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> This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. The thing that mathematicians refuse to admit i

> This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. The thing that mathematicians refuse to admit is that they are extremely sloppy with their notation, terminology and rigor. Especially in comparison to the average programmer. They are conceptually/abstractly rigorous, but in "implementation" are incredibly sloppy. But they've been in that world so long they can't really see it / just expect it. And if you debate with one long enough, they'll eventually concede and say something along the lines of "well math evolved being written on paper and conciseness was important so that took priority over those other concerns." And it leaks through into math instruction and general math text writing. Programming is forced to be extremely rigorous at the implementation level simply because what is written must be executed. Now engineering abstraction is extremely conceptually sloppy and if it works it's often deemed "good enough". And math generally is the exact opposite. Even for a simple case, take the number of symbols that have context sensitive meanings and mathematicians. They will use them without declaring which context they are using, and a reader is simply supposed to infer correctly. It's actually somewhat funny because it's not at all how they see themselves.

rq1 3y ago
This is ridiculous. The average computer scientist (not only "programmer", as a js dev would be) never wrote lean/coq or similar, and is not aware of the Curry-Haskell like theorems and their implications.
- thrwayaistartup 3y ago
  I think you entirely missed the point. GP put it well: >> They are conceptually/abstractly rigorous, but in "implementation" are incredibly sloppy. Maturity in concept-space and the ability to reason abstractly can be achieved without the sort of formal rigor required by far less abstract and much more conceptually simple programming. I have seen this first hand TAing and tutoring CS1. I regularly had students who put off their required programming course until senior year. As a result, some were well into graduate-level mathematics and at the top of their class but struggled deeply with the rigor required in implementation. Think about, e.g., missing semi-colons at the end of lines, understanding where a variable is defined, understanding how nested loops work, simple recursion, and so on. Consider something as simple as writing a C/Java program that reads lines from a file, parses them according to a simple format, prints out some accumulated value from the process, and handles common errors appropriately. Programming requires a lot more formal rigor than mathematical proof writing.
  - rq1 3y ago
    I didn’t miss his point. He’s plain wrong. > Programming requires a lot more formal rigor than mathematical proof writing. This is is just wrong? Syntax rigour has almost nothing to do with correctness.
    - barrkel 3y ago
      > Syntax rigour has almost nothing to do with correctness. Yes, that's the point. You did miss it.
      - rq1 3y ago
        And yet I didn’t miss it. I insist. You need more rigour to prove let’s say Beppo Levy theorem than writing a moderately complex piece of software. Yet you can write it in crappy English, the medium not being the target goal, the ideation process even poorly transcribed in English needs to be perfectly rigorous. Otherwise, you proved nothing.
        simonw 3y ago
        I'm pretty sure the two of you are working from different definitions of the word "rigorous" here. (Which, given the topic of conversation here, is somewhat ironic.)
        rq1 3y ago
        I take it broadly as nothing was clearly defined. :)
    - echion 3y ago
      > Syntax rigour has almost nothing to do with correctness. I see your point: has almost nothing correctness with rigour do to Syntax. Syntax rigor has to do with correctness to the extent that "correctness" exists outside the mind of the creator. Einstein notation is a decent example: the rigor is inherent in the definition of the syntax, but to a novice, it is entirely under-specified and can't be said to be "correct" without its definition being incorporated already...which is the ultimate-parent-posts' point and I think the context in which the post-to-which-you're-replying needs to be taken. And if you're going to argue "This is just wrong?" (I love the passive-aggressive '?'?) while ignoring the context of the discussion...QED.
- nextaccountic 3y ago
  but programmers don't write underspecified notational shortcuts, because those are soundly rejected as syntax errors by the compiler or interpreter this is not about semantics (like dependent types etc) this is just syntax. it works like this in any language. the only way to make syntax accepted by a compiler is to make it unambiguous ... maybe LLMs will change this game and the programming languages of the future will be allowed to be sloppy, just like mathematics
  - rq1 3y ago
    Some compilers can accept “holes” that can be univocally deducted from context for instance. You can think about it as type inference on steroids.
    - nextaccountic 3y ago
