> No way. Mathematicians don't assume things to be true even if there are no counterexamples.

It depends: there do exist some conjectures for which there don't exist any proofs, but a huge amount of results assuming this conjecture (which is very near to assuming that the conjecture is true). Examples are:

- Riemann hypothesis [1]

- Generalized Riemann hypothesis (GRH) [2], and potentially also Extended Riemann hypothesis (ERH) [3]

- Standard conjectures on algebraic cycles [4]

- perhaps Schanuel’s conjecture [5]

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[1] https://en.wikipedia.org/w/index.php?title=Riemann_hypothesi...

[2] https://en.wikipedia.org/w/index.php?title=Generalized_Riema...

[3] https://en.wikipedia.org/w/index.php?title=Generalized_Riema...

[4] https://en.wikipedia.org/w/index.php?title=Standard_conjectu...

[5] https://en.wikipedia.org/w/index.php?title=Schanuel%27s_conj...

Generally, no, but sometimes, yes.

It can be productive to assume something is true and see what happens.

Agree. I give up isn't the MO of many with graduate degrees.