4 ms·

I don't know if I'd predict it, but I agree that it's hard to imagine a more transformative technology than a dramatically improved notation for something as br

by gfodor 12y ago

I don't know if I'd predict it, but I agree that it's hard to imagine a more transformative technology than a dramatically improved notation for something as broad as mathematics. It might mean that "notation" is actually a higher fidelity, more dynamic medium for communicating mathematical ideas only possible within computing. (See Bret Vector, etc.) One has to wonder what the world would be like if physics knowledge (at all levels) were disseminated via a medium beyond static text.

agumonkey 12y ago
Sussman tried to use scheme as a mathematical notation (http://mitpress.mit.edu/sites/default/files/titles/content/sicm/book.html http://mitpress.mit.edu/sites/default/files/titles/content/s...) to avoid ambiguities. I don't know how many people saw this, and what the critics were. The traditional static symbolic notation of math may be the most efficient one, It's constrained, but forces abstraction to emerge. If people had Excel before, we'd never know Gauss sequence sum formula, we'd let the system help us.
- Chinjut 12y ago
  > The traditional static symbolic notation of math may be the most efficient one... If people had Excel before, we'd never know Gauss sequence sum formula, we'd let the system help us. I think some people in this thread must be using "notation" in a different sense than I am accustomed to. I don't think of the computer's ability to automate addition as an innovation in "notation", per se.
  - agumonkey 12y ago
    My analogy was borderline, I just think there's value in terse static glyph, and going away from it would be detrimental.
  - stan_rogers 12y ago
    That wasn't Sussman's point. The problem is that the same notational forms -- the same orthographies, if you will -- can mean different things in different contexts. It doesn't help at all that one must, on occasion, slip from context to context in the same argument, producing adjacent or near-adjacent statements that have similar general forms but wildly different meanings. His solution to the problem was to use a notation that also happens to be valid statements in a computer language he had a large hand in designing. That part isn't necessary, but disambiguating mathematical notation by some means is.
- pakled_engineer 12y ago
  I skimmed that book to figure out how to convert formulas to procedures in Scheme for learning Spivaks Calculus and a few OCW math courses. Also picked up Doing Math w/Python from NoStarch press to see how other people do this, and to learn Python since I've never really used it. It was definitely awesome to debug logic errors in emacs of formulas I translated, it really does give you a structural interpretation of the material you are trying to learn by coding it all. End result is a library with clear procedures I can use for reference whereas I've already forgotten most of the notation I've learned over the years (SICM uses Spivak's Calculus on Manifolds similar notation)
- smorrow 12y ago
  All languages have, at least, nouny things and verby things, and a finite(expandable) number of them composable in infinite ways, and I'd say our brains are hardwired for it. So any notation good for humans would have to have this property, which would mean being "equivalent" (in a way) to the language that already exists. I don't see how any improvement can be anything more than cosmetic.
forkandwait 12y ago
Perhaps Geometric Algebra is a big piece of that in the works? http://en.wikipedia.org/wiki/Geometric_algebra http://en.wikipedia.org/wiki/Geometric_algebra Re Scheme as notation (or APL/ J many years ago) -- when I hear a programming language proposed as a replacement for math, I always wonder "how does that make proving theorems easier?" I haven't read Sussman's book, and he is way smart, but I have listened to lots of APL weanies say "Really, it's just like math" but ... no, it isn't... EDIT: I just glanced at the first part of Sussman's book on Classical Mechanics, and it looked like Lagrangian mathematics to me, no new notation in the first few sections.
- gigigigihkv 12y ago
  GA does not quite do the job.
  - forkandwait 12y ago
    What does that mean? This is hacker news, you can't just say "that sucks" or "that's great" like on reddit...
    - adjaldkjasldj 12y ago
      Oh right, this is the place where you make a crazy claim and back it up by a Wikipedia link instead of an argument! ;) What I meant is that, no, I don't think that GA is a substantial improvement of notation, even though a vocal minority might have made you believe otherwise. One piece of evidence is that physicists would have happily picked up the notation otherwise (and no, this is not an appeal to authority ;). Also, I am not sure what you mean by "in the works", since it is quite old of a concept.
      - jacobolus 12y ago
        GA isn’t really a difference in notation exactly. It’s a difference in model. Anyway, the model is great, and IMO should be taught in high schools. It would reduce a lot of redundant and obnoxiously inconsistent existing models we use instead, and ultimately save students a lot of time. It takes a bit of work to get used to reasoning in GA vs. the standard vector model, or via trigonometry, etc., but there’s a pretty big payoff, I find. Many problems are substantially easier to reason about in GA terms, and many bits of bookkeeping can be avoided compared to the standard approaches. More generally, the study of geometry has in general been systematically deemphasized and devalued over the past 100 years, in favor of analysis. This has IMO caused some serious problems for students’ and scientists’ geometric fluency and reasoning. GA is IMO one way of helping reunify algebraic/analytic with geometric reasoning. By “in the works” he probably means that since David Hestenes has put in a career’s worth of work reintroducing / popularizing the ideas that were mostly ignored after Grassman’s time with the singular exception of Clifford (except in highly technical pure mathematics contexts, and by a few physicists for studying general relativity), and especially in the last 20 years or so since a slightly bigger community has gotten involved, it seems like GA is picking up some steam. Many more people have heard/thought about it today than 10 years ago, there are now a handful of textbooks, etc. In particular some computer graphics / computer vision / robotics folks have found GA very useful to their work. As for why physicists haven’t been champing at the bit.. GA as a model doesn’t allow people to solve any new problems that they couldn’t solve before using existing models, it just makes understanding what’s going on a bit easier. It’s sort of like refactoring all your code: it takes a lot of effort to reframe everything in terms of a new model, and the payoff is mostly for the people who are learning the material for the first time rather than for current professional mathematicians and physicists. Radical changes to low-level models take a long time to propagate through society, or sometimes are altogether impossible (which is why we still use base 10 instead of base 12, why America is stuck with imperial units, why we have a completely hacked together calendar of unequal length months, why we use π as a circle constant instead of either 2π or π/2, why we haven’t normalized English spelling, etc. etc.)