      Yep, but for any notational shortcut the compiler infers a single thing for it. It's still unambiguous as far as computers are concerned (but it may be confusing reading it without context)
      - rq1 3y ago
        It’s not only a notational shortcut as in syntactic sugar though. It can apply a set of rewrite rules given you were able to construct the right type at some point. It’s type inference on steroids because you can brute force the solution by applying the rewrite rules and other tactics on propositions until something (propositional equality) is found, or nothing.
- rq1 3y ago
  s/Curry-Haskell/Curry-Howard/g
dajt 3y ago
This drove me mad when I had to do introductory maths at uni. Maths as written did seem pretty sloppy and not at all like a programming language whose expressions I could parse as I expected. Obv most simple algebra looks about as you'd expect but I clearly recall feeling exactly what you describe in some cases, and commented upon it to the lecturer about it asking why it was that way during a tutorial. He humoured me, was a good guy. But I think mathematicians probably have a point - it did evolve that way over a long time and anyone practicing it daily just knows how to do it and they're not going to do a thorough review and change now. It's us tourists that get thrown for a loop, but so it goes. It's not meant for us.
- nyssos 3y ago
  > Maths as written did seem pretty sloppy and not at all like a programming language whose expressions I could parse as I expected. Look at Lean's mathlib: that's what fully formal mathematics looks like. It's far too verbose to be suitable for communicating with other people; you might as well try to teach an algorithms course transistor by transistor.
ProjectArcturis 3y ago
>Even for a simple case, take the number of symbols that have context sensitive meanings and mathematicians. They will use them without declaring which context they are using, and a reader is simply supposed to infer correctly. Yes!! Like, oh, you didn't know that p-looking thing (rho) means Pearson's correlation coefficient? That theta means an angle? Well just sit there in ignorance because I'm not going to explain it. And those are the easy ones!
pxx 3y ago
Notation and terminology? Sure, some explanations and mechanical manipulations are elided in mathematics because with context they're clear. Rigor? Ha, you have got to be kidding me. Math is rigorous to a fault. Typical computer programming has no rigor whatsoever in comparison. A rigid syntax is not rigor. Syntax, in turn, is certainly not the difficult part of developing software.
- danielmarkbruce 3y ago
  The commenter is quite rigorous about what parts of math are done with rigor and which parts are not.
sidlls 3y ago
My experience with the average programmer is...different from yours. The software development field is exceptionally bad in this regard. Physicists are mathematically sloppy sometimes (why, yes, I will just multiply both sides by `dy` and take as many liberties with operators, harmonics/series, and vector differential operations as I care to, thanks). Mathematics, like any other academic field, has jargon (and this includes notation, customary symbols for a given application, etc.), and students of the field ought to learn the jargon if they wish to be proficient. On the other hand, textbooks meant to instruct ought to teach the jargon. It's been forever since I've opened a mathematics textbook; I don't recall any being terribly bad in this regard.
hasmanean 3y ago
Edit: preface by stating that I agree with you. Well I have a different approach. Sometimes I write and hack it to solve a particular problem. The code might be elegant or not, but if you understand the problem you can probably grok the code. Next I generalize it a bit. Specific variables configurable parameters. Something that happened implicitly or with a single line of code gets handled by its own function. Now it’s general but makes much less sense at first because it’s no longer tied to one problem, but a whole set of problems. It’s a lot less teachable and certainly not self-evident any more. The problem with math education is that we think the solution approach would be inherently superior to the first, and would make a better textbook—because it’s more generic. But that is not how real people learn—they would all “do” math the first way. By taking away the ability of the student to do the generalization themselves we are depriving them of the real pleasure of programming (or math). Maybe back when paper was scarce this approach made sense but not any more. Ideally I would love to present a million specific solutions and let them generalize themselves. That is exactly how we would train a ANN. Not be regurgitating the canned solution but by giving it all sorts of data and letting it generalize for itself. So why don’t we do this for human students? When it comes to education I think people have a blind spot towards how learning is actually done. Education != learning.
BeetleB 3y ago
> The thing that mathematicians refuse to admit is that they are extremely sloppy with their notation, terminology and rigor. Especially in comparison to the average programmer. Not sure why you say that. Mathematicians are pretty open about it. The well known essay On proof and progress on mathematics discusses it. It is written by a Fields medalist.
