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I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. Yes, conventions ha

I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. Yes, conventions have emerged, people tend to use the same sort of notation in a given context, but in the main, the notation should be regarded as an aide memoire, something to guide you. You say that you're struggling because of "the math notations and zero explanation of it in the context." Can you give us some examples? Maybe getting a start on it with a careful discussion of a few examples will unblock the difficulty you're having.

rullopat 5y ago
I was trying to grasp some of the papers linked in the Valhalla DSP block, for example this one: http://www2.ece.rochester.edu/courses/ECE472/resources/Papers/Schroeder_1961.pdf http://www2.ece.rochester.edu/courses/ECE472/resources/Paper... There is a formula with a triangle and I don't get what's that about, for example.
- nickcw 5y ago
  The triangle is a capital Greek Delta and is usually used to indicate a change in something. So ΔT means some change in T.
- ColinWright 5y ago
  You say "There's a formula with a triangle ..." without telling me where. That's not real helpful, and you're making me do the work to find out what you're talking about. If you want assistance to get started, you need to be more explicit. However, I have done that work, so I've looked, and in the second column of page 210 there's a "formula with a triangle": t_c = 5 \middot 10^{-5} \sqrt( V / Dt ) ... where the "D" I've used is where the triangle appears in the formula. But that can't be it, because just two lines above it we have: "For a pulse of width Dt, the critical time ..." So that's stating that "Dt" is the width of the pulse, and should be thought of as a single term. So maybe that's the wrong formula, or maybe it was just a bad example. So trying to be more helpful, the "triangle" is a Greek capital delta and means different things in different places. However, it is often used to mean "a small change in". https://en.wikipedia.org/wiki/%CE%94T https://en.wikipedia.org/wiki/%CE%94T FWIW ... at a glance I can't see where that result is derived, it appears simply to be stated without explanation. I might be wrong, I've not read the rest of the paper.
  - tomerv 5y ago
    I think that this is indeed the formula in GP's question. And indeed sometimes math notation is obtuse like that. It looks like 2 terms, but the triangle goes together with the t as a single term. At other times it might be called "dt" and despite looking like a multiplication of 2 variables (d and t, or triangle and t in this case) it's just a single variable with a named made of 2 characters. The important thing here is that "For a pulse of width Dt" is the definition of this variable, but this can be easily missed if you're not used to this naming convention.
    - enriquto 5y ago
      > it's just a single variable with a named made of 2 characters. I have this same problem with programming, when I have to deal with code written by non-mathematicians. They tend to use all these stupid variables with more than one letter and that confuses the heck out of me.
    - kergonath 5y ago
      That’s because “Δ” means “a change of” or “an interval of”. So, Δt is “an interval of time”. It is like a compound word, really. It conveys more information than giving it an arbitrary, single-letter name. This convention is used in a whole bunch of scientific fields, like quantum mechanics, chemistry, biology, mechanics, thermodynamics, etc. It’s also very useful in how it relates to derivatives, which is a crucial concept in just about any kind of science you could care to mention. So yes, there is a learning curve, but we write things this way for good reasons, most of the time. Multiplication should be represented by a (thin) space in good typography, to avoid this sort of things. Not doing it is sloppy and invites misreading. Same with omitting parenthesis around a function’s argument most of the time (e.g. sin 2πθ instead of sin(2 π θ) ).
  - rullopat 5y ago
    Sorry I didn't mean to make you work for me, but it's a PDF and I didn't know how to explain better the position (maybe I should have told you the first formula on page X). For you it was a D, for me it was a triangle and I didn't get the meaning of that Dt. Maybe it's just a too advanced paper for my knowledge.
- Jensson 5y ago
  That is a physics based paper, it used physics notation. Can find common ones here (including that triangle): https://en.wikipedia.org/wiki/List_of_common_physics_notations https://en.wikipedia.org/wiki/List_of_common_physics_notatio...
- NamTaf 5y ago
