[HN Gopher] Mathematics is hard for mathematicians to understand...
___________________________________________________________________
Mathematics is hard for mathematicians to understand too
Author : mmaaz
Score : 96 points
Date : 2025-11-27 23:05 UTC (5 days ago)
(HTM) web link (www.science.org)
(TXT) w3m dump (www.science.org)
| geomark wrote:
| 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 wrote:
| 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 wrote:
| Of course. I just find it hilarious that someone like von
| Neumann would say that.
| ekjhgkejhgk wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| sorry but that is a dumb quote.
| nyeah wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| I was writing a small article about [Set, Set Builder Notation,
| and Set Comprehension](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 wrote:
| 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 wrote:
| 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 wrote:
| > 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 wrote:
| Someone not educated in programming would not know that a
| given text is Python source code.
| worthless-trash wrote:
| Same problem, but unlike math notation, it is MUCH
| clearer and even my 10 year old newphew can tell python
| from javascript and C.
| nkrisc wrote:
| I would wager that a vast majority of people on the
| planet could not distinguish Python from JavaScript from
| C.
| mzl wrote:
| I wrote about overlapping intervals a while ago, and used what
| I thought was the standard math notation for closed and half-
| open intervals. From comments, I learned that half-open
| intervals are written differently in french mathematics:
| https://lobste.rs/s/cireck/how_check_for_overlapping_interva...
| johngossman wrote:
| 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.
| ekjhgkejhgk wrote:
| > 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 wrote:
| > 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 wrote:
| > 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.
| thaumasiotes wrote:
| > As far as anybody can tell, mathematics is way older than
| literature.
|
| That depends what you mean by "literature". If you want it
| to be written down, then it's very recent because writing
| is very recent.
|
| But it would be normal to consider cultural products to be
| literature regardless of whether they're written down.
| Writing is a medium of transmission. You wouldn't study the
| epic of Gilgamesh _because it 's written down_. You study
| it to see what the Sumerians thought about the topics it
| covers, or to see which god some iconography that you found
| represents, or... anything that it might plausibly tell
| you. But the fact that it was written down is only the
| reason you _can_ study it, not the reason you want to.
| mkl wrote:
| > That depends what you mean by "literature". If you want
| it to be written down
|
| That is what literature means:
| https://en.wiktionary.org/wiki/literature#Noun
| pfortuny wrote:
| Well, then poetry is not literature.
| thaumasiotes wrote:
| No, the argument is even dumber than that. The person who
| writes a poem hasn't created any literature.
|
| The person who hears that poem in circulation and records
| it in his notes _has_ created literature; an anthology is
| literature but an original work isn 't.
| baobun wrote:
| > No, the argument is even dumber than that. The person
| who writes a poem hasn't created any literature.
|
| Sure they have, by virtue of writing it down. It becomes
| literature when it hits the paper (or computer screen, as
| it were).
|
| (Unless you mean to imply that formulating an original
| poem in your mind counts as "writing", in which case I
| guess we illustrate the overarching point of value in
| shared symbols and language and the waste of time in
| stating our original definitions for every statement we
| want to make)
| threatofrain wrote:
| Sure in the context that you mean it's an oral tradition.
| nkrisc wrote:
| If it's not written down, then that's true.
|
| Once someone writes it down, it is.
| Davidzheng wrote:
| 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.
| bell-cot wrote:
| I'll argue for astronomy being the oldest. Minimal knowledge
| would help pre-humans navigate and keep track of the seasons.
| Birds are known to navigate by the stars.
| nkrisc wrote:
| I would argue that some form of mathematics is necessary for
| astronomy, for "astronomy" as defined as anything more than
| simply recognizing and following stars.
| adornKey wrote:
| The desire to hide all traces where a proof comes from is
| really a problem and having more context would often be very
| helpful. I think some modern authors/teachers are nowadays
| getting good at giving more context. But mostly you have to be
| thankful that the people from the minimalist era (Bourbaki,
| ...) at least gave precise consistent definitions for basic
| terminology.
