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.
How about defining it as an instance of functional complexity? And treating the list of properties as following necessarily from this definition? Okay, this is very diamond-in-the-rough (or perhaps zirconium...) but here goes my thoughts on the matter: (Complexity) Many diverse parts (functional) acting for the sake of the whole I think I need to add something like "not as the instrument of another" or something like that. Given that they act together as one whole and given entropy, etc., the organism will need to posses more order than its surroundings (homeostasis). It will therefore need to take in energy (nutrition), which it will use to sustain its readiness to interact with its environment so as to preserve its own being (homeostasis again?), but which will eventually break down (death), so that in order for that life form to continue it will need to duplicate itself (reproduction), which, upon occurring, will involve both development (growth) to maturity and...
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