  What you're looking at is calculus, specifically differentiation. This is pretty core to understanding physics, because so much of physics depends on the time-evolving state of things. That's fundamentally what's happening here. The triangle, for example, is the upper-case greek letter delta, which in calculus represents 'change of'. You might have heard of 'delta-T' with respect to 'change of time'. In calculus, upper-case delta means 'change over a finite time' vs lower-case delta meaning 'instantaneous change'. The practical upshot, for example, is that the lower-case is the instantaneous rate-of-change at an instant in time, whereas the upper-case is the change over a whole time (e.g. the average rate of change per second for time = 0 seconds to time = 3 seconds). If you are trying to grok this, I would suggest an introductory calculus or pre-calculus resource. It doesn't have to be a uni textbook - higher-level high school maths usually teaches this. In this particular case, the Khan Academy would be my recommendation because it is about the right level (we're not talking esoteric higher-level university knowledge here) and it is eminently accessable. For example, this link may be a good starter in this instance: https://www.youtube.com/watch?v=MeU-KzdCBps https://www.youtube.com/watch?v=MeU-KzdCBps
- forgetfulness 5y ago
  Looks like you need to grind through an elementary calculus book. With the exercises, you may think you build intuition by reading just the definitions, but half of the understanding is tacit and you get through the exercises. If you're trying to get into signal processing, it'll involve calculus in complex numbers, and knowledge of that is often gained through plodding through proofs and exercises over and over.
- pfortuny 5y ago
  You probably want to try and read Courant-Johns Calculus (I forget the exact title).
- kens 5y ago
  Everyone is talking about the Δ symbol, but the real problem that you'll encounter will be later in the paper where they start talking about H(ω), which is the Fourier transform of the impulse function (equation 4 and following). You'll need to know a fair bit about Fourier transforms and impulse responses and filter design to get through this section. The notation is the least of the problems. One place to start is https://en.wikipedia.org/wiki/Impulse_response https://en.wikipedia.org/wiki/Impulse_response
  - kergonath 5y ago
    Just like with programming, knowing the meaning of keywords is not enough to understand something.
  - forgetfulness 5y ago
    Wikipedia is truly atrocious for learning math, the articles are like man pages in that they precisely describe the concepts in terms that will only make sense if you already know the thing. They just aren't written for pedagogy. Like in 300 BC today, there's no royal road to geometry.
- BeetleB 5y ago
  A few points: 1. You're reading a journal article. They will assume you know the notation not just of the broader discipline (e.g. physics/electrical engineering), but of the subdiscipline and at times the subsubdiscipline. Journal papers are explicitly written not to be easy to comprehend by beginners.[1] Notation will be only one problem you'll face. 2. As has been pointed out, this is not a mathematics paper. Mathematicians have their own notation, as do physicists and engineers. As I mentioned in the above bullet, they can have their own notation even in subdisciplines (e.g. circuit folks use "j" for the imaginary number, and semiconductor folks use "i"). There is a lot of overlap in notation amongst these parties, but you should never assume because you know one notation that you'll easily understand the math written by other fields. 3. Most introductory textbooks will explain the basic notation. Unfortunately, I often do find gaps where you go to higher level textbooks and they use notation that they don't explain (i.e. they assume you've seen it before), but is not covered in the prior textbooks. 4. Finally, sorry to say this, but "delta" (the triangle) for representing change is used in almost all sciences and engineering. It was heavily used in my high school as well. If you're struggling with this you really need to read some introductory textbooks in, say, physics. [1] I'm not kidding. I've spent time in academia and I've complained how obtuse some articles are, and almost universally the response is "We write for other experts, not for new graduate students". One professor took pride at the fact that in his field, one can comprehend only about one page of a paper per day - and this coming from someone who is an expert. These people have issues.
User23 5y ago
Edsger Dijkstra, who was a mathematician by training, wrote a wonderful little monograph on this subject called The notational conventions I adopted, and why[1]. I am particularly fond of his commented equational proof format. [1] https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/EWD1300.html https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...
cabalamat 5y ago
> I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. One main cause for this belief is that in a programming there is one true noation (or rather, a separate one for each language) that is unambiguous and clearly defined. I dislike maths notation as I find it lacks rigour.
- Jensson 5y ago
