Examples of common false beliefs in mathematics
mathoverflow.net
mathoverflow.net
I'd have to look up what the "/" in "M/L" means in the 5th reply (I didn't)
There are so many _symbols_ used in mathematics. It's all as if they're spells written by wizened greybeard wizards and passed down to apprentices throughout the centuries.
M/L means that M is a field-extension over L. A concrete example would be C/R (the complex numbers over the reals).
Algebraic number theory, in particular Galois theory, studies field extensions by looking at the group of symmetries: the field automorphisms of the larger field that fix the smaller field. For the concrete example above, the Galois group is a group with two elements: the identity function and the function that maps i to -i and keeps real numbers fixed. It's not a coincidence that the dimension of C as an R-vector space is the same as the size of this group (or that the degree of the polynomial that has i (and -i) as roots has degree 2).
That's exactly what they are!
Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why.
M/L could be written Field-extension(M, L). That might make it easier to Google what's going on, rather than having to pick up a group theory book. And indeed in this case it would be extra helpful, because M/L could also mean "quotient group" or even "division" depending on the type of the objects M and L. I personally prefer M:L for field extensions.
But Googling individual concepts is hardly a good way to learn group theory. So I don't think it matters too much. For anyone that spends enough time working in the given field, the notation is by far the easiest thing to figure out.
In general it is nice that mathematics - still being largely a handwritten endeavor - has the freedom to introduce new notation where it makes sense. See https://mathoverflow.net/questions/42929/suggestions-for-goo... for cool examples of new notation improving the lives of everyone.
I want to emphasize this because the weirdness of math notation comes up often. The difficulty of grasping mathematical concepts far outweighs notational concerns. Getting used to notation in a specific field is an absolute drop in the bucket compared to understanding the theorems and proofs, and often the reason notation is so idiosyncratic and inconsistent is because it's meant to try to reflect human intuition, which is idiosyncratic and inconsistent in comparison to the mathematics from which it is derived.
Once you really get into mathematics, notation is so far down the list of things that make learning mathematics hard that I don't think it's worth the associated cost of trying to change it on a large scale.
As a mathematician, notation exists because it's easier to read things with good notation than without. I have some advanced mathematics books in my table and in most of the pages you'll find far more text than notation. Notation is usually explained when it starts being used, usually in the definition. For example, the problem with understanding the notation "Aut(L|k)" is not the notation, because if the author wrote "group of field automorphisms of L fixing k elementwise" I'd be equally lost if I don't understand those words. And once I understand those words, the notation makes it easier to read and understand things whenever the author is talking about that group, because your brain automatically identifies the concept associated with the notation instead of having to read the words and linking them with the concept.
This is simple math.
Start writing formal proofs for your programs. You will then be in the realm of something more complex (and still far lower than the level in a research paper).
Even a proof of the complexity of your algorithm (let alone the correctness) will get ugly without this notation.
So... yes, I do think it would help if at least some of that were already done for me. As it is it takes so damn long that I don't bother unless I have a good reason.
The immediate example for me would be lambda calculus which makes a lot of sense to me when expressed in code (be it lisp, python or Rust) but just looks like vomit in theoretical papers, for instance:
(λy.M)[x := N] = λy.(M[x := N]), if x ≠ y and y ∉ FV(N)
Why is it that software engineers in a few decades have come to the conclusion that proper variable naming is important and abuse of sigils a bad idea but mathematics don't feel the same way?
1. Since you seem to someone who reads papers sometimes, I would encourage you define your "programming notation" and start proving theorems in that notation. Do it for sufficiently complex proofs: the ones that require a few different lemmas, span several pages, and use some non-trivial algebra. Then, attempt to present your proof to someone in your notation. Judge for understanding.
2. You might very well succeed. But if you don't, wonder why out of many many software engineers (or profs at software engineering oriented degrees) over the past few decades who read math/cs papers, no one attempted to use "programming notation" to write a book or notes to communicate mathematics to others?
The big reason for single-letter variable names is that historically multiplication can be denoted by concatenation. It makes formulae shorter, so they don't need line breaks. Personally I don't think that's a huge benefit, particularly when there's more than one sort of product possible so you end up needing to explicitly denote multiplication anyway.
[1] https://eprint.iacr.org/2022/106.pdf (Page 7-8)
The reason for single-letter variable names is that mathematics is best learned by manipulating ideas by hand on paper and pen. And that many mathematical expressions are long and if we started using longer names, we would end up writing only a couple of statements per page, which would be much annoying than just using symbols.
Take matrix multiplication, imagine seeing this:
C = zero(m,p)
for k from 1 to n
for i from 1 to m
for j from 1 to p
C_ij += A_ik * B_kj
C = AB
CB^-1 = ABB^-1
CB^-1 = A
[And this is a small example, algorithmic expressions of algebraic ideas, like plain English expressions of the same, does not scale very well.]
Dijkstra did just that[1]. That same site has hundreds of examples of him proving things in that very programmer friendly notation. Bear in mind that Dijkstra was trained as a “mathematical engineer” for his higher education.
I’m not trying to be controversial, but traditional mathematicians are decades behind the best computing scientists when it comes to crafting formalisms. To be fair the typical practicing programmer is even further behind.
I can only speculate about the appeal of a notation that makes it difficult or impossible to just let symbol manipulation do the heavy lifting. Perhaps mathematicians enjoy the intellectual exercise of holding all those concepts in mind? Perhaps they, like many guilds, appreciate the barriers to outsiders that they feel increase their own prestige? Or perhaps it’s just sheer inertia? I really don’t know.
[1] https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...
To give just one example, when done properly the use of hinting in the multi-line equational proof format will certainly ease the non-expert’s task of following a proof.
