What is the most advanced math subject of all time?

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Economics may use mathematical procedures but it is not mathematics or mathematical....

I have a nitpick here. The common process of "proof-by-induction" on the natural numbers, i.e. mathematical induction, is not an example of inductive reasoning. The rigorous deductive justification of "proof-by-induction" is given in books on axiomatic set theory. In other texts a version of the process follows directly from one of the Peano axioms (Outside of set theory, the Peano axioms are usually stated without proof - set theory allows one to prove them as theorems). In any case, mathematical induction is justified via logical deduction combined with primitive axioms. Inductive reasoning is used in science, not mathematics. Mathematics is 100% deductive.

You're probably looking for differential geometry. It is an advanced course, but probably not the most difficult subject in all of mathematics. It was difficult enough that Einstein needed help from a mathematician in order to apply it to his general theory of relativity. There weren't any good text books on the subject at that time.

Long division, although my maths knowledge doesn't extend past trigonometry, teachers never gave me a chance to go higher, but my career no longer employs the need for maths.

If we're talking about the most advanced humanity has ever faced, I guess it would be a theory of possibly what exists beyond our physical limitations like what's on the other side of a black hole.

The conclusions are deductive, but it still takes inductive reasoning to get to the end result. If it only requires deductive reasoning, it could be solved by a computer, and if it could be solved by a computer, it would already be solved because mathematicians are good with computers.



No, the most advanced math has nothing to do with physical reality.