Posts Tagged 'proofs'

What is a proof, really?

What is a mathematical proof? Way back when I was a college freshman, I could give you a precise answer: A proof of a statement S is a finite sequence of assertions S(1), S(2), … S(n) such that S(n) = S and each S(i) is either an axiom or else follows from one or more of the preceding statements S(1), …,S(i-1) by a direct application of a valid rule of inference.

But I was so much older then, I’m younger than that now.

After a lifetime in professional mathematics, during which I have read a lot of proofs, created some of my own, assisted others in creating theirs, and reviewed a fair number for research journals, the one thing I am sure of is that the definition of proof you will find in a book on mathematical logic or see on the board in a college level introductory pure mathematics class doesn’t come close to the reality.

For sure, I have never in my life seen a proof that truly fits the standard definition. Nor has anyone else.

The usual maneuver by which mathematicians leverage that formal notion to capture the arguments they, and all their colleagues, regard as proofs is to say a proof is a finite sequence of assertions that could be filled in to become one of those formal structures.

It’s not a bad approach if the goal is to give someone a general idea of what a proof is. The trouble is, no one has ever carried out that filling-in process. It’s purely hypothetical. How then can anyone know that the purported proof in front of them really is a proof?

I wrote about this dilemma in my MAA “Devlin’s Angle” column way back in 1996, in an article titled Moment of Truth.

I picked up the theme again in 2003 with my Devlin’s Angle piece When is a Proof?

These days I have a very pragmatic perspective on what a proof is, based on the way people use them in the day-to-day world of mathematics:

Proofs are stories that convince suitably qualified others that a certain statement is true.

If I present you with a proof, and you have the appropriate background knowledge and ability, you can – usually with some time and effort – as a result of reading my story, become convinced that what I claim is true.

But if you take that as your working definition of proof, you have to acknowledge it is fundamentally about communication, not truth. In particular, whether an argument classifies as a proof depends as much on the intended reader as on its creator.

Of course, in order to function in that way, the “story” has to be pretty heavily constrained.

Moreover, the creators and the consumers of those stories have to be familiar with the genre. That part takes time to acquire.

On the other hand, once a person becomes familiar with both the genre and the particular mathematical focus, reading and understanding those stories becomes natural and fluent.

The system works – as any professional mathematician will affirm. It’s how mathematics advances.

To an outsider, however, the whole thing is usually incomprehensible.

Today, many proofs stretch over several pages, not infrequently hundreds of pages. A key feature that allows such proofs to function effectively in the mathematical community is that many steps are left out.

In some cases this is because the step has already been established, either by the same author in a previous piece of work, or by someone else. In such cases, the author simply refers the reader to that source.

In other cases, the author judges that the intended reader should be capable of supplying the missing steps on the fly. The author may provide a hint to help the reader provide the missing steps, but not always.

There is, then, a huge element of audience design in constructing effective proofs. A proof designed for an undergraduate mathematics class is in general very different from one constructed to present at a research seminar.

To the beginner, trying to make the transition from high school mathematics to university level, coming to terms with real proofs is not only difficult, it can be traumatic, with a once comforting illusion of crisp, clean certainty rapidly giving way to a panicked feeling of sinking into shifting quicksand.

At this point, it can be of some comfort to learn that Euclid screwed up big-time when he penned his famous geometry proofs in Elements. Yes, those iconic proofs may seem logically sound, and indeed for two thousand years were held up as models of logically sound reasoning. But as David Hilbert observed in the late Nineteenth Century, Euclid’s arguments are riddled with logical holes.

To give just one example, he often tells you to construct a point by intersecting an arc of a circle with a straight line. But how do you know there is an intersection? Sure, when you draw the arc and the line on a sheet of paper, the arc may cross over the line. But do they actually intersect? That is, do they have an actual (dimensionless) point in common?

That is not only not obvious, it takes a lot of work to answer. (The answer is, it depends on the underlying number system. But it requires some deep machinery not developed until the Nineteenth Century.)

Of course, high school teachers rarely, if ever, tell their students that the geometry proofs they are presented as models are at best sketches of how proofs can be constructed. As a result, those students typically enter university with a totally false impression of what a proof is. In particular, they believe proofs are fundamentally and exclusively about truth, and that they are either right or wrong.

In reality, proofs are about truth, but not fundamentally, and definitely not exclusively. The key property of a proof is not that it is logically correct (it almost certainly is not, but more pertinent, how could you ever be sure it is?), rather that it is expressed in a manner that enables a suitably qualified reader to fill in any holes they notice, to check any steps they have any doubt about, and to correct any errors they find (as they surely will if they dig deep enough).

It’s very much like software engineering, where the most important thing about a program is not that it is bug free (it almost certainly is not), rather that – in addition to working – it is structured and annotated so that someone else can come along later and either fix bugs or else modify the code to do something else.