[1] https://deepblue.lib.umich.edu/bitstream/handle/2027.42/4265...
Could you provide examples? The Dijkstra article doesn't seem like a marked improvement.
> Perhaps mathematicians enjoy the intellectual exercise of holding all those concepts in mind?
Mathematics is not symbol manipulation. I wish it was as easy as that. Mathematics requires you to hold all the concepts in mind while studying and working.
> Perhaps they, like many guilds, appreciate the barriers to outsiders that they feel increase their own prestige?
I am incapable of finding an example where the barrier to outsiders is the notation and not the concepts themselves.
As I said, there are hundreds of examples on that site.
> Mathematics is not symbol manipulation. I wish it was as easy as that. Mathematics requires you to hold all the concepts in mind while studying and working.
Of course concepts must be understood, but that doesn't mean they need to be held in mind for the vast majority of the derivation. In many cases for example it's sufficient to know that an operation is associative to perform some manipulation, without needing to fuss over the specifics of the concept that operation captures. Unless of course you insist on a notational convention where the syntax is semantically ambiguous, as is common practice among traditional mathematicians.
> I am incapable of finding an example where the barrier to outsiders is the notation and not the concepts themselves.
What does the following mean? What concepts must be held in mind to understand it?
sin(s + i + n)
In that article or in the whole site? Because the article just seems like different notation, no great improvements.
> In many cases for example it's sufficient to know that an operation is associative to perform some manipulation
Whenever that's the case, mathematicians have already found a way to use notation to that advantage. See for example the case of groups and rings, where the operations are usually denoted by '+' and '·' (addition and multiplication) even when they might not necessarily be those operations just because it makes operations more intuitive.
> What does the following mean? What concepts must be held in mind to understand it?
Well, you need to know what the sinus is and what is addition. If you don't have those concepts it doesn't matter that I write sin(s + i + n) or 'The sinus of s plus i plus n' (not to mention that there's ambiguity there too, does 'of' refer to 's' or 's plus i plus n).
> On the other hand if one desires to make mathematics more accessible to persons of ordinary intelligence, then reducing the cognitive load by clearly defining the semantics of a formula using an unambigous syntax is certainly necessary. Evidently though that's of little importance to the mathematical guild.
Notation is not the issue when explaining mathematics. The mathematical guild has done a ton of effort in improving math education, and you'll be able to find videos and texts that barely have any notation or equations. Notation exists because once you learn the concepts, it reduces the cognitive load of transmitting those concepts.
Incorrect. In the example I gave one merely needed to understand multiplication, addition, and the distributive property. And you, who are observably skilled in the art of mathematics, failed to follow a grade school level formula. So much for ambiguous syntax not being an impediment to understanding even when all concepts involved are comprehended.
> Notation is not the issue when explaining mathematics. The mathematical guild has done a ton of effort in improving math education, and you'll be able to find videos and texts that barely have any notation or equations.
It's baffling to me that anyone would attempt to argue that ambiguous syntax is not an issue for explaining mathematics. Please understand that the following unflattering comparison is intended merely for clarity and not to insult. To me it feels like trying to argue with a flat earther that the world is round. The absurdity makes argumentation virtually impossible.
> Notation exists because once you learn the concepts, it reduces the cognitive load of transmitting those concepts.
I agree. In fact that's germane to my original point. The mathematics guild is using primitive notation compared to what computing scientists discovered in the latter half of the 20th century. I'm not saying that notation is bad or even that the traditional amgbigous syntax isn't an improvement over just writing everything out in some natural language. I'm saying the mathematics guild is stubbornly ignoring notational advances that further reduces cognitive load.
> The mathematical guild has done a ton of effort in improving math education
Then how do we explain the complete lack of measurable progress in the average American secondary school student's mathematical ability? It appears to me that regardless of how much effort has been expended, little has come of it.
So which notation would you propose that would allow me to understand the formula without knowing those concepts?
> It's baffling to me that anyone would attempt to argue that ambiguous syntax is not an issue for explaining mathematics. Please understand that the following unflattering comparison is intended merely for clarity and not to insult. To me it feels like trying to argue with a flat earther that the world is round. The absurdity makes argumentation virtually impossible.
I am not saying it's not "an issue", I am saying it's not the issue. Ambiguous notation is a problem, yes, but when our language is ambiguous too I don't think it's a problem you can fully solve. And definitely one that will be solved by removing notation.
> I'm saying the mathematics guild is stubbornly ignoring notational advances that further reduces cognitive load.
Such as? Because I still haven't seen those advances.
> Then how do we explain the complete lack of measurable progress in the average American secondary school student's mathematical ability? It appears to me that regardless of how much effort has been expended, little has come of it.
I'd guess that has more to do with local policies and resources than any effort of any mathematician. It seems weird to evaluate the work of a global community based on a measurements of schools in a specific country.
> So which notation would you propose that would allow me to understand the formula without knowing those concepts?
We agree on the importance of understanding concepts. I showed the difficulty for one who does understand the concepts. Surely a student who is learning them and thus by definition doesn’t know them would have even greater difficulties. I propose that a notational convention that forgoes the invisible multiplication operator and that distinguishes between ordering operations and function application rather than using parentheses for both would be considerably more clear. Such as, for example, the one Dijkstra adopted. I’m confident that you would have easily understood either of the following:
sin.(s + i + n)
s*i*n*(s + i + n)
Once the student has achieved proficiency with the requisite concepts using a more sensible notation, supposing they have an interest in further study, they will then need to learn the customary notation on account of the great and valuable body of work that uses it. I think that’s acceptable though since learning new syntax will always be a part of learning math and as you say the real challenge is mastering concepts and not syntax.
> It seems weird to evaluate the work of a global community based on a measurements of schools in a specific country.
