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If you enjoyed this post, you'll probably love "The Elements of Euclid"[1] by Byrne which provides entirely visual proof for ALL the basic proofs of euclidean g

If you enjoyed this post, you'll probably love "The Elements of Euclid"[1] by Byrne which provides entirely visual proof for ALL the basic proofs of euclidean geometry. I actually first came across the book when I saw it mentioned in Beautiful Explanations by Tufte. The beauty of the images is just on another level, the book will just make you feel good when you stare at it and after staring at it you'll absorb a proof accidentally with barely any effort on your part. There is a mistaken belief that visual proofs are less serious than algebraic ones but I believe this is mostly due to a lack of imagination when it comes coming up with good visual proofs. Byrne's book will help you see just how powerful pictures can be. There's lots of good work happening in the Category Theory community to turn diagrams into first class objects in constructing proofs so I'm very optimistic about a boom in visual proof construction. [1] https://www.amazon.com/Byrne-Six-Books-Euclid-Multilingual/dp/3836559382/ref=sr_1_3?keywords=euclid+elements&qid=1571798232&sr=8-3 https://www.amazon.com/Byrne-Six-Books-Euclid-Multilingual/d...

xvilka 7y ago
Could you please provide some links to these efforts for turning diagrams into proofs, please?
- formalsystem 7y ago
  1. This was one of my favorite intro to category theory books, will make you familiar with the general language http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf 2. After skimming 1, check out Algebra Chapter 0 which uses Category Theory and diagrams to prove various theorems from Abstract Algebra (Hard read) https://www.amazon.com/Algebra-Chapter-Graduate-Studies-Mathematics/dp/0821847813 https://www.amazon.com/Algebra-Chapter-Graduate-Studies-Math...
kqr 7y ago
The Taschen reprint of Byrne's Euclid I consider one of the finer books I have. I have spent so much time with it. But there is now also this web version which is made with so much love it in many ways even improves on the reprint in quality: https://www.c82.net/euclid/ https://www.c82.net/euclid/
dwohnitmok 7y ago
One pitfall I'm wary of when introducing visual proofs is not being able to make the leap of how to formalize the proof, i.e. how to turn it into a purely mechanical process that a computer could understand. It can make these sorts of proofs overly convincing. https://math.stackexchange.com/questions/743067/visually-deceptive-proofs-which-are-mathematically-wrong https://math.stackexchange.com/questions/743067/visually-dec.... My favorite is the approximation of the circle one, because it doesn't rely on tricky, underhanded drawing inaccuracies, but instead demonstrates a need to truly formalize what it is you're talking about. For category theory, most people approaching it already have some experience with mathematical proofs and could probably sketch out how to boil a diagram chasing proof down into tedious set of logical statements. If anyone hasn't, I'd recommend doing so for a simple example. Note a version of this can occur for the "algebraic" style of proofs as well. Occasionally students can't really explain why they're "allowed" to cancel out terms (it can be a minor leap to see that really what's being relied on here is injectivity). The other tricky thing about intuitions, visual or otherwise, at least in my experience, is that I often hold multiple mutually incompatible visualizations/intuitions about a mathematical object or process and the most crucial component of my intuition is knowing when to discard one and use the other when they conflict. To actually harmonize all of them requires, well, fully formalizing everything. Otherwise you end up mistaking your intuition for the object itself and going down a logically incoherent path (the evergreen target for this always seems to be Godel's incompleteness theorems). You still need intuition though, because otherwise coming up with the creative spark for a proof is nigh impossible. But it's not a substitute for the formal object itself. More fundamentally, I think both approaches, visual and "algebraic" in the sense of the article make it seem like mathematics is about getting the "correct" answer, when really the part of pure mathematics that resonates most with me is about running wild with "what if" and then rigorously chasing down the implications thereof. For example, the commonly asked playground question "is infinity number?" is not best answered with a "no" or a "yes", but rather an exploration of what no and yes would entail, which first requires the formalization of infinity, which could have many different, mutually incompatible forms! Another fun one is coming up with a world where infinity plus one is larger than infinity (this often leads to an exploration of the ordinals). I also enjoy how relevant your username is.
posterboy 7y ago
diagrams invariably show only 2 dimensions, so you can't reasonably show anything that has complexity in more than three dimensions, which means any problem with three independent variables is out. Animation can add the missing dimension; well color can, too.
- indigochill 7y ago
  I'm cautiously optimistic about VR as a tool for teaching and understanding math up to three dimensions. You may have seen the "Non-euclidian virtual reality" video floating around YouTube (https://www.youtube.com/watch?v=ztsi0CLxmjw https://www.youtube.com/watch?v=ztsi0CLxmjw).

