# Can one branch of mathematics be completely learned from the perspective of another branch of mathematics? [closed]

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?

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I would say, on a practical level, not very much. On a foundational level, there's also the problem that different models of set theory need not be mutually consistent. But then again, there's that whole business with homotopy type theory, which is probably the closest thing to what your friend is describing. –  Willie Wong Aug 14 '13 at 8:32
Though "he blieved there should be a way to transform every result in algebra to calculus and that this should be known to the core mathematicians" sounds awfully lot like a conspiracy theory. :-) –  Willie Wong Aug 14 '13 at 8:33
No, it sounds like somebody whose only exposure to mathematics was a calculus course. –  Goldstern Aug 14 '13 at 9:31
The question atleast makes sense in the context of drawing analogies to prove theorems, may be across branches. The following quote may throw some light: "A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."-Stefan Banach –  Uday Aug 14 '13 at 9:31
In my opinion, the question should be changed a little bit, in its current statement it sounds rhetorical: much work in mathematics is carried out exactly for the purpose of finding relations between different branches. Perhaps, it would be better to ask "which examples of unexpected relations between different disciplies do you know?". Or something like this. –  Sergei Akbarov Aug 14 '13 at 11:23

## closed as primarily opinion-based by David White, Theo Johnson-Freyd, Noah Stein, Steven Landsburg, Karl SchwedeAug 14 '13 at 13:40

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

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.

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