- rigid 3y ago
  I wish teachers would read that.
  - melagonster 3y ago
    Teachers are not mathematician.
    - 12345hn6789 3y ago
      Someone with a PhD in mathematics is not a mathematician?
    - rigid 3y ago
      even more reason to stick to strict, standardized, uniform notation/syntax.
deleted 3y ago
[deleted]
chongli 3y ago
You’re confusing syntax and semantics. Programmers write code for syntax machines (Turing machines). The computers care a lot about syntax and will halt if you make an error. They do not care at all about semantics. A computer is happy to let you multiply a temperature in Fahrenheit times a figure in Australian dollars and subtract the volume of the earth in litres, provided that these numbers are all formatted in a compatible enough way that they can be implicitly converted (this depends on the programming language but many of them are quite liberal at this). If you want the computer to stop you from doing such nonsense, you’ve got to put in a lot of effort to make types or contracts or write a lot of tests to avoid it. But that’s essentially a scheme for encoding a little bit of your semantics into syntax the computer can understand. Most programmers are not this rigorous! Mathematicians, on the other hand, write mathematics for other humans to read. They expect their readers to have done their homework long before picking up the paper. They do not have any time to waste in spelling out all the minutiae, much of which is obvious and trivial to their peers. The sort of formal, syntax-level rigour you prefer, which can be checked by computers, is of zero interest to most mathematicians. What matters to them, at the end of the day, is making a solid enough argument to convince the establishment within their subfield of mathematics.
- agos 3y ago
  But programmers are expected to get the semantics right. Sure, it happens to mismatch temperatures and dollars, but it’s called a bug and you will be expected to fix it
  - chongli 3y ago
    Software in the wild is full of millions of bugs. It’s a gigantic mess! Who is out there fixing it all?
- passion__desire 3y ago
  Why do mathematicians hide their natural way of thinking ? They provide their finished work and everyone is supposed to clap. Why can't they write long articles like about false starts, dead ends and so on. It's only after magazines like Quanta and YouTube channels that we get to feel the thinking process. Math is not hard. Atleast the mathematics we are expected to know.
afiori 3y ago
I believe that any mathematician that took a differential geometry class must have already realized this, the notation is so compressed and implicit that some proofs practically are "by notation" as it can be a dauting prospect to expand a dozen indexes.
akasakahakada 3y ago
Bro I know this feel. Even books teaching algorithms being written by mathematicians are error everywhere. They never state the type, class, no comment, no explanation, read exceed the last index... This list can go endlessly. When they say "lets declare an empty set for variable T", you don't know whether the thing is a list, set, tuple, ndarray, placeholder for a scalar, or a graph. Some even provide actual code, however, never actually run the code to verify their correctness.
- defrost 3y ago
  Try this guy then, he's got a PhD in mathematics from the California Institute of Technology from a thesis Finite Semifields and Projective Planes but he's written a bunch of stuff on algorithms and will write you a check for any errors you find in his work: https://en.wikipedia.org/wiki/Donald_Knuth https://en.wikipedia.org/wiki/Donald_Knuth
  - akasakahakada 3y ago
    Regardless of the name of his degree, dude is a real programmer.
    - defrost 3y ago
      Church organist actually - serious enough to have a two story pipe organ built into his own home. Enough of the True Scotsman .. it's clear as day from the introduction and opening chapters of TAOCP that he approaches programming as a mathematician.
jonnycomputer 3y ago
This, really. Sometimes, when reading math papers, you find that they end up being very hand-wavy with the notation, e.g. with subscripting, because "it's understood". But without extensive training in mathematics, a lot of it is not understood. Programmers will use languages with a lot of syntactic sugar, and without knowing the language, code can be pretty difficult to understand when it is used. But even then, you can't be sloppy, because computers are so damn literal.
c0balt 3y ago
> The thing that mathematicians refuse to admit is that they are extremely sloppy with their notation, terminology and rigor. The refuse part is imo ver dependent on the person. Nearly all of my professors for theoretical cs courses just plainly said that their "unique" notation is just their way because they like it. It's more or less just a simple approach to alter the language to fit your task. This is also not unfamiliar to the programmers who may choose a language based on the task, with, e.g. Fortran for vector based calculus or C for direct hardware access.

