29 ms·

Difficulties, if any, perceived or real, arising in connection with notation, are usually incomparably smaller than those presented with the subject itself. (Pe

Difficulties, if any, perceived or real, arising in connection with notation, are usually incomparably smaller than those presented with the subject itself. (Personally, I only wish mathematical notation were better integrated with software in general and programming languages in particular.)

h-jones 6y ago
I think differential geometry may present the closest exception to this, not only is the notation often incredibly dense and subtle (i.e. spacing between indices when raising and lowering) but often everyone seems to have their personal favorite take on any given notation.
kdmccormick 6y ago
That's a strong assertion. You're implying that difficulties with the subject itself rarely have anything to do with difficulty communicating through some notation. I'm not disagreeing, but I'm curious how you would back up that assertion.
- Koshkin 6y ago
  It’s an experimental fact. (I guess you have to trust me on this one.)
dilap 6y ago
> with the subject itself that's the thing though, you can never grapple with the subject itself, only representations thereof. this is a mix of feelings/images/movements inside your head & mechanical manipulations of the notation; furthermore, the notation itself influences our internal model/feeling of the subject.
- Koshkin 6y ago
  I am not saying that with enough dedication one couldn’t screw up the notation completely.
rytill 6y ago
Not true at all, there are several times I've attempted to read through a textbook only to be stopped by notation because something was introduced prior to being referenced, or notation is overloaded with multiple meanings. I consistently have run into "perceived or real" confusing mathematical notation as an impediment to learning in a way that programming languages have never, ever caused me. Does no one else feel this way? I can't be alone, and like the others responding to you have said, your claim does not seem substantiated.
- Transfinity 6y ago
  I think I agree with you. I would say this trouble comes about in programming though, in the form of DSLs. Anyone who's worked with Rails knows the pain of having to go look up the non-obvious syntax for xyz specific file that you haven't touched in months. Maybe you could say this is a problem with large software systems in general - the problems they solve often aren't that complicated (especially if you've worked on other similar software), but learning all the names and patterns and existing features creates a high barrier to entry.
- jordigh 6y ago
  A lot of people say this, but if I were to sit next to you and tell you what the notation means whenever you asked, would you instantly understand the concept? I don't think so. You still need to do work on your own with the concept before you can get to any understanding. Say, you come across this symbol: ζ(s) And you asked me, what does it mean? Oh, that's probably the Riemann zeta function (it could be other things, but that's probably its most famous meaning). Would you be like, ah, now I understand it all! I sincerely doubt just knowing the English words for the symbols (and all mathematical notation is just shorthand for English, or whatever other natural language; it can all be read out loud) would get you that much further along. I could give you one of several possible equivalent definitions, but you still wouldn't really know much of anything. I could start explaining some things about primes and how they relate to the Riemann zeta function, and you'd still be in the dark. I'd have to talk about the prime number theorem and estimates and complex analysis and a whole bunch of other things before the Riemann hypothesis even started making sense. In fact, people have written entire books just trying to explain in the most accessible way possible what that one symbol is: https://www.wstein.org/rh/ https://www.wstein.org/rh/ I believe what's going on is that a lot of people who find notation to be the first stumbling block are trying to learn mathematics in isolation, without the benefit of a mathematical community to be able to talk to. You pick up a book, see a bunch of weird symbols, blame the notation, you put it down. But people who train for mathematics specifically rarely learn that way. Mathematics is a social activity, you talk to other people, you read the symbols out loud to each other on chalkboards or side-by-side as you scribble notes on paper or on a napkin. You talk to each other a lot, pronouncing the symbols as much as you write them down. The notation is a very superficial obstacle. Notation is very incidental, malleable, almost irrelevant (or as mathematicians might say, non-canonical or not natural). If notation is your biggest obstacle, there's a lot more underneath that you probably don't have enough experience with.
gitgud 6y ago
Mathematical notation finds elegance in brevity. That's why there's such an enormous alphabet of symbols representing important concepts. If programming languages strive for the same elegance via notation, you end up with something like Perl.

h-jones 6y ago

I think differential geometry may present the closest exception to this, not only is the notation often incredibly dense and subtle (i.e. spacing between indices when raising and lowering) but often everyone seems to have their personal favorite take on any given notation.

kdmccormick 6y ago

That's a strong assertion. You're implying that difficulties with the subject itself rarely have anything to do with difficulty communicating through some notation. I'm not disagreeing, but I'm curious how you would back up that assertion.

Koshkin 6y ago
It’s an experimental fact. (I guess you have to trust me on this one.)

dilap 6y ago

> with the subject itself that's the thing though, you can never grapple with the subject itself, only representations thereof. this is a mix of feelings/images/movements inside your head & mechanical manipulations of the notation; furthermore, the notation itself influences our internal model/feeling of the subject.

Koshkin 6y ago
I am not saying that with enough dedication one couldn’t screw up the notation completely.

rytill 6y ago

Not true at all, there are several times I've attempted to read through a textbook only to be stopped by notation because something was introduced prior to being referenced, or notation is overloaded with multiple meanings. I consistently have run into "perceived or real" confusing mathematical notation as an impediment to learning in a way that programming languages have never, ever caused me. Does no one else feel this way? I can't be alone, and like the others responding to you have said, your claim does not seem substantiated.

Transfinity 6y ago
I think I agree with you. I would say this trouble comes about in programming though, in the form of DSLs. Anyone who's worked with Rails knows the pain of having to go look up the non-obvious syntax for xyz specific file that you haven't touched in months. Maybe you could say this is a problem with large software systems in general - the problems they solve often aren't that complicated (especially if you've worked on other similar software), but learning all the names and patterns and existing features creates a high barrier to entry.
jordigh 6y ago
A lot of people say this, but if I were to sit next to you and tell you what the notation means whenever you asked, would you instantly understand the concept? I don't think so. You still need to do work on your own with the concept before you can get to any understanding. Say, you come across this symbol: ζ(s) And you asked me, what does it mean? Oh, that's probably the Riemann zeta function (it could be other things, but that's probably its most famous meaning). Would you be like, ah, now I understand it all! I sincerely doubt just knowing the English words for the symbols (and all mathematical notation is just shorthand for English, or whatever other natural language; it can all be read out loud) would get you that much further along. I could give you one of several possible equivalent definitions, but you still wouldn't really know much of anything. I could start explaining some things about primes and how they relate to the Riemann zeta function, and you'd still be in the dark. I'd have to talk about the prime number theorem and estimates and complex analysis and a whole bunch of other things before the Riemann hypothesis even started making sense. In fact, people have written entire books just trying to explain in the most accessible way possible what that one symbol is: https://www.wstein.org/rh/ https://www.wstein.org/rh/ I believe what's going on is that a lot of people who find notation to be the first stumbling block are trying to learn mathematics in isolation, without the benefit of a mathematical community to be able to talk to. You pick up a book, see a bunch of weird symbols, blame the notation, you put it down. But people who train for mathematics specifically rarely learn that way. Mathematics is a social activity, you talk to other people, you read the symbols out loud to each other on chalkboards or side-by-side as you scribble notes on paper or on a napkin. You talk to each other a lot, pronouncing the symbols as much as you write them down. The notation is a very superficial obstacle. Notation is very incidental, malleable, almost irrelevant (or as mathematicians might say, non-canonical or not natural). If notation is your biggest obstacle, there's a lot more underneath that you probably don't have enough experience with.

gitgud 6y ago

Mathematical notation finds elegance in brevity. That's why there's such an enormous alphabet of symbols representing important concepts. If programming languages strive for the same elegance via notation, you end up with something like Perl.