|
| Mathematics is old, but a lot of basic terminology is
| surprisingly young. Nowadays everyone agrees what an abelian
| group is. But if you look into some old books from 1900 you can
| find authors that used the word abelian for something
| completely different (e.g. orthogonal groups).
|
| Reading a book that uses "abelian" to mean "orthogonal" is
| confusing, at least until you finally understand what is going
| on.
| otoburb wrote:
| >>[...] at least gave precise consistent definitions for
| basic terminology.
|
| Hopefully interactive proof assistants like Lean or Rocq will
| help to mitigate at least this issue for anybody trying to
| learn a new (sub)field of mathematics.
| ikyr9999 wrote:
| Just the other day I was listening to EconTalk on this:
| https://www.econtalk.org/a-mind-blowing-way-of-looking-at-ma...
| MrDrDr wrote:
| Thank you for posting! - I was not aware of this.
| MrDrDr wrote:
| 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]
| segfaultex wrote:
| 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 wrote:
| Gatekeeping, or self-promotion? You don't get
| investors/patents/promotions/tenure by making your knowledge
| or results sound simple and understandable.
| segfaultex wrote:
| 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 wrote:
| Is that really the case or are you just assuming so? Seems
| counter-intuitive to me.
| bncndn0956 wrote:
| 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.
| gjulianm wrote:
| 3blue1brown, while they create great content, they do not
| go as deep into the mathematics, they avoid some of the
| harder to understand complexities and abstractions. Don't
| take me wrong, it's not a criticism of their content, it's
| just a different thing than what you'd study in a
| mathematics class.
|
| Also, an additional thing is that videos are great are
| making people think they understand something when they
| actually don't.
| jules wrote:
| 3blue1brown actually shows the usefulness of formalism. The
| videos are great, but by avoiding formalism, they are at
| least for me harder to understand than traditional sources.
| It is true that you need to get over the hump of
| understanding the formalism first, but that formalism is a
| very useful tool of thought. Consider algebraic notation
| with plus and times and so on. That makes things way easier
| to understand than writing out equations in words (as
| mathematicians used to do!). It is the same for more
| advanced formalisms.
| aleph_minus_one wrote:
| > 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".
|
| OK, challenge accepted: find a way to write one of the
| following papers much more simply:
|
| Fabian Hebestreit, Peter Scholze; A note on higher almost
| ring theory
|
| https://arxiv.org/abs/2409.01940
|
| Peter Scholze; Berkovich Motives
|
| https://arxiv.org/abs/2412.03382
|
| ---
|
| What I want to tell you with these examples (these are, of
| course, papers which are far above my mathematical level) is:
| often what you read in math papers _is_ insanely complicated;
| simplifying even one of such papers is often a huge academic
| achievement.
| gjulianm wrote:
| > I suspect they do so as a means of gatekeeping
|
| I'm surprised at how could you get at this conclusion.
| Formalisms, esoteric language and syntax are hard for
| everyone. Why would people invest in them if their only
| usefulness was gatekeeping? Specially when it's the same
| people who will publish their articles in the open for
| everyone to read.
|
| A more reasonable interpretation is that those fields use
| those things you don't like because they're actually useful
| to them and to their main audience, and that if you want to
| actually understand those concepts they talk about, that
| syntax will end up being useful to you too. And that a lack
| of syntax would not make things easier to understand, just
| less precise.
| MangoToupe wrote:
| > I suspect they do so as a means of gatekeeping.
|
| What, as opposed to using ambiguous language and getting
| absolutely nothing done?
| locknitpicker wrote:
| > 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 think you're confusing "I don't understand this" with "the
| man is keeping me down".
|
| All fields develop specialized language and syntax because a)
| they handle specialized topics and words help communicate
| these specialized concepts in a concise and clear way, b)
| syntax is problem-specific for the same reason.
|
| See for example tensor notation, or how some cultures have
| many specialized terms to refer to things like snow while
| communicating nuances.
