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...)
Understanding new math is hard, and a lot of people don't have a deep understanding of the math they use. Good notation has a lot of understanding already built-in, and that makes math easier to use in certain ways, but maybe harder to understand in other ways. If you understand something well enough, you are either not troubled by the notation, because you are translating it automatically into your internal representation, or you might adapt the notation to something that better suits your particular use case.
> One thing mathematics education is really bad at is motivating the definitions.
I was annoyed by this in some introductory math lectures where the prof just skipped explaining the general idea and motivation of some lemmata and instead just went through the proofs line by line.
It felt a bit like being asked to use vi, without knowing what the program does, let alone knowing the key combinations - and instead of a manual, all you have is the source code.
> 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.
> 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.
They do.
The purpose of papers is to teach working mathematicians who are already deeply into the subject something novel. So of course only novel or uncommon notation is introduced in papers.
Systematic textbooks, on the other hand, nearly always introduce a lot of notation and background knowledge that is necessary for the respective audience. As every reader of such textbooks knows, this can easily be dozens or often even hundreds of pages (the (in)famous Introduction chapter).
> 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.
They already do this. That is how we all learn notation. Not sure what you mean by numerically though, a lot of concepts cannot be defined numerically.
Notation makes a huge difference. I mean, have you TRIED to do arithmetic with Roman numerals?
>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.
Seeing the relationships between objects is partly why math has settled on a terse notation (the other reason being that you need to write stuff over and over). This helps up to a point, but mainly IF you are writing the same things again and again. If you are not exercising your memory in such a way, it is often easier to try to make sense of more verbose names. But at all times there is tension between convenience, visual space consumed, and memory consumption.
I haven't thought about or learned a systematic way to add roman numerals. But, I would argue that the difference is not notation but a fundamental conceptual advance of representing quantities by b (base) objects where each position advances by a power of b and the base objects let one increment by 1. The notation itself doesn't really make a difference. We could call X=1, M=2, C=3, V=4 and so on.
I also don't know what historically motivated the development of this system (the Indian system). Why did the Romans not think of it? What problems were the Indians solving? What was the evolution of ideas that led to the final system that still endures today?
I don't mean to underplay the importance of notation. But good notation is backed by a meaningfully different way of looking at things.
Adding and subtracting Roman numerals is pretty easy because it's all addition and subtraction. A lot of it is just repeating the symbols just like with tally marks. X+X is just XX for example. You do have to keep track of when another symbol is appropriate, but VIIII is technically equivalent to IX. It's all the other operations that get harder. If the Romans had negative numbers, then the digits of a numeral could be viewed as some kind of polynomial with some positive and negative coefficients. But they also didn't have that.
>The notation itself doesn't really make a difference. We could call X=1, M=2, C=3, V=4 and so on.
Technically, the positional representation is part of the notation as well as the symbols used. Symbols had to evolve to be more legible. For example, you don't want to mix up 1 and 7, or some other pairs that were once easily confused.
>Why did the Romans not think of it?
I don't know. I expect that not having a symbol for zero was part of it. Place value systems would be very cumbersome without that. I think that numbers have some religious significance to the Hindus, with their so-called Vedic math, but the West had Pythagoras. I'm sure that the West would have eventually figured it out, as they figured out many impressive things even without modern numerals.
>But good notation is backed by a meaningfully different way of looking at things.
That's just one aspect of good notation. Not every different way of looking at things is equally useful. Notation should facilitate or at least not get in the way of all the things we need to do the most. The actual symbols we use are important visually. A single letter might not be self-describing, but it is exactly the right kind of symbol to express long formulas and equations with a fairly small number of quantities. You can see more "objects" in front of you at once and can mechanically operate on them without silently reading their meaning. On the other hand, a single letter symbol in a large computer program can be confusing and also makes editing the code more complicated.
Considering that post-arithmetic math rarely use numbers at all, and even ancient Greeks use lots of lines and angles instead of numbers, I don't think Roman numerals would really hold math that much.
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.
> 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.
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.
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.
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.
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.
Again, learning notation is part of the process of learning math. No one is gatekeeping anything, at no point you need to do an exam or magically be aware of notation that you never saw. Every book and every class will define new notation at the beginning, in most cases they will do so even when there’s no new notation. I am not sure what your argument is.
That’s a very good gate to keep. Some things are just meant to be gatekept so that the cranks and dilettantes that wastes everyone’s time can stay far outside.