xvilka 7y ago

Could you please provide some links to these efforts for turning diagrams into proofs, please?

formalsystem 7y ago
1. This was one of my favorite intro to category theory books, will make you familiar with the general language http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf 2. After skimming 1, check out Algebra Chapter 0 which uses Category Theory and diagrams to prove various theorems from Abstract Algebra (Hard read) https://www.amazon.com/Algebra-Chapter-Graduate-Studies-Mathematics/dp/0821847813 https://www.amazon.com/Algebra-Chapter-Graduate-Studies-Math...

kqr 7y ago

The Taschen reprint of Byrne's Euclid I consider one of the finer books I have. I have spent so much time with it. But there is now also this web version which is made with so much love it in many ways even improves on the reprint in quality: https://www.c82.net/euclid/ https://www.c82.net/euclid/

dwohnitmok 7y ago

One pitfall I'm wary of when introducing visual proofs is not being able to make the leap of how to formalize the proof, i.e. how to turn it into a purely mechanical process that a computer could understand. It can make these sorts of proofs overly convincing. https://math.stackexchange.com/questions/743067/visually-deceptive-proofs-which-are-mathematically-wrong https://math.stackexchange.com/questions/743067/visually-dec.... My favorite is the approximation of the circle one, because it doesn't rely on tricky, underhanded drawing inaccuracies, but instead demonstrates a need to truly formalize what it is you're talking about. For category theory, most people approaching it already have some experience with mathematical proofs and could probably sketch out how to boil a diagram chasing proof down into tedious set of logical statements. If anyone hasn't, I'd recommend doing so for a simple example. Note a version of this can occur for the "algebraic" style of proofs as well. Occasionally students can't really explain why they're "allowed" to cancel out terms (it can be a minor leap to see that really what's being relied on here is injectivity). The other tricky thing about intuitions, visual or otherwise, at least in my experience, is that I often hold multiple mutually incompatible visualizations/intuitions about a mathematical object or process and the most crucial component of my intuition is knowing when to discard one and use the other when they conflict. To actually harmonize all of them requires, well, fully formalizing everything. Otherwise you end up mistaking your intuition for the object itself and going down a logically incoherent path (the evergreen target for this always seems to be Godel's incompleteness theorems). You still need intuition though, because otherwise coming up with the creative spark for a proof is nigh impossible. But it's not a substitute for the formal object itself. More fundamentally, I think both approaches, visual and "algebraic" in the sense of the article make it seem like mathematics is about getting the "correct" answer, when really the part of pure mathematics that resonates most with me is about running wild with "what if" and then rigorously chasing down the implications thereof. For example, the commonly asked playground question "is infinity number?" is not best answered with a "no" or a "yes", but rather an exploration of what no and yes would entail, which first requires the formalization of infinity, which could have many different, mutually incompatible forms! Another fun one is coming up with a world where infinity plus one is larger than infinity (this often leads to an exploration of the ordinals). I also enjoy how relevant your username is.

posterboy 7y ago

diagrams invariably show only 2 dimensions, so you can't reasonably show anything that has complexity in more than three dimensions, which means any problem with three independent variables is out. Animation can add the missing dimension; well color can, too.

indigochill 7y ago
I'm cautiously optimistic about VR as a tool for teaching and understanding math up to three dimensions. You may have seen the "Non-euclidian virtual reality" video floating around YouTube (https://www.youtube.com/watch?v=ztsi0CLxmjw https://www.youtube.com/watch?v=ztsi0CLxmjw).