Mathematicians know how to admit they’re wrong
One thing I discussed with my students here at HCSSiM yesterday is the question of what is a proof.
They’re smart kids, but completely new to proofs, and they often have questions about whether what they’ve written down constitutes a proof. Here’s what I said to them.
A proof is a social construct – it is what we need it to be in order to be convinced something is true. If you write something down and you want it to count as a proof, the only real issue is whether you’re completely convincing.
Having said that, there are plenty of methods of proof that have been standardized and will help you in your arguments. There are things like proof by contradiction, or the pigeon hole principle, or proof by induction, or taking cases.
But in the end you still need to convince me; if you say there are three cases to consider, and I find a fourth, then I’ve blown away your proof, even if your three cases looked solid. If you try to prove something by induction, but your inductive step argument fails going from the case n=16 to n=17, then it’s not a proof.
Ultimately, then, a proof is a description of why you think something is true. The first half of your training is to problem solve (so, come up with a reason something is true) and construct a really convincing argument.
Coming at it from the other side, how can you check that what you’ve got is really a proof if you’ve written down the reason you think it’s true? That’s when the other half of your training comes in, to poke holes in arguments.
To be a really good mathematician you need to be a skeptic and to walk around with a metaphorical gun to shoot holes in other people’s arguments. Every time you hear a reasoned explanation, you look for the cases it doesn’t cover or the assumptions it’s making.
And you do the same thing with your own proofs to help yourself realize your mistakes before looking like a fool. Because putting out a proof of something is tantamount to asking for other people to shoot holes in your argument.
For that reason, every proof that one of these young kids offers up is an act of courage. They don’t know exactly how to explain their thinking, nor do they yet know exactly how to shoot holes in arguments, including their own. It’s an exercise in being wrong and admitting it. They are being trained to get shot down, to admit their mistake, and then immediately get back up again with better reasoning. The goal is to get so good at being wrong that it doesn’t hurt, that it’s not taken personally, and that it’s even fun to be wrong and to improve your argument.
Not every person gets trained in being wrong and admitting it. I’d wager that most people in the world, for most of their professional lives, are trained to do the opposite in the face of being wrong: namely, to wriggle out of it or deflect criticism. Most disciplines spend more time arguing they’re right, or at least not as wrong, or at least they have different mistakes, than other related fields. In math, you can at the most argue that what you’re doing is more interesting or somehow more important than some other field.
[I’ve never understood why people would think certain math is more important than other math. It’s almost never on the basis of having applications in the real world, or helping people in some way. It’s just some arbitrary snobbery, or at least that’s how it’s seemed. For my part I can’t explain why I love number theory more than analysis, it’s pure sense of smell.]
Most people never even say something that’s provably wrong in the first place. And that makes it harder to prove they’re wrong, of course, but it doesn’t mean they’re always right. Since they’ve not let themselves get pinned down on a provably wrong thing, they tend to stick with their wrong ideas for way too long.
I’m a huge fan of skepticism, and I think it’s generally undervalued. People who run companies, or universities, or government agencies, typically say they like healthy skepticism but actually want people to drink the kool aid. People who are skeptical are misinterpreted as being negative, but there’s a huge difference: negative means you’re not trying to solve the problem, skeptical means you care enough about the problem to want to solve it for real.
Now that I’ve thought about the training I’ve received as a mathematician, though, and that I’m now giving that training to these new students, I’ll add this to my defense of skepticism: I’m also a huge fan of people being able to admit they’re wrong. It’s the flip side of skepticism, and it’s why things get better instead of stay wrong.
By the way, one caveat: I’m not claiming that mathematicians are any better at admitting they’re wrong outside a strictly logical sphere.
mathbabe 
Love this post! I’m teaching a week-long (all day, every day) number theory course right now for in-service middle and high school teachers, and they’re learning to write proofs and thinking about how everything we’re doing fits in with the practice standards from the Common Core. What you’ve written really crystalizes the struggles they’re having and the outlook I want them to take. I’ll be distributing it to them this afternoon.
