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October 30, 2014

Maths, Just in Short Words

Posted by David Corfield

Guest post by David Roberts

How much maths can you talk about if you could just use short words? Late one night, somewhere between waking and sleeping, the cartoon proof of Löb’s theorem and Boolos’ explanation of Gödel’s second incompleteness theorem (Paywall! But the relevant part is all on the first page, if you can see that) teamed up to induce me to produce a proof of Cantor’s theorem of the existence of more than one infinite cardinal, using only words of one syllable (or, “just short words”). I then had the idea that it would be interesting to collect explanations or definitions or proofs that used only words of one syllable, and perhaps publish them in one go. Below the fold, I give my proof of Cantor’s theorem, comments and criticisms welcomed. I was aiming for as complete a proof as I could, whereas a number of other people (more on that below) went for simplicity, or as short as they could, so the style is different to the others I mention.

The fact of Georg C., in short words.

  1. A “set” is made up of things.
  2. The set \mathbb{N} is made up of 1,2,3,4,,n,1,2,3,4,\ldots,n,\ldots
  3. We can add 1 to a thing in \mathbb{N} to get a new thing in \mathbb{N}, not the same as the first.
  4. A “map” from the set AA to the set BB is a rule that takes a thing in AA and gives a thing in BB.
  5. For a set CC and a set DD, the set Map(C,D)Map(C,D), “maps from CC to DD”, is made up of all maps from CC to DD.
  6. The set AA is said to be “not as big as” the set BB if: when you give me a map from AA to BB, I can find a thing in BB that does not come, by the rule for the map, from a thing in AA.
  7. Let M be a map from the set AA to the set Map(A,)Map(A,\mathbb{N}).
  8. This means that MM is a rule that gives a map from AA to \mathbb{N}, for each thing aa in AA. We will call this “aa’s Map”, with the fact this is by the rule for MM in the back of our minds.
  9. Now I will tell you a thing CC in Map(A,)Map(A,\mathbb{N}). It is a map from AA to \mathbb{N}, with rule:

    the thing aa (in AA) goes to the thing that aa’s Map sends aa to (a thing in \mathbb{N}) plus 1.

  10. Read that once more, as it’s a bit hard to grasp.

  11. I claim that CC is a thing that does not come from, by the rule for MM, from a thing in AA.
  12. Why? Since if it did, let us say, come from the thing aa in AA, then CC is aa’s Map for some aa, and so the thing to which aa’s Map sends aa is the same thing to which CC sends aa, which is the thing to which aa’s Map sends aa, now plus 1, which is false, by point 3. So CC is not aa’s Map!
  13. So we were wrong to think that CC was a thing that came from AA (by the rule for MM), and so AA must be not as big as Map(A,)Map(A,\mathbb{N}).
  14. If we take AA to be the set \mathbb{N}, this means that though \mathbb{N} is a big, big set, it is not as big as Map(,)Map(\mathbb{N},\mathbb{N})!

This proof I learned on the homotopy type theory mailing list (this post by Thomas Streicher), and I like it as it uses (almost) minimal logical assumptions: no need for power sets, Axiom of Choice, excluded middle etc. There is of course Lawvere’s proof via his fixed-point theorem, which doesn’t even require function spaces, though it is less clear what this means for cardinalities. To me, though, it feels like an a proof in ‘external logic’, reasoning about the ambient category of discourse from the outside. The proof above works in the internal logic of any Π\Pi-pretopos with natural number object, I believe. Probably it works if we replace the abstract object AA with \mathbb{N} and assume that \mathbb{N}^\mathbb{N} exists, rather than assuming cartesian closedness. But this is off-topic.

After I mentioned this on Google+, Joel Hamkins sent me a different proof of Cantor’s theorem, all in single-syllable words, then a proof of the irrationality of 2\sqrt{2}. Then Tim Gowers wrote another irrationality proof, and then, on my prompting, an explanation and outline of the proof of the Green-Tao theorem on arithmetic progressions in the primes. Toby Bartels wrote out the Peano axioms. Asaf Karagila wrote a proof of Cantor’s theorem without using the letter ‘e’. You can can a number of these in the comments at my original Google+ post, but read on before clicking through.

