In formal classical logic, reductio is acknowledged as valid move, but in intuitionistic logic it is not. This logic was espoused by Brouwer as a rival to Hilberts formalist programme to reduce logic to mathematics via set theory, in fact Brouwer correctly predicted that Hilberts programme would end in contradiction (as eventually shown by Godel).
Intuitionistic logic proceeds by denying the classical law of the excluded middle, that is for any proposition, it must either be true or false. Without this law reductio is no longer valid. Brouwer believed that truth must be justified by a constructive proof, that it actually produces what it claims is true (I imagine he considered that closer to our own intuition). In a sense, he's replacing the idea of truth with the idea of justification.
Though Hilberts formalist programme died, this didn't mean that intuitionistic logic prospered, rather it led something of an underground existance, but recently it has been establishing itself as an important part of mainstream mathematics & logic, via what is called Topos Theory, which is a generalised Set Theory built on structuralist principles. There isn't only a single unique Topos theory, there are many, and each has a so-called internal language/logic built within it, and this logic is intuitionistic; further, and this is important, each topos has a geometry.
A good historical analogy to think about is, how dropping the parallel axiom from euclidean geometry opened up a new world of non-euclidean geometries. Whereas, here we're talking about dropping the excluded middle, and getting a brave new world of non-classical logic, entangled with non-classical set theory & non-classical geometry.
But pure theory is all very well, after all non-euclidean geometry established itself with the success of General Relativity. Some applications I can point to would be:
Smooth Synthetic Geometry, which makes rigorous the idea of the infinitesimal, to be used in pretty much the way Newton & Leibniz did (and so dismissed by Bishop Berkely).
Recent papers by Chris Isham, a respected physicist working in Quantum Gravity, which looks at quantum theory in the context of topos theory. He makes the audacious point, that taking the principle of General Covariance seriously in General Relativity appears to deny the existence of points, that Quantum Field Theory is/has been plagued with nonsense answers for what appears to be the same reason (one of the motivations for String Theory is that point particles are expanded to strings), and his motivation for looking at Topoi, is that in their geometric incarnation, points are also denied.