Is Kant's noumenon infinite and in what sense?

Question

In the Critique of Pure Reason (B306) Kant defines noumenon - the thing in itself:

If, by the term noumenon, we understand a thing so far as it is not an object of our sensuous intuition, thus making abstraction of our mode of intuiting it, this is a noumenon in the negative sense of the word. But if we understand by it an object of a non-sensuous intuition, we in this case assume a peculiar mode of intuition, an intellectual intuition, to wit, which does not, however, belong to us, of the very possibility of which we have no notion — and this is a noumenon in the positive sense.

Is it right that in this second sense: we conceive of a thing in itself by imagining we are not imagining it?

Perhaps an awful gloss. Anyway, it seems that a thing can be said to be infinite in two senses: a gradual process in infinite time; or, something which is infinite at some time.

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Assuming that all made sense:

1. Does the idea of the former "potential infinity" contain its completion?

2. Can a potential infinity never contain "in itself” its completion?

3. And supposing both 1 and 2 which kinda seem a little possible: can a noumenon be potentially infinite?

Answer 1

Critique of Pure Reason:

Kant made a difference between 1) sensible intuition and 2) non-sensible intuition. Here intuition translates the German Anschauung. Kant always uses this term for our mode of processing our sensible input. The output of this first step is structured by space and time.

Kant names the source of our sensible input thing in itself. Kant’s point: We cannot conceive the thing in itself, we cannot conceive any property of the thing-in-itself, we cannot apply any category to the thing-in-itself. The thing- in-itself is a pure hypothesis, necessary as a base to the whole process of cognition. Hence noumenon in the negative sense is the term which fits our manner of cognition.

On the opposite, noumenon in the positive sense would refer to an intuition different from human intuition with time and space. Kant states that we are not acquainted to such type of intuition and that we do not even know whether it is possible.

The whole passage from the Critique of Pure Reason does not relate to the difference between potential infinity and actual infinity.

Potential versus actual infinity:

The standard example to illustrate the difference are the natural numbers 0,1,2,…. You can continue counting without reaching a last number; that’s potential infinity. On the other hand – at least since Cantor’s set theory - you can consider the set of all natural numbers. This set is actual infinite. Aristotle accepted potential infinity but not actual infinity.

To your first question: Paraphrasing the last statement by using your terms I mean that Aristotle’s idea of potential infinity does not contain the idea of completion.

To your second question: What do you mean here with “in itself”? Does it relate to Kant’s term “thing in itself”? In any case, potential infinity and actual infinity are two different terms. Neither contains the other.

To your third question: According to Kant we cannot know anything about a noumenon, notably we cannot know whether it is infinite in any conceivable sense. We cannot even know which concepts apply to a noumenon.

In general, I consider it difficult to speak in abstract terms about infinities. But set theory provides a means for precise terms like finite sets, infinite sets, countable sets, uncountable sets and the whole variety of different infinite cardinalities.

Answer 2

I'm not an expert in textual exegesis of Kant; but the following suggests itself from a first reading of this passage:

He's distinguishing two senses of noumenon - negative and positive:

If, by the term noumenon, we understand a thing so far as it is not an object of our sensuous intuition,

We haven't sensed it (the noumenon) by eye, or by hand.

thus making abstraction of our mode of intuiting it

It's an abstraction of something sensed ie intuited; thus an intellectual idea; like that of the mathematically perfect circle from a physically imperfect circle - like the rim of a wheel.

this is a noumenon in the negative sense of the word.

Because this is from a process of abstraction by the use of the intellect, it is subjective.

But if we understand by it an object of a non-sensuous intuition

I had understood Kants use of the term intuition refers always to sensual intuition; here he is positing another kind; the only possibility that occurs to me is that of Platos Ideas or Forms; they exist in some insensible realm; thus the use of non-sensual intuition.

we in this case assume a peculiar mode of intuition

An assessment that Aristotle would agree with; if one can go by his irritation of this notion in his metaphysics.

an intellectual intuition

He presses the point that this is not sensual Intuition.

to wit, which does not, however, belong to us, of the very possibility of which we have no notion — and this is a noumenon in the positive sense.

This I find hard to decipher; for how can it not belong to us if we use this intuition to clearly and distinctly perceive the mathematically perfect circle? Is he claiming intellectual intuition is not individual and particular to each mind? Possibly - I don't know.

So we concieve of a thing in itself by imagining we are not imagining it?

I think so; so long as your second 'imagining' is interpreted as abstraction; which of course is a process of imagination.

Answer 3

Someone at my university said it's a good point but you can't make those sorts of claims for noumemon.

A. I can imagine that a finite series is an actual infinite

1. but for an actual infinite series to really exist, time must be an actual infinite
2. a potentially infinite series cannot exist if time is an actual infinite
3. any potentially infinite series depends on an actual infinite series

C. no potentially infinite series is real.