  > there is one true noation (or rather, a separate one for each language) that is unambiguous and clearly defined. This is such a disingenuous take. How many of the source code files you write are 100% self contained and well defined? I'd bet not a single one of them are. You reference libraries, you depend on specific compiler/runtime/OS versions, you reference other files etc. If you take a look at any of these scientific papers you call "badly defined", did you really go through all of the referenced papers and look if they defined the things you didn't get? If not then you can't be sure that the paper uses undefined notation. If you argue that it is too much work to go through that many references, well that is what you would have to do to understand one of your program files.
  - throwaway31338 5y ago
    One can look at the source code to a program, the libraries it uses, the compiler for the language, and the ISA spec for the machine language the compiler generates. You can know that there are no hidden unspecified quantities because programs can't work without being specified. When you get down to the microcode of the CPU that implements the ISA you might have an issue if it's ill-specified. You might be talking about an ISA like RISC-V, though, specified at a level sufficient to go down to the gates. You might be talking about an ISA like 6502 where the gate-level implementations have been reverse-engineered. You can take programming all the way down boolean logic if you need to and the tools are readily available. They don't rely on you "just knowing" something.
    - Jensson 5y ago
      > One can look at the source code to a program, the libraries it uses, the compiler for the language, and the ISA spec for the machine language the compiler generates. You can know that there are no hidden unspecified quantities because programs can't work without being specified. I doubt you actually can do that and understand it all. A computer can do it, but I doubt you the human can do that and get a perfect picture of any non trivial program without making errors. Human math is a human language first and foremost, its grammar is human language which is used to define things and symbols. This lets us write things that humans can actually read and understand the entirety of, unlike a million lines of code or cpu instructions. Show me a program written by 10 programmers over 10 years and I doubt anyone really understands all of it. But we have mathematical fields that hundreds of mathematicians have written over centuries, and people still are able to understand it all perfectly. It is true that a computer can easily read a computer program, but since we are arguing about teaching humans you would need to show evidence that humans can actually read and understand complex code well.
bajsejohannes 5y ago
> I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. Just to back up this point: In probably every university-level math book I've read, they introduce and explain all the notation used. In the preface and/or as concepts are introduced. There are lists at wikipedia [1] and other places, but I'm not sure how valuable it is out of context. [1] https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbols https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbo...
- nicoburns 5y ago
  Ha, you've clearly used better books than me. I've read plenty where they glossed over the notation and expected you to guess.
  - ColinWright 5y ago
    Often books assume some prerequisites, the question here is the level of those prerequisites. Some books try to include all the necessary background, others assume a pre-existing base level of knowledge. Different authors, different books, different audiences, and different contexts.
  - bajsejohannes 5y ago
    It's not entirely unlikely that I am remembering just the good stuff :) But I was surprised how many books would define even the most common notation, like ⊂, ∀, and ∃. I guess if you call your book "Introduction to..." you ought to do that. And it seems that all books were called that, regardless of how narrow and advanced the rest of the title was :)
readme 5y ago
came here to say this some people literally make the notation up as they go along
klibertp 5y ago
Why are you telling OP what his problem is? Shouldn't you address his pain points, not your rationalization of them? I wrote it many times already and am bit tired of it, so just a quick summary: - programmers[1] also use cryptic notation and tend to think in concepts rather than syntax - nevertheless, programmers spend a lot of time commenting the code, documenting it, specifying it, and so on. - why can't mathematicians emulate it? What is so wrong about attaching additional few pages to every paper that nobody wants to do it? Pages with explanation of the syntax used, even the common bits. And you know what they could also do? Link to external resources with explanations! But no. This is not happening. Do their PDFs have a size limit or something? Is inserting a link into a paper considered some kind of blasphemy? I don't know the reason, but in all the discussions on this topic mathematicians almost always underestimate the importance of knowing the syntax. It's much more important for comprehension than they tend to admit. And in the end they do exactly nothing to make the syntax more approachable for newcomers. And then newcomers are out-goers in a heartbeat. It's so obvious that I can't help thinking it's premeditated... EDIT: [1] Among many others, of course.