rq1 3y ago

This is ridiculous. The average computer scientist (not only "programmer", as a js dev would be) never wrote lean/coq or similar, and is not aware of the Curry-Haskell like theorems and their implications.

thrwayaistartup 3y ago
I think you entirely missed the point. GP put it well: >> They are conceptually/abstractly rigorous, but in "implementation" are incredibly sloppy. Maturity in concept-space and the ability to reason abstractly can be achieved without the sort of formal rigor required by far less abstract and much more conceptually simple programming. I have seen this first hand TAing and tutoring CS1. I regularly had students who put off their required programming course until senior year. As a result, some were well into graduate-level mathematics and at the top of their class but struggled deeply with the rigor required in implementation. Think about, e.g., missing semi-colons at the end of lines, understanding where a variable is defined, understanding how nested loops work, simple recursion, and so on. Consider something as simple as writing a C/Java program that reads lines from a file, parses them according to a simple format, prints out some accumulated value from the process, and handles common errors appropriately. Programming requires a lot more formal rigor than mathematical proof writing.
- rq1 3y ago
  I didn’t miss his point. He’s plain wrong. > Programming requires a lot more formal rigor than mathematical proof writing. This is is just wrong? Syntax rigour has almost nothing to do with correctness.
  - barrkel 3y ago
    > Syntax rigour has almost nothing to do with correctness. Yes, that's the point. You did miss it.
    - rq1 3y ago
      And yet I didn’t miss it. I insist. You need more rigour to prove let’s say Beppo Levy theorem than writing a moderately complex piece of software. Yet you can write it in crappy English, the medium not being the target goal, the ideation process even poorly transcribed in English needs to be perfectly rigorous. Otherwise, you proved nothing.
      - simonw 3y ago
        I'm pretty sure the two of you are working from different definitions of the word "rigorous" here. (Which, given the topic of conversation here, is somewhat ironic.)
        rq1 3y ago
        I take it broadly as nothing was clearly defined. :)
  - echion 3y ago
    > Syntax rigour has almost nothing to do with correctness. I see your point: has almost nothing correctness with rigour do to Syntax. Syntax rigor has to do with correctness to the extent that "correctness" exists outside the mind of the creator. Einstein notation is a decent example: the rigor is inherent in the definition of the syntax, but to a novice, it is entirely under-specified and can't be said to be "correct" without its definition being incorporated already...which is the ultimate-parent-posts' point and I think the context in which the post-to-which-you're-replying needs to be taken. And if you're going to argue "This is just wrong?" (I love the passive-aggressive '?'?) while ignoring the context of the discussion...QED.
nextaccountic 3y ago
but programmers don't write underspecified notational shortcuts, because those are soundly rejected as syntax errors by the compiler or interpreter this is not about semantics (like dependent types etc) this is just syntax. it works like this in any language. the only way to make syntax accepted by a compiler is to make it unambiguous ... maybe LLMs will change this game and the programming languages of the future will be allowed to be sloppy, just like mathematics
- rq1 3y ago
  Some compilers can accept “holes” that can be univocally deducted from context for instance. You can think about it as type inference on steroids.
  - nextaccountic 3y ago
    Yep, but for any notational shortcut the compiler infers a single thing for it. It's still unambiguous as far as computers are concerned (but it may be confusing reading it without context)
    - rq1 3y ago
      It’s not only a notational shortcut as in syntactic sugar though. It can apply a set of rewrite rules given you were able to construct the right type at some point. It’s type inference on steroids because you can brute force the solution by applying the rewrite rules and other tactics on propositions until something (propositional equality) is found, or nothing.
rq1 3y ago
s/Curry-Haskell/Curry-Howard/g