You make a good point here. It would be better to make a global comparison using something like PISA math scores and seeing if didactic innovation has resulted in any improvement in the places it has been implemented. I’m not aware of any such improvement, but the world is a big place so that’s hardly evidence that there isn’t any.
But a student who is learning them will not start by looking at the equation, but at the concept, and that concept will disambiguate between sin() and s·i·n.
The example you use is interesting. Yes, that would remove ambiguity between sin() and sin. But how would that notation evolve? If in most instances it's clear when you have function application and when it's multiplication, people will stop writing it. Same with the multiplication operator.
Not to mention that notation also introduces ambiguity, because the dot is part of the written language, so you'll have instances where it isn't clear whether the dot means "apply function" and when it means "sentence stop".
Even assuming that the notation stays and doesn't devolve to something that's faster to write and to read, what did we actually achieve? We wouldn't have removed the complexity of learning trigonometric functions. You wouldn't stop a student from doing sin.(a + b) = sin.a + sin.b, for example, or trying to use the same formulas for sin and cos.
My point is that while you will have some instances where notation could be improved (and naming too, for example closed and open sets are confusing because they are not inverse properties) because most of the time what's difficult is the concept itself, notation is like an extra step after having gone up four flights of stairs.
Some symbolic (broadly construed, including drawings, vocalizations, etc) representation is going to be required to communicate any concept. Why not use one that's minimally ambiguous? I get your point that we can hope the student will not struggle too much with the meaning of sin on the day the day the sine function is being taught. Even so, concepts once introduced usually appear elsewhere. We shouldn't be surprised if after learning both the sine and the invisible multiplication operator, the student might be confused as to whether or not some string is a sequence of multiplications or a function name. This isn't just hypothetical either, I've been in enough math classes to see the students of ordinary intelligence struggle with this.
> The example you use is interesting. Yes, that would remove ambiguity between sin() and sin. But how would that notation evolve? If in most instances it's clear when you have function application and when it's multiplication, people will stop writing it. Same with the multiplication operator.
I consider using invisible operators to be generally unwise. I wouldn't consider adding ambiguity to save a key or pen stroke a wise trade-off. I'm aware that many mathematicians do, and all I can say to that is that it baffles me. In humility I'm willing to allow that they know something I don't, so perhaps my bafflement is a personal defect. Even so I don't think I want to repair it. In my own work I appreciate the clarity too much. And since that work is just reasoning about programs I want to write and amounts to personal notes and not something I have any interest in publishing, it doesn't much matter to anyone else what notation I use.
> Not to mention that notation also introduces ambiguity, because the dot is part of the written language, so you'll have instances where it isn't clear whether the dot means "apply function" and when it means "sentence stop".
LaTeX and other comparable typesetting software adequately solve for this. As for manuscript, there are also ways to indicate whether a portion thereof is a formula or explanatory text. What I really think is important though isn't the choice of the glyph "." but avoiding using the same symbol for two completely unrelated concepts like precedence and function application.
> Even assuming that the notation stays and doesn't devolve to something that's faster to write and to read, what did we actually achieve? We wouldn't have removed the complexity of learning trigonometric functions. You wouldn't stop a student from doing sin.(a + b) = sin.a + sin.b, for example, or trying to use the same formulas for sin and cos.
Reading speed is by chunk and not character count. I challenge the notion that f(x+y) is faster to read than f.(x+y). As for being slower to write, I doubt that any mathematics beyond the most basic arithmetic are constrained by typing speed. I accept that there may be a stronger argument for some kind of shorthand in manuscript, but I still doubt the savings are worth it.
As an aside I find this pleasantly parseable:
sin.(a + b) = 1/csc.(a + b)
> My point is that while you will have some instances where notation could be improved (and naming too, for example closed and open sets are confusing because they are not inverse properties) because most of the time what's difficult is the concept itself, notation is like an extra step after having gone up four flights of stairs.
I think a better analogy is that it's like going up a set of stairs where the occasional step is false and drops into a pit. Once one gets used to the pits one can navigate the stairs virtually as well as if they weren't there, but that's hardly an argument in favor of booby trapping the stairs. It certainly will make things considerably harder for first time climbers.
Nevertheless, I continue to agree that learning concepts is the more challenging and interesting part of mathematics. I also welcome improvements in clarifying concepts. Sadly, making a complicated concept easier to understand is a much greater challenge than making an ambiguous and muddled syntax unambiguous and clear. My preference is that we pursue both, because they're complementary.
I honestly have not seen that. Maybe some minor confusion between sin and asin maybe, but most of the time it's clear what it's meant.
> I consider using invisible operators to be generally unwise. I wouldn't consider adding ambiguity to save a key or pen stroke a wise trade-off. I'm aware that many mathematicians do, and all I can say to that is that it baffles me. In humility I'm willing to allow that they know something I don't, so perhaps my bafflement is a personal defect. Even so I don't think I want to repair it. In my own work I appreciate the clarity too much.
I have seen tons of times authors omitting notation to make it less cumbersome. It's usually preceded by something like "we omit X for brevity/simplicity in the following". The reason is that symbolic notation exists for density of information and focus. When clarity, details and specifics are required, mathematicians use text.
> LaTeX and other comparable typesetting software adequately solve for this.
Funnily enough, they also solve for sin() and sin if you use \sin (or \mathrm{sin}).
> Reading speed is by chunk and not character count. I challenge the notion that f(x+y) is faster to read than f.(x+y)
The dot is short enough to not change things too much, but compare "xy + yz + zy + xyz" to "xy + yz + zy + xy*z". And this happens a lot, because often you want only symbols for things that matter and remove the redundant things. For example, if you're doing calculus you'll often write down the arguments for the functions, but in differential equations you'll omit them because they're not really important.
