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> 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.

by User23 5y ago

> 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. 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/EWD1300.html https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...

abdullahkhalids 5y ago
I don't see anything in that link that is markedly different from math notation used by one math community or the other. Sure, there are difference that are "improvements" in some sense of the word, and I think some of those can be adopted. But nothing major, that will somehow massively reduce the complexity of understanding mathematics by non-experts.
- User23 5y ago
  > But nothing major, that will somehow massively reduce the complexity of understanding mathematics by non-experts. 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.
  - abdullahkhalids 5y ago
    The two-column proof has been a thing in math education for over a century [1]. I have been told by math educators that it is almost universally detested by school students, because of how it restricts their thinking. Nevertheless, it would be useful at the research level to include more of this hinting. But hinting is not a notational issue at all. [1] https://deepblue.lib.umich.edu/bitstream/handle/2027.42/42653/10649_2004_Article_5096042.pdf;jsessionid=010CB25D780CF8F8D1734F50743ED5AB?sequence=1 https://deepblue.lib.umich.edu/bitstream/handle/2027.42/4265...
    - User23 5y ago
      The equational and two column proof formats aren’t the same. Perhaps you should read Dijkstra’s monograph’s section on it to understand the differences. And to say they are somehow isomorphic is to miss the point entirely. Sure “x is equal to y” means the same as “x = y”, but even the traditionalists recognize the merits of Recorde’s notational device, even if they fail to follow the greater implications thereof.
gjulianm 5y ago
> I’m not trying to be controversial, but traditional mathematicians are decades behind the best computing scientists when it comes to crafting formalisms. 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.
- User23 5y ago
  > Could you provide examples? The Dijkstra article doesn't seem like a marked improvement. 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) Even the average graduate mathematician is absurdly above the general population average for raw cognitive ability. Because of this, the frankly primitive approach to syntax is less of an impediment to doing interesting work than it would otherwise be. 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.
  - gjulianm 5y ago
    > As I said, there are hundreds of examples on that site. 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.
throw10920 5y ago
Heck, I think that Dijkstra invented his own notation for numbers for a presentation/teaching system that he was developing. (I can't find any references to it - I'm desperately hoping that someone else who might have any idea what vague thing I'm alluding to has actually seen it and can link it here)

by User23 5y ago

abdullahkhalids 5y ago

I don't see anything in that link that is markedly different from math notation used by one math community or the other. Sure, there are difference that are "improvements" in some sense of the word, and I think some of those can be adopted. But nothing major, that will somehow massively reduce the complexity of understanding mathematics by non-experts.

User23 5y ago
> But nothing major, that will somehow massively reduce the complexity of understanding mathematics by non-experts. 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.
- abdullahkhalids 5y ago
  The two-column proof has been a thing in math education for over a century [1]. I have been told by math educators that it is almost universally detested by school students, because of how it restricts their thinking. Nevertheless, it would be useful at the research level to include more of this hinting. But hinting is not a notational issue at all. [1] https://deepblue.lib.umich.edu/bitstream/handle/2027.42/42653/10649_2004_Article_5096042.pdf;jsessionid=010CB25D780CF8F8D1734F50743ED5AB?sequence=1 https://deepblue.lib.umich.edu/bitstream/handle/2027.42/4265...
  - User23 5y ago
    The equational and two column proof formats aren’t the same. Perhaps you should read Dijkstra’s monograph’s section on it to understand the differences. And to say they are somehow isomorphic is to miss the point entirely. Sure “x is equal to y” means the same as “x = y”, but even the traditionalists recognize the merits of Recorde’s notational device, even if they fail to follow the greater implications thereof.

gjulianm 5y ago

> I’m not trying to be controversial, but traditional mathematicians are decades behind the best computing scientists when it comes to crafting formalisms. 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.

User23 5y ago
> Could you provide examples? The Dijkstra article doesn't seem like a marked improvement. 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) Even the average graduate mathematician is absurdly above the general population average for raw cognitive ability. Because of this, the frankly primitive approach to syntax is less of an impediment to doing interesting work than it would otherwise be. 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.
- gjulianm 5y ago
  > As I said, there are hundreds of examples on that site. 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.

throw10920 5y ago

Heck, I think that Dijkstra invented his own notation for numbers for a presentation/teaching system that he was developing. (I can't find any references to it - I'm desperately hoping that someone else who might have any idea what vague thing I'm alluding to has actually seen it and can link it here)