Number words come in various categories. There are the cardinal numbers (in English: one, two, three...), the ordinal numbers (first, second, third...) and the adverbial numbers (once, twice, thrice). There are also other number-word categories, but my special interest here is in the adverbials.
On the face of it, the cardinal numbers seem more linguistically primitive in the sense that they constitute the basic form upon which the adverbials are built. For example, the Middle English ones (= once) is an inflected form (genitive) of the Middle English word on (= one).
In Latin the situation is slightly different. The first few adverbials (semel, bis, ter, quater) are not obviously derived from - though all but the first are related to - the equivalent cardinals. Semel comes from the Proto-Indo-European *sem.
But irrespective of which category of number represents the earliest form linguistically, a case could be made that mathematically (and logically) the adverbial form is the basic one.
This is just a preliminary idea and I don't want to make too much of it. Also, I am aware of the use of cardinals and ordinals in set theory which makes it more difficult to make my point clearly. I am not talking set theory here.
The idea that the adverbials are somehow basic attracts me because it seems to provide a way of looking at numbers which tends to undermine (or at least not encourage) mathematical Platonism.
Focusing on the cardinals encourages mathematical Platonism because, even though it was through counting actual things that cardinal number words no doubt arose, their usefulness lies in their not being tied to any one kind of thing and so being applicable to anything countable. Inevitably, the cardinal numbers came to be seen as objects themselves, existing in an abstract realm.
If, on the other hand, we see numbers as being based on, or deriving from, the iteration process, then our focus moves from static objects and a timeless Platonistic realm to the ordinary world we all inhabit, of processes or actions which may (or may not) be repeated.
Interestingly, not only the Latin word semel (= once) but also the English word 'same' derives ultimately from the Proto-Indo-European *sem.
It's odd to think of certain modern English expressions as having such an ancient lineage.
"Same again," for example. (Licensing the re-execution of a previous order, a particular drink at a bar, say, and thus requiring the barman to go through roughly the same motions twice (or thrice...).)
This way of conceptualizing number is quite as natural as counting apples or oranges, and may, as I suggested, provide a good basis for a non-Platonistic and altogether more satisfactory way of understanding mathematics.
Modern mathematical Platonism is a long way from Plato, but it shares with Plato a static view of (mathematical) reality. It is at odds not only with the dynamic character of ordinary life and experience but also with the new ways of looking at things which the digital revolution of the last century has encouraged.
(I am currently looking at how the ideas of one of the great twentieth-century logicians, Alonzo Church, may relate to this notion of number as iteration and to mathematical Platonism. More later, perhaps. It's heavy stuff and may not be worth the trouble!)
