In (applied and pure) math study, do we only discuss and need material equivalence, not logical equivalence?

Question

I am a mathematics major student and interested in logic. I have some questions, in math(both pure and applied aspects) study and research, do we clearly distinguish between logical equivalence and material equivalence as what we do in the pure logic study, or do we only discuss and require material equivalence (and roughly think that the two equivalences are the same meaning), because I find that in my daily math study we only discuss and require the material equivalence while ignoring logical equivalence. I want to know the answer and why, I am very grateful!

Answer 1

Two observations from the propositional perspective:

• If $A$ and $B$ are already established propositions and their connection is already well understood, one usually does not have to view this as a deduction each time, $A\vdash B$, and may carry on with material implication $A\rightarrow B$ instead.
• More generally, the deduction theorem allows one to pass from one view of matters to another (from the logical point of view): $$\Gamma,A\vdash B\Longleftrightarrow\Gamma\vdash A\rightarrow B$$

The same goes for material equivalence, $\leftrightarrow$, and logical equivalence, $\dashv\vdash$.

It should be remarked that logic is not a description of working mathematicians' cognitive processes, nor of communicative practices. Such logical distinctions exhibit themselves when mathematical arguments are needed on some purpose to be regimented and rearranged into a logical form.