“That's How the Light Gets In"
Tyler J. Jarvis
July 9, 2013 –
BYU Devotional
Tyler J. Jarvis was a BYU professor in the
Department of Mathematics when this devotional address was given on 9 July
2013.
An
important problem that arises in many settings is the Traveling Salesman
Problem.
A
traveler must visit many destinations to sell her goods or make her deliveries.
Her problem is this: what route will be the fastest way to get to all the
destinations? A poor choice could mean she travels many times farther than she
would if she made a good choice.
Obviously
this problem is important to companies like UPS, the U.S. Postal Service,
Walmart, and Amazon. For example, according to Wired magazine, UPS has roughly 55,000 delivery trucks running each day.
If each driver could choose a route that shaves just one mile off the daily
trip, that would save the company $30 million each year.1
And this
problem is not only interesting to companies involved in delivery. The
Traveling Salesman Problem also has applications in computer chip
manufacturing, DNA sequencing, and many other areas.
Consider
the situation for three destinations—call them A, B, and C. One of the possible
routes our traveler could take to these three destinations would be to first go
to city C, then to city B, then to city A, and then home again.
Altogether
there are six possible routes in this situation, so to solve the Traveling
Salesman Problem with three destinations, I need only compare the six routes
and see which is shortest.
With four
destinations I must check a bit more: twenty-four possible routes. I can do
that. With five destinations we have 120 routes. I am too lazy to check all
those, but it is not hard to write some computer code to do it for me. You may
have noticed that the number of routes to check is growing rapidly. For ten
destinations we have 3,628,800 possible routes to check—a lot, but not
impossible.
For
twenty destinations it grows to 2,432,902,008,176,640,000. This is a little too
big for my laptop to check in any reasonable amount of time. In fact, if my
computer could check a billion routes per second, it would still take
seventy-seven years to check all the possible routes. At current energy prices,
just the electricity for the computation would cost roughly $675,000.
Now let
me reassure you. It may look like I am going all mathy on you, but don’t
worry—I won’t make you compute anything, you don’t need to remember any of
these numbers, and there won’t be a quiz at the end. So stay with me for just a
bit longer.
The point
is, even for just twenty destinations we have way too many possible routes to
check in any reasonable amount of time. The bad news is that in many real-life
situations we have a lot more destinations than just twenty—for example, a UPS
driver makes an average of 120 deliveries each day.
For 120
deliveries there are so many possible routes we couldn’t store them all in the
memory of any computer in existence, not even if our computer’s memory
comprised all the atoms in the universe—or even all the atoms in a googol2 universes as big as ours (that’s a one with a
hundred zeros after it).
If we
could find a way to deal with the memory problem, we’d still have to check all
the routes. If we had as many processors as particles in the universe and if
each processor could check a trillion routes per second, it would still take
more than a googol years—far, far more than the age of the universe.
The Clay
Mathematics Institute in Cambridge, Massachusetts, has even offered a
one-million-dollar prize for the first person to find an algorithm to solve the
Traveling Salesman Problem in a reasonable amount of time.
Many
other important mathematical problems suffer from similar difficulties—we can’t
seem to solve them exactly because they are just way, way too big.
These
sorts of problems show up in many aspects of our lives: curing diseases,
preventing accidents, reducing pollution and traffic jams, and building smarter
robots. They are also important for better understanding how the world works:
modeling important biological and chemical processes; modeling populations of
people, animals, and bacteria; understanding how galaxies form; and many other
aspects of our world.
With all
these problems, as long as we insist on getting a perfect answer—the one and
only very best route—we are utterly paralyzed by the size and complexity of the
problem. You could say we are paralyzed
by perfection.
But
despite their complexity and size, we still need to solve these problems. So
let me tell you how to become unparalyzed. And because this is a devotional
talk, you know that I will also use this as a metaphor for something spiritual.
Step 1.
Admit and Accept Imperfection
The first
step is to admit and accept imperfection.
For many
of these hard problems, like the Traveling Salesman Problem, if we really want
a good answer in a reasonable amount of time, we must make a compromise; we
must make do with an approximation and admit some chance of error. I am
something of a perfectionist, so this is difficult for me.
But if I
am willing to accept an answer that is only close to the perfect one—a good
answer but not the perfect answer, an answer with some error in it—as soon as I
give up on perfection, something amazing happens. We can get a very good
approximate solution to the Traveling Salesman Problem very quickly. In fact,
not just very quickly, but blazingly, astoundingly fast.
Now this
solution is not a perfect solution. It will not win the Clay Math
million-dollar prize because it is not exact. But it is a very good solution.
And if you need a better one, we can do that too. Not perfect, but better.
Similarly,
in our own lives, to avoid being paralyzed by perfection we must admit and
accept imperfection. This requires honesty and humility. We can’t try to cover
up our ignorance or our mistakes. We must admit them and learn from them.
