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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?

dineshdileep
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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 08:32
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    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 08:33
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    No, it sounds like somebody whose only exposure to mathematics was a calculus course. – Goldstern Aug 14 '13 at 09:31
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    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 09:31
  • 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
  • @every one. After reading every answers and comments, I realize what my question should have been. What we (non-mathematicians) were curious about is that, can one branch of mathematics can be completely interpreted in terms of other? Or is there any particular branch which is more fundamental than other? – dineshdileep Aug 14 '13 at 18:43
  • My own feeling is that the question should be more about student education, and here a good way for research students to learn to mathematicise is to try to rewrite topic A (or results, ... ) in terms of the language of B and to decide if the result is an improvement, or has value in other ways. This gets students writing with some kind of guide (from A) but with something to do (from B), and also asks for judgement. A standard choice for B is of course category theory, especially if this is somewhat neglected in A. A step towards "complete learning" is of course writing it out clearly. – Ronnie Brown Aug 16 '13 at 09:45
  • @WillieWong I ahd a friend once who insisted that a circle really had 361 degrees and that the government was hiding that extra degree for its own purposes. He would not explain further and, as far as I could tell, he was completely serious. – Steven Alexis Gregory Mar 17 '16 at 04:42

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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.

fedja
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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.

Goldstern
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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
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    @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. –  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
  • @SergeiAkbarov I think the only difference is what we mean by "set theory" in this context. Say, if somebody wants to explain me something about their research on I don't know say Brauer groups I might ask them to spell out some of the defintions/terminology they typically would use casually. But I do know what a group and a field are and various other things. If they'd need to explain me that a group is a set with a binary operation with... and a binary operation is a function...and a function is...down to "set theory" then this just won't work, in my opinion. –  Aug 14 '13 at 14:27
  • @SergeiAkbarov I had not seen your second comment. Just to avoid a misunderstanding, differential geometry was chosen pretty randomly among fields I do not know much about and that involve real numbers, I could eqaully well have used dynamical systems, PDEs, and various other things. –  Aug 14 '13 at 14:30
  • @quid: In your example it is helpful for you to know the definition of a group, is it? Goldstern writes that "it is often neither necessary nor even helpful to know the set-theoretic details...". So in this example you disagree with Goldstern, and agree with me, who claims that it is always helpful to know the set-theoretic details, right? If yes, then I don't see controversy between us. If no, then I don't understand your point. (Just in case: the downvote here was not mine.) – Sergei Akbarov Aug 14 '13 at 15:05
  • @SergeiAkbarov if one looks at it in this way then, well, yes. But I understood the answer more as something like if one wants to understand something new it is often neither necessary nor even helful to engage with this matrial only or primarily in a way reduced down to a purely set theoretic formulation. And, because I do know what a group is, I do not think about it as some set with some subsets of the cartesian prodduct with itself that has some list of properties. So while it is likely helpful that I do know this, it is often not helpful to actively think about it in this way. –  Aug 14 '13 at 15:26
  • @quid: OK, so the problem is just that we understand this answer differently. I think the author should clarify his point. If he will write that OFTEN IT IS NOT SUFFICIENT TO KNOW THE SET-THEORETIC DETAILS..., then I would not object! But up to this moment I would think, that my objections are correct... – Sergei Akbarov Aug 14 '13 at 15:47
  • @SergeiAkbarov yes I agree that this different understanding seems our only point of dissent. Thank you for the interesting and clarifying exchange. –  Aug 14 '13 at 15:53
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    @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
  • To clarify my point: Of course set theory is useful as a common language when you want to explain a basic concept, such as a group: "A set with a binary function..." But if you want to explain a higher level - simple, nilpotent, etc, it is more reasonable to use intermediate level concepts, such as "normal subgroup", commutator, etc, rather than go all the way down to very basic notions. – Goldstern Aug 15 '13 at 21:45