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?

  • "reveal" ? In what sense ? An imemdiate illumination of the mind that "see the truth" ? – Mauro ALLEGRANZA Jul 3 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 3 at 15:59
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    There are formulations of logic that have no axioms at all, only rules of manipulation, 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? – Conifold Jul 3 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 4 at 7:46
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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 5 at 10:35

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