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Proofs Without Words

artofproblemsolving.com

111 pointssquircle26 comments

There's a book called "Proofs without words". Fun to have a glance. (https://ia801405.us.archive.org/24/items/proofs-without-word...) It also has a sequel.

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.

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.

Yeah, I don’t get how you distinguish between a correct visual proof and a visual proof that looks right but doesn’t actually prove what it’s trying to prove. You could probably make a pretty convincing-looking visual proof that the limit of the sum of the harmonic series is below some finite number, that 0.9 repeating is less than 1, that there are more rationals than integers, that there are the same number of reals and rationals, and that sort of thing.

On the linked page, a lot of the proofs are essentially proofs by induction that stop at some (pretty small) n. Maybe there’s a way to make it rigorous by visually showing the induction step that proves n+1 given n, but if there is, it’s not shown.

This can be great for building intuition for a statement known to be true by other means, but I wouldn’t consider them to be proofs.

> Yeah, I don’t get how you distinguish between a correct visual proof and a visual proof that looks right but doesn’t actually prove what it’s trying to prove.

This problem exists not only for visual proofs, but for standard written ones too.

Not in the same way. For a written proof it can be hard but with effort and sufficient background knowledge you can figure out if it actually proves the statement or not. If the proof doesn’t prove the statement there will be a step that doesn’t follow from the rest of it. You may not be able to spot it but it can at least theoretically be spotted.

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.

I've found Dudley's A Budget of Trisections pretty fun in this regard: the book is filled with depictions of various people's attempts to trisect an angle with straightedge and compass. It turns out that a few steps can get you within arcseconds of the correct result, easily enough to fool the eye.

I'm the opposite. I am not convinced until I "see it". Probably has to do with our innate talents.

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.

Nice. There's an entire book like this for geometric statements. Every picture is a fact, proofs are supplied by the reader:

https://users.mccme.ru/akopyan/papers/EnGeoFigures.pdf

Caution: proofs of some of the statements in it are difficult.

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.

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

Here’s a proof with just a few words that got published in a serious math journal: https://fermatslibrary.com/s/shortest-paper-ever-published-i...

Notice that one of the authors is John H. Conway. Serious badass, among many many other things known for:

- The game of life

- The fractran language (a Turing-complete programming language consisting only of fractions)

- Surreal numbers

- The “Conway base 13 function” (a brain-scramblingly hideous function that is everywhere discontinuous and yet somehow takes on every real number on every interval - invented as an analysis counterexample to prove that a function can satisfy the intermediate value property and yet not be continuous).

- A lot of work on sporadic simple groups. The three groups Co1, Co2 and Co3 are named after John H. Conway, and he was co-author of the “Monstrous Moonshine” paper and conjecture that Richard Borcherds won the Fields medal for proving.

… and a bunch of other whacky stuff, such as inventing an algorithm to figure out what day of the week any given date in history was (he used to do this in his head)

Sadly he died of complications from Covid. https://en.wikipedia.org/wiki/John_Horton_Conway

Anyone who enjoys this should read David Bessis’s Mathematica.

Thanks for the mention. I loved the book [1], and it started me off on a journey to spark intuition, and sensory (visual) connection.

On another note, I was shocked to find that some members of my family have aphantasia which is a complete inability to visually imagine geometric figures or pictures, and yet, they were good at math. So, there are faculties beyond visual imagination which are invoked, and even within visual imagination, there is a spectrum among people as to its strength, and quality.

[1] https://www.amazon.com/Mathematica-Secret-World-Intuition-Cu...

Another great site is https://theoremoftheday.org/ with a neat one-pager overview of each theorem

There is also this youtube channel called 'Mathematical Visual Proofs' on similar theme: https://www.youtube.com/@MathVisualProofs

I made a game out of creating proofs without words: https://brianberns.github.io/Tactix/

I know a nice proof of volume of tetrahedron being 1/3 of the corresponding paralellepiped. You split it into smaller tetrahedra by midpoints and count them.

Also there is a nice visual proof that in an equilateral triangle, for every point in it, the sum of distances from all the sides is constant.

The second one https://en.wikipedia.org/wiki/Viviani%27s_theorem

"The sum of the first $n$ positive integers is ${n+1 \choose 2}$" is beautiful! for anyone lacking the background to get it, the right hand side is "(n + 1) choose 2", the number of ways of selecting 2 elements out of a set of (n + 1). and if you look at the picture, selecting any two balls in the bottom row uniquely identifies a ball in the triangle, and vice versa (selecting a ball in the triangle picks a unique pair of balls in the bottom row). so the sum of all the balls in the first n rows is indeed the number of ways of choosing two balls from the bottom row!

and also how to lie with visual proofs: https://www.youtube.com/watch?v=VYQVlVoWoPY

I’m not a huge fan of these, but this time I noticed that the best ones feel a lot like naturality arguments. As in, moving structural bits in a way that makes it clear that we’re not touching anything that ought to be universally quantifiable.

I still don’t love this sort of thing being presented as “proof”, but I thought that idea is interesting. Is there a way to formalize naturality into technical diagrams? Probably!

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)

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