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“I’m not claiming that mathematicians are any better at admitting they’re wrong outside a strictly logical sphere.”
I would like to think that our experience of what it means to _actually_ prove something means we tend to be more honest about the unreliability of our assertions outside that sphere.
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Don’t under-estimate how well people can compartmentalize parts of their lives. In my own experience, many mathematicians (or scientists for that matter) manage to learn their trade quite well without necessarily exporting their professional skepticism to the rest of their thinking.
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A wonderful post (and sadly, the only one other than the t.p. rant I’ve been able to understand since you went to math camp).
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In my (relatively limited) experience dealing with students who are new to proofs, the bigger problem is less that the arguments are “incomplete” or “wrong,” but rather that they just don’t make sense. That is, the student has written down a bunch of stuff, all of which is “true” (though potentially vague), but little or none of which constitutes any sort of argument. Or the “gap” is not a missing case or some bad reasoning, but rather the omission of the entire crux of the argument. This sort of thing happens frequently when proving “obvious” things. But your summer kids are smart, so maybe that doesn’t happen as much.
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Nice post! I too find my students arrive at college with too rigid a notion of proof and work at getting them to think of a proof as an attempt at convincing. However, I think they also have too rigid a sense of what it means to have poked a hole in an argument — they think then its just wrong and should be scrapped and replaced by someone else’s “correct” proof. I like to give them examples of proofs that, when a hole was poked in it that couldn’t be fixed, could still have important ideas salvaged from it. For example, Lame’s “proof” of Fermat’s Last Theorem failed because unique factorization into primes does not necessarily hold when you leave the setting of the integers. This hole led Kummer to invent ideal numbers, which in turn led to all kind of fantastic ideas but in particular proofs of special cases of Fermat’s Last Theorem. Even Lame’s original proof could be used to give a simple proof of cases of Fermat’s Last Theorem.
I always interpret E. H. Moore’s phrase “sufficient unto the day is the rigor thereof” as a beautiful way to express the fluidity in what constitutes a convincing argument. It describes not only the evolution of mathematics itself but also to the evolution of a student’s understanding of mathematical argument.
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As the fiancee of a mathematician (and I’m not one myself), my experience is that they certainly ARE better at admitting they’re wrong outside of the mathematical sphere. And he’s not afraid to point out when I’m wrong (which can often be very frustrating, especially since he’s usually right!). On the whole, it’s helped me to grow into a better person, and a more honest one, too. ❤ mathematicians.
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100% agree with the idea that organizations really don’t want skeptics. I think it is partly because even those people who understand how to do a proof (a small minority of the population) often don’t think to apply that type of thinking outside of mathematics. It is a rare few who do (and annoy everyone else).
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Hello Cathy! Thank you for writing this.
You’re right that submitting a proof is an act of courage. For teenagers it’s a fine line between maintaining a healthy dose of confidence and the ability to remain humble and unfortunately many teachers are not trained to deal with this or, worse, don’t believe that it’s something they need to understand.
It’s interesting that you bring up how being a skeptic is not actually valued by many industrie / organizations. We as teachers ideally want to instill a healthy sense of creativity of skepticism in our students, but I wonder if we\’re actually doing them any harm employment-wise.
Anyway, it’s great to here that the young yellow pigs are in good hands. =)
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not sure a mathematician like Hilbert would agree that a proof is a ‘social’ construct. ‘What we need’ to be convinced something is true often falls short of formal proof. A proof starts from the axioms of your formal system and uses logic to derive other propositions which can be then used to prove further propositions. A proof means you can believe something is true regardless of social constructs or anything else.
the hardest part is getting people to formulate and think in terms of well-formed statements that can be tested, logically reasoned with and proved true or false.
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Hilbert would be full of shit then.
