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There are axioms, and then there are well established methods of working with them. It is clear that these methods are nothing but logical operations (rules of manipulation of symbols) on previous statements.

Let us say we are required to prove a particular result. All we have at our disposal are axioms, and rules of manipulation. What I wish to ask is the following:

Since all we will be doing is just logical operations on axioms, it means that the result resides in the axiom itself. It therefore seems that: Axiom dictates what is the case, and also what is not the case.

  1. What value does the logical operation adds to the whole process? Why axiom itself does not reveal the status of conjecture (true or false)?

  2. In general, what happens when a mathematical proposition transitions from one expression to another?

Ajax
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  • "reveal" ? In what sense ? An imemdiate illumination of the mind that "see the truth" ? – Mauro ALLEGRANZA Jul 03 '19 at 15:54
  • @MauroALLEGRANZA By can't we bypass proof and still return true status of the conjecture (through intuition of course)? After all *axioms* decides all that is true. – Ajax Jul 03 '19 at 15:59
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    There are formulations of logic that have no axioms at all, only rules of manipulation, [natural deduction](https://en.wikipedia.org/wiki/Natural_deduction) is one. So the "dictating" is split between rules and axioms even when mixed formulations are used, it does not reside in axioms alone. As for the "value" of what happens, one view is that the depth information is brought to the surface, see [What is the difference between depth and surface information?](https://philosophy.stackexchange.com/q/59391/9148) – Conifold Jul 03 '19 at 17:01
  • @Conifold Can you give an example to demonstrate how "dictating" does not reside solely in axioms, but also in method/rules? Also, can there be more than one correct method in arithmetic? – Ajax Jul 04 '19 at 07:46
  • Natural deduction simply has no axioms, only introduction/elimination rules for connectives, so any natural derivation in it is an example. Wikipedia gives an example of deriving A⊃(B⊃(A∧B) from zero premises in the Hypothetical Derivations section. I am not sure what "correct method in arithmetic" is, there are plenty of different axiomatizations, not even all equivalent. – Conifold Jul 04 '19 at 09:32
  • @Conifold But proof by contradiction works because nothing is questionable except the assumption. If rules of manipulation also shape the result, we shouldn't be directly pointing finger at assumption, but also on applicability of rules on the assumption. – Ajax Jul 04 '19 at 19:29
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    Everything is always questionable, rules, assumptions, application of rules and assumptions, whether we point our finger at them or not. Wittgenstein questioned the determinacy of rule-following even after the rules are spelled out. The point is that there is no meaningful separation between the role of "axioms" and "rules", it is purely a matter of convenience. If people disagree with either the "proof" isn't a proof to them, and so intuitionists reject proofs by contradiction. – Conifold Jul 05 '19 at 10:35

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