> Nevertheless, I continue to agree that learning concepts is the more challenging and interesting part of mathematics. I also welcome improvements in clarifying concepts. Sadly, making a complicated concept easier to understand is a much greater challenge than making an ambiguous and muddled syntax unambiguous and clear. My preference is that we pursue both, because they're complementary.
Yes, but my point is that while notation can sometimes be improved, the relation effort/gains is usually small. For starters, notation is not the hardest things one faces when learning mathematics. Then, you have the issue of improvements in one aspect of notation causing problems in other aspects because the set of symbols we have is limited (for example, dot is used as the dot product in vector spaces too). And of course, the problem of changing notation that is already written. Sometimes the gains are worth the effort, such as the ceiling/floor notation of Iverson (and the bracket, although I don't think it's as standard). But that's reasoning mathematicians use for/against notation changes. It's not because having difficult notation is enjoyable or because it acts as gatekeeping.
You'll be hard pressed to find any symbol/notation that isn't overloaded with other operators in math. Things like '*' and the 'x' symbols have different meanings in different contexts, and when you're doing those things, and need multiplication, you run into problems.
Personally, and I know I'm not alone, when I convert physics equations to code, the '*' symbol is one of the ones that make reading code challenging.
For function application, I can see the problem, and I'm sure most have gotten confused at one point or other where they're not sure if it's a function application vs multiplication. But as others have pointed out, the context makes it clearer.
I think one of the key differences between many here and those who favor the notation is that in SW, the wish is the formalized version (i.e. the code) is readable enough to understand, without much prose (i.e. comments). In mathematics, it usually is not the case - in fact, one of my math professors often had to point out to students that it's better if they wrote part of their answer in prose rather than purely logic symbols. In that sense, there often is little ambiguity. It really should be clear from the context whether you're dealing with the sine function or multiplying symbols. If it isn't, the problem isn't the notation, but the lack of understanding of the context. Having a guide to the notation will not elucidate much.
And for the one who uses a lot of math (some engineers, physicists, and of course mathematicians) - having to write (any) symbol for multiplication is crazy tedious - even if it is just a dot. There is a reason they opt not to put it.
What I disapprove of is using these very terse syntax in definitions like, say, in Wikipedia articles.
In other words it's like how I have no issue using a variable named "int i;" locally in a function, but I'd consider it a very bad practice if a library exported a global "extern int I;" in their public interface.
On the other hand, notation in mathematics is used to represent only a limited set of common concepts for the field, you don't have autocomplete and a single line can contain a lot of concepts. For example, compare two expressions of Stokes' theorem (given HN typesetting limitations):
∫_A dω = ∫_∂A ω
integral(A, differential(ω)) = integral(boundary(A), ω)
Edit: Also, the second one is only "easy" if you're familiarized with function calling in programming. One could argue that to go full notation-less you should only write in proper sentences, and that would make it even more complex.
I had this problem when I started higher level maths and the Greek alphabet was used. I couldn’t ask what a symbol meant when I couldn’t say what the symbol was. I couldn’t write down a symbol whose name was read out when I didn’t know what it should look like.
I tried learning the Greek alphabet but flash cards and spaced repetition didn’t work for my ADHD brain and I just had to stop doing maths, which was a shame, because up to that point I was good at it.
Usually by looking at the words that surround the equation. Most math texts will introduce notation in words, and it's common to repeat something in words and notation (e.g. 'let the field extension L/k...'). I have studied math for five years and I don't think I ever had the problem of not knowing how to search for something. The only problem I had with notation was finding the LaTeX code for a specific symbol, but other than that the bigger problem were the concepts.
There are also glossaries at the end of books and in some specialized pages too (https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbo...)
> I couldn’t ask what a symbol meant when I couldn’t say what the symbol was. I couldn’t write down a symbol whose name was read out when I didn’t know what it should look like.
In my classes these problems were usually solved by asking "what's that wiggly thingy" or "how do you write 'alpha'".
And the greek alphabet is not used because mathematicians like to annoy people. There are certain customs for how objects are notated. x,y,z for unknowns/variables; f,g,h for variables; α,β for angles; A,B for matrices; G, H for groups... They are used because, even though they might be harder for beginners, they aren't harder than the concepts they're describing and help a lot in reducing the cognitive load when reading things.
The internet, including this forum, is chock-full of people making useless comments like "well I like it" or "it works for me", just asserting their views or experiences while advancing no argument and offering no discussion.
The post you just replied to is not doing that.
The poster you replied to is taking the time to describe and explain his experiences and their implications. He's contributed a lot more to the conversation than you have, and doesn't deserve sarcasm.
You're clearly capable of learning at least one alphabet, why was the second so much harder?
Learning the Greek alphabet shouldn't take more than 10 hours. Not that one needs to. I never did. However, in a given discipline, practitioners tend to be consistent on which Greek letters they use for which concepts. As long as you spend time on a given topic, the Greek symbols should become ingrained - just as knowing how to write a for loop in C would become if you do it often enough.
And I would hope people studying math spend a lot more than hundreds of hours on it. If someone came and complained about the arcane syntax used in programming languages and hadn't spent, say, 100 hours programming, I don't think you would give much weight to their views.
Reading Newton’s principia is humbling. He does that the whole book; it takes him pages to explain concepts we now learn before university and that can be condensed in a couple of equations. Finding words to describe the maths he uses is difficult, you can almost feel the insane amount of work just to translate the concepts into sentences. It’s a wonderful example, and of course a masterpiece. But reading it is also very difficult because of all the noise around the concepts. Words are fine to give an intuitive understanding, but at some point you need equations.
Same for Maxwell’s equations, for example. You’d spend less time learning basic calculus and algebra than trying to understand what the hell he was meaning.