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Sounds you’re a social constructivist and Hilbert was a formalist
http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Formalism
http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Social_constructivism_or_social_realism
Certainly, which parts of mathematics get attention and how completely and correctly mathematical knowledge is developed depends on what problems society is trying to solve, on the society’s values, hierarchies, etc.
Euclidean geometry might be important at one time, and a non-Euclidean geometry might be important later, when people are understanding a relativistic universe. One geometry is not more true than the other, but one might be more advantageous (Pirsig).
But what is provable in a given formal system is the part that does not change according to a society’s ability to stomach it.
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You are 100% wrong. Hilbert was not a formalist. He was presented by others as one, based on a hearsay report by Blumenthal. Read the recent books released by Springer on his lectures, and recant.
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It’s absolutely a social construct. Most proofs do not start from the axioms. They would be unweildy, lengthy an fairly unintelligible. Any time you cite a previous result or something you know (even if that is how to add or multiply) you are using socially accepted “truths” that are based on that axiomatic system but really are just agreed upon as true to the writer and his/her intended audience.
Also, there are things in mathematics like the Continuum Hypothesis that are social constructs. The truth or falsity of the Continuum Hypothesis cannot be established using the foundational set of axioms we use, therefore it has been taken as true. It’s also how we get to non-Euclidean geometries – by deciding that the fifth postulate isn’t true.
Thanks to Cathy for posting this. My dissertation research is on students’ transition to proof, and this is a great distillation of some of the issues. I think I will share it with my students in the future.
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It’s true that in practice starting from the axioms and building up from them using strict logic is prohibitively hard and time consuming, so in practice mathematicians justify their inferences in a slightly vaguer way, and generally everybody gets away with. But the “gold standard” set by things like using the Peano axioms to build up rules of arithmetic is important, and when mistakes or apparent contradictions or paradoxes emerge (as they do all the time) the way to resolve them is to go over the ground you thought you already covered, but more carefully (approaching the “gold standard” more closely).
Of course a perfect proof is an unattainable ideal, and in the end even the ideal is, as you say, a social construct. But the aim is to come as close to possible to universal, noncontingent knowledge. To the extent to which that is achieved (this extent is easily exaggerated by pompous people but still not zero) we’re doing something more than adhering to social convention: we’re discovering truth.
On a tangential note I’ve found that the mathematical writers I respect and admire the most are in many cases among those who come closest to this ideal. At each step of the proof, it is made as explicit as possible which assumptions and which prior results are being used and in precisely what way.
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Right, the axiomatic method is a beautiful thing and gets us as close as we can to that ideal. However, when looking at a proof of a theorem whether or not it is a valid proof depends on the context in which the proof is places. I write proofs differently as a teacher of mathematics, as a student in a class, and as a student of mathematics verifying results for myself. The level of support and justification I feel I need to provide depends on the audience.
That’s what I mean by calling it a social construct. At the same time, I will disagree with Cathy in saying that a proof is what we need to be convinced that something is true because, as has been pointed out earlier, something can be convincing without being a proof (and without even being correct). Also, proof novices sometimes do not find proofs to be convincing and find a need to test the proof with empirical evidence.
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I’d like to offer an alternative view of the continuum hypothesis. Almost all mathematics, as it is practiced today, is insensitive to the validity of CH. That is, mathematicians rarely take a position on this question in their work, because they don’t need to, and their proofs work whether CH is true or false. When a proof uses CH or not-CH in an essential way, it is conventional to state that assumption explicitly.
On the other hand, axioms like replacement and choice are typically tacitly assumed for convenience, even though they are unnecessary for “most” mathematical work. This is mostly because the benefits of eliminating axioms are limited, while the costs in time and headaches can be big.
