This arose from a discussion with a friend (people involved are two engineers) who argued that every result in mathematics should be transformable into another branch. For example, he argued that Pythagoras theorem can be proved using tools of probability. Another example is that, he believed there should be a way to transform every result in algebra to calculus and that this should be known to the core mathematicians. Us being engineers may not be able to appreciate the breadth and depth of it. How much truth is there in this?
closed as primarily opinion-based by David White, Theo Johnson-Freyd, Noah Stein, Steven Landsburg, Karl Schwede Aug 14 at 13:40
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Proving the Pythagorean theorem from the viewpoint of probability would be somewhat tough because it would require being extremely cautious to avoid a vicious circle and not to use the Descartes' description of the plane as $\mathbb R\times\mathbb R$ or anything similar anywhere (once you use it, it becomes unclear why you should invoke probability at all, when simple algebra that is unavoidable is already enough) or to fall into the trap of merely using one of the standard geometric proofs but calling area "probability", etc. My own attitude towards any such claim is an immediate "Show me!". Most of the time it finishes the discussion but in the cases when the opponent is up to the task, I learn something new.
What is true, however, is that most, if not all, results in one field can be interpreted in another one either directly (through showing that some object satisfies the assumptions of the theorem (continuous functions form an algebraic ring, etc.) or indirectly (through applying the same ideas in a different situation) and that bringing tools from another area into a problem often turns out to be extremely beneficial and illuminating.
As to the "core mathematicians" (I have no idea what exactly this group of people is) having some esoteric knowledge, I have to disappoint you: there is none to talk about except, perhaps, a few tricks related to how to think out of the box and to see connections between things described in totally different languages. Any decent engineer knows these tricks as well and uses them every day.
That's all I can say about the general philosophy. As to the practical matters, I can also engage into a discussion with a friend about whether it would be possible to make an engine out of ice using nothing except water and sunlight as the source of power or whether you can drill a well with nothing but controlled electrical discharges, but, while either of those can be viewed as a challenging mental exercise, it has about as little to do with your everyday work as your question has with mine or that of almost any other mathematician.
Theoretically, most or even all of mathematics can be formulated and proved in set theory. But this is not done, for a good reason: to understand a high-level concept in some other branch of mathematics (say, stiffness of differential equations, to pick something on the other side of the mathematical universe), it is often neither necessary nor even helpful to know the set-theoretic details behind it.
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