by gfodor 12y ago

agumonkey 12y ago

Sussman tried to use scheme as a mathematical notation (http://mitpress.mit.edu/sites/default/files/titles/content/sicm/book.html http://mitpress.mit.edu/sites/default/files/titles/content/s...) to avoid ambiguities. I don't know how many people saw this, and what the critics were. The traditional static symbolic notation of math may be the most efficient one, It's constrained, but forces abstraction to emerge. If people had Excel before, we'd never know Gauss sequence sum formula, we'd let the system help us.

Chinjut 12y ago
> The traditional static symbolic notation of math may be the most efficient one... If people had Excel before, we'd never know Gauss sequence sum formula, we'd let the system help us. I think some people in this thread must be using "notation" in a different sense than I am accustomed to. I don't think of the computer's ability to automate addition as an innovation in "notation", per se.
- agumonkey 12y ago
  My analogy was borderline, I just think there's value in terse static glyph, and going away from it would be detrimental.
- stan_rogers 12y ago
  That wasn't Sussman's point. The problem is that the same notational forms -- the same orthographies, if you will -- can mean different things in different contexts. It doesn't help at all that one must, on occasion, slip from context to context in the same argument, producing adjacent or near-adjacent statements that have similar general forms but wildly different meanings. His solution to the problem was to use a notation that also happens to be valid statements in a computer language he had a large hand in designing. That part isn't necessary, but disambiguating mathematical notation by some means is.
pakled_engineer 12y ago
I skimmed that book to figure out how to convert formulas to procedures in Scheme for learning Spivaks Calculus and a few OCW math courses. Also picked up Doing Math w/Python from NoStarch press to see how other people do this, and to learn Python since I've never really used it. It was definitely awesome to debug logic errors in emacs of formulas I translated, it really does give you a structural interpretation of the material you are trying to learn by coding it all. End result is a library with clear procedures I can use for reference whereas I've already forgotten most of the notation I've learned over the years (SICM uses Spivak's Calculus on Manifolds similar notation)
smorrow 12y ago
All languages have, at least, nouny things and verby things, and a finite(expandable) number of them composable in infinite ways, and I'd say our brains are hardwired for it. So any notation good for humans would have to have this property, which would mean being "equivalent" (in a way) to the language that already exists. I don't see how any improvement can be anything more than cosmetic.

