5 ms·

I think it's worth linking to the 'Notation as a tool of thought' thread from the other day, as it gave me quite a lot to read, watch and think about [0]. It a

by Zhyl 6y ago

I think it's worth linking to the 'Notation as a tool of thought' thread from the other day, as it gave me quite a lot to read, watch and think about [0]. It also reminds me of Graphical Linear Algebra [1] which I occasionally see mentioned here. And, as included in my comment in that thread, the notion of using Tau as the circle constant in equations [2]. Notation is a weird topic to tackle. Like with new technologies or languages on HN, there seem to be those who get [a new notation when it is proposed] and evangelise it, and those who see it as pointless and vocally dismiss it. Posts like the article where you're weighing up and exploring benefits and limitations of notation seem rare - and even those that do exist seem to be pitching for their new notation to be a global replacement rather than as a pedagogical or epistemological tool. [0] https://news.ycombinator.com/item?id=25249563 https://news.ycombinator.com/item?id=25249563 [1] https://graphicallinearalgebra.net/about/ https://graphicallinearalgebra.net/about/ [2] https://tauday.com https://tauday.com

sriku 6y ago
While a notation for this case makes for an interesting discussion (which the post is), the "blast radius" of this situation is so small that (to me) it wouldn't be worth adopting any of the proposals (not saying anyone there pushed them). For a notation to be worth adoption,its impact must be far reaching. The "graphical linear algebra" makes a good case with a larger blast radius, for example. However even that has nowhere near the impact of, say, Feynman diagrams. The x^y notation is repurposed elsewhere - ex: power set of a set S is written 2^S .. with no implication that the "logarithm" of the power set to the base 2 is S. Same for matrices raised to a power, where matrices are usually not thought of as a base for doing logarithms. Same for operators in calculus (ex: laplacian .. now that would be confusing to club with a triangle!)
- alanbernstein 6y ago
  I'm not sure what you mean by blast radius, but if you were to think about the amount of people learning these concepts - as explained in the SE post here - the order would obviously be the reverse of what you suggest (exponents most important).
  - sriku 6y ago
    By "blast radius" I meant the impact of the notation beyond the originating case into other areas. I wasn't referring to the number of people it would impact. The notation in the OP doesn't play well with "nearby" areas like powerset and matrices .. which are well served by the conventional exponentiation notation.
- umvi 6y ago
  Speaking of Feynman, I remember when he was in high school he invented his own notation[0] for sin/cos/tan to use little angular lines, but then the issue he ran into was that nobody could decipher his math notation and he had trouble helping others learn, so he conformed to the sin/cos/tan notation. [0] https://tex.stackexchange.com/questions/274463/feynman-trig-notation-creating-custom-characters https://tex.stackexchange.com/questions/274463/feynman-trig-...
- OscarCunningham 6y ago
  > The "graphical linear algebra" makes a good case with a larger blast radius, for example. However even that has nowhere near the impact of, say, Feynman diagrams. These are sort of the same example! They're both string diagrams (https://en.wikipedia.org/wiki/String_diagram https://en.wikipedia.org/wiki/String_diagram), just in different monoidal categories.
chrisweekly 6y ago
Kudos for the insightful comment and good links. Tangent: I first stumbled upon the Tau manifesto several years ago, and though it made sense to me, I couldn't tell whether anyone took it seriously. Still wondering.
- Zhyl 6y ago
  I was a Math undergrad when I first read it and I think the main reaction I got from faculty was 'dismissal' and the main reaction I got from peers was 'indifference' and, in one case 'annoyance'. I think it took root in the pop-math education space [0,1], but overall I think most of the Mathematical world has dismissed it as trivial at best and unnecessarily disruptive at worst. It's a shame, because I feel that understanding why Tau makes equations and concepts easier for people to grasp and build intuitions around is key to understanding why people generally find maths difficult and frustrating, and can give some guidance on how to bring down that barrier/activation energy. [0] Vi Hart, Pi is Wrong (https://www.youtube.com/watch?v=jG7vhMMXagQ https://www.youtube.com/watch?v=jG7vhMMXagQ) [1] Numberphile, Pi vs Tau Smackdown (https://www.youtube.com/watch?v=ZPv1UV0rD8U https://www.youtube.com/watch?v=ZPv1UV0rD8U)
  - threatofrain 6y ago
    People are looking at a bigger delta of improvement than that, IMO, but that also means they only want big commits from big names. The Tau thing really is too trivial, which means it may be outweighed by friction alone.
    - Zhyl 6y ago
      Yeah, the circle constant is considered 'solved' and has been a well understood concept since... (googles) .. oh wow, 2560 BCE [0], with pi being used as the symbol for the last 400 odd years [1]. It's understandable that there wouldn't be much appetite to tamper with that if there isn't a gosh darned good reason. [0] https://en.wikipedia.org/wiki/Pi#History https://en.wikipedia.org/wiki/Pi#History [1] https://en.wikipedia.org/wiki/Pi#Adoption_of_the_symbol_%CF%80 https://en.wikipedia.org/wiki/Pi#Adoption_of_the_symbol_%CF%... (Interesting to note the original coining of Pi was π/δ for what we now call Pi (circumference over diameter) and π/ρ for what we are calling Tau (circumference over radius).)
WorldMaker 6y ago
It also reminds me of the Calculus itself. It's often considered one of the best outcomes of Newton and Liebniz discovering the Calculus nearly simultaneously and with little to no coordination until after the fact is that as people discussed the results and compared the two efforts what we ultimately ended up with was a lot of Newton's insights and pedagogy, but based more upon Liebniz's notation. That has long stuck with me that Liebniz developed a better, more generally useful notation as a tool of thought for following mathematicians, but Newton still developed a lot more lasting insights into the Calculus despite a generally more "inferior" notation (though certainly a notation presumably specifically more useful to Newton's own thought processes).

