On infinites, esp. Words

Aristotle distinguishes two accounts of the infinite

a.)  that which has nothing outside

b.)  that which always has something outside.

He says (b) is the correct account.

Proof: if “infinite” were “an X which had nothing outside”, we could think of an X that had nothing greater – namely the infinite one. But calling numbers or spaces or time infinite means recognizing there is no number, time, or space for which a greater is inconceivable.

I want to tie this familiar sort of infinite to another one that is just as infinite, namely, the possible uses of a term. We can’t enumerate all possible uses of… (I’m pulling out a random book and scanning for the first noun)… “School”. Miriam-Webster gives five noun meanings, but others were left out and others will arise, and this is before we include all the ways the term might be used metaphorically, ironically, euphemistically, in puns, malapropisms, idioms, Freudian slips…

My point is not that the uses of the term could actually go on forever, but the word as such has no definite limit.  So, for example, the fact that languages constantly melt into dialects into new languages sets an extrinsic limit to how many uses a term can have, but the word as such is as infinite as numbers or space, since no matter how many uses one enumerates there will always be another possible use outside them. Moreover, time and space are infinite in exactly this way: there will never be an actual infinite, but the mind in knowing these things knows it can never conceive of a last one, but, as we said, there will always be another outside. 

While sentience is infinite in the sense that we can always see one more blue thing or taste one more salty one, words, numbers, and space are not infinite in this way. To see one blue thing after another is not the same as seeing an equivocal meaning of a term, forming another species of number, or a space that contains another and so bestows a form on it (i.e. place). Carving out this new sense of the infinite points to its spirituality, though describing that will have to wait for later.


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