Talking in Tuples

Mathematics is the *Most* Different Language.

In Can we make mathematics intelligible?, R P Boas jokes:

There is a test for identifying some of the future professional mathematicians at an early age. These are the students who instantly comprehend a sentence beginning “Let X be an ordered quintuple (a, T, πσ, 𝔅), where …”

I’ll try to explain what mathematicians mean when they write this way.


Think about a set containing the letters {A, B, C, D, E, F, G}. As written it’s equivalent to the set {F, C, E, G, A, B}, so the set doesn’t communicate the order information we know “should” go along with these letters. To express that, we should talk about the pair ( {A,B,C,D,E,F,G}, 𝓞) where 𝓞 is the ordering A < B < C < D < E < F < G.

Would it have been clearer if I’d pasted the definition of the ordering into the interior of the pair, instead of using 𝓞 as a shorthand? I’m not sure. Part of the way you have to learn to read mathematics papers is mentally substituting shorthands for definitions wherever they appear.

Since no one reading this has to look up the enumerative definition of the alphabet, let me just use the shorthand 𝓪 for the set containing each of the letters and 𝓞ʹ for the well-known ordering of the letters A < B < C … < X < Y < Z (remember, I already used 𝓞 so now I have to add a prime to differentiate this new, larger ordering). Now I can just write (𝓪, 𝓞ʹ) for the ordered alphabet.

The Next Letter

So what if I wanted to talk about “the letter after Q” ? Using the current pair (𝓪, 𝓞ʹ) this concept is undefined. In order to include “after” as a concept in the “space” I am developing, I need to expand the pair to a triple.

Now, how should I add in the concept of “after”? I could parsimoniously add only the +1 operation. But I may want to talk about “the fourth letter after Q” as well. Should that be four iterations of +1 (i.e., 𝔰∘𝔰∘𝔰∘𝔰 where 𝔰 is the successor function)? It will be annoying enough to write out a definition of 𝔰 that clearly states “C = B+1,   S = R+1,   ” and so on. Deary me. I wouldn’t want to have to enumerate that for +3, +13, and so on. I don’t have an infinite amount of either ink or patience.

I’ll leave it to function composition  and just define the output (image) of the +1 operator 𝔰 as “The letter to the right of the input, under the ordering 𝓞ʹ.” That doesn’t sound formal enough to be correct, but I’ll stop there.

An Ordered Triple

Now “we have” a triple (𝓪, 𝓞ʹ, 𝔰) containing a set 𝓪, an order 𝓞ʹ, and a function 𝔰. This is still the alphabet we’re trying to talk about here, right? In fact I’m starting to doubt if even a triple is enough, because the triple doesn’t contain either < or ∘, and those are symbols I’ve used so they’d better belong to the universe.

So I’ll say the alphabet is defined as a quintuple (𝓪, 𝓞ʹ, 𝔰, <,) containing a set, an order, a unary function 𝔰, and two binary functions > and . Phew. Please tell me I’m done!

You know what, I just thought of something else. What about the letter before? Argh! It’s so simple (the alphabet) and yet so difficult (defining an appropriate k-tuple). Alrighty, I learned a shortcut in school for this: I’ll define an inverse operation 𝔰¹. And everybody knows what I mean before I say it so I’ll just stop here.

Now, consider the alphabet. The alphabet is defined as a quintuple (𝓪 , 𝓞ʹ , 𝔰 , < , , 𝔰 ⁻¹) . Or maybe I should say it’s a triple? (𝓪 , (𝓞ʹ, <) , (𝔰, ∘, 𝔰⁻¹) ). One has options.

I still haven’t captured everything we know about the alphabet. I couldn’t do Excel spread sheets with Z < AA < AB < AC < … < AZ < BA < BB < BC < … BZ < …. Nor could I take account of cryptographic rules where Z loops around: Z+1=A, and A < B < C < … X < Y < Z < A (←a non-wellfounded set). I didn’t include the Alphabet Song or the pronunciation of the letters (have you been saying “zed” or “zee”?), nor did I include vowel/consonant classification, rhyming info, or an IPA-style breakdown of the phonemes each letter can make (and in English there are many phonemes per letter). But I did include a bunch of known information about the alphabet into the logical universe .

So here’s the point of this example. Even to express a simple concept that everyone knows — the alphabet — as well as what are normally implicit mappings and relationships — you have to explicitly include those facts in a tuple to be logically complete.

Mathematics is a totally different language than English. It’s more different from English than is Mandarin, Pormpuraaw, Tagalog, Aymara, Farsi, or Pirahã. That means you can think different thoughts once you learn mathematics. You can fathom what was unfathomable. Conceive what was inconceivable. See what was invisible. It also means that learning to “speak” this way sounds very strange.

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