Are there still mathematicians who don't accept proof by contradiction? When I was a kid, I read popular scientific texts about the different philosophies of mathematics; formalism, intuitionism, constructivism and many others. I learned that there existed mathematicians who did not accept proofs by contradiction and some others who did not consider proof of existence of solutions important, but required proof of how to actually construct solutions. Are such stances still common among mathematicians?

Constructive mathematics is alive and well, including at MSE; see e.g., tags constructive-mathematics and intuitionistic-logic. Some of the most active authors in this area are Douglas Bridges and Fred Richman, who authored a combined total of over 300 publications between them. One should also mention Michael Beeson and others.

One of the most recent articles in this area is

> Beeson, Michael. Brouwer and Euclid. Indag. Math. (N.S.) 29 (2018), no. 1, 483–533.

I believe this is supposed to be a constructive re-writing of Euclid.