What I’d really like to see is two things. First: people who have proved amazing theorems explain said theorems in this manner (I was hoping to get, for instance, Terry Tao or Ben Green to write up the Green-Tao theorem). Or perhaps see how many Fields Medallists we could get to do this exercise, on the result/concept of their choice. Second, and this is a more local challenge to Café patrons: write about some category-theoretic concept using monosyllabic words. Otherwise, anyone can chip in with their favourite proof, fact, concept or definition in the comments! It is best to give it a go before reading others’ attempts, especially if you happen to have chosen the same thing to write about as someone else.

So get those pens or keys and go: write maths in just short words!

Posted at October 30, 2014 12:41 PM UTC

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27 Comments & 0 Trackbacks

Re: Maths, Just in Short Words

Six plus one is… hmm.

Posted by: Tom Leinster on October 30, 2014 12:56 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

… one more than six.

Posted by: Jonathan Kirby on October 30, 2014 1:14 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

…the fourth prime?

A secondary school teacher friend of mine was lamenting the difficulty of fulfilling a requirement that she has to use only words of at most two syllables when speaking to students with low academic achievement.

Posted by: David Corfield on October 30, 2014 1:20 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

I suppose that this requirement is to ensure that they continue to have low academic achievement?

Posted by: Toby Bartels on November 3, 2014 7:44 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

Right! It’s like not speaking to babies with words as they don’t understand any yet.

Posted by: David Corfield on November 3, 2014 12:57 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

I think I just had an ultrafinitist spell there :-) ‘though I can’t say that…

Posted by: David Roberts on October 31, 2014 12:04 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

This xkcd suggests a different constraint (arguably a more reasonable one, if the goal is human-readability).

Posted by: Mark Meckes on October 30, 2014 3:03 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

There is a spell checker for these ten hundred words. There are also standardized languages:Basic English and Simplified Technical English.

Posted by: Bas Spitters on October 30, 2014 10:18 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

Randall Monroe created his own checker for the purposes of writing a book in this format and made it available with an in-browser interface.

Posted by: David Roberts on September 23, 2015 8:57 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

I feel the ABC conjecture deserves a short words proof

Posted by: Mark Callaghan on October 30, 2014 8:28 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

At the camp I used to work at, we had a game with the name “The Game of Four”. The rule is that you may only use a word if, when you mark it on a page, you use only four (or less!) bits (I mean, bits of a word, like “ay”, “bee”, or “cee”).

At the time, I was very good at this game. I am not any more as able to say a full idea, but I do like to try — The Game of Four is a nice test of your wit. The best is to try it at real time, and see if you can have a long talk such that your folk that you talk with don’t hear the fact that you are in a game, well, don’t hear that fact when you set out — they will get it by the end.

Once you get good at the Game, you can try to give a math talk at the same time that you play the Game. It is like your rule that each word must have only one oral bit, but I feel it is a bit more hard.


Oh, by the way, the “Way of Two and One” has a bit too … erm … it is too bad for me.


Another, slightly stricter lingual requirement is to force all sayings to have prime length (“length” in the sense of writing). As it turns out, there are fewer words (at least, words in regular use) meeting the stricter requirement.

Posted by: Theo on October 31, 2014 2:12 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

Mike also told us of the Rule of Four. It it not so easy!

Posted by: David Roberts on October 31, 2014 3:31 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

One-syllable words? Quite a constraint!

Another type of constraint which a lot of people find fun is to set math to rhyme; check out for example the proof that the halting problem is undecidable by Geoffrey Pullum (of Language Log fame). Perhaps even better, try making math into song!

Maybe I can try to get the ball rolling though. Category theory is heavy on definitions, so here’s one.

(0) We bet you know this, but just in case: we speak of “sets” and “rules” twixt sets. A rule from a set A to a set B, when fed a thing in A, spits out just one thing in B.

(0)’ We bet you know this as well, but we also talk of “laws” when there is some thing, some phrase, we need or ask to hold or be true.

(1) A “cat” for short has two sets, one of things we will call “nodes” and the other of things we will call “maps”. It has some more “rules”, and some laws of cats, that you need to know. Here they are:

(2) There is a rule which, when fed a map f, spits out a node we call the head of f. You can call that the head map if you like. And there is a rule which, when fed a map of f, spits out a node we call the tail of f. Call that the tail map if you like.

(3) There is a rule which, when fed a node x, spits out a map we will call “same at x” whose head and tail are both x.

