Sunday, December 30, 2012

Reconciling Rationalism and Empiricism: the Infinite, Finite and Indefinite

Considering the relation between the infinite, finite and indefinite can help overcome the difference between the Rationalists and the Empiricists. This is a rough exploration of this possibility.
A well known way of thinking about the indefinite is to consider a series that has no end, for example, the series of all natural numbers. We can imagine counting this series without having to stop. What does this "without having to stop" consist in?
In some way, it is a characteristic of a natural number is that they are finite. Because of this quality finitude, when we think a number we can always think a larger which will also have the quality of finitude. Since finitude is figured into every natural number in advance, it would be a contradiction to suppose any largest in the series.
From the above, Empiricism concludes that there can not be an absolute largest in the series of natural numbers. We are restricted to experience in our discovery of natural numbers, but we can think no reason the series could end, so we can say the series of natural numbers is indefinite, but not infinite.
This can appear to be a sufficient refutation of various Rationalist arguments that employ the infinite to prove the existence of absolute substances, causes, ideas, &c. We have no justification for saying that the series terminates in an absolute, since the series does not admit of any absolute member.
For example, Leibniz' Principle of Sufficient Reason demands a ground (reason) for everything. Leibniz uses this to argue that in order to have something there had to be an absolute (infinite) ground. From the perspective of a series, this is faulty reasoning, since we cannot see how any particular thing supposes a ground that is absolute rather than finite, and so an Empiricist would reject Leibniz' Principle of Sufficient Reason.
However, if we adjust our view we might see a different way of understanding Leibniz' principle.
When we count our series of natural numbers we can understand that we can count on forever without finding a largest. What grounds this feeling of being able to count on forever? If we turn to the individual numbers, and say that with a natural number we already understand that they are finite, and that we can count higher, we are not in a better position, since we can still ask: what grounds are there for thinking this quality of finitude?
If we understand the discussion of the infinite (absolute) as trying to understand how finite things attain this quality of finitude rather than how they appear in a series, then we can understand the discourse on the infinite as simply saying, we think the finite in contrast with the infinite. In this case, we are not thinking infinite as the largest finite thing - which is a contradiction - but rather we think the infinite strictly as the ground for thinking the quality of finitude distinctly (distinctness has to do with thinking something with a specific different).
If this possibility is employed in understanding the Principle of Sufficient Reason, then we see that the absolute (infinite) ground does not need to be thought in terms of the cause furthest in the past, or first in the series, but rather the absolute ground is that through which we comprehend the finite aspects of the series.
Kant thinks along similar lines in the Critique of Pure Reason.In the resolution to the Third Antinomy - which concerns the thesis, "there is a first cause", and the anti-thesis, "there is no first cause" - Kant concludes that both are true, if you allow yourself to think of them as operating in different ways.
Kant thinks cause in the following way: whenever something happens, something is always presupposed that came before. From this it is easy to see the necessity of an always prior happening. However, Kant asks if we are required to think of the prior thing in terms of a happening. His conclusion is that there is no such requirement, and so we can think the prior thing in a different way. Whenever we think the coming before in terms of happening, we also suppose a further coming before. But, when we do not think the coming before in terms of happening, then we do not suppose an earlier happening, but an intelligible ground. In the first case, cause is thought temporally as happening, and in the second as ground, which is not temporal.
To illustrate thinking a prior thing without thinking it as a happening, consider a basic natural law, such as gravity. When we see an object move towards the Earth, we see it in terms of the series of happenings, this series is an expression of the law of gravity. But in thinking gravity as a ground, we think it as 'something' spontaneous - not in time.Gravity, as a ground, is not an appearance, or a happening; we can't posit existence for it, but it is intelligible.
Applying this back to Leibniz, we can see that his manner of thinking the absolute (infinite) ground could be seen in terms of a law thought as an intelligible something, and not in terms of a most prior element in the series of happenings. According to Leibniz, then, in order to think any element in a series in its particular dependent way, we also think of merely intelligible thing which grants it this dependent status. An element of a series is dependent upon its prior cause in one sense, but in another sense it is dependent on the particular way it is conceived of as dependent.
(Generally, I think a resolution between the Empiricists and Rationalists using a syllogism as a model:
Major Premise
Minor Premise
Empiricists seem generally concerned with the validity of conclusions. Rationalists seem to be concerned with how the major premise (rule) can apply to the minor premise (case).
Asking into the validity of this or that conclusion is asking into the a posteriori. The accounting for the connection of the rule to the particular is asking into the a priori. The former considers cause in terms of the series of events, the latter considers the cause from the perspective of how the rules are applicable at all a priori - in terms of 'logical' ground.)


