Saturday, 22 September 2012

Why do we have to learn proofs!?

That’s right.  You are going to have to endure proofs. Like many of my students, per- haps you are asking yourself (or me), why do I have to learn proofs? Aren’t they just some esoteric, jargon-filled, technical writing that  only a professional mathematician would care about?
Well, no. And I’d like to offer a short justification  of this claim. My argument is three-fold:  (1) proofs are all around  you, (2) it’s quite possible to get better  at them by practice and by benefiting from the accumulated  knowledge of two thousand years of mathematicians, and (3) this will really help you in “real life,” whether you go into mathematics, carpentry,  or child-rearing.
First  and  foremost,  it  is important to  know what  a proof is.   It  is not  one of those  horrible  two-column  tables  of axioms and  deductions  you saw in 10th  grade geometry.     That is a bizarre  (though  sometimes  useful)  invention  of mathematics  educators  which constitutes a particular way  to  write  down a very special kind of proof in a very narrow area.  No mathematician would be caught dead writing such a thing1.  A proof is not some long sequence of equations  on a chalk board,  nor is it a journal article.  These things  are ways that mathematician communicate  proofs, but the truth is, proof is in your head.
A proof is an  argument,  a justification,  a reason  that  something  is true.   It’s got to be a particular  kind of reasoning   logical to be called a proof.  (There  are certainly plenty of other, equally valid forms of reasoning. And some of them are even used in “doing” mathematics. But they’re not proofs.) A proof is just the answer to the question “Why?”,  when the person asking the question wants  an argument that  is indisputable, in the  sense that  any person of normal  intelligence who has enough time  could be convinced of it.  “Why is the  sky blue? is a question  any answer to which could conceivably be wrong:  perhaps  we will all wake up tomorrow,  and find it orange.   It’s  possible, both  physically and metaphysically.  (In fact, in New York City,  this  happens  quite  frequently.)   On  the  other  hand,  if one asks,  “Why  does
2 + 2 = 2 × 2,” or, “Why is the area of a rectangle the product  of its side lengths?”, I can give you an answer that’s  impossible to argue with.  If we agree on the premises the definitions of the terminology being used and we agree that,  if A implies B, and A is true, then B is true, then we agree on the conclusion. There’s no way around it.2
In fact,  you use “proofs”  all the  time.   When  you reason with  yourself that it will be cheaper  to buy the larger  cans of beans,  you are proving something  about the respective prices. When you play that terribly  addictive game Minesweeper, and you see that, oh, yeah, there  has to be a mine under  this block, well, you’ve proven something to yourself. When you up the ante because you’ve got three kings, and you know who’s got the aces, you’re proving something about the possible hands that your



friends hold. When you know you can’t turn  right on red in at a certain  intersection,  you’ve constructed a proof of the illegality of this move. (It goes something like this: “Normally,  I have to  stop at a red light.  But  I’m in the right  lane, so that  rule is superceded  by the state  law that right-on-red  is okay.  However, there’s a sign right there  that says it’s not  okay at this intersection.  So it wouldn’t be legal to turn.3 ) Many, many real-life situations  require reasoning that is, at least in part,  something that would qualify as a mathematical proof.  At the  very least,  mathematical-type reasoning is a powerful addition  to anyone’s critical thinking toolbox, applicable in a wide variety  of settings.  Mathematics is also a cornerstone  of the sciences, which in turn  provide a profoundly  useful way to understand  and interpret the world around us.
Okay, so what?  Why are you, a Drama major, being subjected to the requirement that you write  down a so-called “proof in some crazy mathematical hieroglyphics about  constructs like integrals and binomial coefficients that you’ll never see again? The reason is that mathematicians have spent  more than  two  millenia working out what proofs are and distilling them out of the complex and unruly real world. The idea is that studying the concept of proof will make you better at it in all sorts of situations. We could muck it up by giving you problems involving other types of reasoning, too, by  why not  just  boil it down to the  mathematics itself ?  (Word problems are sort of a middle-ground:   they  invite  you to figure out  what  sorts  of argument  you’re going to  need,  and  you’re supposed  to  produce  a proof and  then  apply  it  to  the ridiculously idealized circumstances  described.)  And the reason that  the proofs you are dealing with involve mathematical  constructs is that mathematical constructs  are things  that  have precise definitions  one can  reason  about.   Notwithstanding what your mathematics-obsessed philosophy professor told you4, you can’t prove anything about  thoughts,  lightning,  or yodelling.  They’re literally impossible to encapsulate in mathematical language, though often it is possible to “model” them by idealizing the situation  first.
Well, that’s the reason that practicing proving things is a good idea, but why must you write it  down?  No, it’s not because I’m sadistic  and proofs are my instrument of torture.  (Not to say that  all math  professors aren’t  sadistic,  as this  is decidedly not  true.)   Here are a few real reasons.  First of all, because otherwise  I can’t tell if you know why something is true!  You haven’t  learned anything  if you get the final answer by luck (or by good peripheral  vision).   If I could read  your  mind,  I could check if you thought through the problem “correctly,”  i.e., how clear your reasoning is.   But  I can’t,  so you would you please just  write  down your  thinking  so I can see if you get it?   Second, the very act  of writing  it  down serves as an  extremely effective way to cut away the chaff,  straighten out your thinking,  and force you to work out  the  details   which often are at the  heart  of the  problem instead of just producing some sort of vague, hand-wavy,  it-could-be-turned-into-a- proof.  Third,  a written representation of a proof, even if it’s just a sequence of equations,  is a way to check and see if everything  is kosher with your argument.   Finally, and this applies


particularly to people in the  sciences or engineering, you might have to write down proofs or pseudo-proofs in subsequent coursework or even (gasp!)  real life.
Now, given that I’ve convinced you (haven’t  I?) that  proofs are useful, let me assure you that you can do them.  You do them  all the time.  I’ve literally never met a student who couldn’t prove things, although  I’ve met plenty of students  unwilling to or lacking the time to learn how to.  Perhaps you will need some practice  when it comes to reasoning about  such abstract objects  that  come up in a math  class, but that’s  just  a matter  of doing your problem sets.  Furthermore there  are lots of proof “techniques”  that  can guide the  way and make the work almost  formulaic once you get the hang of it.  (Induction comes to mind here.)  And getting the hang of it means practice, as annoying as it is. Believe me, I sympathize  with your urge to say, “I get it, why do I have to do it fifty times? But the truth is, you don’t know if you get it until you’ve done it at least a few times, and you’re unlikely to retain  it for very long unless you do it a few more. Very often you think  you know what’s going on, until a problem comes up with a twist in it, and you realize that you haven’t quite grokked the big picture  yet.
I hope that  explains why you’re being tormented so with proofs. Written proofs are a record of your understanding, and  a way to communicate  mathematical ideas with others.  “Doing mathematics is all about finding proofs. And real life has a lot to do with “doing” mathematics, even if it doesn’t look that  way very often.

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