Or 19th-century chemists, who used fancy and complicated names for compounds that are described much more clearly and succinctly with a chemical formula. Yes, you need to learn the formalism, but it pays off.
Some counter examples are Einstein and Feynman, who could communicate complex concepts with words in an understandable way. Though even them need some abstract notation at some point (lots of it, actually).
If yo read code in standard libraries, you'll see a lot of either single-letter variables or extremely generic variable names.
Most mathematics is dealing with things at least one level of abstraction higher than a standard library.
In most programs outside of things like standard libraries, a variable usually stands for something concrete and specific. A customer. An order. A specific type of element in a UI. Etc.
In theory papers, a variable usually stands in for something generic and general. An arbitrary program. An arbitrary finite set. Etc. Sometimes even an arbitrary program in a programming language that is not defined in particular but only in general (e.g., "any language with parametric polymorphism", "any language with a specific sort of binding structure between things in these two syntactic categories", etc.).
Again, standard libraries already start using more generic variable names, and most theory papers are dealing with an abstraction level higher than standard libraries.
1. A byte is pretty damn concrete. The Java byte implementation [1] uses almost exclusively single-letter names (b,s) or names that are so generic that they might as well be single-letter names (e.g., anotherByte instead of b). When more meaningful names are used, it's because they are public type names (String, int), which is, again, pretty damn concrete.
2. The next level of abstraction is Generics. Here, even Java -- a language whose verbosity is a long-standing joke -- starts using single-letter variable names for both types and values [2].
3. Finally, we dive into things that abstract over generics [3] and start seeing weird sigils in addition to single-letter names. (What does Predicate<? super E> mean?!)
And, again, this is a strong-manned example, since Java is famously verbose and I'm choosing some of the most-used and therefore most verbosely documented .java files in the world.
Notice, btw, that natural language documentation increases as the verbosity of names decreases. This is the same in math, where those symbols are small pieces of 20+ page papers full of english prose explaining the meaning of the symbols.
And, again most mathematics is dealing with things at least one level of abstraction higher than anything you find in a standard library. Sometimes several levels of abstraction.
[1] Byte.java
[2] Dequeue.java
[3] Collection.java
As someone who spent a lot of time in mathy subjects, this is very readable to me - even though I don't know lambda calculus. I'd posit that if you have trouble with this, it is merely due to not spending much time in math.
Imagine someone who spent all his time in BASIC and he suddenly reads a Java codebase, and complains about the syntax.
> Why is it that software engineers in a few decades have come to the conclusion that proper variable naming is important and abuse of sigils a bad idea but mathematics don't feel the same way?
Because they've been doing it for a few hundred years longer than SW engineers have. I hear this refrain often here on HN. I would love to see someone write a textbook on electromagnetics or quantum mechanics using this verbose notation. The derivation of a harmonic oscillator (without ladder operators) takes a few pages of this concise notation. I shudder how lengthy it would be when more verbose.
The common way you learn mathematics, the notation comes automatically, and new mathematics is often understood as a variation on the examples you learn.
Some of these conventions you probably know, like having i for the varying number in a sum or product (or loop in programming) which goes up to an integer n.
If you called an integer f or a complex number n it would make it much harder to read.
For other people, the function of math in the school system seems to be mainly to sort them by math ability -- and for that, artificial barriers aren't a problem. So there's little demand for better UX design for nonspecialists. Sucks if you think society would be better off if more people appreciated/understood/applied more math.
Having done consulting, I frankly disagree. Except pathological cases most of the codebases I've been involved in it's possible to open a file at random at quickly make sense of what happens even without prior exposure. With math papers my experience is that you have to go 3/4 references deep before startit to understand where the notation comes from which wastes hours and hours. They should provide normalized names that could be searched for in e.g. a coq-based database of math concepts or something like that.
Most of the time you'd need even more hours to understand not the notation but the concept themselves. In this list there isn't that much weird notation compared to weird words. For example, 'The Krull dimension of a noetherian integral domain is finite'. Even without notation, I'd expect a non-math major to have to go read 8-10 wikipedia pages just to understand "Krull dimension".
In math it is a strange symbol in a PDF and you cannot search for at all
In code someone would write "lightspeed" and that can be searched. In a paper, they write "c" and when you search for it you find thousands of unrelated things.
I think about this analogy from time to time, when I get frustrated by maths. Myself not being mathematician but on occasion needing to read a research paper, my frustration is typically begins with the raw syntax of maths. It’s so obviously optimized for the combined limitations of the human brain’s limited short term memory and the human hand’s speed of writing/typing. Which probably is the correct optimization for high level practitioners.
For communication with non experts, it’s dreadful, but I think fixable post-facto with a auto-notation tool. Think black.py for maths.
A terse equation or statement in mathematics represents what you will understand, the rest of the paper is to get you there.
Wanting to read the main equations and understand them, so as to use that knowledge reading the paper, is backward. If the main equation was ax^2+bx+c=0, you wouldn't need to read the paper, right? It's about the quadratic equation, it's aimed at primary students.
But that's how it works, we keep pounding away at the paper: taking notes, following references, thinking a lot about each of the definitions, then eventually, you look at the main equation and say "oh. ok. I know what this says now".
Software engineers have built careers off of making code more readable, both in terms of the language itself and the codebase it applies to. Mathematicians seem to go out of their way to make their work as obtuse as possible.
Can you imagine if software engineers had stopped at C? If we were all still trying to apply the spiral rule [0] to read everything? That's what reading math papers feels like. It feels like you stopped at C and are unwilling to even consider that there could be a better way to express things.
Mathematicians go out of their way to make their work as understandable as possible. If they wanted to make it as obtuse as possible, you'd just see a picture of chaotic scribbles on a blackboard.