by Zhyl 6y ago

sriku 6y ago

While a notation for this case makes for an interesting discussion (which the post is), the "blast radius" of this situation is so small that (to me) it wouldn't be worth adopting any of the proposals (not saying anyone there pushed them). For a notation to be worth adoption,its impact must be far reaching. The "graphical linear algebra" makes a good case with a larger blast radius, for example. However even that has nowhere near the impact of, say, Feynman diagrams. The x^y notation is repurposed elsewhere - ex: power set of a set S is written 2^S .. with no implication that the "logarithm" of the power set to the base 2 is S. Same for matrices raised to a power, where matrices are usually not thought of as a base for doing logarithms. Same for operators in calculus (ex: laplacian .. now that would be confusing to club with a triangle!)

alanbernstein 6y ago
I'm not sure what you mean by blast radius, but if you were to think about the amount of people learning these concepts - as explained in the SE post here - the order would obviously be the reverse of what you suggest (exponents most important).
- sriku 6y ago
  By "blast radius" I meant the impact of the notation beyond the originating case into other areas. I wasn't referring to the number of people it would impact. The notation in the OP doesn't play well with "nearby" areas like powerset and matrices .. which are well served by the conventional exponentiation notation.
umvi 6y ago
Speaking of Feynman, I remember when he was in high school he invented his own notation[0] for sin/cos/tan to use little angular lines, but then the issue he ran into was that nobody could decipher his math notation and he had trouble helping others learn, so he conformed to the sin/cos/tan notation. [0] https://tex.stackexchange.com/questions/274463/feynman-trig-notation-creating-custom-characters https://tex.stackexchange.com/questions/274463/feynman-trig-...
OscarCunningham 6y ago
> The "graphical linear algebra" makes a good case with a larger blast radius, for example. However even that has nowhere near the impact of, say, Feynman diagrams. These are sort of the same example! They're both string diagrams (https://en.wikipedia.org/wiki/String_diagram https://en.wikipedia.org/wiki/String_diagram), just in different monoidal categories.

chrisweekly 6y ago

Kudos for the insightful comment and good links. Tangent: I first stumbled upon the Tau manifesto several years ago, and though it made sense to me, I couldn't tell whether anyone took it seriously. Still wondering.

Zhyl 6y ago
I was a Math undergrad when I first read it and I think the main reaction I got from faculty was 'dismissal' and the main reaction I got from peers was 'indifference' and, in one case 'annoyance'. I think it took root in the pop-math education space [0,1], but overall I think most of the Mathematical world has dismissed it as trivial at best and unnecessarily disruptive at worst. It's a shame, because I feel that understanding why Tau makes equations and concepts easier for people to grasp and build intuitions around is key to understanding why people generally find maths difficult and frustrating, and can give some guidance on how to bring down that barrier/activation energy. [0] Vi Hart, Pi is Wrong (https://www.youtube.com/watch?v=jG7vhMMXagQ https://www.youtube.com/watch?v=jG7vhMMXagQ) [1] Numberphile, Pi vs Tau Smackdown (https://www.youtube.com/watch?v=ZPv1UV0rD8U https://www.youtube.com/watch?v=ZPv1UV0rD8U)
- threatofrain 6y ago
  People are looking at a bigger delta of improvement than that, IMO, but that also means they only want big commits from big names. The Tau thing really is too trivial, which means it may be outweighed by friction alone.
  - Zhyl 6y ago
    Yeah, the circle constant is considered 'solved' and has been a well understood concept since... (googles) .. oh wow, 2560 BCE [0], with pi being used as the symbol for the last 400 odd years [1]. It's understandable that there wouldn't be much appetite to tamper with that if there isn't a gosh darned good reason. [0] https://en.wikipedia.org/wiki/Pi#History https://en.wikipedia.org/wiki/Pi#History [1] https://en.wikipedia.org/wiki/Pi#Adoption_of_the_symbol_%CF%80 https://en.wikipedia.org/wiki/Pi#Adoption_of_the_symbol_%CF%... (Interesting to note the original coining of Pi was π/δ for what we now call Pi (circumference over diameter) and π/ρ for what we are calling Tau (circumference over radius).)

WorldMaker 6y ago

It also reminds me of the Calculus itself. It's often considered one of the best outcomes of Newton and Liebniz discovering the Calculus nearly simultaneously and with little to no coordination until after the fact is that as people discussed the results and compared the two efforts what we ultimately ended up with was a lot of Newton's insights and pedagogy, but based more upon Liebniz's notation. That has long stuck with me that Liebniz developed a better, more generally useful notation as a tool of thought for following mathematicians, but Newton still developed a lot more lasting insights into the Calculus despite a generally more "inferior" notation (though certainly a notation presumably specifically more useful to Newton's own thought processes).