(Cross-posted at M-Phi)
In his commentary on Euclid, the 5th century Greek philosopher Proclus defines indirect proofs, or ‘reductions to impossibility’, in the following way (I owe this passage to W. Hodges, from this paper):
Every reduction to impossibility takes the contradictory of what it intends to prove and from this as a hypothesis proceeds until it encounters something admitted to be absurd and, by thus destroying its hypothesis, conﬁrms the proposition it set out to establish.
Schematically, a proof by reduction is often represented as follows:
It is well know that indirect proofs pose interesting philosophical questions. What does it mean to assert something with the precise goal of then showing it to be false, i.e. because it leads to absurd conclusions? Why assert it in the first place? What kind of speech act is that? It has been pointed out that the initial statement is not an assertion, but rather an assumption, a supposition. But while we may, and in fact do, suppose things that we know are not true in everyday life (say, in the kind of counterfactual reasoning involved in planning), to suppose something precisely with the goal of demonstrating its falsity is a somewhat awkward move, both cognitively and pragmatically.
Even in the relevant circles of specialists, quite a few people have issues with indirect proofs, most famously intuitionists who reject double-negation elimination – the crucial step which goes from the rejection of ~A to the assertion of A. It is also often said that Frege’s account of inference as going from true statements to true statements leaves no room for indirect proofs (but here is a recent paper by Ivan Welty countering this claim). So even within mathematics and logic, indirect proofs are somewhat controversial.
If we accept that indirect proofs are a bit of an oddity even within mathematics, it makes sense to ask how on earth this argumentative strategy might have emerged and established itself as one of the most common ways to prove mathematical theorems. Now, as some readers may recall, my current research project focuses on ‘the roots of deduction’, adopting the hypothesis that we need to go back to deduction’s dialogical origins to make sense of the whole thing (as discussed here, for example). And here again, it seems that the dialogical, multi-agent perspective offers fresh insight into the nature of indirect proofs.
Assume a dialectical context in which two participants are disputing on a certain topic, and let us call them 1 and 2 and B to keep it neutral. Then imagine that 1 wants to convince 2 of proposition A; how can she go about? Well, she can propose ~A and see if 2 takes the bait. It is important that ~A be put forward in the form of a question (which is indeed how such disputations often began in ancient Greece, as attested for example by Aristotle’s Topics), so that by accepting ~A, 2 commits to its truth but not 1; 1 has merely put it forward as a question and thus has herself not endorsed ~A. 1 can now proceed to show that something absurd follows from the acceptance of ~A, because this is not her position; it is 2’s position. By showing that something absurd follows from ~A, 1 in fact shows that it was a bad idea for 2 to accept ~A in the first place. There is still the contentious last step which goes from ‘accepting ~A is a bad idea’ to ‘accepting A is a good idea’. But 1 has not done anything pragmatically incoherent because she herself never committed to ~A.
In legal contexts, reductio arguments are used in much the same way. The prosecution may claim A (the defendant was at the crime scene), and the defense may then show that, given additional background information, A leads to absurdity (say, to the possibility of traveling between Paris and London in less than 30 min). (Welty’s paper has a similar legal example.) So what you show as entailing absurdity in a reductio argument is in fact the position of your opponent, not your own position (not even your own assumption). The adversarial component is crucial to understand what it means to prove something indirectly; it makes the postulation of the strange speech-act of supposing precisely that which you want to prove to be false superfluous. In a purely mono-agent context, in contrast, she who formulates an indirect proof has to play awkwardly conflicting roles. (Naturally, it is perfectly possible to formulate an indirect proof on your own, but this is a consequence of what I describe as the ‘internalization of opponent’ by the method itself.)
I think that this multi-agent, dialogical account of indirect proofs is conceptually appealing on its own, but within the Roots of Deduction project, we (Matthew Duncombe, Leon Geerdink and myself) are also investigating the historical plausibility of the hypothesis. For now, it is interesting to notice that, in the Prior Analytics, Aristotle makes extensive use of indirect proofs, as is well known, but also that he often uses dialectical vocabulary to explain the concept of an indirect proof. (In fact, he uses dialectical vocabulary throughout the text.) UPDATE: here is a subsequent post I wrote on indirect proof in the Prior Analytics.
(A cool coincidence is that just yesterday Mic Detlefsen invited me to present at his PhilMath Intersem colloquium in Paris in June, precisely on the topic of the history of indirect proofs. So there is much to be done on the topic for me, but for now this is my starting point.)