Caution: proofs of some of the statements in it are difficult.
JadeNB 10 hours ago [-]
I think that is rather different. The traditional meaning of "proofs without words" is that the picture is the proof, or at least, if you believe that a proof can only be in words, that the picture should convey the idea so transparently that anyone with reasonable mathematical skill can routinely translate it into words.
fiforpg 2 hours ago [-]
You are correct, after posting I realized the difference. The book is rather "theorems [formulated] without words".
Which is why I added that the proofs are left to the reader :P
I've never really been a fan of proofs without words; they've always felt way too slippery to me, for lack of a better term. A well worded proof with nice explanatory diagrams hits the spot for me instead.
perlgeek 6 hours ago [-]
For me, the visual proofs of simple sums (like The sum of the first n odd natural numbers is n²) works pretty well for me.
For the more geometry-based ones where you have move triangles around and so, it's often not obvious to me that two angles that look the same really always are the same, and that things that add up to rectangle do so reliably, independently of the actual angles used in the examples.
I guess in these cases, a more parameterized, interactive version would work better, where you can use sliders to adjust some of the angles and lengths used. That should make it much more obvious that it's not just an artifact of particular angles used in an example.
seanhunter 2 hours ago [-]
Feel the same way. It’s way too close to the infamous proof by “just look at it”. Our visual intuition is way too easy to trick especially in three dimensions, and our intuition for any dimension higher than that is basically zero.
paufernandez 6 hours ago [-]
I'm the opposite. I am not convinced until I "see it". Probably has to do with our innate talents.
seanhunter 2 hours ago [-]
The problem is not whether you (or anybody) can be convinced by seeing something that is true. Mathematics study involves a lot of drawing curves etc so you can develop geometric/visual intuition about things, and of course that is a good idea.
The problem is that it is far too easy to convince someone of something which is not true via visual means.
See also O. Byrne, "The First Six Books of the Elements of Euclid, in Which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners", https://www.c82.net/euclid/ (reproduction in CSS by Nicholas Rougeux)
https://users.mccme.ru/akopyan/papers/EnGeoFigures.pdf
Caution: proofs of some of the statements in it are difficult.
Which is why I added that the proofs are left to the reader :P
For the more geometry-based ones where you have move triangles around and so, it's often not obvious to me that two angles that look the same really always are the same, and that things that add up to rectangle do so reliably, independently of the actual angles used in the examples.
I guess in these cases, a more parameterized, interactive version would work better, where you can use sliders to adjust some of the angles and lengths used. That should make it much more obvious that it's not just an artifact of particular angles used in an example.
The problem is that it is far too easy to convince someone of something which is not true via visual means.