- Jtsummers 5y ago
  Mathematicians do document and comment, that's what papers and textbooks are: commentary on the math. They don't throw out formulae and equations and call it a day. Attaching a full tutorial for every level of reader is tantamount to attaching Stroustrup's C++ books to every C++ program, or K&R to every C program. You wouldn't do that, you'd expect the reader to ask you for references or to seek them out themselves.
  - klibertp 5y ago
    > or K&R to every C program. That's actually doable... ;) K&R is rather terse, what, 1/5 of Stroustrup or something like that. But I digress. More on topic: there's also a class of programs that DO come with a book attached - or rather, multiple books, for every level; if not included outright in the distribution then at least linked to in the "learn" tab on a homepage. They're called programming languages. So, it can be done. That's all I want to say.
Grustaf 5y ago
There pretty much is one true notation. There could be some slight variations, like bolding vectors, putting an arrow over them or not distinguishing them at all from scalars. But 95% of the time everyone uses the same notation.
- ColinWright 5y ago
  I don't know your background, but I wonder how broad it is in terms of mathematical topics. The notations used in Algebraic Topology vs Category Theory vs Algebraic Number Theory vs Analytical Combinatorics vs Complex Analysis. This isn't a criticism, it's just that notations vary wildly in those areas, and there's lots of cross-over of notations, not all of which agree with each other. I'm not an expert, but I've had some exposure to the problem(s).
  - Grustaf 5y ago
    I studied diff geo at phd level and met stat at undergrad level, plus a sprinkling of category theory, some discrete mathematics and some physics, so I’ve been exposed to most of these. I presumed we were talking about basic mathematics here since new notation is the least of your worries when your thinking about fibre bundles and cohomologies, but I can’t really think of any significant overlap in notation that would be different between the fields I’ve come across. Could you give some examples?
    - ColinWright 5y ago
      I'm trying to be more general than specific questions at the mid-undergrad level, because looking in from the outside, people seem to thing that if only the notation weren't so mysterious then they could understand everything. But this comment -- https://news.ycombinator.com/item?id=29344238 https://news.ycombinator.com/item?id=29344238 -- gives a flavour, talking about coming across "π" in different contexts and having to give different interpretations. But I remember sketching an algorithm to someone and just inventing notations on the fly as I did so, knowing that they would simply be ways to remember the underlying ideas. Even so, at 1st year undergrad the notations used in Mathematical Physics vary from those used in Introductory Graph Theory, and again from Real Analysis. But once the reader knows the underlying semantics, the actual notation is mostly a non-issue (as you know).
kazinator 5y ago
> I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. No, that belief isn't the problem; that actual status quo itself is obviously the problem. There are numerous notations and authors don't explain what they are using, assuming everyone has recursively read all of their references depth-first before reading their paper.
- hansvm 5y ago
  Why does RTFM not apply to mathematics dependencies?
  - layer8 5y ago
    Because there usually isn’t anything resembling an actual manual.
    - Jensson 5y ago
      Sure there is, read these and you will understand most math papers people here struggle with: https://mathblog.com/mathematics-books/ https://mathblog.com/mathematics-books/ Of course you still wont be able to understand most math papers written by pure mathematicians, but it should be fine for whatever you need in CS. I know all the topics on that page, it is just a very fleshed out math major.
      - kazinator 5y ago
        That's "read all the fine manuals", not "read the fine manual". In the end you might eventually read all of them, but it's not the efficient thing now.
        Jensson 5y ago
        But why are you reading research papers with math without having studied math? If you want to understand them fully then you need to do the relevant courses, people spend years learning these things. You don't have to read them all, just the branch relevant to the paper.

