You have a child. What do you name them? David or Jason to be willfully non-descript? Brittany or Simone to grab a bit of aspirational French classiness? Emily or Enid in an attempt to be quirkily old-fashioned in the vain hope that everyone else in your cohort isn’t thinking the same thing? Maybe you go with Rachel because that was your grandmother’s name. No name is a blank slate–each trails connotations behind it. Don’t overthink it though, because your child’s personality will soon overwhelm any baggage the name comes with. Meaning is not destiny. What is essential is that the name you give be distinct. A running gag in the second Newhart series was that two brothers in a backwoods New England family were both named Daryl. “I’m Larry, and this is my brother Daryl, and this is my other brother Daryl,” was the recurring line from the only one of them who spoke, the joke being what parent would subject themselves to such needless confusion?
You have a bank account. In this account you have $1373.56. Then you use your debit card to buy a taco and coffee for $5.23. Now you have $1368.33 left. It is crucial for the functioning of the economy that we be able to distinguish between these two financial states. So what do we call them? How about we call $1373.56 “David” and $1368.33 “Emily”? You see the problem? Along comes another guy with $1368.32 in his account. What do we call that– “Ringo”?
No. A better system is to call $1373.56 “1373.56”, $1368.33 “1368.33”, and $1368.32 “1368.32”. Numbers are names. If your society is simple enough to tolerate a bit of “Daryl”/“Daryl” ambiguity then you can get by without them, but at a certain point you will require a deep common well of distinction. Numbers fit the bill. “1” is different from “2” is different from “243” is different from “136823” (which is just “$1368.23” in disguise). The well is deep–infinite, in fact. There will always be a name for each new thing.
But is different all that numbers are? Do they, like people’s names, come with baggage? Certainly the first few do. Concepts like unity and duality have significance outside mathematics, and also line up neatly with the numbers “1” and “2”. Say “3” and people will think triple play, menage à trois, Father/Son/Holy Ghost, what have you. We could play this game for a while with the small integers, but the connotation atmosphere thins rapidly as you ascend. Quick, what cultural associations does “2,732,409” bring to mind?
So then do big numbers achieve a kind of semantic Zen state, blanched of meaning? Are they pure distinction, unencumbered by connotation?1 Not entirely. There are, for examples, statements you can make about 2,732,409 besides “2,732,409 is not x” where x is any number not equal to 2,732,409. For starters, 2,732,409 is 1653 squared and is itself the square root of 7,466,058,943,281. It has prime factorization 32 ·192 ·292. It is smaller than 2,800,000 and larger than 53. And so forth. If you set out to create an exhaustive list of facts about this and all other numbers, the list would be called “Arithmetic” and you would never finish.
Even if it lacks semantics, arithmetic has structure.2 It’s easy enough to create a formal system that defines an infinite set of numbers but willfully ignores any arithmetic facts about them, but that’s just putting your hands over your eyes.3 A system of pure distinction without any imputable structure could maybe be achieved by inventing a new nonsense sound every time you wanted to name something, but that would quickly get confusing. “I forget–is the guy with the hat a weegle or a blurgh?” Infinite arbitrariness is undermined by finite working memory. If you want infinity (or even a moderately large number of things) a bit of structure is the price you pay. Everything has baggage.
1 I almost asked if they were denotation without connotation, though arguably a number that denoted only itself–along with perhaps the fact that it wasn’t any other number–could not be said to be properly denoting anything, though that is a controversial sidetrack I want to keep safely quarantined in a footnote.
2 Let’s also duck the question of whether that’s a valid distinction.
3 “I will countenance Peano’s axioms but go no farther!”