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A proof is a series of truth-preserving steps, each of them appealing to some generally valid deductive method. For example, if you have X, and X->Y, then you can say Y, because whenever A is true and A->B is true, B is true. Suppose I prove that 2=1 as follows:
x = y = 1
x^2 = xy
x^2 – y^2 = xy – y^2
(x+y)(x-y) = y(x -y)
x+y = y
2 = 1
In each of these steps, we’re trying to do things that, in general, yield true equations from true equations. If it is in fact true that x = y, then we are allowed to multiply both sides of the equation by the same value, including the value y, and end up with a true equation after that. But if you actually substitute in 1s for the values of X and Y and try to follow along this proof, you’ll find that there’s a particular line where it goes from a true equation to a false equation. Clearly, whichever step we used to go from a true equation to a false equation must not be generally valid, which is to say, truth-preserving; we must’ve done something that can yield falsity from truth. A further bit of staring will tell the student what it is.
It’s also important to realize that if, in fact, x does not equal y, then the conclusions will also probably be wrong; logic is not there to create truth, but to preserve truth. Since tautologies are valid in every possible world, observing the validity of a logical implication never tells you which possible world you live in. Rather, logic tells you the implications of what is already known, observed, or induced; it tells you that if you already believe in gravity on the basis of previous empirical observations, you also ought to believe that apples will fall at a certain rate and that Mercury will appear in the sky in a certain place. Should the apple fail to fall, we can check to see if our logic was invalid; but if our logic is valid, then it is the assumption of gravity that is falsified, not logic itself.
Such is the nature of proofs.
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It is good that young mathematicians hear these arguments. In the computer science context, a classic paper on this from 1979 was written by deMillo, Lipton and Perlis at http://dl.acm.org/citation.cfm?id=359106
As a mathematically-trained researcher on the theory side of computer systems topics, I agree with the view expressed so nicely by Cathy. It differs from what Eliezer describes in coment 15, but really this disagreement is a matter of confusion between different notions that use the same term. There is “proof” as studied in logic or implemented in various fomal methods software programs; that sort of proof is a syntactically-checkable sequence of steps (or more generally a tree of steps), and there is “proof” as practiced by mathematicians and written in math books (part of a social process, aimed at increasing conviction). The mathematical logic version is intended as an idealization of the practice, but (like other idealizations in physics, economics etc) it has travelled rather far from the reality it was modelled on!
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I think that willingness to admit failure is integral to being a sceptic. To be sceptical of something is to admit that you do not know if it is wrong or right, but you question the assumption that it is one or the other. Hopefully you will have an idea about an approach that would go some way to arriving at a correct decision, but it is the rejection of assumption that is most important. It is the sceptic’s willingness to admit they cannot make the decision that leaves them open to attack, but the defence of course is that it is better to know you don’t know, than to think you know but be wrong.
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Beautifully said. When my students struggle with the notion of proof (they often just don’t know what is expected from them), I tell them that they should pretend they are trying to convince a skeptical reader their understanding of the issue is correct. That’s what a proof is, not the strict adherence to any given logical format or precise notation – as they often mistakenly assume, or are sometimes led to believe from bad teachers.
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Alan, although the idealization has travelled far from the reality, it is still connected, and this is critical. Ultimately mathematicians CAN come to reliable agreement as to whether something is true or not, and this is due to the efforts of great mathematicians like Boole, Frege, Russell, Hilbert, Zermelo, and Godel, who nailed down the relationship between formal and informal proofs so well that 99% of all published “proofs” can be straightforwardly (i.e. with much effort and time but no significant difficulty) expanded and filled-in to a formalized machine-checkable proof (while the other 1%, it is generally believed, can be NON-straightforwardly converted to such a proof).
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Oops, my previous comment should have said that mathematicians can come to reliable agreement as to whether something is “proved”, not whether it it is “true”, please fix.
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Take a look at the International Tournament of Young Mathematicians (www.itym.org). Participants review and critique each other’s work, including proofs, and they get feedback on their critiques. You might be able to use some of the past written materials to do the same with your students as a way of helping them understand what constitutes a proof (and maybe even get into fun style questions, too).
Other than being a course grader, I can’t think of a formal activity in my math education that was directed at evaluating someone else’s arguments. Of course, that wasn’t intentional as a part of the grader experience either!
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