Many mathematicians do in fact teach the rules of the game in numerous introductory texts. However, you don't expect to have to explain the rules every time you play the game with people who you've established know the game. Any session would take ages if so, and in many cases the game only become more fun the more fluent the players are.
I'm not fully convinced the article makes the claim that jargon, per se, is what needs to change nor that the use of jargon causes gatekeeping. I read more about being about the inherent challenges of presenting more complicated ideas, with or without jargon and the pursuit of better methods, which themselves might actually depend on more jargon in some cases (to abstract away and offload the cognitive costs of constantly spelling out definitions). Giving a good name to something is often a really powerful way to lower the cognitive costs of arguments employing the names concept. Theoretics in large part is the hunt for good names for things and the relationships between them.
You'd be hard pressed to find a single human endeavor that does not employ jargon in some fashion. Half the point of my example was to show that you cannot escape jargon and "gatekeeping" even in something as silly and fun as a card game.
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.
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".
> 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?
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.
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.
>
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
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.
These papers are actually great examples of what is problematic with mathematics, just as what is problematic with papers in any other specialised field: how do you judge if this could be ever useful to you?
If you want to understand what is going on there, what is the most effective way to build a bridge from what you know, to what is written there?
If you are in a situation where the knowledge of these papers could actually greatly help, how do you become aware of it?
I think if AI could help solve these two issues, that would be really something.
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.
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.
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.
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.
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.
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.
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.
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 ℵ₀, ℵ₁, ♯, ♭, or Σ mean, and you can ask it to read the arxiv pdf and have it explain it to you.
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..
It's like saying that learning Arabic is the easy part of writing a great Saudi novel. True, but you have to understand that being literate is the price of admission. Clearly you consider yourself very facile with mathematical notation but you might have some empathy for the inumerate. Not everyone had the good fortune of great math teachers or even the luxury of attending a good school. I believe there is valid frustration borne out of poor mathematical education.
Not at all. Over and over I find really intimidating math notation actually represents pretty simple concepts. Sigma notation is a good example of this. Hmm, giant sigma or sum()?
Imagine how much unnecessary time would be added to a course about series if the lecturer had to write sum() instead of ∑ every time. If you find it hard to remember that ∑ means sum, math might not be for you, and that’s fine.
Wait until you learn about integration. Measures, limits and the quirks of uncountable spaces don't become simpler once you call the operation integrate().
The fact that there is a precise analogy between how Ax + s = b works when A is a matrix and the other quantities are vectors, and how this works when everything is scalars or what have you, is a fundamental insight which is useful to notationally encode. It's good to be able to readily reason that in either case, x = A^(-1) (b - s) if A is invertible, and so on.
It's good to be able to think and talk in terms of abstractions that do not force viewing analogous situations in very different terms. This is much of what math is about.
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?)
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.
This is funny. “Mathematics notation is confusing to me because I refuse to learn it. I refuse to learn it because mathematics notation is confusing to me.” Okay sure, be happy with yourself.
> This is so wrong it can only come from a place of inexperience and ignorance.
Thanks for the laughs :D
> Show a child a matrix numerically and they can understand it, show them Ax+s=b, and watch the confusion.
Show a HN misunderstood genius Riemann Zeta function as a Zeta() and they think they can figure out it's zeros. Show it as a Greek letter and they'll lament how impossible it is to understand.
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..
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.
1. I work in finance and here people sometimes write math using words as variable names. I can tell you it gets extremely cumbersome to do any significant amount of formula manipulation or writing with this notation. Keep in mind that pen and paper are still pretty much universally used in actual mathematical work and writing full words takes a lot of time compared to single Greek letters.
Large part of math notation is to compress the writing so that you can actually fit a full equation in your vision.
Also, something like what you want already exists, see e.g. Lean: https://lean-lang.org/doc/reference/latest/. It is used to write math for the purpose of automatically proving theorems. No-one wants to use this for actually studying math or manually proving theorems, because it looks horrible compared to conventional mathematics notation (as long as you are used to the conventional notation).
(1) I always tell my students that if they don't understand why things are done a certain way, that they should try to do it in the way most natural to them and then iterate to improve it. Eventually they will settle on something very similar to most common practice.
(2) Higher-level proofs are using so many ideas simultaneously that doing this would be tantamount to writing Lean code from scratch: painful.
