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I haven't previously come across the idea of trying to make wordless proofs. I guess this is the mathematical equivalent of code golf: a clever game for inside

by zazen 8y ago

I haven't previously come across the idea of trying to make wordless proofs. I guess this is the mathematical equivalent of code golf: a clever game for insiders to play, but basically the opposite of good practice from a readability perspective.

laurentl 8y ago
I didn’t get this feeling from the proof in the article. I thought there was a kind of poetry to it. The diagram and colors make it reasonably easy to understand and they emphasize the geometrical nature of the proof, which uses only elementary math: the formula for the area of a triangle and Thales’ theorem (If my memory is correct, I haven’t used that name since high school!). On the other hand, a “proof with words” would likely have derived the explicit formulas for I_n and C_n in order to prove the relationship and would have been much more clanky. If I had to make a comparison, this felt like reading good functional code. Concise, a bit mysterious at first, but once you understand what it does you feel enlightened. (Vs eg. wading through for loops and index arithmetic, which also does the job but is more tedious to read)
- whatshisface 8y ago
  I think it wasn't so much a proof as it was an image that made you realize the proof with some thought. Still, that's about what all published proofs are in the end, isn't it?
agumonkey 8y ago
Same for education purposes, I deeply believe that most terms cloud the learning process. Most of the time a pattern can be witnessed and learned without giving too many names if at all. Maybe then, after the structure/topology[1] has set in the mind, you can label things to accelerate communication. But words first feels like cart before the wheel. [1] in the fuzzy sense, not the mathematical field
coldtea 8y ago
>I haven't previously come across the idea of trying to make wordless proofs. I guess this is the mathematical equivalent of code golf: a clever game for insiders to play, but basically the opposite of good practice from a readability perspective. There are lots of geometrical proofs for e.g. the pythagorean theorem that need no words. It's not like code golf any more than a text-based proof is. Words still need to get interpreted just as images do. If anything, a succinct geometric proof can be much easier to grasp and verify than a text-based proof.
jonahx 8y ago
> a clever game for insiders to play, but basically the opposite of good practice from a readability perspective. This is truly, deeply the wrong lesson to take away. I'd say that, in a spirit similar to the Feynman technique, you don't really understand proofs whose central idea you can't explain simply -- and a wordless proof is good evidence that you have in fact grasped the proof's essence. Of course, this doesn't mean you can do away with formal rigor. Mathematics needs that too. But I'd argue that is "the clever game for insiders to play" (albeit an important game), while intuitive understanding is what you should really be after, and is what generates new ideas and new proofs. More beautiful proofs without words: https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain#734790 https://math.stackexchange.com/questions/733754/visually-stu... https://math.stackexchange.com/questions/1337750/easy-math-proofs-or-visual-examples-to-make-high-school-students-enthusiastic-ab?noredirect=1&lq=1 https://math.stackexchange.com/questions/1337750/easy-math-p... https://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias%3Daps&field-keywords=proofs+without+words https://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias%3...

by zazen 8y ago

laurentl 8y ago

I didn’t get this feeling from the proof in the article. I thought there was a kind of poetry to it. The diagram and colors make it reasonably easy to understand and they emphasize the geometrical nature of the proof, which uses only elementary math: the formula for the area of a triangle and Thales’ theorem (If my memory is correct, I haven’t used that name since high school!). On the other hand, a “proof with words” would likely have derived the explicit formulas for I_n and C_n in order to prove the relationship and would have been much more clanky. If I had to make a comparison, this felt like reading good functional code. Concise, a bit mysterious at first, but once you understand what it does you feel enlightened. (Vs eg. wading through for loops and index arithmetic, which also does the job but is more tedious to read)

whatshisface 8y ago
I think it wasn't so much a proof as it was an image that made you realize the proof with some thought. Still, that's about what all published proofs are in the end, isn't it?

agumonkey 8y ago

Same for education purposes, I deeply believe that most terms cloud the learning process. Most of the time a pattern can be witnessed and learned without giving too many names if at all. Maybe then, after the structure/topology[1] has set in the mind, you can label things to accelerate communication. But words first feels like cart before the wheel. [1] in the fuzzy sense, not the mathematical field

coldtea 8y ago

>I haven't previously come across the idea of trying to make wordless proofs. I guess this is the mathematical equivalent of code golf: a clever game for insiders to play, but basically the opposite of good practice from a readability perspective. There are lots of geometrical proofs for e.g. the pythagorean theorem that need no words. It's not like code golf any more than a text-based proof is. Words still need to get interpreted just as images do. If anything, a succinct geometric proof can be much easier to grasp and verify than a text-based proof.

jonahx 8y ago

> a clever game for insiders to play, but basically the opposite of good practice from a readability perspective. This is truly, deeply the wrong lesson to take away. I'd say that, in a spirit similar to the Feynman technique, you don't really understand proofs whose central idea you can't explain simply -- and a wordless proof is good evidence that you have in fact grasped the proof's essence. Of course, this doesn't mean you can do away with formal rigor. Mathematics needs that too. But I'd argue that is "the clever game for insiders to play" (albeit an important game), while intuitive understanding is what you should really be after, and is what generates new ideas and new proofs. More beautiful proofs without words: https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain#734790 https://math.stackexchange.com/questions/733754/visually-stu... https://math.stackexchange.com/questions/1337750/easy-math-proofs-or-visual-examples-to-make-high-school-students-enthusiastic-ab?noredirect=1&lq=1 https://math.stackexchange.com/questions/1337750/easy-math-p... https://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias%3Daps&field-keywords=proofs+without+words https://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias%3...