forkandwait 12y ago

Perhaps Geometric Algebra is a big piece of that in the works? http://en.wikipedia.org/wiki/Geometric_algebra http://en.wikipedia.org/wiki/Geometric_algebra Re Scheme as notation (or APL/ J many years ago) -- when I hear a programming language proposed as a replacement for math, I always wonder "how does that make proving theorems easier?" I haven't read Sussman's book, and he is way smart, but I have listened to lots of APL weanies say "Really, it's just like math" but ... no, it isn't... EDIT: I just glanced at the first part of Sussman's book on Classical Mechanics, and it looked like Lagrangian mathematics to me, no new notation in the first few sections.

gigigigihkv 12y ago
GA does not quite do the job.
- forkandwait 12y ago
  What does that mean? This is hacker news, you can't just say "that sucks" or "that's great" like on reddit...
  - adjaldkjasldj 12y ago
    Oh right, this is the place where you make a crazy claim and back it up by a Wikipedia link instead of an argument! ;) What I meant is that, no, I don't think that GA is a substantial improvement of notation, even though a vocal minority might have made you believe otherwise. One piece of evidence is that physicists would have happily picked up the notation otherwise (and no, this is not an appeal to authority ;). Also, I am not sure what you mean by "in the works", since it is quite old of a concept.
    - jacobolus 12y ago
      GA isn’t really a difference in notation exactly. It’s a difference in model. Anyway, the model is great, and IMO should be taught in high schools. It would reduce a lot of redundant and obnoxiously inconsistent existing models we use instead, and ultimately save students a lot of time. It takes a bit of work to get used to reasoning in GA vs. the standard vector model, or via trigonometry, etc., but there’s a pretty big payoff, I find. Many problems are substantially easier to reason about in GA terms, and many bits of bookkeeping can be avoided compared to the standard approaches. More generally, the study of geometry has in general been systematically deemphasized and devalued over the past 100 years, in favor of analysis. This has IMO caused some serious problems for students’ and scientists’ geometric fluency and reasoning. GA is IMO one way of helping reunify algebraic/analytic with geometric reasoning. By “in the works” he probably means that since David Hestenes has put in a career’s worth of work reintroducing / popularizing the ideas that were mostly ignored after Grassman’s time with the singular exception of Clifford (except in highly technical pure mathematics contexts, and by a few physicists for studying general relativity), and especially in the last 20 years or so since a slightly bigger community has gotten involved, it seems like GA is picking up some steam. Many more people have heard/thought about it today than 10 years ago, there are now a handful of textbooks, etc. In particular some computer graphics / computer vision / robotics folks have found GA very useful to their work. As for why physicists haven’t been champing at the bit.. GA as a model doesn’t allow people to solve any new problems that they couldn’t solve before using existing models, it just makes understanding what’s going on a bit easier. It’s sort of like refactoring all your code: it takes a lot of effort to reframe everything in terms of a new model, and the payoff is mostly for the people who are learning the material for the first time rather than for current professional mathematicians and physicists. Radical changes to low-level models take a long time to propagate through society, or sometimes are altogether impossible (which is why we still use base 10 instead of base 12, why America is stuck with imperial units, why we have a completely hacked together calendar of unequal length months, why we use π as a circle constant instead of either 2π or π/2, why we haven’t normalized English spelling, etc. etc.)