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.
> ... 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.)
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.
We do often find add(a, b, c), just written as Σ(a, b, c). Similar for mul and Π. 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 Σ(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.
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 thing I didn't appreciate the first time I read Spencer-Brown's book is that he actually cites Sheffer's 01913 paper, and proves Sheffer's postulates within his system in an appendix. This situates him significantly closer to the mathematical mainstream than I had thought previously, however flawed his proof of the four-color theorem may have been.
His first statement of the first axiom in the book is a little more general than the version I reproduced earlier and which is inscribed on his gravestone; rather than his "form of condensation"
[][] = []
his "law of calling" is general idempotence, i.e.,
AA = A
although the two statements are equipotent within the system he constructs. Similarly, before stating his "form of cancellation"
[[]] =
he phrases it as the "law of crossing", which I interpret as
When you render it for proper typesetting, do the parentheses around dy/dx disappear? (Oh, I guess you've removed them in your edit.)
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.
Supposing that the parentheses aren't necessary, as implied by your edit: how does AsciiMath determine that e^y isn't in the numerator in "e^y dy/dx", or (worse) in the denominator in "d/dx e^y"?
It seems somewhat less noisy than the LaTeX version, but not much; assuming I can insert whitespace harmlessly:
The rules are basically the same as LaTeX, with saner symbol names, support for fractions, \ is not needed before symbols and () can be used instead of {}.
> Supposing that the parentheses aren't necessary, as implied by your edit: how does AsciiMath determine that e^y isn't in the numerator in "e^y dy/dx"
It seems to me that dx,dy,dz,dt behave like numbers, single letter variables and symbols (probably they are symbols, but not listed for some reason). Just as LaTeX doesn't need {} parentheses for numbers, single letter variables and symbols, AsciiMath allows omitting them too.
So `/` captures a single number/symbol/variable left to it, and that is `dy`. But if there was `du` for example it would only capture u, and you would need to put du between parentheses.
Thanks! It does better than I expected on tricky input like [0, 1/2). It seems like there are a lot of special cases, though. It does indeed remove parentheses from the output in some cases but not others.
Probably figuring out how to write things in AsciiMath is more trouble than copying and pasting them from Wikipedia though. (The alt text on equation images is the LaTeX source preceded with \displaystyle.)
How do you do \bigg(\big((4x + 2)x + 1\big)x - 3\bigg)x + 5 in AsciiMath? (((4x + 2)x + 1)x - 3)x + 5 makes all the parens the same size.
Why would you want to manually set the sizes of parens? I always use \left \right when writing LaTeX (and having to do it is one of the reasons I hate LaTeX math notation).
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.
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.
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.
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 φ^* ω.
> 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.
φ^* ω is just 2 weird symbols. There are tons of symbols to learn. I gave up trying to learn the Russian alphabet after a few days so why do u think I am capable of memorizing the Greek one?
The theories are learnable, making sense of all the weird symbols is what's breaking my brain. I tried to get into set theory thrice now, not happening with all the math lingo, hieroglyphs and dry ass content. Learning can be incredibly fun if it was designed fun. Math is a dry and slow process. Make it fun, make it readable and people will be capable to learn it easier.
> why do u think I am capable of memorizing the Greek one
No one memorizes the Greek alphabet. We just learn it as we go because it’s useful to have different types of letters to refer to different types of objects. That’s it.
> I tried to get into set theory thrice now, not happening with all the math lingo, hieroglyphs and dry ass content.
That sounds like you’re trying to learn a specific field without actually having any of the prerequisites to learn it. I don’t know what you’re specifically referring to when you say “set theory” as that’s an incredibly wide field, and depending on what you’re trying to learn it can be quite technical.
> Learning can be incredibly fun if it was designed fun. Math is a dry and slow process.
This sounds like someone complaining that getting to run a marathon is tiresome and hard. Yes, teaching mathematics can always be improved and nothing is perfect, but it will still be hard work.
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.
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.
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.
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.
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.
"
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...
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
"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
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.
> 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.
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.
>>[...] 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.
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.
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.
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.
> 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.
> 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.
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.
> 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)
> Unless you mean to imply that formulating an original poem in your mind counts as "writing"
You're close. I'm making the point that, in modern English, no other verb is available for the act of creating a poem.
