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Mathematicians prove equivalencies. Because A holds -> B holds, etc. To be honest, I don't consider their 'theorems' to be 'theories'.
Also, mathematics doesn't have theories in a Popperian sense. The axioms are, per definition, assumed. This makes the 'theorems' completely insulated. What is then the difference with Freud's theory of the subconscious?
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Alrite...if you insist to reason in this way...
1. Mathematics is all about equivalence?.....
Then John nash's 28 pages of equaivalent relations are certianly not Game "Theory" and his few lines of Galois extention were just unneccessary formalization, and will not lead to your so called "goal". (but Nobel committee had a lower definition of goal than you do)
Then Martingale "theorems" are a collection of mathematics "equivalences", certainly does not make any sense in economics (according to your reasoning).
It also follows from your logic that Joseph Doob's Optional Stopping Time "Theorem" is just merely another collection of equivalences which "says" one cannot devise a "strategy" to beat the market when the stock prices follows a martingale. Surely one needs to know what a stopping time is
(in addition to filtration).
2.Watereddown mathematic suffice, since economists only uses it as a tool?
In financial econometrics, how do one estimate Stochastic Volatility Models with Levy jumps without knowing what acutally an Ito Integral is, the "theories and theorems" behind stochastic "Differential" Equation (as opposed to "difference" equations frequently adopted in economics)
and Levy Process with jumps, what about Levy without cadlag sample paths? how to derive the density for the first hitting time of Brownian motion?
With only a comprehension of waterdowned level mathematics, one's research in economics would be confined to a subset of topics, areas like Stochastic Differential Games, Mathematical Finance would be out of one's league. And it is very hard to see how ideas come into play with a "watered down toolbox" to develop or even follow one's so called "idea".
Mathematics is all about ideas, just that when one lost comprehension , one only sees a densed set of Greeks flying around.....hence your so called "formalization".