> That's what reading math papers feels like. It feels like you stopped at C and are unwilling to even consider that there could be a better way to express things.
Do you have any suggestions?
Mathematics is a hard subject because it deals with concepts in several levels deep of abstraction. That's a complexity that you can't escape. You can't expect to read a math paper (aimed at researchers in the same field) and understand everything, the same way it would be unreasonable to ask you to write your code in a way that someone who isn't familiar with either programming or the concepts your code works with to understand it if they went to read it.
Even something as simple as the set of integers being the letter Z requires you to understand German to know why we chose Z (and not I) in the first place. Or d/dx being totally unintuitive for anyone not well-versed in differential calc. I'm quite certain that every student learning calc thinks "Well the d's just cancel out there". Granted, I've heard the intuition behind the notation and I understand why it's tolerated. Still, it seems less helpful than it could be if mathematicians weren't married to tradition.
Believe me, there are a lot of people trying to find better ways to teach and communicate mathematics. There's inertia, of course, but it wouldn't be the first time people change notation because it's better.
> Even something as simple as the set of integers being the letter Z requires you to understand German to know why we chose Z (and not I) in the first place.
I mean, you'd just move from confusing people who don't speak German to confusing people who don't speak English.
> Or d/dx being totally unintuitive for anyone not well-versed in differential calc.
You are not going to find any notation for a derivative that makes the concept intuitive for people that don't know differential calculus. If anything, it's more intuitive than f'.
> Still, it seems less helpful than it could be if mathematicians weren't married to tradition.
Or maybe the tradition has been built because after decades of mathematics nobody has found anything better.
I've basically abandoned all languages/ecosystems that aren't, at the very least, extremely grep-friendly. Static types preferred so I can get that sweet, sweet auto-refactoring, jump-to-references/definition, docs-on-hover, et c.
If we expected people to read code like they're expected to read math, I'd have taken the huge pay hit and left the profession by now. It'd be a choice between that or a spiral into depression ending in tragedy. I don't know how math-loving folks can stand it.
Variables, as opposed to notation, tend to be local in scope, so 1) it isn't vital to have a memorable name and 2) it wouldn't really help to have a 'proper' name, either. As an example, if we're talking about two elements from a group, we usually talk about two elements, a and b, in a group, G. Could you call them elements 1 and 2 from an ExampleGroup? Maybe, but there's no clarity provided since these variables are only scoped for a given section. Additionally, longer equations means that it takes longer to read through and check steps like algebraic manipulation in a proof, or application of a lemma or theorem.
What is globally scoped are things like theorems, lemmas, definitions, etc., and those are easy to track down since they are by convention numbered after chapter and section.
And stuff that used to be simple algebra is now explained as something like 'Abelian group linear transformations'.
I haven't heard of "smash products on pointed spaces" for example, and idk why I would think them to be associative. Fun language though.
> And stuff that used to be simple algebra is now explained as something like 'Abelian group linear transformations'.
"Simple algebra" means different things in different professions? For example, Dummit&Foote's Abstract Algebra[0] would be considered "elementary algebra" by most mathematicians. For another example, Principles of Mathematical Analysis by Walter Rudin[1] is "elementary analysis".
[0] https://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/...
[1] https://www.amazon.com/Principles-Mathematical-Analysis-Inte...
On the note of notation, mathematics repurposes notation all the time, hopefully in a way that helps the person reading to understand by analogy or at least not be mislead by the ananlogy.
You said M/L and the / I think of then is quotient group, but that's apparently not what it means (I wouldn't know without reading it in context).
If X and Y are vector spaces (e.g., a line or a plane through the origin), then X+Y is the sum of all elements of each.
If you think about this example (dim(X)=1 and dim(Y)=2) you can easily recover the first formula intutively. There are two cases for the formula
dim(X+Y) = dim(X) + dim(Y) - dim(X ∩ Y)
Case 2: X intersects Y at the origin. In that case X+Y is a three-dimensional space and X∩Y is 0, thus the formula says 3=1+2-0.
I think I must understand when you say "X + Y" is the sum of elements that you're not using a formal sum {x} + {y} of each element x, and y but you mean sort of like the union of elements in X and Y?
To add to the confusion, in some contexts the sum is the same as ("isomorphic to") the product!
I've done mathematics in secondary school (coordinate systems, Pythagorean wotsit), but I can still visualize those and put them into practical use. Later on I had statistics and linear algebra - that last one I had to redo and really work on, it was only after I did a minor in game design that I could finally map that math to a practical application.
But I just don't have the background - or interest! - in anything else related to math. And I don't miss it either, else I would force myself to learn about it.
There's a couple of professions/qualities I have a lot of respect for, mathematicians is one of them. I appreciate them for three reasons: their ability to deal with the sheer complexity correctly and calmly, the fact it is unbiased, like science used to be (or rather: a couple of sciences aren't). The third reason is the one outlined above, generally: its a quality I've been unable to possess.
> “When you see a for, while, or do, all you know is that some kind of loop is coming up. To acquire even the faintest idea of what that loop does, you have to examine it. Not so with algorithms. Once you see a call to an algorithm, the name alone sketches the outline of what it does.”[0]
It's why in my last C++ job, I spent a lot of time convincing colleagues to use more of the functions in the algorithms library - so that it's easier to see what they're trying to do, and less likely to insert a bug.
Consider languages that have the ~sum~ function. Would you recommend they instead write a for loop? If not, just replace ~sum~ with the Sigma symbol and you have the mathematical notation.
[0] http://www.drdobbs.com/stl-algorithms-vs-hand-written-loops/...
It's a misconception commonly held by non-mathematicians that the symbols are somehow the key to mathematics. I often see complaints, particularly on HN, of it being "impossible to read" research mathematics due to "all the dense symbols".