dajt 3y ago

This drove me mad when I had to do introductory maths at uni. Maths as written did seem pretty sloppy and not at all like a programming language whose expressions I could parse as I expected. Obv most simple algebra looks about as you'd expect but I clearly recall feeling exactly what you describe in some cases, and commented upon it to the lecturer about it asking why it was that way during a tutorial. He humoured me, was a good guy. But I think mathematicians probably have a point - it did evolve that way over a long time and anyone practicing it daily just knows how to do it and they're not going to do a thorough review and change now. It's us tourists that get thrown for a loop, but so it goes. It's not meant for us.

nyssos 3y ago
> Maths as written did seem pretty sloppy and not at all like a programming language whose expressions I could parse as I expected. Look at Lean's mathlib: that's what fully formal mathematics looks like. It's far too verbose to be suitable for communicating with other people; you might as well try to teach an algorithms course transistor by transistor.

ProjectArcturis 3y ago

>Even for a simple case, take the number of symbols that have context sensitive meanings and mathematicians. They will use them without declaring which context they are using, and a reader is simply supposed to infer correctly. Yes!! Like, oh, you didn't know that p-looking thing (rho) means Pearson's correlation coefficient? That theta means an angle? Well just sit there in ignorance because I'm not going to explain it. And those are the easy ones!

pxx 3y ago

Notation and terminology? Sure, some explanations and mechanical manipulations are elided in mathematics because with context they're clear. Rigor? Ha, you have got to be kidding me. Math is rigorous to a fault. Typical computer programming has no rigor whatsoever in comparison. A rigid syntax is not rigor. Syntax, in turn, is certainly not the difficult part of developing software.

danielmarkbruce 3y ago
The commenter is quite rigorous about what parts of math are done with rigor and which parts are not.

sidlls 3y ago

My experience with the average programmer is...different from yours. The software development field is exceptionally bad in this regard. Physicists are mathematically sloppy sometimes (why, yes, I will just multiply both sides by `dy` and take as many liberties with operators, harmonics/series, and vector differential operations as I care to, thanks). Mathematics, like any other academic field, has jargon (and this includes notation, customary symbols for a given application, etc.), and students of the field ought to learn the jargon if they wish to be proficient. On the other hand, textbooks meant to instruct ought to teach the jargon. It's been forever since I've opened a mathematics textbook; I don't recall any being terribly bad in this regard.

hasmanean 3y ago

Edit: preface by stating that I agree with you. Well I have a different approach. Sometimes I write and hack it to solve a particular problem. The code might be elegant or not, but if you understand the problem you can probably grok the code. Next I generalize it a bit. Specific variables configurable parameters. Something that happened implicitly or with a single line of code gets handled by its own function. Now it’s general but makes much less sense at first because it’s no longer tied to one problem, but a whole set of problems. It’s a lot less teachable and certainly not self-evident any more. The problem with math education is that we think the solution approach would be inherently superior to the first, and would make a better textbook—because it’s more generic. But that is not how real people learn—they would all “do” math the first way. By taking away the ability of the student to do the generalization themselves we are depriving them of the real pleasure of programming (or math). Maybe back when paper was scarce this approach made sense but not any more. Ideally I would love to present a million specific solutions and let them generalize themselves. That is exactly how we would train a ANN. Not be regurgitating the canned solution but by giving it all sorts of data and letting it generalize for itself. So why don’t we do this for human students? When it comes to education I think people have a blind spot towards how learning is actually done. Education != learning.