|
| > "wow, this could be written a LOT more simply"
|
| That's fine. A big part of research is to digest findings. I
| mean, we still see things like novel proofs for the
| Pythagoras theorem. If you can express things clearer, why
| aren't you?
| zozbot234 wrote:
| Statistics is a weird special case where major subfields of
| applied statistics (including machine learning, but not
| only) sometimes retain wildly divergent terminology for the
| exact same concepts, for no good reason at all except the
| vagaries of historical development.
| beng-nl wrote:
| My opinion on this is that in mathematics the material can be
| presented in a very dry and formal way, often in service of
| rigor, which is not welcoming at all, and is in fact
| unnecessarily unwelcoming.
|
| But I don't believe it to be used as gatekeeping at all. At
| worst, hazing ("it was difficult for me as newcomer so it
| should be difficult to newcomers after me") or intellectual
| status ("look at this textbook I wrote that takes great
| intellectual effort to penetrate"). Neither of which should
| be lauded in modern times.
|
| I'm not much of a mathematician, but I've read some new and
| old textbooks, and I get the impression there is a trend
| towards presenting the material in a more welcoming way, not
| necessarily to the detriment of rigor.
| TimPC wrote:
| If it's actually in the service of rigor then it's not
| unnecessaryily unwelcoming. If it's only nominally in the
| service of rigor than maybe, but Mathematics absolutely
| needs extreme rigor.
| zozbot234 wrote:
| The upside of a "dry and formal" presentation is that it
| removes any ambiguity about what exactly you're discussing,
| and how a given argument is supposed to flow. Some steps
| may be skipped, but at least the overall structure will be
| clear enough. None of that is guaranteed when dealing with
| an "intuitive" presentation, especially when people tend to
| differ about what the "right" intuition of something ought
| to be. That can be even more frustrating, precisely when
| there's insufficient "dry and formal" rigor to pin
| everything down.
| fragmede wrote:
| In this modern era of easily accessible knowledge, how gate
| keepy is it though? It's inscrutable at first glance, but
| ChatGPT is more than happy to explain what the hell 0, 1, #,
| , or S mean, and you can ask it to read the arxiv pdf and
| have it explain it to you.
| ncfj76 wrote:
| I say the same thing about the universe. There is some gate
| keeping going on there. My 3 inch chimp brain at the age of 3
| itself was quite capable of imagining a universe. No quantum
| field equations required. Then by 6 I was doing it in
| minecraft. And by 10 I was doing it with a piano. But then
| they started wasting my time telling me to read Kant.
| Someone wrote:
| > 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.
|
| That assumes it's the language that makes it hard to understand
| serious math problems. That's partially true (and the reason
| why mathematicians keep inventing new language), but IMO the
| complexity of truly understanding large parts of mathematics is
| intrinsic, not dependent on terminology.
|
| Yes, you can say _"A monad is just a monoid in the category of
| endofunctors"_ in terms that more people know of, but it would
| take many pages, and that would make it hard to understand,
| too.
| voidhorse wrote:
| Precisely. Think of mathematics like a game.
|
| Players of magic the gathering will say a creature "has
| flying" by which they mean "it can only be blocked by other
| creatures with reach or flying".
|
| Newcomers obviously need to learn this jargon, but once they
| do, communication is greatly facilitated by not having to
| spell out the definition.
|
| Just like games, the definitions in mathematics are ethereal
| and purely formal as well, and it would be a pain to spell
| them out on every occasion. It stems more from efficient
| communication needs then from gatekeeping.
|
| You expect the players of the game to learn the rules before
| they play.
| racl101 wrote:
| Well said.
|
| I'd say the ability to take complicated definitions and to
| not have to through a rigorous definition every time the
| ideas are referenced are, in a sense a form of abstraction,
| and a necessary requirement to be able to do advanced Math
| in the first place.
| matheme wrote:
| My entire being is anthithetical to this type of
| gatekeeping.
