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I'm curious about what math (mathematical logic, metamath) says about the way we reason. This is going to be a vague question because I have not yet explored mathematical logic myself. My question: can we somehow develop, or construct, all of the valid rules of inference ways we reason about proofs (by contradiction, implication, and so forth) in some mathematical way? Can it be done from some simple principles?

Like, if you had some sort of model, and these ways of reasoning, as objects in that model, that could be derived. What would be the model?

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Do you mean philosophically, mathematically or both? – Lays Apr 22 '13 at 4:33
Well, preferably in a mathematical way. Like, if you had some sort of model, and these ways of reasoning, as objects in that model, could be derived. What would be the model? – user73738 Apr 22 '13 at 4:41
Here does this link help or maybe this and possibly this? – Lays Apr 22 '13 at 4:47
This question is quite vague. What do you mean by "reason about proofs"? – Qiaochu Yuan Apr 22 '13 at 4:48
I meant the rules of inference. The way we infer new truths from old ones. Can we derive the rules of inference? I guess "reason about proofs" didn't really make sense. – user73738 Apr 22 '13 at 4:51

Regarding what mathematical logic & metamathematics say about the way we reason, I think that you can see ML as a mathematical model of reasoning in Mathematics : Proof Theory, Model Theory, Computability Theory, all are about ways the mathematician (an idealize one) makes proofs and computing. All these activities are performed with language: so ML has to start with some idealized models of language (formal systems for First Order Calculus, and so on); those model are very much simplified, but they are very useful.

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