rullopat 5y ago

I was trying to grasp some of the papers linked in the Valhalla DSP block, for example this one: http://www2.ece.rochester.edu/courses/ECE472/resources/Papers/Schroeder_1961.pdf http://www2.ece.rochester.edu/courses/ECE472/resources/Paper... There is a formula with a triangle and I don't get what's that about, for example.

nickcw 5y ago
The triangle is a capital Greek Delta and is usually used to indicate a change in something. So ΔT means some change in T.
ColinWright 5y ago
You say "There's a formula with a triangle ..." without telling me where. That's not real helpful, and you're making me do the work to find out what you're talking about. If you want assistance to get started, you need to be more explicit. However, I have done that work, so I've looked, and in the second column of page 210 there's a "formula with a triangle": t_c = 5 \middot 10^{-5} \sqrt( V / Dt ) ... where the "D" I've used is where the triangle appears in the formula. But that can't be it, because just two lines above it we have: "For a pulse of width Dt, the critical time ..." So that's stating that "Dt" is the width of the pulse, and should be thought of as a single term. So maybe that's the wrong formula, or maybe it was just a bad example. So trying to be more helpful, the "triangle" is a Greek capital delta and means different things in different places. However, it is often used to mean "a small change in". https://en.wikipedia.org/wiki/%CE%94T https://en.wikipedia.org/wiki/%CE%94T FWIW ... at a glance I can't see where that result is derived, it appears simply to be stated without explanation. I might be wrong, I've not read the rest of the paper.
- tomerv 5y ago
  I think that this is indeed the formula in GP's question. And indeed sometimes math notation is obtuse like that. It looks like 2 terms, but the triangle goes together with the t as a single term. At other times it might be called "dt" and despite looking like a multiplication of 2 variables (d and t, or triangle and t in this case) it's just a single variable with a named made of 2 characters. The important thing here is that "For a pulse of width Dt" is the definition of this variable, but this can be easily missed if you're not used to this naming convention.
  - enriquto 5y ago
    > it's just a single variable with a named made of 2 characters. I have this same problem with programming, when I have to deal with code written by non-mathematicians. They tend to use all these stupid variables with more than one letter and that confuses the heck out of me.
  - kergonath 5y ago
    That’s because “Δ” means “a change of” or “an interval of”. So, Δt is “an interval of time”. It is like a compound word, really. It conveys more information than giving it an arbitrary, single-letter name. This convention is used in a whole bunch of scientific fields, like quantum mechanics, chemistry, biology, mechanics, thermodynamics, etc. It’s also very useful in how it relates to derivatives, which is a crucial concept in just about any kind of science you could care to mention. So yes, there is a learning curve, but we write things this way for good reasons, most of the time. Multiplication should be represented by a (thin) space in good typography, to avoid this sort of things. Not doing it is sloppy and invites misreading. Same with omitting parenthesis around a function’s argument most of the time (e.g. sin 2πθ instead of sin(2 π θ) ).
- rullopat 5y ago
  Sorry I didn't mean to make you work for me, but it's a PDF and I didn't know how to explain better the position (maybe I should have told you the first formula on page X). For you it was a D, for me it was a triangle and I didn't get the meaning of that Dt. Maybe it's just a too advanced paper for my knowledge.
Jensson 5y ago
That is a physics based paper, it used physics notation. Can find common ones here (including that triangle): https://en.wikipedia.org/wiki/List_of_common_physics_notations https://en.wikipedia.org/wiki/List_of_common_physics_notatio...
NamTaf 5y ago