|
| > You expect the players of the game to learn the rules
| before they play.
|
| TFA is literally from a 'player' who has 'learned the
| rules' complaining that the papers remain indecipherable.
|
| > You expect the players of the game to learn the rules
| before they play.
|
| Actually, _I_ expect to have to _teach_ rules to new
| players before they play. We are different.
| gjulianm wrote:
| The article does not complain about notation. It
| describes how the different fields of mathematics are so
| deep and so abstract that it's hard to understand them as
| a professional mathematician in a different field. That's
| a hard problem worthy of discussion, but as the article
| says, it's not as much a problem of notation or of
| explanations, rather than it's just intrinsically
| difficult and complex because these are abstract and deep
| fields.
|
| It's not gatekeeping. It's just hard.
| matheme wrote:
| I was calling you a gatekeeper rather than notation, but
| feel free to keep stuffing that man with your straw.
|
| The sentence I called out, independent of the article's
| content: "You expect the players of the game to learn the
| rules before they play."
|
| Is you explicitly stating _your_ goal is gatekeeping.
| gjulianm wrote:
| The only thing that sentence says is that it's impossible
| to understand math without understanding the language of
| math and how it is constructed. Not sure how that is
| controversial or gatekeeping. If you are annoyed at that
| comment saying "learn" instead of "be taught", I think
| that's a pedantic argument because the argument wasn't
| about that at all.
| matheme wrote:
| "Can I enter your gate?"
|
| "In order to enter this gate you must know what this
| symbol means."
|
| "I am unfamiliar with that symbol."
|
| "Well, I _expect_ you to _learn_ what it means _before_ I
| allow you to enter this gate. Now go away. "
| dr_dshiv wrote:
| See Brett Victor's: Kill Math https://worrydream.com/KillMath/
|
| He separates conceptual understanding from notational
| understanding-- pointing out that the interface of using math
| has a major impact on utility and understanding. For instance,
| Roman numerals inhibit understanding and utilization of
| multiplication.
|
| Better notational systems can be designed, he claims.
| isolli wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| Yea because hieroglyphs are more understandable than the
| name of a function
| Jensson wrote:
| 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.
| SetTheorist wrote:
| There is an inherent complexity in a lot of mathematics.
| The compact notation makes it much easier (or even
| possible) to understand what is going on.
|
| Compare something like
|
| equals(integral(divide(exponentiate(negate(divide(square(
| var),2))),sqrt(multiply(2,constant_pi))),var,negate(infin
| ity),infinity),1)
|
| vs
|
| $$\int_{-\infty}^{\infty}\frac{e^{-x^2/2}}{\sqrt{2\pi}}dx
| = 1$$
|
| (imagine the actual generated mathematical formula here
| :-/ )
|
| it is infinitely easier to grok what is going on using
| symbolic notation after a minimal amount of learning.
| abraxas wrote:
| There has to be a happy medium between the tersness of the
| current notation systems and the verbosity of code-like
| expressions. We just need to rethink this so more people
| can learn it. Math still stands a bit like writing did in
| ancient culture. It's a domain reserved for a few high
| priests inducted into the craft and completely inaccessible
| to everyone else.
| gjulianm wrote:
| I wonder why so many people are under the impression that
| the notation is what is keeping them away and if only the
| notation was easier then the underlying concepts would be
| clear. For example, if you don't know what the pullback
| of a differential form is, it doesn't matter if I write
| it in clear text or if I write the common notation ph^*
| o.
|
| > It's a domain reserved for a few high priests inducted
| into the craft and completely inaccessible to everyone
| else.
|
| It's a domain reserved for people who want to learn it,
| and there's ton of resources to learn it. Expecting to
| understand it without learning it does not make any
| sense.
| nyrikki wrote:
| The problem is that math is not some universal language,
| it is a broad field with various sub domains with their
| own conventions, assumptions, and needs.
|
| Polysemy vs Homonymy vs Context Dependency will always be
| a problem.