Here's a quote from the fantasy novel The Way of Kings that always appealed to me:
>> "Many of our nuatoma -- this thing, it is the same as your lighteyes, only their eyes are not light--"
>> "How can you be a lighteyes without light eyes?" Teft said with a scowl.
>> "By having dark eyes," Rock said, as if it were obvious. "We do not pick our leaders this way. Is complicated. But do not interrupt story."
For an example from reality, I am forced to tell people who ask me that the English translation of 姓 is "last name", despite the fact that the 姓 comes first.
Similarly, the word for writing a poem is "write", whether this creates a written artifact or not. And the poem is literature whether a written artifact currently exists, used to exist, or never existed.
(Though you've made me curious: if the Iliad wasn't literature until someone wrote it down, do you symmetrically believe that Sophocles' Sisyphus is no longer literature because it is no longer written down?)
> 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.
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.
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."
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?)
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.
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".
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.
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.
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!"
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.
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.
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.
> 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.
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.
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...)
Understanding new math is hard, and a lot of people don't have a deep understanding of the math they use. Good notation has a lot of understanding already built-in, and that makes math easier to use in certain ways, but maybe harder to understand in other ways. If you understand something well enough, you are either not troubled by the notation, because you are translating it automatically into your internal representation, or you might adapt the notation to something that better suits your particular use case.
> One thing mathematics education is really bad at is motivating the definitions.
I was annoyed by this in some introductory math lectures where the prof just skipped explaining the general idea and motivation of some lemmata and instead just went through the proofs line by line.
It felt a bit like being asked to use vi, without knowing what the program does, let alone knowing the key combinations - and instead of a manual, all you have is the source code.
> 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.
> 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.
They do.
The purpose of papers is to teach working mathematicians who are already deeply into the subject something novel. So of course only novel or uncommon notation is introduced in papers.
Systematic textbooks, on the other hand, nearly always introduce a lot of notation and background knowledge that is necessary for the respective audience. As every reader of such textbooks knows, this can easily be dozens or often even hundreds of pages (the (in)famous Introduction chapter).
> 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.
They already do this. That is how we all learn notation. Not sure what you mean by numerically though, a lot of concepts cannot be defined numerically.
Notation makes a huge difference. I mean, have you TRIED to do arithmetic with Roman numerals?
>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.
Seeing the relationships between objects is partly why math has settled on a terse notation (the other reason being that you need to write stuff over and over). This helps up to a point, but mainly IF you are writing the same things again and again. If you are not exercising your memory in such a way, it is often easier to try to make sense of more verbose names. But at all times there is tension between convenience, visual space consumed, and memory consumption.
I haven't thought about or learned a systematic way to add roman numerals. But, I would argue that the difference is not notation but a fundamental conceptual advance of representing quantities by b (base) objects where each position advances by a power of b and the base objects let one increment by 1. The notation itself doesn't really make a difference. We could call X=1, M=2, C=3, V=4 and so on.
I also don't know what historically motivated the development of this system (the Indian system). Why did the Romans not think of it? What problems were the Indians solving? What was the evolution of ideas that led to the final system that still endures today?
I don't mean to underplay the importance of notation. But good notation is backed by a meaningfully different way of looking at things.
Adding and subtracting Roman numerals is pretty easy because it's all addition and subtraction. A lot of it is just repeating the symbols just like with tally marks. X+X is just XX for example. You do have to keep track of when another symbol is appropriate, but VIIII is technically equivalent to IX. It's all the other operations that get harder. If the Romans had negative numbers, then the digits of a numeral could be viewed as some kind of polynomial with some positive and negative coefficients. But they also didn't have that.
>The notation itself doesn't really make a difference. We could call X=1, M=2, C=3, V=4 and so on.
Technically, the positional representation is part of the notation as well as the symbols used. Symbols had to evolve to be more legible. For example, you don't want to mix up 1 and 7, or some other pairs that were once easily confused.
>Why did the Romans not think of it?
I don't know. I expect that not having a symbol for zero was part of it. Place value systems would be very cumbersome without that. I think that numbers have some religious significance to the Hindus, with their so-called Vedic math, but the West had Pythagoras. I'm sure that the West would have eventually figured it out, as they figured out many impressive things even without modern numerals.
>But good notation is backed by a meaningfully different way of looking at things.