What you're looking at is calculus, specifically differentiation. This is pretty core to understanding physics, because so much of physics depends on the time-evolving state of things. That's fundamentally what's happening here. The triangle, for example, is the upper-case greek letter delta, which in calculus represents 'change of'. You might have heard of 'delta-T' with respect to 'change of time'. In calculus, upper-case delta means 'change over a finite time' vs lower-case delta meaning 'instantaneous change'. The practical upshot, for example, is that the lower-case is the instantaneous rate-of-change at an instant in time, whereas the upper-case is the change over a whole time (e.g. the average rate of change per second for time = 0 seconds to time = 3 seconds). If you are trying to grok this, I would suggest an introductory calculus or pre-calculus resource. It doesn't have to be a uni textbook - higher-level high school maths usually teaches this. In this particular case, the Khan Academy would be my recommendation because it is about the right level (we're not talking esoteric higher-level university knowledge here) and it is eminently accessable. For example, this link may be a good starter in this instance: https://www.youtube.com/watch?v=MeU-KzdCBps https://www.youtube.com/watch?v=MeU-KzdCBps
forgetfulness 5y ago
Looks like you need to grind through an elementary calculus book. With the exercises, you may think you build intuition by reading just the definitions, but half of the understanding is tacit and you get through the exercises. If you're trying to get into signal processing, it'll involve calculus in complex numbers, and knowledge of that is often gained through plodding through proofs and exercises over and over.
pfortuny 5y ago
You probably want to try and read Courant-Johns Calculus (I forget the exact title).
kens 5y ago
Everyone is talking about the Δ symbol, but the real problem that you'll encounter will be later in the paper where they start talking about H(ω), which is the Fourier transform of the impulse function (equation 4 and following). You'll need to know a fair bit about Fourier transforms and impulse responses and filter design to get through this section. The notation is the least of the problems. One place to start is https://en.wikipedia.org/wiki/Impulse_response https://en.wikipedia.org/wiki/Impulse_response
- kergonath 5y ago
  Just like with programming, knowing the meaning of keywords is not enough to understand something.
- forgetfulness 5y ago
  Wikipedia is truly atrocious for learning math, the articles are like man pages in that they precisely describe the concepts in terms that will only make sense if you already know the thing. They just aren't written for pedagogy. Like in 300 BC today, there's no royal road to geometry.
BeetleB 5y ago
A few points: 1. You're reading a journal article. They will assume you know the notation not just of the broader discipline (e.g. physics/electrical engineering), but of the subdiscipline and at times the subsubdiscipline. Journal papers are explicitly written not to be easy to comprehend by beginners.[1] Notation will be only one problem you'll face. 2. As has been pointed out, this is not a mathematics paper. Mathematicians have their own notation, as do physicists and engineers. As I mentioned in the above bullet, they can have their own notation even in subdisciplines (e.g. circuit folks use "j" for the imaginary number, and semiconductor folks use "i"). There is a lot of overlap in notation amongst these parties, but you should never assume because you know one notation that you'll easily understand the math written by other fields. 3. Most introductory textbooks will explain the basic notation. Unfortunately, I often do find gaps where you go to higher level textbooks and they use notation that they don't explain (i.e. they assume you've seen it before), but is not covered in the prior textbooks. 4. Finally, sorry to say this, but "delta" (the triangle) for representing change is used in almost all sciences and engineering. It was heavily used in my high school as well. If you're struggling with this you really need to read some introductory textbooks in, say, physics. [1] I'm not kidding. I've spent time in academia and I've complained how obtuse some articles are, and almost universally the response is "We write for other experts, not for new graduate students". One professor took pride at the fact that in his field, one can comprehend only about one page of a paper per day - and this coming from someone who is an expert. These people have issues.