|
| There are lots of areas to improve, but one of the
| reasons learning math is hard is that in the elementary
| forms we pretend that there is a singular ubiquitous
| language, only to change it later.
|
| That is why books that try to be rigorous tend to
| dedicate so much room at the start to definitions.
|
| Abstract algebra is what finally help it click for me,
| but it is rare for people to be exposed to it.
| gjulianm wrote:
| I'm not sure that symbols are the thing actually keeping you
| away. Clear text functions might not be as clear, as it will
| be harder to scan and it will still contain names that you
| might not be familiar with. Those "weird symbols" are not
| there because people liked to make weird symbols. No one
| likes them, it's just that it makes things easier to
| understand.
| kragen wrote:
| Probably not. The conventional math notation has three major
| advantages over the "[n]o superscripts or subscripts or [G]reek
| letters and weird symbols" you're proposing:
|
| 1. It's more human-readable. The superscripts and subscripts
| and weird symbols permit preattentive processing of formula
| structures, accelerating pattern recognition.
|
| 2. It's familiar. Novel math notations face the same problem as
| alternative English orthographies like Shavian
| (https://en.wikipedia.org/wiki/Shavian_alphabet) in that,
| however logical they may be, the audience they'd need to appeal
| to consists of people who have spent 50 years restructuring
| their brains into specialized machines to process the
| conventional notation. Aim t3mpted te rait qe r3st ev q1s
| c0m3nt 1n mai on alterned1v i6gl1c orx2grefi
| http://canonical.org/~kragen/alphanumerenglish bet ai qi6k ail
| rez1st qe t3mpt8cen because, even though it's a much better way
| to spell English, nobody would understand it.
|
| 3. It's optimized for rewriting a formula many times. When you
| write a computer program, you only write it once, so there
| isn't a great burden in using a notation like (eq (deriv x (pow
| e y)) (mul (pow e y) (deriv x y)) 1), which takes 54 characters
| to say what the conventional math notation1 says in 16
| characters3. But, when you're performing algebraic
| transformations of a formula, you're writing the same formula
| over and over again in different forms, sometimes only slightly
| transformed; the line before that one said (eq (deriv x (pow e
| y)) (deriv x x) 1), for example2. For this purpose, brevity is
| essential, and as we know from information theory, brevity is
| proportional to the logarithm of the number of different weird
| symbols you use.
|
| We could _certainly_ improve conventional math notation, and in
| fact mathematicians invent new notation all the time in order
| to do so, but the direction you 're suggesting would not be an
| improvement.
|
| People do make this suggestion all the time. I think it's
| prompted by this experience where they have always found math
| difficult, they've always found math _notation_ difficult, and
| they infer that the former is _because of_ the latter. This
| inference, although reasonable, is incorrect. Math is
| inherently difficult, as far as anybody knows (an observation
| famously attributed to Euclid) and the difficult notation
| actually makes it _easier_. Undergraduates routinely perform
| mental feats that defied Archimedes because of it.
|
| ______
|
| 1 \frac d{dx}e^y = e^y\frac{dy}{dx} = 1
|
| 2 \frac d{dx}e^y = \frac d{dx}x = 1
|
| 3 See
| https://nbviewer.org/url/canonical.org/~kragen/sw/dev3/logar...
| for a cleaned-up version of the context where I wrote this
| equation down on paper the other day.
| zozbot234 wrote:
| > ... It's optimized for rewriting a formula many times.
|
| It's not just "rewriting" arbitrarily either, but rewriting
| _according to well-known rules_ of expression manipulation
| such as associativity, commutativity, distributivity of
| various operations, the properties of equality and order
| relations, etc. It 's precisely when you have such strong
| identifiable properties that you tend to resort to operator-
| like notation in any formalism (including a programming
| language) - not least because that's where a notion of
| "rewriting some expression" will be at its most effective.