That's just one aspect of good notation. Not every different way of looking at things is equally useful. Notation should facilitate or at least not get in the way of all the things we need to do the most. The actual symbols we use are important visually. A single letter might not be self-describing, but it is exactly the right kind of symbol to express long formulas and equations with a fairly small number of quantities. You can see more "objects" in front of you at once and can mechanically operate on them without silently reading their meaning. On the other hand, a single letter symbol in a large computer program can be confusing and also makes editing the code more complicated.
Considering that post-arithmetic math rarely use numbers at all, and even ancient Greeks use lots of lines and angles instead of numbers, I don't think Roman numerals would really hold math that much.
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]
> 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.
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.
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.
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.
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.
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.
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.
"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."
Again, learning notation is part of the process of learning math. No one is gatekeeping anything, at no point you need to do an exam or magically be aware of notation that you never saw. Every book and every class will define new notation at the beginning, in most cases they will do so even when there’s no new notation. I am not sure what your argument is.
Every good mathematical textbook introduces the notation it’s using.
That’s a very good gate to keep. Some things are just meant to be gatekept so that the cranks and dilettantes that wastes everyone’s time can stay far outside.
Many mathematicians do in fact teach the rules of the game in numerous introductory texts. However, you don't expect to have to explain the rules every time you play the game with people who you've established know the game. Any session would take ages if so, and in many cases the game only become more fun the more fluent the players are.
I'm not fully convinced the article makes the claim that jargon, per se, is what needs to change nor that the use of jargon causes gatekeeping. I read more about being about the inherent challenges of presenting more complicated ideas, with or without jargon and the pursuit of better methods, which themselves might actually depend on more jargon in some cases (to abstract away and offload the cognitive costs of constantly spelling out definitions). Giving a good name to something is often a really powerful way to lower the cognitive costs of arguments employing the names concept. Theoretics in large part is the hunt for good names for things and the relationships between them.
You'd be hard pressed to find a single human endeavor that does not employ jargon in some fashion. Half the point of my example was to show that you cannot escape jargon and "gatekeeping" even in something as silly and fun as a card game.
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.
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.
> 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?
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.
> 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.
> 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.
These papers are actually great examples of what is problematic with mathematics, just as what is problematic with papers in any other specialised field: how do you judge if this could be ever useful to you?
If you want to understand what is going on there, what is the most effective way to build a bridge from what you know, to what is written there?
If you are in a situation where the knowledge of these papers could actually greatly help, how do you become aware of it?
I think if AI could help solve these two issues, that would be really something.
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.
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.
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.
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.
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.
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.
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.
> I suspect they do so as a means of gatekeeping.
What, as opposed to using ambiguous language and getting absolutely nothing done?
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 ℵ₀, ℵ₁, ♯, ♭, or Σ mean, and you can ask it to read the arxiv pdf and have it explain it to you.
Gatekeeping, or self-promotion? You don't get investors/patents/promotions/tenure by making your knowledge or results sound simple and understandable.
Is that really the case or are you just assuming so? Seems counter-intuitive to me.
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.
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..
It's like saying that learning Arabic is the easy part of writing a great Saudi novel. True, but you have to understand that being literate is the price of admission. Clearly you consider yourself very facile with mathematical notation but you might have some empathy for the inumerate. Not everyone had the good fortune of great math teachers or even the luxury of attending a good school. I believe there is valid frustration borne out of poor mathematical education.
Indeed, confused people say things that don't make sense.
Not at all. Over and over I find really intimidating math notation actually represents pretty simple concepts. Sigma notation is a good example of this. Hmm, giant sigma or sum()?
You think changing sigma to sum() would make it easier to understand the 5 paper, 1000 page proof of the geometric Langlands conjecture?
Imagine how much unnecessary time would be added to a course about series if the lecturer had to write sum() instead of ∑ every time. If you find it hard to remember that ∑ means sum, math might not be for you, and that’s fine.
Wait until you learn about integration. Measures, limits and the quirks of uncountable spaces don't become simpler once you call the operation integrate().
> 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.
The fact that there is a precise analogy between how Ax + s = b works when A is a matrix and the other quantities are vectors, and how this works when everything is scalars or what have you, is a fundamental insight which is useful to notationally encode. It's good to be able to readily reason that in either case, x = A^(-1) (b - s) if A is invertible, and so on.
It's good to be able to think and talk in terms of abstractions that do not force viewing analogous situations in very different terms. This is much of what math is about.
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?)
> a bunch of operations.
Sorry, the notation is bit confusing. The 'A' here is a matrix.
