**Catarina Dutilh Novaes**

(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.

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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.

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.[1] 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...)

[1] “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)

This is very interesting. It occurred to me that if one is the sort of logical pluralist that I (sometimes) am---holding roughly that different consequence relations are relevant in different contexts---then reductios may not always be valid outside of classical mathematics. The thought then is that, especially in the case of `the gap at the end', we have noticed that this argument form fails in other contexts, and thus are not convinced that it should apply in the mathematical case either

ReplyDeleteApologies if this is something you will be covering in subsequent posts.

An account of logical pluralism is also one of the things that the dialogical perspective of logic and deduction that I endorse can deliver. But more specifically (and this is coming up in a post soon), the last step presupposes that the enumeration of cases is exhaustive, something that doesn't seem obvious at all in a number of circumstances, perhaps including portions of mathematics.

DeleteMy instinct when reading what Maria has to say is to want to tell her that her conceptual problems could be avoided if she expressed essentially the same proof in a different way. The less confusing version goes like this. Suppose that a and b are two numbers and that ab=0. (Note, I am not assuming any kind of impossible imaginary world here -- it's perfectly possible for two numbers to have product zero.) Now suppose in addition that b is not zero. In that case, we can divide both sides by b and deduce that a = 0/b = 0. So if b is not zero, then a must be zero.

ReplyDeleteFrom here we have to get to the conclusion that one of b and a is zero. In other words, we have to accept a deduction of the form (not p implies q) implies (p or q). If we use the formal definition of "implies", we might say that the meaning of (not p implies q) is (not not p or q). So the one step that remains is the idea that not not p is the same as p. While some people may find that philosophically suspicious, I don't think it is counterintuitive.

Actually, when discussing what's counterintuitive and what isn't, I prefer not to take this last step and go back to the deduction of (p or q) from (not p implies q). I think this falls into Dennett's category of types of reasoning we go in for the whole time. For example, if I say, "If Djokovic doesn't win the US open, then Federer will," it's absolutely clear that I'm claiming that either Djokovic or Federer will win the US open.

This doesn't invalidate what you are saying, since there are other examples of arguments where reductio ad absurdum is used in a more essential way. I just think that Maria's conceptual difficulties with this particular example may result more from a not very well expressed proof than from fundamental philosophical problems with the proof itself.

Another thing I'd like to say to Maria is that the rules that apply in the absurd world are the same as the rules that apply in the actual world. Her difficulty here seems to me to be bound up with the word "blatant". In order even to think about how to prove that there are no non-trivial zero divisors, you have to get out of your head the idea that it's just plain obvious. I think her problem is mainly the familiar one that the statement seems too obvious to need a proof, which again is not a fault with reductio ad absurdum. Someone with more mathematical experience will think of examples like the integers mod 6, where you do have non-trivial zero divisors. Simple examples like that help to dispel the idea that there's nothing to prove. It would be interesting to see how these students feel about reductio proofs of statements that they find less obviously true in the first place.

Yes, it's pretty clear that Maria hasn't been much helped by how reductio has been explained to her... The general point of the paper is to argue for the dialogical conception, which at the end I note also has some practical pedagogical applications. The hypothesis would be that, if students are told that a reductio proof is like a dialogue with an opponent, then many of these issues and confusions would not arise. However, the last step, the 'act of faith' remains problematic to my mind, not just a matter of modes of presentation. The problem remains that we need to be sure that the enumeration of cases is exhaustive.

DeletePerhaps some readers may be interested in this discussion by mathematicians of the use of reductio in mathematical proofs: http://mathoverflow.net/q/12342/1946.

ReplyDeleteThanks!

DeleteThis is a bit tangential to the post, but I wonder about the *pedagogical* gap that is behind some of the student's confusion. One thing I noticed is that both students from your examples weren't majoring in math or philosophy, so perhaps their exposition to the subject was a bit too quick? Do you also experience this type of confusion when teaching undergrads, Catarina?

ReplyDeleteAlso, one thing that occurred to me is if the distinction between perfect and imperfect information couldn't be of help here. It seems to me that part of the problem is that the student's are trying to force a situation which trades on imperfect information into a situation which trades on perfect information. Generally speaking, when attempting a proof we have only imperfect information available to use (those are encoded in the premises, axioms, and rules of inference); in particular, in the case of RAA, we don't know the status of the premise we're trying to prove. Of course, outside the proof environment, we do know this status, but *this* piece of information is not available *inside* the proof environment (I got this distinction between perfect-imperfect information from Kaye's *The Mathematics of Logic*, which gives as a helpful example König's Lemma).

In my experience teaching intro to logic, students do find reductio proofs a bit puzzling. But you are right, it is related to the kind of explanation they get (see my point above on how adopting a dialogical conception could have some positive pedagogical implications).

DeleteAnd thanks for the reference to the distinction between perfect and imperfect information; I hadn't thought of it in this context, will think more about it!

In my experience teaching intro to logic, students do find reductio proofs a bit puzzling. But you are right, it is related to the kind of explanation they get (see my point above on how adopting a dialogical conception could have some positive pedagogical implications).

DeleteAnd thanks for the reference to the distinction between perfect and imperfect information; I hadn't thought of it in this context, will think more about it!

Can't help but think that educator's own difficulty in explaining the proof adds an "illusory" perception of complication. There is something resembling the writing of antiquity in a typical educator's account, that wordiness... It should flow more naturally. I swear I could get students to give every-day reductios of basic propositions. The most well known case might be the alibi, which even 5 year olds can do. Why should "I was somewhere else" be proof that you didn't commit a crime? Is the answer easier because we are used to reasoning about people and their doings?

ReplyDeleteThe idea is not that we KNOW something's true, but we're wondering if it's true. Often students are told briefly "Prove P." I like when authors phrase it "P? Prove your result." Perhaps that could fill the "first gap," the discomfort of assuming a lie; you're not 'supposed' to know it's a 'lie.'

When you write it out, it seems long and involved, but the narrative structure is so common that Popper thought to found science on it ("falsification"). Perhaps emphasizing an experimental frame of mind could be a move towards filling these gaps.

Agreed! (See point above about pedagogical implications.)

Delete