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Slightly related, but I've always wondered if there are similar (preferably successful) explorations in maths. "Mainstream" notations are clearly optimized for

Slightly related, but I've always wondered if there are similar (preferably successful) explorations in maths. "Mainstream" notations are clearly optimized for handwriting, but there's just so much room with current tech for optimizing for e.g. visual intuitiveness. And let's not even talk about all the historical burden that's made these notations the convoluted mess they are today. (Uncreative, probably very bad) examples: Replace brackets with boxes, use colors (think syntax highlighting), icons, or even basic text-like formatting. And that's still being very conservative in relation to what could be done, as exemplified by the OP. Any pointers/ideas?

eru 8y ago
Mathematicians play around with new notation all the time. But yes, mostly hand-written. The standard notation is mostly for communicating with other people mediated via papers. But we don't actually use that many brackets in math---that's more of a programming thing. (Because programmers can't really use visual structure all that much, and mostly rely on linearization of their syntax trees. Languages with significant whitespace like Haskell (and to a lesser extent Python) see less parens and braces.)
filleokus 8y ago
I have always wondered how professional mathematicians work day to day. It depends on their field of course, but I guess most still use pen and paper (or a digital version, i.e iPad Pro + Apple Pencil / Wacom). I have at least never seen any tools for "explorative" manipulation of symbolic expressions. TeX etc works great for final type setting, but when going from mind to _something_ I guess that something is still paper, or an equivalent. Would be really interesting to see the equivalent of an IDE / good text editor for math, perhaps with pen input for speed.
- xelxebar 8y ago
  I'm a Ph.D. student in a pure math field, and just coming from personal experience, the overall relationship with TeX is a pretty strained one. The overwhelming majority of working out is done by hand on paper or a chalk board. That said, as an experiment, I did about half of my Master's thesis work in TeX. Without quick feedback on the rendered formulae, I had a much harder time reasoning through things, so having a comfortable text editing environment seems crucial. That said, I wouldn't recommend using TeX as a scratchpad. It just doesn't map well to the sets of tools that we tend to use: mini diagrams, arrows between parts of the page, strikeout and circling text, etc. What might work is something like e-paper that renders a handwritten document into text. Ideally, we could turn letters into Unicode and diagrams into vector graphics, providing tools to edit and embellish the rendered versions if desired. It's extremely common to edit equations and diagrams on the fly, so I would probably feel friction with anything other that doesn't seem to just magically "prettify" whatever I handwrite at it. Seems like a really tall order though.
norepicycle 8y ago
3Blue1Brown has a YouTube video called the "Triangle of Power" which uses a clever bit of notation to unify exponents and logarithms: https://www.youtube.com/watch?v=sULa9Lc4pck https://www.youtube.com/watch?v=sULa9Lc4pck A bit lowbrow, perhaps (you'd want Penrose notation or such things in that case).
OscarCunningham 8y ago
There's the Penrose Graphical Notation[0] for linear algebra, and (much more generally) string diagrams[1] for representing and manipulating morphisms in n-categories. [0] https://en.m.wikipedia.org/wiki/Penrose_graphical_notation https://en.m.wikipedia.org/wiki/Penrose_graphical_notation [1] https://en.m.wikipedia.org/wiki/String_diagram https://en.m.wikipedia.org/wiki/String_diagram

eru 8y ago

Mathematicians play around with new notation all the time. But yes, mostly hand-written. The standard notation is mostly for communicating with other people mediated via papers. But we don't actually use that many brackets in math---that's more of a programming thing. (Because programmers can't really use visual structure all that much, and mostly rely on linearization of their syntax trees. Languages with significant whitespace like Haskell (and to a lesser extent Python) see less parens and braces.)

filleokus 8y ago

I have always wondered how professional mathematicians work day to day. It depends on their field of course, but I guess most still use pen and paper (or a digital version, i.e iPad Pro + Apple Pencil / Wacom). I have at least never seen any tools for "explorative" manipulation of symbolic expressions. TeX etc works great for final type setting, but when going from mind to _something_ I guess that something is still paper, or an equivalent. Would be really interesting to see the equivalent of an IDE / good text editor for math, perhaps with pen input for speed.

xelxebar 8y ago
I'm a Ph.D. student in a pure math field, and just coming from personal experience, the overall relationship with TeX is a pretty strained one. The overwhelming majority of working out is done by hand on paper or a chalk board. That said, as an experiment, I did about half of my Master's thesis work in TeX. Without quick feedback on the rendered formulae, I had a much harder time reasoning through things, so having a comfortable text editing environment seems crucial. That said, I wouldn't recommend using TeX as a scratchpad. It just doesn't map well to the sets of tools that we tend to use: mini diagrams, arrows between parts of the page, strikeout and circling text, etc. What might work is something like e-paper that renders a handwritten document into text. Ideally, we could turn letters into Unicode and diagrams into vector graphics, providing tools to edit and embellish the rendered versions if desired. It's extremely common to edit equations and diagrams on the fly, so I would probably feel friction with anything other that doesn't seem to just magically "prettify" whatever I handwrite at it. Seems like a really tall order though.

norepicycle 8y ago

3Blue1Brown has a YouTube video called the "Triangle of Power" which uses a clever bit of notation to unify exponents and logarithms: https://www.youtube.com/watch?v=sULa9Lc4pck https://www.youtube.com/watch?v=sULa9Lc4pck A bit lowbrow, perhaps (you'd want Penrose notation or such things in that case).

OscarCunningham 8y ago

There's the Penrose Graphical Notation[0] for linear algebra, and (much more generally) string diagrams[1] for representing and manipulating morphisms in n-categories. [0] https://en.m.wikipedia.org/wiki/Penrose_graphical_notation https://en.m.wikipedia.org/wiki/Penrose_graphical_notation [1] https://en.m.wikipedia.org/wiki/String_diagram https://en.m.wikipedia.org/wiki/String_diagram