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.
You must be correct, because this interaction is completely devoid of any confusion between the two people attempting to communicate clearly.
I do not have any confusion with the notation, I am confused about what the argument you’re trying to convey with English words.
Ceci n'est pas une pipe.
This is funny. “Mathematics notation is confusing to me because I refuse to learn it. I refuse to learn it because mathematics notation is confusing to me.” Okay sure, be happy with yourself.
> This is so wrong it can only come from a place of inexperience and ignorance.
Thanks for the laughs :D
> Show a child a matrix numerically and they can understand it, show them Ax+s=b, and watch the confusion.
Show a HN misunderstood genius Riemann Zeta function as a Zeta() and they think they can figure out it's zeros. Show it as a Greek letter and they'll lament how impossible it is to understand.
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..
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.
1. I work in finance and here people sometimes write math using words as variable names. I can tell you it gets extremely cumbersome to do any significant amount of formula manipulation or writing with this notation. Keep in mind that pen and paper are still pretty much universally used in actual mathematical work and writing full words takes a lot of time compared to single Greek letters.
Large part of math notation is to compress the writing so that you can actually fit a full equation in your vision.
Also, something like what you want already exists, see e.g. Lean: https://lean-lang.org/doc/reference/latest/. It is used to write math for the purpose of automatically proving theorems. No-one wants to use this for actually studying math or manually proving theorems, because it looks horrible compared to conventional mathematics notation (as long as you are used to the conventional notation).
(1) I always tell my students that if they don't understand why things are done a certain way, that they should try to do it in the way most natural to them and then iterate to improve it. Eventually they will settle on something very similar to most common practice.
(2) Higher-level proofs are using so many ideas simultaneously that doing this would be tantamount to writing Lean code from scratch: painful.
Go ahead. Write a math paper with your proposed new notation with hyperlinks and submit it to a journal somewhere.
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.
> ... 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.)
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.
We do often find add(a, b, c), just written as Σ(a, b, c). Similar for mul and Π. 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 Σ(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.
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:
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:
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) but not, to my eye, any less confusing.A thing I didn't appreciate the first time I read Spencer-Brown's book is that he actually cites Sheffer's 01913 paper, and proves Sheffer's postulates within his system in an appendix. This situates him significantly closer to the mathematical mainstream than I had thought previously, however flawed his proof of the four-color theorem may have been.
Also, the axioms I cited above are written in his notation on his gravestone: https://en.wikipedia.org/wiki/G._Spencer-Brown#/media/File:G... but I have evidently reversed left and right in my rendering of the DNF rewrite rule above. It should be:
His first statement of the first axiom in the book is a little more general than the version I reproduced earlier and which is inscribed on his gravestone; rather than his "form of condensation" his "law of calling" is general idempotence, i.e., although the two statements are equipotent within the system he constructs. Similarly, before stating his "form of cancellation" he phrases it as the "law of crossing", which I interpret asAsciiMath makes easy equations read easy.
1 and 2 would be
edit: edited, first got them wrongWhen you render it for proper typesetting, do the parentheses around dy/dx disappear? (Oh, I guess you've removed them in your edit.)
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.
Supposing that the parentheses aren't necessary, as implied by your edit: how does AsciiMath determine that e^y isn't in the numerator in "e^y dy/dx", or (worse) in the denominator in "d/dx e^y"?
It seems somewhat less noisy than the LaTeX version, but not much; assuming I can insert whitespace harmlessly:
Here is an online renderer and the description: https://asciimath.org/
The rules are basically the same as LaTeX, with saner symbol names, support for fractions, \ is not needed before symbols and () can be used instead of {}.
> Supposing that the parentheses aren't necessary, as implied by your edit: how does AsciiMath determine that e^y isn't in the numerator in "e^y dy/dx"
It seems to me that dx,dy,dz,dt behave like numbers, single letter variables and symbols (probably they are symbols, but not listed for some reason). Just as LaTeX doesn't need {} parentheses for numbers, single letter variables and symbols, AsciiMath allows omitting them too.
So `/` captures a single number/symbol/variable left to it, and that is `dy`. But if there was `du` for example it would only capture u, and you would need to put du between parentheses.
Thanks! It does better than I expected on tricky input like [0, 1/2). It seems like there are a lot of special cases, though. It does indeed remove parentheses from the output in some cases but not others.