Anonymous said...

It seems you are assuming that laws are static (not implying temporality). But if laws are formulated as "if x [happens], then y [happens]," then this argument loses force. Furthermore it is problematic to discuss the meaning (or ontological status, as it were) of a law in absence of its operation. For instance, what is a mathematical limit if not a law "of movement?" The same could be said of infinity. If we conceive of natural numbers as the output of a "+1 generator," then I don't think an empiricist needs to posit a limited number of them. (He can continue counting even after all the jelly beans are exhausted from the jar.) Perhaps he needs to abstract from temporality in order to make that "leap" from indefinite to infinite, but that is not the same as saying that the finite meaningfully "depends" on the infinite.

Erik Christianson said...

Thanks for your comment. I hope I can clarify some of this and revise in light of it.

1) I'm happy formulating laws the way you do as explaining a temporal connection. I am trying to consider how the 'lawfulness' is also thought in terms of the ground for thinking the connection. So, we experience gravitation (temporally), but we also think gravity in a way where it is not given (as an appearance) in the sequence. Gravity (as opposed to gravitation) is only intelligible (we don't know how to tell if it is a possible or impossible object of experience, since we just think it). Gravitation is experienced as the sequence comprehended in a certain orderly way.

2) I didn't mean to suggest that an empiricist considers himself limited to however high he has actually counted. We have a sense of being able to go on; it was this sense that I was calling into question.

Empiricists do say there is no ground for deducing the existence of the infinite from the finite, just as you cannot deduce a necessary connection of events. The Empiricist does abstract a notion of infinity, but we need to consider how one performs such an abstraction. I am basically suggesting that the manner in which this abstraction works for empiricists is best explained by the Rationalist treatment of the infinite. I am also suggesting that the limits the Empiricists recognize for deduction of the infinite be applied to the understanding of the Rationalists.

Anonymous said...

I understand you to be saying that we need to comprehend infinity in order to count to 1 (i.e., finitude = not infinity, or finitude = infinity + limits, or some such derivation). Of course this will not appeal to an empiricist who claims it is the other way around.

One reason is that the argument seems to equate finitude with countability. It is easy to say that "infinite" is the opposite of "finite," but uncomfortable to say it is the opposite of 1 (and every other natural number). Even assuming we need to comprehend finitude "distinctly," that seems like a category error.

Erik Christianson said...

I think I am better understanding the difficulty now.

When I said, "we think the finite in contrast with the infinite", I didn't mean to suggest that the infinite is the opposite of every individual natural number in terms of a negation. I apologize for the confusion. I am suggesting that Rationalists think natural numbers as finite relative to a ground, and not to the series.

Using the Principle of Sufficient Reason (PSR) we could argue that there must be a ground (reason) for why natural numbers are finite. No natural number has a relation to another as ground, so it must be something else outside the sequence that grounds it - something not limited by the series, something in-finite. The reality (or lack thereof) attributed to this ground I can leave aside for now.

An Empiricist will argue, correctly, that no matter how far we count we feel like we can count further, and that we cannot justify the infinite a priori with our inability to reach the end. The emphasis here is on the series as indefinitely great, and not the nature of the members of the series. The PSR tries to address a concern with the nature of any member of the series.

We can ask, and should ask, what grounds the PSR, or other similar Rationalist procedures. Here I was just concerned with opening up the possibility that a great deal of difference perceived between Rationalists and Empiricists is simply the result of failing to understand the strategies employed by either side. Perhaps the Rationalists didn't even understand their own procedure as well as they could have.