|
| (This is generally true in reverse too; it's why e.g. text-
| like operators such as fadd() and fmul() are far better
| suited to the actual low-level properties of floating-point
| computation than FORTRAN-like symbolic expressions, which are
| sometimes overly misleading.)
| kragen wrote:
| Hmm, I'm not sure whether operator-like notation has any
| special advantage for commutativity and distributivity
| other than brevity. a + b and add(a, b) are equally easy to
| rewrite as b + a and add(b, a).
|
| Maybe there _is_ an advantage for associativity, in that
| rewriting add(a, add(b, c)) as add(add(a, b), c) is harder
| than rewriting a + b + c as a + b + c. Most of the time you
| would have just written add(a, b, c) in the first place.
| That doesn 't handle a + b - c (add(a, sub(b, c)) vs.
| sub(add(a, b), c)) but the operator syntax stops helping in
| that case when your expression is a - b + c instead, which
| is not a - (b + c) but a - (b - c).
|
| Presumably the notorious non-associativity of floating-
| point addition is what you're referring to with respect to
| fadd() and fmul()?
|
| I guess floating-point multiplication isn't quite
| commutative either, but the simplest example I could come
| up with was 0.0 * 603367941593515.0 *
| 2.9794309755910265e+293, which can be either 0 or NaN
| depending on how you associate it. There are also examples
| where you lose bits of precision to gradual underflow, like
| 8.329957634267304e-06 * 2.2853928075274668e-304 *
| 6.1924494876619e+16. But I feel like these edge cases
| matter fairly rarely?
|
| On my third try I got 3.0 * 61.0 * 147659004176083.0, which
| isn't an edge case at all, and rounds differently depending
| on the order you do the multiplications in. But it's an
| error of about one part in 10-16, and I'd think that
| algorithms that would be broken by such a small amount of
| rounding error are mostly broken in floating point anyway?
|
| I am _pretty_ sure that both operators are commutative.
| zozbot234 wrote:
| We do often find add(a, b, c), just written as S(a, b,
| c). Similar for mul and P. The binary sub operator can be
| simply rewritten in terms of add and unary minus; the
| fact that we write (a - b) instead of (a + [-b]) or
| perhaps S(a, [-b]) is ultimately a matter of notational
| convenience, but comes at some cost in mathematical
| elegance. Considering operators that are commutative yet
| not associative is not very useful; ultimately we want
| more from our expression rewriting than just flipping
| left and right subexpressions within an expression tree
| while keeping the overall complexity unchanged.
| kragen wrote:
| Usually you'd have to write that as \sum_{v \in \\{a, b,
| c\\}} v; one of the ways I think conventional math
| notation could in fact be improved would be by separating
| the aggregate function of summation from the generation
| of the items, allowing you to write \sum \\{a, b, c\\},
| at the minor cost of having to write \sum_{i = 1}^N i^2
| as something like \sum |_{i=1}^N i^2.
|
| It's not conventional to write commutative-but-not-
| associative functions as infix operators, but I don't
| think that's due to some principled reason, but just
| because they're not very common; non-associative
| operators such as subtraction and function application
| are almost universally written with infix operators, even
| the empty-string operator in the case of function
| application. The most common one is probably the Sheffer
| stroke for NAND (although Sheffer himself used it to mean
| NOR in his 01913 paper: https://www.ams.org/journals/tran
| /1913-014-04/S0002-9947-191...).
|
| You can go a bit further in the direction of logical
| manipulability, as George Spencer Brown did with "Laws of
| Form" (LoF): his logical connective, the "cross", is an
| N-ary negation function whose arguments are written under
| the operation symbol without separators between them, and
| he denotes one of the elementary boolean values as the
| empty string (let's call it false, making the cross NOR).
| ASCII isn't good at reproducing his "cross" notation, but
| if we use brackets instead, we can represent his two
| axioms as: [][] = [] (not false or not
| false is not false) [[]] = (not not false is
| false)
|
| In this way Spencer Brown harnesses the free monoid on
| his symbols: the empty string is the identity element of
| the free monoid, so appending it to the arguments of a
| cross doesn't change them and thus can't change the
| cross's value. Homomorphically, false is the identity
| element of disjunction, which is a bounded semilattice,
| and thus a monoid.
