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I’ve seen a lot of people enamored with graphical notations like this one, but nobody ever using them for anything nontrivial. In general, people just use them

by knzhou 7y ago

I’ve seen a lot of people enamored with graphical notations like this one, but nobody ever using them for anything nontrivial. In general, people just use them to laboriously reproduce a few results from undergrad courses, nothing more. If ordinary notation is like C, these graphical notations are like code golf languages, a neat gimmick that performs excellently at the few things you hardcoded it to do, and not much else.

wodenokoto 7y ago
A Nobel Price winner seems to disagree with you. From the article: > the American physicist Frank Wilcjek, who worked with Feynman in the 1980s, once wrote: “The calculations that eventually got me a Nobel Prize in 2004 would have been literally unthinkable without Feynman diagrams.“
- improbable22 7y ago
  Sure, but that's an odd reading of GP's "graphical notations like this one". Feynman diagrams are very useful, and made respectable lots of other graphical notations. There's a nice book on this: https://www.press.uchicago.edu/ucp/books/book/chicago/D/bo3534300.html https://www.press.uchicago.edu/ucp/books/book/chicago/D/bo35... But diagrams as a replacement for writing out tensor indices, like in TFA, which was what I presume GP meant... these have not proven widely useful. I guess people doing tensor network things use them.
- knzhou 7y ago
  Diagrammatic intuition is good. I use Feynman diagrams every day. But nobody computes with Feynman diagrams, they use them as an intuition aid to tell them what to compute, and that computation is done using standard notation.
  - wodenokoto 7y ago
    > ... and that computation is done using standard notation. I really thought you would have ended that sentence with "Matlab" or "python", not "standard notation"
joppy 7y ago
I don't think that analogy holds at all. A better analogy is comparing some boolean function written purely in terms of AND, OR, XOR, etc with its corresponding circuit diagram. From the diagram, the information flow is much more obvious, and common constructions (building adders, as a random example) can be visualised. The diagrammatics give a different intuition on the problem than purely symbolic algebra. Of course, both the formula and the diagram encode exactly the same information. However when attempting to prove something about a formula, it can be very helpful to translate the formula into a diagram and use the diagram to guide algebraic manipulations (manipulations which perhaps we would not have known to make, if it weren't for the diagram). I know that Feynman diagrams are a quite famous example - even if results do not use them directly, the intuition they give on a problem are indispensable. There are other examples in pure mathematics where new proofs have been given not in terms of diagrams, but the development of suitable diagrams have enabled people to come up with a proof.
- knzhou 7y ago
  There’s nothing wrong with using graphical intuition, my point is that nobody ever uses graphical notations to compute. For example, Feynman diagrams are a great way to tell you what to compute, but then the computation itself is always done in standard notation.

by knzhou 7y ago

wodenokoto 7y ago

A Nobel Price winner seems to disagree with you. From the article: > the American physicist Frank Wilcjek, who worked with Feynman in the 1980s, once wrote: “The calculations that eventually got me a Nobel Prize in 2004 would have been literally unthinkable without Feynman diagrams.“

improbable22 7y ago
Sure, but that's an odd reading of GP's "graphical notations like this one". Feynman diagrams are very useful, and made respectable lots of other graphical notations. There's a nice book on this: https://www.press.uchicago.edu/ucp/books/book/chicago/D/bo3534300.html https://www.press.uchicago.edu/ucp/books/book/chicago/D/bo35... But diagrams as a replacement for writing out tensor indices, like in TFA, which was what I presume GP meant... these have not proven widely useful. I guess people doing tensor network things use them.
knzhou 7y ago
Diagrammatic intuition is good. I use Feynman diagrams every day. But nobody computes with Feynman diagrams, they use them as an intuition aid to tell them what to compute, and that computation is done using standard notation.
- wodenokoto 7y ago
  > ... and that computation is done using standard notation. I really thought you would have ended that sentence with "Matlab" or "python", not "standard notation"

joppy 7y ago

I don't think that analogy holds at all. A better analogy is comparing some boolean function written purely in terms of AND, OR, XOR, etc with its corresponding circuit diagram. From the diagram, the information flow is much more obvious, and common constructions (building adders, as a random example) can be visualised. The diagrammatics give a different intuition on the problem than purely symbolic algebra. Of course, both the formula and the diagram encode exactly the same information. However when attempting to prove something about a formula, it can be very helpful to translate the formula into a diagram and use the diagram to guide algebraic manipulations (manipulations which perhaps we would not have known to make, if it weren't for the diagram). I know that Feynman diagrams are a quite famous example - even if results do not use them directly, the intuition they give on a problem are indispensable. There are other examples in pure mathematics where new proofs have been given not in terms of diagrams, but the development of suitable diagrams have enabled people to come up with a proof.

knzhou 7y ago
There’s nothing wrong with using graphical intuition, my point is that nobody ever uses graphical notations to compute. For example, Feynman diagrams are a great way to tell you what to compute, but then the computation itself is always done in standard notation.