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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
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closed as primarily opinion-based by David White, Theo Johnson-Freyd, Noah Stein, Steven Landsburg, Karl Schwede Aug 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.

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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.

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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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"...to understand a high-level concept in some other branch of mathematics... it is often neither necessary nor even helpful to know the set-theoretic details behind it", -- this doesn't sound convincing for me. I would say, on the contrary, usually it is impossible to understand what people in one branch of mathematics have in mind, when explaining something, if you are not a specialist in this field, and you can't guess the set-theoretic backgroud that lies behind their explanations. –  Sergei Akbarov Aug 14 '13 at 11:00
@SergeiAkbarov I know little about, e.g., differential geometry. I doubt it would help me understand if a differential geometer were to explain things to me going down to the contruction of reals from the rationals those from the integers and so on until we are down to settheoretic in a pure form at some point. Would you consider this helpful? The "definition" for set-theoretic details I use is likely different from yours, but possibly closer to the intended one in the answer than yours. –  quid Aug 14 '13 at 13:35
@quid: Maybe I misunderstood, but my point is that there are common languages that allow people to understand each other, and it is not wise to ignore this. I used to think that for mathematicians set theory is one of those common laguages (by the way, category theory is another). As a corollary this can't be useless to translate (from time to time) what you say into those common languages. –  Sergei Akbarov Aug 14 '13 at 14:04
I agree with you that differential geometry is a branch where specialists abuse their intuition, ignore the necessity to attach their field closer to the rest of mathematics. In my opinion, this is not because of the lack of will to explain everything directly from rationals, etc. I would say if they used category theory more actively, that would be much more clear. I don't want to scold anybody, but this is indeed a bright example. –  Sergei Akbarov Aug 14 '13 at 14:09
@quid: I see here an illustration of how mathematicians can understand each other when arguing about something not quite mathematical. Thank you for this useful contribution to my theory. :) –  Sergei Akbarov Aug 14 '13 at 16:01
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