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What I've realized is that I like to prove ideas in math that are indeed applied to other fields. In other words, I dislike pure math such as topology but I love to prove things in other types of math such as statistics/calculus. But I've heard that the applied math don't care much for proofs and that they use intuition rather than formalizing everything as a pure mathematician does. This view of applied mathematics scares me, and it cannot be right because I have never imagined mathematics to be devoid of proof! But for example, I've heard that engineers for example, are often oblivious to "proof-based work" by mathematicians, even if it pertains to engineering. So is there any place for proof-based work in applied mathematics? Surely there must be! But the attitude of the engineers I have met is discouraging me so much.

As an example, I am hugely fascinated by things such as Arrow's Impossibility Theorem, the Minimax theorem, things like that... which are indeed based on mathematical proof, but which could never be considered as "pure mathematics". But the economists I met seem largely oblivious to the "proof-based mindset of a mathematician". So there is this disconnect I am observing which I cannot account for!

So do applied mathematicians actually do proof-based work like the pure mathematicians or do they eventually rely on simulations/experiments like the engineers and natural scientists? Because what I want is proof-based work which can be applied in real life.

BiP