By Catarina Dutilh Novaes
(Cross-posted at M-Phi)
As some readers may recall, I ran a couple of posts on reductio proofs from a dialogical perspective quite some time ago (here and here). I am now *finally* writing the paper where I systematize the account. In the coming days I'll be posting sections of the paper; as always, feedback is most welcome! The first part will focus on what seem to be the cognitive challenges that reasoners face when formulating reductio arguments.
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For philosophers and mathematicians having been suitably ‘indoctrinated’ in the relevant methodologies, the issues pertaining to reductio ad absurdum arguments may not become immediately apparent, given their familiarity with the technique. And so, to get a sense of what is problematic about these arguments, let us start with a somewhat dramatic but in fact quite accurate account of what we could describe as the ‘phenomenology’ of producing a reductio argument, in the words of math education researcher U. Leron:
We begin the proof with a declaration that we are about to enter a false, impossible world, and all our subsequent efforts are directed towards ‘destroying’ this world, proving it is indeed false and impossible. (Leron 1985, 323)
In other words, we are first required to postulate this impossible world (which we know to be impossible, given that our very goal is to refute the initial hypothesis), and then required to show that this impossible world is indeed impossible. The first step already raises a number of issues (to be discussed shortly), but the tension between the two main steps (postulating a world, as it were, and then proceeding towards destroying it) is perhaps even more striking. As it so happens, these are not the only two issues that arise once one starts digging deeper.
To obtain a better grasp of the puzzling nature of reductio arguments, let us start with a discussion of why these arguments appear to be cognitively demanding – that is, if we are to believe findings in the math education literature as well as anecdotal evidence (e.g. of those with experience teaching the technique to students). This will offer a suitable framework to formulate further issues later on.

