Because this blog post
covers both mountain biking and proving theorems, it is being simultaneously
published in Devlin’s more wide ranging blog profkeithdevlin.org.
In my post last month, I described my efforts to ride a
particularly difficult stretch of a local mountain bike trail in the hills just
west of Palo Alto. As promised, I will now draw a number of conclusions for
solving difficult mathematical problems.
Most of them will be familiar to anyone who has read George
Polya’s classic book How to Solve It. But my main conclusion may come as a surprise unless you
have watched movies such as Top Gun
or Field of Dreams, or if you follow
professional sports at the Olympic level.
Here goes, step-by-step, or rather
pedal-stroke-by-pedal-stroke. (I am assuming you have recently read my last
post.)
BIKE: Though bikers with extremely strong leg muscles can
make the Alpine Road ByPass Trail ascent by brute force, I can't. So my first
step, spread over several rides, was to break the main problem—get up an
insanely steep, root strewn, loose-dirt climb—into smaller, simpler problems,
and solve those one at a time.
MATH: Breaking a large problem into a series of smaller ones is a
technique all mathematicians learn early in their careers. Those
subproblems may still be hard and require considerable effort and several
attempts, but in many cases you find you can make progress on at least some of
them. The trick is to make each subproblem sufficiently small that it requires
just one idea or one technique to solve it.
In particular, when you break the overall problem down
sufficiently, you usually find that each smaller subproblem resembles another
problem you, or someone else, has already solved.
When you have managed to solve the subproblems, you are left
with the task of assembling all those subproblem solutions into a single whole.
This is frequently not easy, and in many cases turns out to be a much harder
challenge in its own right than any of the subproblem solutions, perhaps
requiring modification to the subproblems or to the method you used to solve
them.
BIKE: Sometimes there are several different lines you can
follow to overcome a particular obstacle, starting and ending at the same
positions but requiring different combinations of skills, strengths, and
agility. (See
my description last month of how I managed to negotiate the steepest section
and avoid being thrown off course—or off the bike—by that troublesome
tree-root nipple.)
MATH: Each subproblem takes you from a particular starting
point to a particular end-point, but there may be several different approaches
to accomplish that subtask. In many cases, other mathematicians have solved
similar problems and you can copy their approach.
BIKE: Sometimes, the approach you adopt to get you past one
obstacle leaves you unable to negotiate the next, and you have to find a
different way to handle the first one.
MATH: Ditto.
BIKE: Eventually, perhaps after many attempts, you figure
out how to negotiate each individual segment of the climb. Getting to this
stage is, I think, a bit harder in mountain biking than in math. With a math
problem, you usually can work on each subproblem one at a time, in any order.
In mountain biking, because of the need to maintain forward (i.e., upward)
momentum, you have to build your overall solution up in a cumulative fashion—vertically!
But the distinction is not as great as might first appear.
In both cases, the step from having solved each individual subproblem in
isolation to finding a solution for the overall problem, is a mysterious one
that perhaps cannot be appreciated by someone who has not experienced it. This
is where things get interesting.
Having had the experience of solving difficult (for me)
problems in both mathematics and mountain biking, I see considerable
similarities between the two. In both
cases, the subconscious mind plays a major role—which is, I presume, why
they seem mysterious. This is where this two-part blog post is heading.
BIKE: I ended my previous post by promising to
"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…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."
BOTH: How does the human mind make a breakthrough? How are
we able to do something that we have not only never done before, but failed
many times in attempts to do so? And why does the breakthrough always seem to occur
when we are not consciously trying
to solve the problem?
The first thing to note is that we never experience the
process of making that breakthrough. Rather, what we experience, i.e., what we
are conscious of, is having just made
the breakthrough!
The sensation we have is a combined one of both elation and surprise. Followed almost
immediately by a feeling that it wasn’t
so difficult after all!
What are we to make of this strange process?
Clearly, I cannot provide a definitive, concrete answer to
that question. No one can. It’s a mystery. But it is possible to make a number of relevant observations,
together with some reasonable, informed speculations. (What follows is a
continuation of sorts of the thread I developed in my 2000 book The Math Gene.)
The first observation is that the human brain is a result of
millions of years of survival-driven, natural selection. That made it supremely
efficient at (rapidly) solving problems that threaten survival. Most of that
survival activity is handled by a small, walnut-shaped area of the brain called
the amygdala, working in close conjunction with the body’s nervous system and motor
control system.
In contrast to the speed at which our amydala operates, the
much more recently developed neo-cortex that supports our conscious thought,
our speech, and our “rational reasoning,” functions at what is comparatively
glacial speed, following well developed channels of mental activity—channels
that can be built up by repetitive training.
Because we have conscious access to our neo-cortical thought
processes, we tend to regard them as “logical,” often dismissing the actions of
the amygdala as providing (“mere,” “animal-like”) “instinctive reactions.” But
that misses the point that, because that “instinctive reaction organ” has
evolved to ensure its owner’s survival in a highly complex and ever changing
environment, it does in fact operate in an extremely logical fashion, honed by
generations of natural selection pressure to be in sync with its owner’s environment.
Which leads me to this.
Do you want to identify that part of the brain that makes
major scientific (and mountain biking) breakthroughs?
I nominate the amygdala—the “reptilian brain” as it is
sometimes called to reflect its evolutionary origin.
