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Speculation and the fertility of axioms

Whitehead says many times in Modes of Thought that his conception of philosophy is tied to the discussion about the scope and the adequacy of principles. I tried to explain this today resorting to ways we understand axiomatic procedures to expand conclusions (from axioms and inferences thereof). From a point of view that privileges certainty and avoidance of error, a set of axioms has to be appropriate in itself independently of the theorems it yields. To be sure, if axioms cannot be self-evident, at least reasons ought to be given in favor of each of the axioms picked. However, if we say to a mathematician engaged in axiomatization of her field that in case of doubt the axiom of choice shouldn't be assumed and ZF should be preferred to ZFC, she will most certainly complain that without the axiom of choice she cannot prove too many things - it is fertile and that fertility is shown in the field, that is, in the practice of demonstration. If we try and persuade any mathematician to go intuitionistic - maybe to play safer - the answer will be similar: intuitionism reduces to much the scope of what can be proved, and this is to its detriment. From the point of view that privileges certainty and avoidance of error, such responses could prompt outrage: "Why, if you just prefer to prove more, your efforts lead to no more than those of the players of a game like chess!". At which point the mathematician (in both cases) would be offended. And the speculative approach could explain why she is right to be offended. It is, to be sure, an answer somehow reminiscent of what Penelope Maddy once called naturalism in mathematics.

The speculative explanation would go as follows. ZFC is better (or classic mathematic is better) because it enables one to see broadly by proving more. It sort of surveys more ground. If we're not focused on certainty and avoidance of error, we're interested in axioms that are fertile, that can give us more insight about how things are articulated. Proofs are instruments to give us insight, more than they are advances into certainty. To be sure, it is doubtful they can be advances into certainty if the axioms they start with are themselves less than certain. But they can enjoy a surveying capacity. They can enable us to see how, say, different areas of mathematics relate together or how different materials get together to enable a proof given some assumptions. This is why proofs are important, not because of what they prove, but because of what they go through from the axioms in order to reach what they reach. Axioms are good if they are fertile. It is not enough, clearly, to be fertile, they have to enjoy other features, for instance they can be part of a set that coalesce, that get together in an insightful manner. They also have to have some prima facie plausibility - which is not to say that they are self-evident. They have to be, to use a perhaps vague term, worth pursuing. They have to be intriguing, intriguing enough. And it is better if they prove that by proving things that happen to be intriguing also. There is no choice of axioms that are independent of what we want to prove and how much we want to cover. A good choice of axioms is one that illuminates without flying in the face of what is already taken to be known.

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