up a framing for me whenever I go and read a proof involving groups. The framing helps tremendously because it can serve as a foil too—i.e. if some surprising theorem contravene's the summary "oh, maybe groups aren't just about inversions" that allows for an intellectual development and expansion that I find more intuitive. I sometimes think of myself as a scientist examining a world of abstract creatures (the various models (individuals) of a particular theory (species)) 5. Contextualize. Nearly all of mathematics grew out of certain lines of investigation, and often out of concrete technical needs. Understanding this history is a surprisingly effective way to make many initially mysterious aspects of a theory more obvious, more concrete, and more related to other bits of knowledge about the world, which really helps bolster understanding.

youoy 10mo ago

> As Venkatesh concludes in his lecture about the future of mathematics in a world of increasingly capable AI, “We have to ask why are we proving things at all?” Thurston puts it like this: there will be a “continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true.” This type of resoning becomes void if instead of "AI" we used something like "AGA" or "Artificial General Automation" which is a closer description of what we actually have (natural language as a programming language). Increasingly capable AGA will do things that mathematitians do not like doing. Who wants to compute logarithmic tables by hand? This got solved by calculators. Who wants to compute chaotic dynamical systems by hand? Computer simulations solved that. Who wants to improve by 2% a real analysis bound over an integral to get closer to the optimal bound? AGA is very capable at doing that. We just want to do it if it actually helps us understand why, and surfaces some structure. If not, who cares it its you who does it or a machine that knows all of the olympiad type tricks.

karmakurtisaani 10mo ago

A lot of people here suggesting they'd be great mathematicians if only it wasn't for the pesky notation. What they are missing is that the notation is the easy part..

nh23423fefe 10mo ago
Indeed, confused people say things that don't make sense.
matheme 10mo ago
> What they are missing is that the notation is the easy part. This is so wrong it can only come from a place of inexperience and ignorance. Mathematics is flush with inconsistent, abbreviated, and overloaded notation. Show a child a matrix numerically and they can understand it, show them Ax+s=b, and watch the confusion.
- gjulianm 10mo ago
  Well, obviously they will be confused because you jumped from a square of numbers to a bunch of operations. They’d be equally confused if you presented those operations numerically. I am not sure what it is you want to prove with that example. I am also not sure that a child can actually understand what a matrix is if you just show them some numbers (i.e., will they actually understand that a matrix is a linear transformer of vectors and the properties it has just by showing them some numbers?)
  - matheme 10mo ago
    > a bunch of operations. Sorry, the notation is bit confusing. The 'A' here is a matrix.
    - gjulianm 10mo ago
      I know it is a matrix, the notation is not confusing at all. I am saying that the concept of a matrix as a set of numbers arranged in a rectangles and the concept of operations on a matrix are very different things, the confusion will not come from notation.

zkmon 10mo ago

The views quoted are just as cryptic as modern mathematics. Did mathematicians lose the ability to convey stuff tin plain simple ways? Probably they are trying to romanticize something that may not sound good if told plainly. Face it. Mathematics is one of fields strongly affected by AI, just like programming. You need to be more straight forward about it rather than beating around the bush. To simply put, it appears to be a struggle for redefining new road map, survival and adoption in AI era.

matheme 10mo ago

> Venkatesh argued that the record on this is terrible, lamenting that “for a typical paper or talk, very few of us understand it.” > "few of us" You see, if you plebs are unable to understand our genius its solely due to your inadequacies as a person and as an intellect, but if we are unable to understand our genius, well, that's a lamentable crisis. To make Mathematics "understandable" simply requires the inclusion of numerical examples. A suggestion 'the mathematics community' is hostile to. If you are unable to express numerically then I'd argue you are unable to understand.

xigoi 10mo ago
A lot of math is not about numbers, so not everything can have a numerical example.

assemblyman 10mo ago

I find software engineers spend too much time focused on notation. Maybe they are right to do so and notation definitely can be helpful or a hindrance, but the goal of any mathematical field is understanding. It's not even to prove theorems. Proving theorems is useful (a) because it identifies what is true and under what circumstances, and (b) the act of proving forces one to build a deep understanding of the phenomenon under study. This requires looking at examples, making a hypothesis more specific or sometimes more general, using formal arguments, geometrical arguments, studying algebraic structures, basically anything that leads to better understanding. Ideally, one understands a subject so well that notation basically doesn't matter. In a sense, the really key ingredient are the definitions because the objects are chosen carefully to be interesting but workable. If the idea is that the right notation will make getting insights easier, that's a futile path to go down on. What really helps is looking at objects and their relationships from multiple viewpoints. This is really what one does both in mathematics and physics. Someone quoted von Neumann about getting used to mathematics. My interpretation always was that once is immersed in a topic, slowly it becomes natural enough that one can think about it without getting thrown off by relatively superficial strangeness. As a very simple example, someone might get thrown off the first time they learn about point-set topology. It might feel very abstract coming from analysis but after a standard semester course, almost everyone gets comfortable enough with the basic notions of topological spaces and homeomorphisms. One thing mathematics education is really bad at is motivating the definitions. This is often done because progress is meandering and chaotic and exposing the full lineage of ideas would just take way too long. Physics education is generally far better at this. I don't know of a general solution except to pick up appropriate books that go over history (e.g. https://www.amazon.com/Genesis-Abstract-Group-Concept-Contribution/dp/0486458687 https://www.amazon.com/Genesis-Abstract-Group-Concept-Contri...)

matheme 10mo ago
> If the idea is that the right notation will make getting insights easier, that's a futile path to go down on. I agree whole heartedly. What I want to see is mathematicians employ the same rigor of journalists using abbreviations: define (numerically) your notation, or terminology, the first time you use it, then feel free to use it as notation or jargon for the remainder of the paper.

carlCarlCarlCar 10mo ago

Applied math is little more than semantics compression. This fundamental truth is embedded in the common symbols of arithmetic... + ... one line combined with another ...linear...line wee - ...opposite of + one line removed x ...eXponential addition, combining groups •/• ... exponential breaking into groups ...also hints at inherent ratio From there it's symbols that describe different objects and how to apply the fundamental arithmetic operations; like playing over a chord in music The interesting work is in physical science not the notation. Math is used to capture physics that would be too verbose to describe in English or some other "human" language. Which IMO should be reserved for capturing emotional context anyway as that's where they originate from. Programming languages have senselessly obscured the simple and elegant reality of computation, which is really just a subset of math; the term computer originated to describe humans that manually computed. Typescript, Python, etc don't exist[1]. They are leaky abstractions that waste a lot of resources to run some electromagnetic geometry state changes. Whether it's politics, religion or engineering, "blue" language, humans seem obsessed with notation fetishes. Imo it's all rather prosaic and boring [1] at best they exist as ethno objects of momentary social value to those who discuss them

moi2388 10mo ago

I think notation and motivation matters greatly, in the context of learning. Awesome that for mathematicians notation does not matter, an every solved problem is trivial.. But for a student this is not the case yet. Take the simple pi vs tau debate. Of course it doesn’t matter which you use once you understand them. But if you don’t understand it yet, and learn about it for the first time, tau makes everything a lot more intuitive.