Thursday, June 3, 2010

Leibniz and Cantor on the infinite

If you're in an auditorium and you notice that no one is standing but all the seats are taken, it makes sense to conclude that there are exactly as many people in the room as there are seats. Leibniz concluded the same thing about infinite numbers, should they exist. Assume an exact one-to-one matching between the natural numbers (1, 2, 3, 4 ...) and the even numbers (2, 4, 6, 8...) such that:

1 2 3 4...
| | | |
2 4 6 8...

For each even number, e.g., 234,168, there is a corresponding natural number, viz., 117,084. And yet, the set of even numbers forms a part of the set of the natural numbers (the odd numbers and the even numbers). If a one-to-one matching is possible, then the number of all natural numbers is not larger than the number of all even numbers. The whole is not larger than the part.

According to Martin Davis in Machines of Logic:
Cantor reasoned much as Leibniz had and faced the same dilemma: either it makes no sense to speak of the number of elements in an infinite set or some infinite sets will have the same number of elements as one of its subsets. However, while Leibniz had chosen one horn of this dilemma, Cantor chose the other. He went on to develop a theory of number that would apply to infinite sets and just accepted the consequence that an infinite set could have the same number of elements as one of its parts.
I don't know who was right in this, Leibniz or Cantor. Cantor had his detractors and still does. It's far from a settled dispute. What intrigues me is the option of revising logic. Few things are more self-evident than that the whole is greater than the part. And yet Cantor felt compelled—by the subject-matter of mathematics itself—to reject this assumption. What does this imply about the relation between mathematics and logic? Which is a branch of which? And if it's the case that one is a branch of the other, how can they be in such contradiction?

I felt a similar tension the first time I learned about Bell's inequality. I know others have different intuitions, but I couldn't help but feel there was something flawed in classical logic. It's as though the logic of the whole were somehow independent of the logic of the parts. And assuming transcendental grammar projects transcendental semantics, then there is a metaphysical independence and priority of the whole from and over the parts.

There are a lot of interpretations of mathematics and quantum mechanics. I wouldn't claim anything definitive or with certainty. Nevertheless it seems to me that these examples show that logic (and hence metaphysics) is not a mere given but at least at the moment is a matter of interpretation. And furthermore it is not merely a priori but is subject to experimental result and historical accomplishment. Meaning we may ascend to some point in the future where our transcendental grammar and transcendental semantics are radically different from what they are now.


Anonymous said...

What I like most about this aspect of number theory is the existence of different infinities, of different sizes. For instance, Cantor also demonstrated that there is no one-to-one correspondence between real numbers and natural numbers. Cantor concluded that there are infinities of different sizes, with the infinity of real numbers is larger than the infinity of natural numbers. What does this say about our very notion of "the whole" if we have unequal sets of infinity? Is "the whole" even a meaningful concept in number theory?


der Augenblick said...

If Cantor is right, then it makes sense to speak of a completed infinity and to speak of the size of one whole infinite set versus the size of another. Leibniz and Gauss - and pretty much everyone since the time of Aristotle - held that there was no such thing as a completed infinity, and so it makes no sense to speak of the whole set of natural numbers or real numbers. One can only speak of increasing a set indefinitely, by adding a unit, and then another, and then another, etc. So (according to Leibniz and Gauss) there are indefinitely large sets but no completed infinite sets. But this is precisely the assumption Cantor believed he was refuting. So yes, I think from Cantor's point of view, it does make sense to speak of the "whole" as applied to infinite.

Jim Macdonald said...

If I read Leibniz correctly, he is not arguing that the set of numbers is "indefinite." Like Cantor - and unlike Aristotle - he is arguing that infinity is actual. But, unlike Cantor - and like Aristotle - infinity is not therefore a number (that is, there is no "set of numbers"). It isn't that we can't say how high the numbers go; it's that the numbers actually never stop.

Countining numbers is an infinitey recursive algorithmic process; it is quite actual. Yet, you cannot express it by a number. There is a fascinating paper on this by a scholar named Richard Arthur. See

der Augenblick said...

Thank you for the thoughtful comment. I'll be sure to look at the paper.

My synopsis was based on information in Martin Davis' book Engines of Logic. You may want to forward your remarks to him. He was kind enough to return my e-mails when I asked him about his book.

Best regards...