|
| This allows not only the associative axiom but also the
| identity axiom to be simple string identity, which seems
| like a real notational advantage. (Too bad there isn't
| any equivalent for the commutative axiom.) It allows
| Spencer Brown to derive all of Boolean logic from those
| two simple axioms.
|
| However, so far, I haven't found that the LoF notation is
| an actual improvement over conventional algebraic
| notation. Things like normalization to disjunctive normal
| form seem much more confusing: a(b + c)
| - ab + ac (conventional notation, rewrite rule
| towards DNF) [[a][bc]] - [[a][b]][[a][c]] (LoF
| notation)
|
| It's a little less noisy in Spencer Brown's original two-
| dimensional representation (note that the vertical breaks
| between the U+2502 BOX DRAWINGS LIGHT VERTICAL characters
| are not supposed to be there; possibly if you paste this
| into a text editor or terminal it will look better)
| +----- +----+---- |+-+-- - |+-+-|+-+-
| ||a|bc ||a|b||a|c
|
| but not, to my eye, any less confusing.
| bmacho wrote:
| AsciiMath makes easy equations read easy.
|
| 1 and 2 would be 1) d/dx e^y = e^y*(dy/dx) =
| 1 2) d/dx e^y = d/dx x = 1
| kragen wrote:
| When you render it for proper typesetting, do the
| parentheses around dy/dx disappear?
|
| If they do, it seems like an error-prone way to write your
| math.
|
| If they don't, it seems like it will make your math look
| terrible.
|
| It seems somewhat less noisy than the LaTeX version, but
| not much.
| youoy wrote:
| Notation an symbology comes out of a minmax optimisation.
| Minimizing complexity maximizing reach. As with every local
| critical point, it is probably not the only state we could have
| ended at.
|
| For example, for your point 1: we could probably start there,
| but once you get familiar with the notation you dont want to
| keep writing a huge list of parameters, so you would probably
| come up with a higher level data structure parameter which is
| more abstract to write it as an input. And then the next
| generation would complain that the data structure is too
| abstract/takes too much effort to be comunicated to someone new
| to the field, because they did not live the problem that made
| you come with a solution first hand.
|
| And for you point 2: where do you draw the line with your
| hyperlinks. If you mention the real plane, do you reference the
| construction of the real numbers? And dimensionl? If you reason
| a proof by contradiction, do you reference the axioms of logic?
| If you say "let {xn} be a converging sequence" do you reference
| convergence, natural numbers and sets? Or just convergence? Its
| not that simple, so we came up with a minmax solution which is
| what everybody does now.
|
| Having said this, there are a lot of articles books that are
| not easy to understand. But that is probably more of an issue
| of them being written by someone who is bad at communicating,
| than because of the notation.
| sfpotter wrote:
| Go ahead. Write a math paper with your proposed new notation
| with hyperlinks and submit it to a journal somewhere.
| fithisux wrote:
| 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 wrote:
| "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 wrote:
| 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 wrote:
| > 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 wrote:
| 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 wrote:
| Indeed, confused people say things that don't make sense.
| matheme wrote:
| > 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 wrote:
| 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 wrote:
| > a bunch of operations.
|
| Sorry, the notation is bit confusing. The 'A' here _is_ a
| matrix.
| gjulianm wrote:
| 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.
| matheme wrote:
| You must be correct, because this interaction is
| completely devoid of any confusion between the two people
| attempting to communicate clearly.
| gjulianm wrote:
| I do not have any confusion with the notation, I am
| confused about what the argument you're trying to convey
| with English words.
| matheme wrote:
| Ceci n'est pas une pipe.
| zkmon wrote:
| 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 wrote:
| > 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.
| assemblyman wrote:
| 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-Contri...)
| matheme wrote:
| > 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.
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