User23 5y ago

Edsger Dijkstra, who was a mathematician by training, wrote a wonderful little monograph on this subject called The notational conventions I adopted, and why[1]. I am particularly fond of his commented equational proof format. [1] https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/EWD1300.html https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...

cabalamat 5y ago

> I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. One main cause for this belief is that in a programming there is one true noation (or rather, a separate one for each language) that is unambiguous and clearly defined. I dislike maths notation as I find it lacks rigour.

Jensson 5y ago
> there is one true noation (or rather, a separate one for each language) that is unambiguous and clearly defined. This is such a disingenuous take. How many of the source code files you write are 100% self contained and well defined? I'd bet not a single one of them are. You reference libraries, you depend on specific compiler/runtime/OS versions, you reference other files etc. If you take a look at any of these scientific papers you call "badly defined", did you really go through all of the referenced papers and look if they defined the things you didn't get? If not then you can't be sure that the paper uses undefined notation. If you argue that it is too much work to go through that many references, well that is what you would have to do to understand one of your program files.
- throwaway31338 5y ago
  One can look at the source code to a program, the libraries it uses, the compiler for the language, and the ISA spec for the machine language the compiler generates. You can know that there are no hidden unspecified quantities because programs can't work without being specified. When you get down to the microcode of the CPU that implements the ISA you might have an issue if it's ill-specified. You might be talking about an ISA like RISC-V, though, specified at a level sufficient to go down to the gates. You might be talking about an ISA like 6502 where the gate-level implementations have been reverse-engineered. You can take programming all the way down boolean logic if you need to and the tools are readily available. They don't rely on you "just knowing" something.
  - Jensson 5y ago
    > One can look at the source code to a program, the libraries it uses, the compiler for the language, and the ISA spec for the machine language the compiler generates. You can know that there are no hidden unspecified quantities because programs can't work without being specified. I doubt you actually can do that and understand it all. A computer can do it, but I doubt you the human can do that and get a perfect picture of any non trivial program without making errors. Human math is a human language first and foremost, its grammar is human language which is used to define things and symbols. This lets us write things that humans can actually read and understand the entirety of, unlike a million lines of code or cpu instructions. Show me a program written by 10 programmers over 10 years and I doubt anyone really understands all of it. But we have mathematical fields that hundreds of mathematicians have written over centuries, and people still are able to understand it all perfectly. It is true that a computer can easily read a computer program, but since we are arguing about teaching humans you would need to show evidence that humans can actually read and understand complex code well.

bajsejohannes 5y ago

> I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. Just to back up this point: In probably every university-level math book I've read, they introduce and explain all the notation used. In the preface and/or as concepts are introduced. There are lists at wikipedia [1] and other places, but I'm not sure how valuable it is out of context. [1] https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbols https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbo...

nicoburns 5y ago
Ha, you've clearly used better books than me. I've read plenty where they glossed over the notation and expected you to guess.
- ColinWright 5y ago
  Often books assume some prerequisites, the question here is the level of those prerequisites. Some books try to include all the necessary background, others assume a pre-existing base level of knowledge. Different authors, different books, different audiences, and different contexts.
- bajsejohannes 5y ago
  It's not entirely unlikely that I am remembering just the good stuff :) But I was surprised how many books would define even the most common notation, like ⊂, ∀, and ∃. I guess if you call your book "Introduction to..." you ought to do that. And it seems that all books were called that, regardless of how narrow and advanced the rest of the title was :)

readme 5y ago

came here to say this some people literally make the notation up as they go along

klibertp 5y ago

Why are you telling OP what his problem is? Shouldn't you address his pain points, not your rationalization of them? I wrote it many times already and am bit tired of it, so just a quick summary: - programmers[1] also use cryptic notation and tend to think in concepts rather than syntax - nevertheless, programmers spend a lot of time commenting the code, documenting it, specifying it, and so on. - why can't mathematicians emulate it? What is so wrong about attaching additional few pages to every paper that nobody wants to do it? Pages with explanation of the syntax used, even the common bits. And you know what they could also do? Link to external resources with explanations! But no. This is not happening. Do their PDFs have a size limit or something? Is inserting a link into a paper considered some kind of blasphemy? I don't know the reason, but in all the discussions on this topic mathematicians almost always underestimate the importance of knowing the syntax. It's much more important for comprehension than they tend to admit. And in the end they do exactly nothing to make the syntax more approachable for newcomers. And then newcomers are out-goers in a heartbeat. It's so obvious that I can't help thinking it's premeditated... EDIT: [1] Among many others, of course.

