(Cross-posted at NewAPPS)
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.
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.
Whether reductio ad absurdum arguments are cognitively demanding is obviously by and large an empirical question. Going beyond anecdotal evidence that “people” seem to be using this argumentative strategy “all the time”, the level of difficulty encountered by those learning to follow and produce such arguments should tell us something about the cognitive challenges involved. Indeed, although there have been few systematic studies of how people fare when reasoning by means of reductios, there is a small but interesting literature in math education that is highly relevant for the present investigation. There is general consensus among specialists that students receiving mathematical education tend to find proofs in general difficult, and some kinds of proofs specifically, including reductio proofs, exceptionally difficult. In particular, students often seem to experience a lack of conviction in reductio ad absurdum proofs, even if they can produce them.
These findings seem to contradict the claim that “people use [reductios] all the time” (Dennett 2014, 29); if they do, why is it so hard for students to accept and internalize this argumentative approach in the context of mathematical instruction? And what exactly is so hard about reductios? To address these questions, we may want to pay attention to what students themselves say about their experiences with such proofs (even if there are limits to how much introspection can tell us about cognitive processes).
In (Antonini& Mariotti 2008), two exemplar protocols are discussed, where two university students report on their experiences and attitudes towards reductio proofs, which tell us much about what seems to be going on when people formulate such proofs. One student, Maria (majoring in pharmaceutical sciences, final year, and having familiarity with mathematical proofs), offers the following remarks while discussing with the interviewer the possibility of formulating a concrete proof by reductio at absurdum (a proof that, if ab = 0, then either a = 0 or b = 0):
[...] well, assume that ab = 0 with a different from 0 and b different from 0... I can divide by b... ab/b = 0/b... that is a = 0. I do not know whether this is a proof, because there might be many things that I haven’t seen.
Moreover, so as ab = 0 with a different from 0 and b different from 0, that is against my common beliefs [Italian: ‘‘contro le mie normali vedute’’] and I must pretend to be true, I do not know if I can consider that 0/b = 0. I mean, I do not know what is true and what I pretend it is true.
[Interviewer: Let us say that one can use that 0/b = 0.]
It comes that a = 0 and consequently … we are back to reality. Then it is proved because … also in the absurd world it may come a true thing: thus I cannot stay in the absurd world. The absurd world has its own rules, which are absurd, and if one does not respect them, comes back.
But my problem is to understand which are the rules in the absurd world, are they the rules of the absurd world or those of the real world? This is the reason why I have problems to know if 0/b = 0, I do not know whether it is true in the absurd world. […]
In the case of the zero-product, I cannot pretend that it is true, I cannot tell myself such a lie and believe it too! (Antonini & Mariotti 2008, 406; emphasis added)
Maria raises a number of issues that for her are cognitive/epistemic issues, but which capture much of what also appears to be philosophically suspicious about reductio proofs more generally. For example, she considers the possibility that she may have overlooked other options; as we’ll discuss shortly, a reductio argument only works on the assumption that the enumeration of cases is exhaustive and all alternative possibilities have been considered (each leading to absurdity). She seems particularly bothered by the cognitive dissonance of having to assume that which she knows to be false – “I cannot tell myself such a lie and believe it too!” Maria also wonders whether in the absurd world that is postulated at the beginning of the proof (the world where ab = 0 but a ≠ 0 and b ≠ 0), the usual rules of the actual world still hold, so that the reasoning can proceed in the usual way. Indeed, once one accepts such a blatant absurdity, what guarantee do we have that other absurdities will not arise?
Another student, Fabio (majoring in physics, final year), offers equally insightful considerations:
Yes, there are two gaps, an initial gap and a final gap. Neither does the initial gap is comfortable: why do I have to start from something that is not? […] However, the final gap is the worst, […] it is a logical gap, an act of faith that I must do, a sacrifice I make. The gaps, the sacrifices, if they are small I can do them, when they all add up they are too big.
My whole argument converges towards the sacrifice of the logical jump of exclusion, absurdity or exclusion… what is not, not the direct thing. Antonini & Mariotti 2008, 407; emphasis added)
As Maria, Fabio feels uncomfortable with the idea of starting from something that he knows is not the case. But to him, the most disturbing aspect of a reductio proof is the last step, from absurdity to the contradictory of the initial hypothesis; he describes this step as an ‘act of faith’. He also notes the cumulative effect of the cognitive ‘sacrifices’ he has to make; each of them individually is not so bad, but there are just too many of them involved in a reductio proof.
Naturally, these are the testimonies of just two students: for all we know, they are not representative of how reasoners in general view reductio ad absurdum. However, there is rather strong support in the math education literature for the general idea of cognitive difficulties related to reductio proofs, and so others may well be facing the same issues. In particular, it is interesting to notice that, even if they master the technique in terms of being able to produce reductio proofs themselves, students may still feel that these proofs are not entirely trustworthy from an epistemic perspective; they are left unconvinced (as Maria and Fabio).
Additional empirical investigation of reasoning abilities with reductio ad absurdum is required to further confirm these findings; but for our purposes, what Maria and Fabio tell us about these proofs provide exactly the right starting point to formulate some of the philosophical issues arising in connection with reductio arguments.
(To be continued...)
 “Research into students’ ability to follow or produce proofs ... confirms that students find proof difficult, with proofs by (mathematical) induction and proofs by contradiction presenting particular difficulties.” (Robert & Schwarzenberger 1991, 130)