How mathematical operations require the infinite: they don't require an explicit affirmation of the infinite any more than reasoning requires an explicit affirmation of the principle of non-contradiction. But they do require an openness to the applicability of the form of the operation to more numbers. And if numbers had a limit, X+1, where the operation would not work, then our knowledge about the form of the proposition would not be genuine knowledge. We wouldn't even be able to know how to apply the operation to known numbers IF we denied that it could apply to others. An openness to an unbounded applications is necessary in principle for us to be able to know what we do about numbers within the bounds that we have found them so far.
Here is a summary and comments on the essay Freedom and Resentment by PF Strawson. He makes some great points, and when he is wrong, it is in such a way as to clarify things a great deal. My non-deterministic position is much better thanks to having read this. I’ll summarize it in this post and respond in a later one. In a nutshell: PFS first argues that personal resentment that we may feel toward another for having failed to show goodwill toward us would have no problem coexisting with the conviction that determinism is true. Moral disapprobation, as an analog to resentment, is likewise capable of coexisting with deterministic convictions. In fact, it would seem nearly impossible for a normally-constituted person (i.e., a non-sociopath) to leave behind the web of moral convictions, even if that person is a determinist. In this way, by arguing that moral and determinist convictions can coexist in the same person, PFS undermines the libertarian argument ...
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