Jtsummers 5y ago
Mathematicians do document and comment, that's what papers and textbooks are: commentary on the math. They don't throw out formulae and equations and call it a day. Attaching a full tutorial for every level of reader is tantamount to attaching Stroustrup's C++ books to every C++ program, or K&R to every C program. You wouldn't do that, you'd expect the reader to ask you for references or to seek them out themselves.
- klibertp 5y ago
  > or K&R to every C program. That's actually doable... ;) K&R is rather terse, what, 1/5 of Stroustrup or something like that. But I digress. More on topic: there's also a class of programs that DO come with a book attached - or rather, multiple books, for every level; if not included outright in the distribution then at least linked to in the "learn" tab on a homepage. They're called programming languages. So, it can be done. That's all I want to say.

Grustaf 5y ago

There pretty much is one true notation. There could be some slight variations, like bolding vectors, putting an arrow over them or not distinguishing them at all from scalars. But 95% of the time everyone uses the same notation.

ColinWright 5y ago
I don't know your background, but I wonder how broad it is in terms of mathematical topics. The notations used in Algebraic Topology vs Category Theory vs Algebraic Number Theory vs Analytical Combinatorics vs Complex Analysis. This isn't a criticism, it's just that notations vary wildly in those areas, and there's lots of cross-over of notations, not all of which agree with each other. I'm not an expert, but I've had some exposure to the problem(s).
- Grustaf 5y ago
  I studied diff geo at phd level and met stat at undergrad level, plus a sprinkling of category theory, some discrete mathematics and some physics, so I’ve been exposed to most of these. I presumed we were talking about basic mathematics here since new notation is the least of your worries when your thinking about fibre bundles and cohomologies, but I can’t really think of any significant overlap in notation that would be different between the fields I’ve come across. Could you give some examples?
  - ColinWright 5y ago
    I'm trying to be more general than specific questions at the mid-undergrad level, because looking in from the outside, people seem to thing that if only the notation weren't so mysterious then they could understand everything. But this comment -- https://news.ycombinator.com/item?id=29344238 https://news.ycombinator.com/item?id=29344238 -- gives a flavour, talking about coming across "π" in different contexts and having to give different interpretations. But I remember sketching an algorithm to someone and just inventing notations on the fly as I did so, knowing that they would simply be ways to remember the underlying ideas. Even so, at 1st year undergrad the notations used in Mathematical Physics vary from those used in Introductory Graph Theory, and again from Real Analysis. But once the reader knows the underlying semantics, the actual notation is mostly a non-issue (as you know).

kazinator 5y ago

> I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. No, that belief isn't the problem; that actual status quo itself is obviously the problem. There are numerous notations and authors don't explain what they are using, assuming everyone has recursively read all of their references depth-first before reading their paper.

hansvm 5y ago
Why does RTFM not apply to mathematics dependencies?
- layer8 5y ago
  Because there usually isn’t anything resembling an actual manual.
  - Jensson 5y ago
    Sure there is, read these and you will understand most math papers people here struggle with: https://mathblog.com/mathematics-books/ https://mathblog.com/mathematics-books/ Of course you still wont be able to understand most math papers written by pure mathematicians, but it should be fine for whatever you need in CS. I know all the topics on that page, it is just a very fleshed out math major.
    - kazinator 5y ago
      That's "read all the fine manuals", not "read the fine manual". In the end you might eventually read all of them, but it's not the efficient thing now.
      - Jensson 5y ago
        But why are you reading research papers with math without having studied math? If you want to understand them fully then you need to do the relevant courses, people spend years learning these things. You don't have to read them all, just the branch relevant to the paper.