Probably figuring out how to write things in AsciiMath is more trouble than copying and pasting them from Wikipedia though. (The alt text on equation images is the LaTeX source preceded with \displaystyle.)
How do you do \bigg(\big((4x + 2)x + 1\big)x - 3\bigg)x + 5 in AsciiMath? (((4x + 2)x + 1)x - 3)x + 5 makes all the parens the same size.
Why would you want to manually set the sizes of parens? I always use \left \right when writing LaTeX (and having to do it is one of the reasons I hate LaTeX math notation).
Because \left( ... \right) doesn't give very readable results in cases like that; all the parens end up the same size.
What’s wrong with Greek letters? Would the number π be any easier to understand if it was written differently?
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.
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.
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.
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 φ^* ω.
> 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.
φ^* ω is just 2 weird symbols. There are tons of symbols to learn. I gave up trying to learn the Russian alphabet after a few days so why do u think I am capable of memorizing the Greek one?
The theories are learnable, making sense of all the weird symbols is what's breaking my brain. I tried to get into set theory thrice now, not happening with all the math lingo, hieroglyphs and dry ass content. Learning can be incredibly fun if it was designed fun. Math is a dry and slow process. Make it fun, make it readable and people will be capable to learn it easier.
> why do u think I am capable of memorizing the Greek one
No one memorizes the Greek alphabet. We just learn it as we go because it’s useful to have different types of letters to refer to different types of objects. That’s it.
> I tried to get into set theory thrice now, not happening with all the math lingo, hieroglyphs and dry ass content.
That sounds like you’re trying to learn a specific field without actually having any of the prerequisites to learn it. I don’t know what you’re specifically referring to when you say “set theory” as that’s an incredibly wide field, and depending on what you’re trying to learn it can be quite technical.
> Learning can be incredibly fun if it was designed fun. Math is a dry and slow process.
This sounds like someone complaining that getting to run a marathon is tiresome and hard. Yes, teaching mathematics can always be improved and nothing is perfect, but it will still be hard work.
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.
Yea because hieroglyphs are more understandable than the name of a function
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(infinity),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.
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.
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.
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...
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. "
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.
> 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.
Someone not educated in programming would not know that a given text is Python source code.
Same problem, but unlike math notation, it is MUCH clearer and even my 10 year old newphew can tell python from javascript and C.
I would wager that a vast majority of people on the planet could not distinguish Python from JavaScript from C.
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...
We don’t talk about the French notation for intervals. Let it stay in France.
Yep, I agree on that. But still, interesting to see that such a "standard" thing can be so different in different dialects of mathematical notation.
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
"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
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.
> 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.
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.
>>[...] 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.
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.
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.
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.
> 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.
> 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.
> 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.
> 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
Well, then poetry is not literature.
If it’s not written down, then that’s true.
Once someone writes it down, it is.
Sure in the context that you mean it’s an oral tradition.
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.
> 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)
> Unless you mean to imply that formulating an original poem in your mind counts as "writing"
You're close. I'm making the point that, in modern English, no other verb is available for the act of creating a poem.
Here's a quote from the fantasy novel The Way of Kings that always appealed to me:
>> "Many of our nuatoma -- this thing, it is the same as your lighteyes, only their eyes are not light--"
>> "How can you be a lighteyes without light eyes?" Teft said with a scowl.
>> "By having dark eyes," Rock said, as if it were obvious. "We do not pick our leaders this way. Is complicated. But do not interrupt story."
For an example from reality, I am forced to tell people who ask me that the English translation of 姓 is "last name", despite the fact that the 姓 comes first.
Similarly, the word for writing a poem is "write", whether this creates a written artifact or not. And the poem is literature whether a written artifact currently exists, used to exist, or never existed.
(Though you've made me curious: if the Iliad wasn't literature until someone wrote it down, do you symmetrically believe that Sophocles' Sisyphus is no longer literature because it is no longer written down?)
> 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.
Just the other day I was listening to EconTalk on this: https://www.econtalk.org/a-mind-blowing-way-of-looking-at-ma...
Thank you for posting! - I was not aware of this.
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.
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."
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?)
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.
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".
Of course. I just find it hilarious that someone like von Neumann would say that.
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.
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.
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.
sorry but that is a dumb quote.
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!"
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.
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.
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.
> 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.
A lot of math is not about numbers, so not everything can have a numerical example.
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.