The indefinite and the infinite

- Yesterday, while explaining the possible translations of Latin nouns, all of us fell into a collective astonishment that the indefinite article cannot be plural. You can’t have “a forests”, though this is not because the article is itself singular (there is no mystery to why you can’t say “that forests” or “this planets”, but the impossibility of “a planets” is not the same).

(N.B. see the comments for a discussion of “some” which is indefinite while being both singular and plural.)

-The first explanation anyone ventured was that the article was somehow infinite, and there is only one infinite thing.  Someone else objected with a second-hand account of Cantor’s multiple infinities, though everyone saw that we had to give some account of why the idea “infinities” made sense, and so we had to allow that they could be somehow many.

-The indefinite and the infinite are not the same. Mathematics gives a particularly sharp division: it is not the same thing for things to be infinite as it is for them to be undefined. “Undefined” means something like “there is no way to incorporate the thing into an operation” (say, the idea of dividing X by zero, since we’re not sure if the result should be X, or infinite, or zero). Whatever infinite means, it’s not this – even though we don’t know how to deal with “infinity” as though it were exactly the same as a finite quantity (applying basic algebra to ∞ + 1 = ∞ gives us 1=0).

-The infinite is the negation of a terminus or limit; the indefinite is the incompletion of the first act of the understanding. Modernspeak: infinite is either metaphysical or a broad category including the metaphysical and the epistemological; the indefinite is epistemological.

-For St. Thomas, two infinites bookend reality: on the one hand there is an infinite which is such by having no potency at all; on the other hand there is an infinite that is purely potential. David of Dinant identified them.

-This meshed with a text I was reading from Giovanni Gentile, arguing that consciousness was infinite, since for it to recognize anything is to place it within its ambit, and so any possible limit would be placed within it. The point can be generalized to all that exists: no living thing has a terminus that is a part of its being – death is not a part of life; and the corruption or analysis of any form is not a moment in the existence of that form.

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5 Comments

1. everydayasceticism said,

September 18, 2013 at 9:11 am

No to nitpick, but what about “some”? I guess it’s not an article, but it is an indicator of indefinite plural.

• James Chastek said,

September 18, 2013 at 10:15 am

I hadn’t thought of “some”, which can be said of both one individual and many (certainly the Latin “quidam” can be translated as either “a” for the singular and “some” for the plural). Significantly, this comes up in St. Thomas’s discussion of persons, where a person is some man.

To adjust the point though, I think the amazement was more over the idea of the opposition between the indefinite and the definite, rather than the opposition between the singular and plural. I don’t think either I or the kids saw this, but I think this is what it comes to. Taken in this sense, the definite marks out our coming to a totality or an actuality, while the indefinite is our not coming to a totality or an actuality. By this account, God (at least to his own intellect) is a definite infinite whereas matter or potential (even to the divine intellect) is an indefinite infinite.

• everydayasceticism said,

September 18, 2013 at 11:04 am

According to Google:

determiner
determiner: some

1.
an unspecified amount or number of.
“I made some money running errands”
2.
used to refer to someone or something that is unknown or unspecified.
“she married some newspaper magnate twice her age”

I’m not sure how “determiners” are distinct from articles (I admit to never having heard of such a part of speech), but it is interesting that it seems to preserve the possibility of being taken singular or plural that you mention with regards to Aquinas.

2. rigadoon said,

September 18, 2013 at 9:42 am

“Indefinite” does not mean “undefined” in mathematics. It means something that has an arbitrary element. An indefinite integral is defined within a constant so the constant part is arbitrary.

• James Chastek said,

September 18, 2013 at 10:18 am

“Definition” can mean more than one thing, as can its negation.