Could we take certain results as axioms and prove the original axioms using our new ones?

Question

Axioms are statements that are simply taken as true. We prove certain theorems using these axioms. What if we now forget about the original axioms, take a set of these theorems and pronounce them new axioms, that is, just assume them to be true? Would we be able to prove the original axioms using the new ones?

For example, we derive the Pythagorean Theorem from the axioms of Euclidean geometry, which we assumed to be true. We then remove the "axiom-status" from the original axioms - they are now mere hypotheses. Now we say that the Pythagorean theorem is true, that is, we make it an axiom. Could we now prove things such as "That all right angles are equal to one another"?

(I know that we would need more than that but this is just an example.)

Is this theoretically possible if we chose the right theorems? (Not just in Euclidean geometry; in any theory.) Has anybody ever attempted this?

Answer 1

Yes, this has been intensely studied in a number of contexts. We pick some very small set of axioms, basically all the uninteresting ones; we then look at what implications this "base theory" can prove between old axioms and theorems. Early geometers, for instance, were interested in which theorems implied the parallel postulate, over the other four, and set theorists looked at theorems equivalent to the axioms of choice, over ZF. for a more organized approach, with a computational flavor, check out reverse mathematics.

Answer 2

Yes. A famous example is the parallel postulate in geometry. There are many results in geometry that depend on it, but it feels less obvious than the rest of the axioms. People worked hard for centuries to prove it from the other axioms. In the process they found many other statements that were logically equivalent to the parallel postulate in the presence of the other axioms. You can assume any one of them as an axiom and prove all the rest. We now know that the parallel postulate is independent of the other axioms, so you need to assume one of the equivalents.

Answer 3

This is exactly the program of reverse mathematics. To quote from the Wikipedia article: the program was founded by Harvey Friedman (1975, 1976) and brought forward by Steve Simpson.

Answer 4

What if we now forget about the original axioms, take a set of these theorems and pronounce them new axioms, that is, just assume them to be true? Would we be able to prove the original axioms using the new ones?

Not in general. Simple example. My 'axiom' is $P \land Q$, and I derive 'theorem' $P$. Well, from $P$ alone I cannot go back to $P \land Q$.

OK, you say, but I could also derive $Q$ from $P \land Q$, and from $P$ and $Q$ I can go back to $P \land Q$. Sure, but now the question becomes: what is a theorem? Because if a theorem is simply whatever logically follows from the axioms, then all axioms themselves (or at the very least something logically equivalent) will be theorems, and so of course we can derive the old axioms from the old theorems/new axioms. So if we look at theorems that way, we would obtain a trivial result.

OK, so it would be more interesting if we derive some small set of theorems from a small set of axioms, and are then able to reverse that. Do we have examples of that? Sure, we have many different axiomitizations of propositional logic, for example, each consisting of a small set of axioms (e.g the Hilbert system, Rosser system, Frege's system, etc.), and we can derive each axiom set from each other.

Indeed, if you can reverse the process in the way you indicate, then wat that means is that apparently you can have multiple different 'bases' for your theory, and that could certainly be very interesting, as it would 'swap' our conceptual understanding of what is more 'fundamental'. As another example, we can use Newtonian laws to derive that heavier objects don't fall faster to Earth than lighter ones (disregarding friction) ... but what if we take that result as our basis of physical mechanics? Then suddenly we may obtain quite a different outlook on the physical world, even if the results are the same in the end.