Ridding high school graduates of the “proofs are about logical correctness” misconception is generally a difficult (for both instructor and student) and painful (for the student) process. Just what it entails has been a focus of a study I have been making in my MOOC Introduction to Mathematical Thinking, currently being offered for the fifth time. I describe my most recent observations in a new post on my other blog, MOOCtalk.org,

where this account continues…

Want to learn how to prove a theorem? Go for a mountain bike ride

Because this blogpost covers both mountain biking and proving theorems, it is being simultaneously published by the Mathematical Association of America in my Devlin’s Angle series.

Mountain biking is big in the San Francisco Bay Area, where I live. (In its present day form, using specially built bicycles with suspension, the sport/pastime was invented a few miles north in Marin County in the late 1970s.) Though there are hundreds of trails in the open space preserves that spread over the hills to the west of Stanford, there are just a handful of access trails that allow you to start and finish your ride in Palo Alto. Of those, by far the most popular is Alpine Road.

My mountain biking buddies and I ascend Alpine Road roughly once a week in the mountain biking season (which in California is usually around nine or ten months long). In this post, I’ll describe my own long struggle, stretching over many months, to master one particularly difficult stretch of the climb, where many riders get off and walk their bikes.

[SPOILER: If your interest in mathematics is not matched by an obsession with bike riding, bear with me. My entire account is actually about how to set about solving a difficult math problem, particularly proving a theorem. I’ll draw the two threads together in a subsequent post, since it will take me into consideration of how the brain works when it does mathematics. For now, I’ll leave the drawing of those conclusions as an exercise for the reader! So when you read mountain biking, think math.]

Alpine Road used to take cars all the way from Palo Alto to Skyline Boulevard at the summit of the Coastal Range, but the upper part fell into disrepair in the late 1960s, and the two-and-a-half-mile stretch from just west of Portola Valley to where it meets the paved Page Mill Road just short of  Skyline  is now a dirt trail, much frequented by hikers and mountain bikers.

Alpine Road. The trail is washed out just round the bend

Alpine Road. The trail is washed out just round the bend

A few years ago, a storm washed out a short section of the trail about half a mile up, and the local authority constructed a bypass trail. About a quarter of a mile long, it is steep, narrow, twisted, and a constant staircase of tree roots protruding from the dirt floor. A brutal climb going up and a thrilling (beginners might say terrifying) descent on the way back. Mountain bike heaven.

There is one particularly tricky section right at the start. This is where you can develop the key abilities you need to be able to prove mathematical theorems.

So you have a choice. Read Polya’s classic book, or get a mountain bike and find your own version of the Alpine Road ByPass Trail. (Better still: do both!)

My mountain bike at the start of the bypass trail

My mountain bike at the start of the bypass trail

When I first encountered Alpine Road Dirt a few years ago, it took me many rides before I managed to get up the first short, steep section of the ByPass Trail.

What lies around that sharp left-hand turn?

What lies around that sharp left-hand turn?

It starts innocently enough – because you cannot see what awaits just around that sharp left-hand turn.

The short, narrow descent

The short, narrow descent

After you have made the turn, you are greeted with a short narrow downhill. You will need it to gain as much momentum as you can for what follows.

I’ve seen bikers with extremely strong leg muscles who can plod their way up the wall that comes next, but I can’t do it that way. I learned how to get up it by using my problem-solving/theorem-proving skills.

The first thing was to break the main problem – get up the insanely steep, root strewn, loose-dirt climb – into smaller, simpler problems, and solve those one at a time. Classic Polya.

But it’s Polya with a twist – and by “twist” I am not referring to the sharp triple-S bend in the climb. The twist in this case is that the penalty for failure is physical, not emotional as in mathematics. I fell off my bike a lot. The climb is insanely steep. So steep that unless you bend really low, with your chin almost touching your handlebar, your front wheel will lift off the ground. That gives rise to an unpleasant feeling of panic  that is perhaps not unlike the one that many students encounter when faced with having to prove a theorem for the first time.

Steep. If you are not careful, your front wheel will lift off the ground.

Steep. If you are not careful, your front wheel will lift off the ground.

The photo above shows the first difficult stretch. Though this first sub-problem is steep, there is a fairly clear line to follow to the right that misses those roots, though at the very least its steepness will slow you down, and on many occasions will result in an ungainly, rapid dismount. And losing momentum is the last thing you want, since the really hard part is further up ahead, near the top in the picture.

Also, do you see that rain- and tire-worn groove that curves round to the right just over half way up – just beyond that big root coming in from the left? It is actually deeper and narrower than it looks in the photo, so unless you stay right in the middle of the groove you will be thrown off line, and your ascent will be over. (Click on the photo to enlarge it and you should be able to make out what I mean about the groove. Staying in the groove can be tricky at times.)

Still, despite difficulties in the execution, eventually, with repeated practice, I got to the point of  being able to negotiate this initial stretch and still have some forward momentum. I could get up on muscle memory. What was once a series of challenging problems, each dependent on the previous ones, was now a single mastered skill.

[Remember, I don’t have super-strong leg muscles. I am primarily a road bike rider. I can ride for six hours at a 16-18 mph pace, covering up to 100 miles or more. But to climb a steep hill I have to get off the saddle and stand on the pedals, using my body weight, not leg power. Unfortunately, if you take your weight off the saddle on a mountain bike on a steep dirt climb, your rear wheel will start to spin and you come to a stop – which on a steep hill means jump off quick or fall. So I have to use a problem solving approach.]