This isn't how it works. You learn the mathematics. That's hard. As you go along, you realize that some things that require a lot of words can be condensed. So you make up symbols, or, more likely, the source that you're using teaches you some commonly-agreed-upon symbols. Now, once you've learned the math at hand, the symbol isn't the issue anymore.
(Sure, it's a bit of a pain when people disagree about symbols for concepts that everyone agrees should be abbreviated, but in the big picture it's minor)
In contrast, the language of mathematical notation is used for expressing concepts that are utterly unfamiliar and unintuitive. As a language, mathematical notation is nowhere near the complexity of a full human language, or even a programming language. It's much simpler than that. But the concepts behind it don't come naturally and have to be studied carefully.
Understanding what an algebraic group "really is" (with variations and extensions e.g. Lie groups) is more than one lifetime's work, but if you know what "group" means in English, well, you won't be disappointed. "Ring", even "field" kind of work this way.
If I ever really develop an intuition for what a magma is, and I doubt it, this will not resemble the molten core of the earth. Or will it?
On the other hand, you cannot use non-standard symbols to represent a sound in English. For example, nobody reading an English text will pronounce the letter "k" as "d" whereas mathematicians might do something like that, but they'll let you know in the opening pages of their book.
It’s about whether Euclid’s proof that there’s no finite set of primes is a proof by contradiction or not. The fact that this is disputed at all shows a certain unwillingness to use original sources — maybe each of them only looked at a different textbook’s restatement of the proof. Because no matter whether a proof of the theorem can be stated without contradiction, it took me all of 30 seconds to find a translation of the original proof to show that Euclid did in fact use one:
> I say that G is not the same with any of the numbers A, B, and C.
> If possible, let it be so. Now A, B, and C measure DE, therefore G also measures DE. But it also measures EF. Therefore G, being a number, measures the remainder, the unit DF, which is absurd.
(http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX...)
Within constructive logic, there are different non-equivalent definitions of infiniteness (that are all equivalent in classical logic):
- A set is non-finite if its finiteness leads to a contradiction.
- A set is infinite if any finite subset of it can be extended.
Within the context of the MathOverflow answer, the claim that Euclid's proof establishes infiniteness as opposed to non-finiteness, is correct. And the proof is therefore not "by contradiction", because that would only show non-finiteness.
[edit] Edited out some impoliteness.
Edit: The above added a nit-picky side-point to your original response, which was a lot shorter. Afterwards, you edited your reply to be a lot more exhaustive and a lot less... friendly? I don't think that's the best way to use HN's edit window. But see my grandchild comment below for my response to all the points you edited in afterwards.
And this construction can be carried out algorithmically: Just find the prime factors of ABC+1.
Obviously, this is not computationally efficient, but constructive logic doesn't care. Constructive logic only cares that a construction can be carried out, and that its worst case time complexity has some explicit bound. The existence of this bound for finding larger primes implies a weak bound on the density of the primes.
Also, I made some edits to my original comment.
Some points I’d like to clarify:
1. Multiple contradictory edits in the answer is weird by Stack Exchange standards regardless of the merits of the discussion, that’s the main thing I observed.
2. Whether Euclid uses contradiction or not is not really open to interpretation: the word “absurd” appears right in the proof. If people want to split hairs about whether that contradiction is in the lemma or the main proof, I guess fine.
3. There are of course multiple subschools of constructivism that accept and don’t accept different things. Not all of them ignore computational complexity. For example, some ultrafinitists reject the unique prime factorization theorem on the basis that you can’t really execute the factorization algorithm even for relatively modest numbers. To be clear, that’s not exactly my view, but I would for example agree that something like “just find the prime factors” sweeps a bit of relevant subtlety under the rug: you can only do that conceptually, not actually.
Sorry. Point taken. I'm way too quick to annoy at the moment, and I retroactively edit too much. I probably should keep my comments relatively unchanged after making them.
> If people want to split hairs
This entire discussion is about splitting hairs. And the MO answer acknowledges that there's a contradiction somewhere, and only argues over where it is. See the last edit on the answer.
That makes two of us. No offense taken.
(1) Given a finite set of primes {p_1, ... p_n}, then there exists another prime not in that set.
However, turning that into the following statement does require proof by contradiction (or, equivalently, the law of the excluded middle):
(2) There are an infinite number of primes
Euclid himself made statement (1) and proved it without contradiction, so looking at the original source gives the impression that you don't need it. But (2) is so similar-looking that many people incorrectly make the deduction that it doesn't need proof by contradiction either.
If A is the set of all primes, then A is an infinite set.
Contrapositive proof: Suppose A is a finite set containing only primes. By (1) we know there exists some p not in A and therefore A is not the set of all primes.
I don't believe contrapositive needs law of excluded middle but I'm honestly not sure. Logic is not my area
Let's say you know A and ~B=>~A, then you can deduce B. Proof by contradiction: assume otherwise, i.e. ~B, then by ~B=>~A you have ~A, but that contradicts A.
So if you have ~B=>~A then you have A=>B.
Contrapositive:
A-> B == A or ~B == ~B or A ==* ~B or ~(~A) == ~B -> ~A
Contradiction:
~(~A and B) == ~~A or ~B ==* A or ~B == A -> B
[1] A > B(assumption)
[2] ~B (assumption)
[3] A (assumption)
[4] B (modus ponens on 1 and 3)
[5] ~A (proof of negation on 3, 4 and 2)
Proof by contradiction has the following form: ~A -> false |- A. But the quoted proof has the form A -> false |- ~A.
This is FUD designed to discredit intuitionistic logic.