BeetleB 3y ago

> The thing that mathematicians refuse to admit is that they are extremely sloppy with their notation, terminology and rigor. Especially in comparison to the average programmer. Not sure why you say that. Mathematicians are pretty open about it. The well known essay On proof and progress on mathematics discusses it. It is written by a Fields medalist.

rigid 3y ago
I wish teachers would read that.
- melagonster 3y ago
  Teachers are not mathematician.
  - 12345hn6789 3y ago
    Someone with a PhD in mathematics is not a mathematician?
  - rigid 3y ago
    even more reason to stick to strict, standardized, uniform notation/syntax.

deleted 3y ago

[deleted]

chongli 3y ago

You’re confusing syntax and semantics. Programmers write code for syntax machines (Turing machines). The computers care a lot about syntax and will halt if you make an error. They do not care at all about semantics. A computer is happy to let you multiply a temperature in Fahrenheit times a figure in Australian dollars and subtract the volume of the earth in litres, provided that these numbers are all formatted in a compatible enough way that they can be implicitly converted (this depends on the programming language but many of them are quite liberal at this). If you want the computer to stop you from doing such nonsense, you’ve got to put in a lot of effort to make types or contracts or write a lot of tests to avoid it. But that’s essentially a scheme for encoding a little bit of your semantics into syntax the computer can understand. Most programmers are not this rigorous! Mathematicians, on the other hand, write mathematics for other humans to read. They expect their readers to have done their homework long before picking up the paper. They do not have any time to waste in spelling out all the minutiae, much of which is obvious and trivial to their peers. The sort of formal, syntax-level rigour you prefer, which can be checked by computers, is of zero interest to most mathematicians. What matters to them, at the end of the day, is making a solid enough argument to convince the establishment within their subfield of mathematics.

agos 3y ago
But programmers are expected to get the semantics right. Sure, it happens to mismatch temperatures and dollars, but it’s called a bug and you will be expected to fix it
- chongli 3y ago
  Software in the wild is full of millions of bugs. It’s a gigantic mess! Who is out there fixing it all?
passion__desire 3y ago
Why do mathematicians hide their natural way of thinking ? They provide their finished work and everyone is supposed to clap. Why can't they write long articles like about false starts, dead ends and so on. It's only after magazines like Quanta and YouTube channels that we get to feel the thinking process. Math is not hard. Atleast the mathematics we are expected to know.

afiori 3y ago

I believe that any mathematician that took a differential geometry class must have already realized this, the notation is so compressed and implicit that some proofs practically are "by notation" as it can be a dauting prospect to expand a dozen indexes.

akasakahakada 3y ago

Bro I know this feel. Even books teaching algorithms being written by mathematicians are error everywhere. They never state the type, class, no comment, no explanation, read exceed the last index... This list can go endlessly. When they say "lets declare an empty set for variable T", you don't know whether the thing is a list, set, tuple, ndarray, placeholder for a scalar, or a graph. Some even provide actual code, however, never actually run the code to verify their correctness.

defrost 3y ago
Try this guy then, he's got a PhD in mathematics from the California Institute of Technology from a thesis Finite Semifields and Projective Planes but he's written a bunch of stuff on algorithms and will write you a check for any errors you find in his work: https://en.wikipedia.org/wiki/Donald_Knuth https://en.wikipedia.org/wiki/Donald_Knuth
- akasakahakada 3y ago
  Regardless of the name of his degree, dude is a real programmer.
  - defrost 3y ago
    Church organist actually - serious enough to have a two story pipe organ built into his own home. Enough of the True Scotsman .. it's clear as day from the introduction and opening chapters of TAOCP that he approaches programming as a mathematician.

jonnycomputer 3y ago

This, really. Sometimes, when reading math papers, you find that they end up being very hand-wavy with the notation, e.g. with subscripting, because "it's understood". But without extensive training in mathematics, a lot of it is not understood. Programmers will use languages with a lot of syntactic sugar, and without knowing the language, code can be pretty difficult to understand when it is used. But even then, you can't be sloppy, because computers are so damn literal.

c0balt 3y ago

> The thing that mathematicians refuse to admit is that they are extremely sloppy with their notation, terminology and rigor. The refuse part is imo ver dependent on the person. Nearly all of my professors for theoretical cs courses just plainly said that their "unique" notation is just their way because they like it. It's more or less just a simple approach to alter the language to fit your task. This is also not unfamiliar to the programmers who may choose a language based on the task, with, e.g. Fortran for vector based calculus or C for direct hardware access.