It is a fine thing to have a paradox to ponder.  This is one that I heard as gossip when I was on the math competition circuit in my high school days.

For today, I mean “number” in the sense of natural number.  0, 1, 2, 3, and so on.  All the things that might be the answer to a “How many” question.

Consider all the ways to name numbers with English words.  So there’s “four”, of course, for 4, but there’s also “seven minus three” and “two squared” and “the smallest number that is not a factor of thirty” and lots of other things.

Maybe we even accept things like “the number of legs on a healthy dog” or “the number of sides that a trapezoid has”.  The most important things are that names consist of strings of English letters and spaces, and must unambiguously refer to a single natural number.  We could make various decisions about what kinds of operations are allowed, and what context is assumed, whether time-specific things such as “the number of letters in the middle name of the president” count, etc.  The exact details of the conventions we take here are not important, so let’s just suppose we’ve laid down what it means to have a “name” of a number.  A number can have many, many names, and a string of words may or may not actually be the name of any number.

Now let’s think about what happens if we limit the length of our names, say, to 60 letters maximum.  Then the total number of combinations of at most 60 letters from the English alphabet is a finite (enormous) number.  Even if you factor in where the spaces go, it is still finite.  You could make a list of all the strings of spaces and letters containing at most 60 letters and no two consecutive spaces.  (It would be a long list, but a finite one.)  Now, the vast majority of these strings are unpronounceable garbage like “akljta df grt ar sfg rt dfg”.  Of those consisting of bona fide words, the vast majority are nonsense like “whereupon cat clavichord”.  Of those that are discernibly a noun phrases, the vast majority do not name a number, like “a cloudy day”.  Of those that do name numbers, the many are duplicates.  When all is said and done we have only found names for finitely many different numbers.  If you had time enough, you could actually figure out which numbers they are, and you could figure out which is the first number not among them.  In other words, you could, in principle, identify the first number which cannot be named with sixty letters or fewer.  What an interesting number that must be, huh?

But then “the smallest number that cannot be named with sixty letters or fewer” is a name for that very number, and we have done it with just 57 letters.

Discuss.

(There seems to be some ambiguity over who first suggested this paradox; I’ll shout if I can find something definitive.)

### 3 Responses to The Paradox of Naming

2. mouse says:

In a non-numerical sense, you just described “Octospeak”. When people hit ~80 years, it becomes more difficult for them to call to mind the “proper” name for particular objects or phenomena. This leaves them free to describe life with particularly colorful and/or poetic phrasings.
We all do this all the time, but it becomes more pronounced when you’re old. It’s fun to decipher “second head of hair” (wig) and “things that spiral and create devastation” (tornadoes).
There is a town somewhere in Europe where people developed an alternate speech form for fun; “knitting” could be referred to as “Mousing”, or a knit object as a “Mouse Courtois”. People continually change, expand, and replace signifying words, forcing the others to problem solve and decipher to keep up. It’s English, but an unintelligible English to the vast majority.
Naming variation is incredibly human; I guess that’s why we have so many languages, and why they get bigger at such an alarming rate. Unlike computers, we can make certain experience specific leaps of logic and judgment that would allow a particular group of people to understand “Garth and Sally’s Children” as ten, or “consecutive S-names in direct descent” as five.
“Whereupon cat clavichord” could be shorthand for “whereupon I found that dirty rotten cat sleeping on my clavichord”, which could be a named phenomenon like “Feline Sopor Obstructus”. Or it could be 930, or 9,310 depending on whether “Clavichord” represents “multiply by ten” or simply “ten”. 313 or 7, syllabic structure? 212 or 5 for signifiers that can be logically separated, depending on groupings?
All in all I’m not really sure I’m helping your argument, or even on topic. “The smallest number that cannot be named with sixty letters or fewer” is impractical and hinders productivity. There is too much room for error. Most people will declare “the number of legs on a healthy dog” to be four, but what about the person whose dog lost a leg? The missing leg does not determine the overall health of the dog. This is the problem with alternate names; it is hard to make them totally universal. On the other hand, isn’t the general assumption that most dogs have four legs part of why we can apply math to life?

• Cap Khoury says:

Eek! There’s a mouse in my blog! (Two, if purple mouse was around when you made the comment.)

Welcome, cousin. I see you’ve brought a whole lot of things into the picture.

And here I thought I was the only one who used the term Mousing to mean knitting!

You’re absolutely right about the way these poetic and elaborate names for things enter the language when we don’t recall (or don’t wish to use) some certain word. This happens often to me, not so much with old people, but with foriegn mathematicians whose command of English is exceptional in the realm of number theory but more limited in the realm of where to buy an umbrella.

I like what you have to say about naming variation. This is not only why we have different languages, but why the structure and hierarchy of words and ideas is not the same in each, why learning a new language is not just word-for-word translation. Different groups of people (even within a language group) conceive of the way words map to ideas and vice versa in very different ways.

This is a big topic, and an interesting one, and one worth pursuing further (but probably not in this blog, since it’s diverging rapidly from the mathematical core of the post).

Getting back to the mathematics, you have demonstrated why we need a lot more side conventions to make what I’m saying make sense. We would need to agree on an ad hoc, unambiguous “dictionary”, just for this purpose. Assume we’ve done that, and assume that the dictionary is simple enough to rule out ambiguity but large enough to include basic mathematical operationas and concepts, and the definition of “name” as used here. It’s not hard to imagine such a thing Then I’ve argued pretty convincingly that “the smallest number which cannot be named with sixty letters or fewer” is a legitimate name for some number. Figuring out what that number is would take preposterously long, but in principle, it seems, it could be done. And whether or not I know what number it actually is, we have a problem, since this number defined by the fact that it doesn’t have a short name, does in fact have a short name.

> On the other hand, isn’t the general assumption that most dogs have four legs part of why we can apply math to life?

Something like that. It’s certainly true that mathematical theorems require assumptions when applied to life, and the simplest theorems usually require substantial simplifying assumptions. It’s even more certainly true that applying mathematics requires a certain consistency of how we interpret words. This issue is a big part of why applying mathematics to real life can be confusing. Maybe I use “healthy dog” to necessarily mean four-legged dog and you do not. Then “in a cluster of healthy dogs, you can find out how many there are by counting legs and dividing by four” is an absolutely true statement for me. But not for you. This isn’t a logical contradiction or anything, because it’s not really the same statement when I say it as it is when you say it. If we were having a serious discussion on the subject, say because we were collaborating on a paper for the prestigious Journal of Animal Counting, we would just set up a convention to fix this. Either you would agree that “healthy” as applied to dogs implies “four-legged”, just for the purposes of talking to me, or else we would agree to use a different word to mean “healthy in the Mouse sense and having four legs”, or something. (This happens every day for mathematicians, by the way, when they collaborate with colleagues from other countries or institutions who use language slightly differently.)

This is also why real-life discussions are so much more fraught than mathematical ones. To even have a rational conversation, to interact as reasoners, we must agree on the terms, at least as they are being used in the conversation, “for the sake of argument”, as they say. But in, say, the gay marriage debate, the various parties do not use certain key terms in the same way, and explicitly refuse to do so. So it’s not much of a surprise that very reasonable, very intelligent, very coherent people can fail utterly to have any kind of meaningful discussion that doesn’t degenerate.