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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.