(4) We will say that a map f chains to a map g if the head of f is the tail of g. Then, there is a rule which, when fed a pair of maps f and g where f chains to g, spits out a map we name “f then g”. A law of cats is that the tail of “f then g” is the tail of f. And there is a law of cats that the head of “f then g” is the head of g.

(5) Note that if we call the head of a map f by the name ‘x’, then f chains to “same at x”. A law of cats is that the map we call “f then same at x” is the same as f.

(6) Then too, if we call the tail of a map f by the name ‘y’, then “same at y” chains to f. A law of cats is that the map we call “same at y then f” is the same as f.

(7) One last law of cats is this. Say we have maps f, g, and h where f chains to g and g chains to h. Then the map we call “(f then g) <air quotes> then h” is the same as the map we call “f then (g then h) <air quotes>”.

Posted by: Todd Trimble on October 31, 2014 2:37 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

I’m impressed! I had vague dreams about explaining Yoneda, but this is much more achievable.

Posted by: David Roberts on October 31, 2014 3:26 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

We had the same dream. But I realized that I wasn’t going to get anywhere with fancy words like “functor” and natural transformation getting in the way. :-) So first things first. Feel free to define “fun” between cats.

I made a slight slip by using the word ‘also’. It could just as well have been left out.

There used to be this French group Oulipo who carried out similar types of writing exercises. Quite a few illustrious names were members.

Posted by: Todd Trimble on October 31, 2014 3:36 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

Used to be?

Posted by: Charles Waldman on November 3, 2014 4:55 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

Claim: The list of whole primes does not end.
What if the list of primes does end? We build a new one! Let a, b, c, and on to p be a list of all the primes. Take a times b times c and on to p, and add 1. Call this Z. If Z is prime, it is new to the prime list, and we are done. Else, there is q, a prime, and r, a whole, such that q times r is Z. If q were on the list, then Z is q times r and Z less one is q times s, where s is a whole. Then 1 is q times a whole (r less s), which makes no sense. So q is new to the prime list, and we are done.
Posted by: Chris Rasmussen on November 1, 2014 10:45 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

Nice one!

Posted by: David Roberts on November 3, 2014 6:52 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

Isn’t step 3 insufficient, due to not ruling out cycles? It might be an idea to try to render this into Anna Wierzbicka’s Natural Semantic Metalanguage, since that’s a definite collection of resources which arguably appear in some form in all languages.

Posted by: Avery Andrews on November 14, 2014 2:13 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

de.wikipedia.org seems to say that “Georg” has two syllables.

Posted by: Chris Grant on November 21, 2014 6:35 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

Yes, I knew that, but was cheating slightly. There are words in English for which it is a little ambiguous, or dependent of accent, whether there are one or two syllables, for instance ‘while’. Georg is a bit like that. The English ‘George’ clearly has one, though, so one could substitute it.

Posted by: David Roberts on November 21, 2014 11:27 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

Christopher Rasmussen wrote a proof of the first Noether isomorphism theorem on Google+. Or, as Tim Gowers wrote:

Or, as one might say, this was the first big fact proved for groups by one who was well known for such things and, for what it is worth, not of the same sex as me.

Evelyn Lamb wrote a proof of Rolle’s theorem.

Keep ‘em coming!

Posted by: David Roberts on November 26, 2014 5:44 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

Nice! Though I would change the wording in 6:

when you give me a map from A to B, I can find a thing in B that does not come, by the rule for the map, from a thing in A.

To read:

for any map you give me from A to B …

To me it reads:

there exists a map you can give me …

Which of course makes no sense.

Posted by: AP on May 20, 2016 7:31 PM | Permalink | Reply to this

Re: Maths, Just in Short Words

Ah, but ‘any’ has two syllables. :-)

Posted by: David Roberts on July 4, 2016 1:03 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

Make it ‘each’, then. :-)

Posted by: Todd Trimble on July 4, 2016 7:53 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

Vietnamese would be useful here…

Posted by: Cosmia Nebula on July 3, 2016 9:35 AM | Permalink | Reply to this

Re: Maths, Just in Short Words

Could you supply one, Cosmia? Or know anyone who could?

Posted by: David Roberts on July 4, 2016 1:02 AM | Permalink | Reply to this

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