Once I’d mastered the first sub-problem, I could address the next. This one was much harder. See that area at the top of the photo above where the trail curves right and then left? Here is what it looks like up close.

The crux of the climb/problem. Now it is really steep.

The crux of the climb/problem. Now it is really steep.

(Again, click on the photo to get a good look. This is the mountain bike equivalent of being asked to solve a complex math problem with many variables.)

Though the tire tracks might suggest following a line to the left, I suspect they are left by riders coming down. Coming out of that narrow, right-curving groove I pointed out earlier, it would take an extremely strong rider to follow the left-hand line. No one I know does it that way. An average rider (which I am) has to follow a zig-zag line that cuts down the slope a bit.

Like most riders I have seen – and for a while I did watch my more experienced buddies negotiate this slope to get some clues – I start this part of the climb by aiming my bike between the two roots, over at the right-hand side of the trail. (Bottom right of picture.)

The next question is, do you go left of that little tree root nipple, sticking up all on its own, or do you skirt it to the right? (If you enlarge the photo you will see that you most definitely do not want either wheel to hit it.)

The wear-marks in the dirt show that many riders make a sharp left after passing between those two roots at the start, and steer left of the nobbly root protrusion. That’s very tempting, as the slope is notably less (initially). I tried that at first, but with infrequent  success. Most often, my left-bearing momentum carried me into that obstacle course of tree roots over to the left, and though I sometimes managed to recover and swing  out to skirt to the left of that really big root, more often than not I was not able to swing back right and avoid running into that tree!

The underlying problem with that line was that thin looking root at the base of the tree. Even with the above photo blown up to full size, you can’t really tell how tricky an obstacle it presents at that stage in the climb. Here is a closer view.

The obstacle course of tree roots that awaits the rider who bears left

The obstacle course of tree roots that awaits the rider who bears left

If you enlarge this photo, you can probably appreciate how that final, thin root can be a problem if you are out of strength and momentum. Though the slope eases considerably at that point, I – like many riders I have seen – was on many occasions simply unable make it either over the root or circumventing it on one side – though  all three options would clearly be possible with fresh legs. And on the few occasions I did make it, I felt I just got lucky – I had not mastered it. I had got the right answer, but I had not really solved the problem. So close, so often. But, as in mathematics, close is not good enough.

After realizing I did not have the leg strength to master the left-of-the-nipple path, I switched to taking the right-hand line. Though the slope was considerable steeper (that is very clear from the blown-up photo), the tire-worn dirt showed that many riders chose that option.

Several failed attempts and one or two lucky successes convinced me that the trick was to steer to the right of the nipple and then bear left around it, but keep as close to it as possible without the rear wheel hitting it, and then head for the gap between the tree roots over at the right.

After that, a fairly clear left-bearing line on very gently sloping terrain takes you round to the right to what appears to be a crest. (It turns out to be an inflection point rather than a maximum, but let’s bask for a while in the success we have had so far.)

Here is our brief basking point.

AD8

The inflection point. One more detail to resolve.

As we oh-so-briefly catch our breath and “coast” round the final, right-hand bend and see the summit ahead, we come – very suddenly – to one final obstacle.

AD9

The summit of the climb

At the root of the problem (sorry!) is the fact that the right-hand turn is actually sharper than the previous photo indicates, close to a switchback. Moreover, the slope kicks up as you enter the turn. So you might not be able to gain sufficient momentum to carry you over one or both of those tree roots on the left that you find your bike heading towards. And in my case, I found I often did not have any muscle strength left to carry me over them by brute force.

What worked for me is making an even tighter turn that takes me to the right of the roots, with my right shoulder narrowly missing that protruding tree trunk. A fine-tuned approach that replaces one problem (power up and get over those roots) with another one initially more difficult (slow down and make the tight turn even tighter).

And there we are. That final little root poking up near the summit is easily skirted. The problem is solved.

To be sure, the rest of the ByPass Trail still presents several other difficult challenges, a number of which took me several attempts before I achieved mastery. Taken as a whole, the entire ByPass is a hard climb, and many riders walk the entire quarter mile. But nothing is as difficult as that initial stretch. I was able to ride the rest long before I solved the problem of the first 100 feet. Which made it all the sweeter when I finally did really crack that wall.

Now I (usually) breeze up it, wondering why I found it so difficult for so long.

Usually? In my next post, I’ll use this story to talk about strategies for solving difficult mathematical problems. In particular, I’ll look at the role of the subconscious in being able to put together a series of mastered steps in order to solve a big problem. For a very curious thing happened after I took the photos to illustrate this post. I walked back down to collect my bike from the ByPass sign where I’d left it, and rode up to continue my ride.

It took me four attempts to complete that initial climb!

And therein lies one of the biggest secrets of being able to solve a difficult math problem.

To be continued …


I'm Dr. Keith Devlin, a mathematician at Stanford University, an author, the Math Guy on NPR's Weekend Edition, and an avid cyclist. (Yes, that's me cycling on the Marin Headland.)

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