I should acknowledge that I am not the first person to make
this suggestion. Well, for mathematical breakthroughs, maybe I am. But in
sports and the creative arts, it has long been recognized that the key to truly
great performance is to essentially shut down the neo-cortex and let the
subconscious activities of the amygdala take over.
Taking this as a working hypothesis for mathematical (or
mountain biking) problem solving, we can readily see why those moments of great
breakthrough come only after a long period of preparation, where we keep
working away—in conscious fashion—at trying to solve the problem or perform
the action, seemingly without making any progress.
We can see too why, when the breakthrough (or the great
performance) comes, it does so instantly and surprisingly, when we are not actively trying to achieve the goal, leaving our
conscious selves as mere after-the-fact observers of the outcome.
For what that long period of struggle does is build a
cognitive environment in which our reptilian brain—living inside and being
connected to all of that deliberate, conscious activity the whole time—can
make the key connections required to put everything together. In other words,
investing all of that time and effort in that initial struggle raises the internal,
cognitive stakes to a level where the amygdala can do its stuff.
Okay, I’ve been playing fast and loose with the metaphors
and the anthropomorphization here. We’re
really talking about biological systems, simply operating the way natural
selection equipped them. But my goal is not to put together a scientific
analysis, rather to try to figure out how to improve our ability to solve novel
problems. My primary aim is not to be “right” (though knowledge and insight are
always nice to have), but to be able to improve performance.
Let’s return to that tricky stretch of the ByPass section on
the Alpine Road trail. What am I consciously focusing on when I make a
successful ascent?
BIKE: If you have read my earlier account, you will know
that the difficult section comes in three parts. What I do is this. As I
approach each segment, I consciously think about, and fix my eyes on, the
end-point of that segment—where I will be after I have negotiated the
difficulties on the way. And I keep my eyes and attention focused on that
goal-point until I reach it. For the whole of the maneuver, I have no conscious
awareness of the actual ground I am cycling over, or of my bike. It’s total
focus on where I want to end up, and nothing else.
So who—or what—is controlling the bike? The mathematical control problem involved in getting a person-on-a-bike up a steep,
irregular, dirt trail is far greater than that required to auto-fly a jet
fighter. The calculations and the speed with which they would have to be
performed are orders of magnitude beyond the capability of the relatively slow
neuronal firings in the neocortex. There is only one organ we know of that
could perform this task. And that’s the amygdala, working in conjunction with
the nervous system and the body’s motor control mechanism in a super-fast
constant feedback loop. All the neo-cortex and its conscious thought has to do
is avoid getting in the way!
These days, in the case of Alpine Road, now I have “solved”
the problem, the only things my conscious neo-cortex has to do on each occasion
are switching my focus from the goal of one segment to the goal of the next. If
anything interferes with my attention at one of those key transition moments,
my climb is over—and I stop or fall.
What used to be the hard parts are now “done for me” by
unconscious circuits in my brain.
MATH: In my case at least, what I just wrote about mountain
biking accords perfectly with my experiences in making (personal) mathematical
problem-solving breakthroughs.
It is by stepping back from trying to solve the problem by putting together everything I know and
have learned in my attempts, and instead simply focusing on the problem
itself—what it is I am trying to show—that I suddenly find that I have the
solution.
It’s not that I arrive at the solution when I am not
thinking about the problem. Some mathematicians have expressed their
breakthrough moments that way, but I strongly suspect that is not totally true.
When a mathematician has been trying to solve a problem for some months or
years, that problem is always with them. It becomes part of their existence.
There is not a single waking moment when that problem is not “on their mind.”
What they mean, I believe, and what I am sure is the case
for me, is that the breakthrough comes when the problem is not the focus of our
thoughts. We really are thinking about something else, often some mundane
detail of life, or enjoying a marvelous view. (Google “Stephen Smale beaches of
Rio” for a famous example.)
This thesis does, of course, explain why the process of
walking up the ByPass Trail and taking photographs of all the tricky points made
it impossible for me to complete the climb. True, I did succeed at the fourth
attempt. But I am sure that was not because the first three were “practice.”
Heavens, I’d long ago mastered the maneuvers required. It was because it took
three failed attempts before I managed to erase the effects of focusing on the
details to capture those images.
The same is true, I suggest, for solving a difficult
mathematical problem. All of those techniques Polya describes in his book, some
of which I list above, are essential to prepare the way for solving the
problem. But the solution will come only when you forget about all those
details, and just focus on the prize.
This may seem a wild suggestion, but in some respects it may
not be entirely new. There is much in common between what I described above and
the highly successful teaching method of R. L. Moore. For sure you have to do a fair amount of translation from
his language to mine, but Moore used to demand that his students not
clutter their minds by learning stuff, rather took each problem as it came and
then try to solve it by pure reasoning, not giving up until they found the
solution.
In terms of training future mathematicians, what these
considerations imply, of course, is that there is mileage to be had from
adopting some of the techniques used by coaches and instructors to produce
great performances in sports, in the arts, in the military, and in chess.
Sweating the small stuff will make you good. But if you want
to be great, you have to go beyond that—you have to forget the small stuff
and keep your eye on the prize.
And if you are successful, be sure to give full credit for
that Fields Medal or that AMS Prize where it is rightly due: dedicate it to
your amygdala. It will deserve it.