An instance of LEM is not "a logical error"; it is merely a claim which requires justification. For the claim at hand -- i.e. some natural number G is a member of a finite, constructed set, or it is not -- the corresponding LEM instance is perfectly true [and proven]. We say that the claim is decidable. So (given that we have a proof the claim is decidable) there would be no problem in intuitionistic logic with this part of Euclid's proof even if it took the form of a proof by contradiction (which of course it doesn't).
To make a more general rebuttal of the FUD, proofs by contradiction are not automatically invalid in intuitionistic logic; they just require one extra piece of evidence.
Furthermore the claim that whether something is a proof by contradiction depends on one's position on the logical validity of LEM is ridiculous on its face. There is a fact of the matter as to whether any given proof is by contradiction. There may be disagreements as to the definition of "proof by contradiction", but the definition which includes proof of negation is completely useless. In mathematics we usually treat useless definitions as in some sense "false" in order to facilitate communication - e.g., 1 could be prime depending on definitions, but that would be useless so we facilitate communication by treating "1 is prime" as false. Likewise here we should treat as false the claim that the quoted section of Euclid's proof is by contradiction.
Source for claims about LEM/proof by contradiction: https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016...
All I mean by saying it's a "logical error" is that it's not an argument which is always valid. Of course particular instances can still be correct.
I think a definition of "proof by contradiction" which includes proof of negation is useful if you are reasoning classically (and this doesn't have to be something you always do, or always avoid; you can reason classically some of the time and intuitionistically some of the time). Although the ideal situation to avoid any confusion would be to have three separate terms, one covering both types of proof and two for the individual types.
For example, the answers mention the belief that if 3 distributions are pairwise independent then they're jointly independent. Another false belief is that if 2 distributions are independent then they are conditionally independent. Also, people might think that you can pick a real number from R uniformly.
This one seems to stand out though - is there a specific "non-math" situation where this is going to cause problems ?
> If f is a smooth function with df = 0, then f is constant.
I assume the smoothness and df = 0 criteria are said to hold on the whole domain, right? Yet the claim is that f fails to be constant over the same domain?
This happens in engineering and software all the time. Time and time again, issues being resolved usually revolves around clarifying assumptions.
There is a similar thing in graph theory, where the kernel of the incidence matrix counts how many connected components a graph has.
(-1,2) are all number between -1 and 2, but not including -1 and 2. [-1,2] however are the same numbers, but including -1 and 2. Slightly different sets, where the former is open and the latter is closed.
Another way to think about it is for any point in (-1,2), we can find some number that is "closer to the boundary" but still in the set. This is not true for [-1,2] if we pick -1, because any number below -1 is suddenly outside the set. -1 is a boundary point, and therefore that side of the interval is not open!
Why do we need this? Well, lots of calculus is about sequences that converge in smaller and smaller distances. One can for example define whether a function "jumps" by checking whether the inputs have boundary points or not (etc.).
But for math people, this is not general enough, because it requires things like distances. Hence, they have a more general definition: There's a collection of stuff, and open sets are sets of this stuff that follow some rules.
Now here is the important part: Closed sets do not have an extra set of rules. Instead, they are just defined as the complement (or opposite) of an open set.
So R - the set of all real numbers - has a complement which is the empty set. The opposite of "all numbers" is nothing! Since R is open, by definition, the empty set is closed.
Cool. But then, if we take our rules, we see the the empty set is also open. Whoops. Then, again by definition - the opposite of the empty set - which is R, so all numbers - is closed.
Confusion conclusion: Both R and its complement, the empty set, are both closed and open at the same time.
Note that sets are always closed or open relative to some other specified (often implied) set S.
In some courses, a closed set (in S) is defined to be a set which is the complement S-T for some open set T. In others, a closed set (in S) is defined to be a set which contains all its limit points (in S). And then whichever isn't the definition gets proven as a theorem.
The misconception is about not knowing that the generalization to arbitrary sets has some unexpected properties.
(Half-open intervals are neither open nor closed, by the way)
Wow, pretty sure I read this in New Scientist magazine in the early 2000s, quoting some "cryptography expert". Guess he wasn't such an expert.
Why would this be wrong? If the derivative is zero everywhere, how would the function not be constant? Or does this mean something completely different?
EDIT: Someone already provided the answer here https://news.ycombinator.com/item?id=30146426
To the question of how advanced these are, with only a few exceptions, the majority of things there are covered in undergraduate analysis, algebra, and topology courses.
Things like "Many students believe that 1 plus the product of the first primes is always a prime number" (a misunderstandment of Euclid's proof that there exist infinitely many prime numbers) or "If A implies B then B implies A" (difference between a necessary and a sufficient condition) are literally middle-school math.
Things like "If 𝑓(𝑥,𝑦) is a polynomial with real coefficients, then the image of 𝑓 is a closed subset of ℝ" are high-school level.
The top answer about dimensions of vector spaces is a well-known undergraduate-level "trap".
"The Krull dimension of a noetherian integral domain is finite." is, to be brutally honest, not a sentence I understand in the slightest.
Also
> "If 𝑓(𝑥,𝑦) is a polynomial with real coefficients, then the image of 𝑓 is a closed subset of ℝ"
I don't think that this is high school level. Secondary school level math rarely touches anything related to multivariate polynomials (and usually doesn't have a notion of "closed sets" beyond closed intervals, but that's not as relevant apart from how the problem is stated).
I don't doubt that what you're saying is true, it's just something far outside what I personally am familiar with.
At the University of Waterloo, CS is strictly in the faculty of Math, so I'd expect students to know much of this (I think? I'm in Computer Engineering at UW, which is a distinctively different program).
OTOH, I'd imagine things are different in universities where CS is thought of as its own major, with no Software Engineering/Computer Engineering as an alternative.