In fact,
I know only a few things perfectly: among those, that the Father and the Son
live and love me and you, that the Book of Mormon is what it claims to be, and
that this church is where the Lord wants me to be.
I also
know, with a perfect knowledge, that for now, on this earth, we are all
imperfect, both in knowledge and in performance, but Christ’s Atonement can
bring us to perfection if we allow it to.
The first
step to applying the Atonement is to admit and accept our imperfection.
We are
all imperfect, but it is not always easy to admit that. Being a mathematician
often forces me to admit what I don’t know. Julia Robinson, the first woman
elected to the National Academy of Sciences and past president of the American
Mathematical Society, was once required to submit a description of what she did
each day to her university’s personnel office. This is what she wrote:
Monday—tried
to prove theorem
Tuesday—tried to prove theorem
Wednesday—tried to prove theorem
Thursday—tried to prove theorem
Friday—theorem false3
Now,
while a star like Julia Robinson may find her mistakes after only a week,
people like me usually need a lot more time than that to figure out our
mistakes—and even more time to get the courage to admit them.
And it is
not only our knowledge that is imperfect. Just as the computer has limited
ability to execute the programs given it, so we have limited ability to execute
what we know we should. To survive and succeed in this life we must admit and
accept that imperfection and be patient and understanding with imperfection in
ourselves and in others.
As Elder
Jeffrey R. Holland told us at general conference this April:
Be kind
regarding human frailty—your own as well as that of those who serve with you in
a Church led by volunteer, mortal men and women. Except in the case of His only
perfect Begotten Son, imperfect people are all God has ever had to work with.
That must be terribly frustrating to Him, but He deals with it. So should we.4
And it is
not only human weaknesses we must accept. Sometimes perfection just isn’t
possible in our finite, imperfect world. When my wife was a missionary in
Germany, she and her companion decided they were going to keep all of the
mission rules—exactly. They got out their white handbooks and the additional
list of rules for their mission and sat down with their blue planners to
schedule everything into the week: wake up at 6:30 and then spend one-half hour
for personal scripture study, one-half hour for companion scripture study, one-half
hour for exercise, and so on throughout the day. But somehow they couldn’t make
it all fit before their required bedtime of 10:30. They tried and tried, but
they just couldn’t get it to work. They only figured out why it was so hard to
schedule when they added up all the hours necessary to keep every rule each
day—twenty-five.
The
realities of living in our limited, imperfect world mean that we have no choice
but to make do with an approximation—to admit and accept imperfection.
Step 2.
Work Hard to Get Your Best Approximation
The
second step is to work hard to get your best approximate solution to your
problem.
As I
mentioned before, accepting imperfection transforms many important mathematical
and computational problems from being unsolvable in the lifetime of the
universe to being solvable now on current, actual computers—but the solutions
still require deep thought and hard work.
In the
same way, admitting and accepting imperfection allows us to find imperfect but
workable solutions to our personal and spiritual problems—but these solutions
also require deep thought and hard work.
As the
Lord told Oliver Cowdery after he had tried unsuccessfully to translate the
plates: “You took no thought save it was to ask me. . . . You must study it out
in your mind.” 5
And
Brigham Young said: “Whatever duty you are called to perform, take your minds
with you, and apply them to what is to be done.” 6
Not long
ago I had a student who told me he hated math and that he was no good at it. He
was sure he would not understand the math in my class, and he was a little
angry because he needed my class for some requirement or other. But with a lot
of encouragement he reluctantly promised me he would do something he had not
done before. He would not only answer all the homework exercises but would also
really make sure he understood all the steps in each problem, why that step was
the right thing to do, and why it worked. He agreed to study it out in his
mind.
At first
this was painful to him because he had never done math this way before. He
really had to fight his frustration and impatience. He wanted to say, “Just
tell me what to do and let me get this over with.” But he kept his promise. He
read and reread the explanations in the book. He came to my office hours and asked
me lots of questions—questions about how and why things worked. He asked the TA
many similar questions. And as he continued to work at it, he began to
understand, for the first time, a little bit about how math works and why it is
the way it is. Partway through the semester he confessed to me that he actually
liked the class. By the end of the semester he not only earned a good grade in
the class but was really excited about what he had learned. He even wanted to
take another math class. He was no longer “bad at math.” His hard work and deep
thought had transformed not only his attitude but also his ability.
But this
is not easy. My former neighbor Eliot Butler said it well:
To learn
is hard work. It requires discipline. And there is much drudgery. When I hear
someone say that learning is fun, I wonder
if that person has never learned or if he has just never had fun. There are
moments of excitement in learning: these seem usually to come after long
periods of hard work, but not after all long periods of hard work.7
But, like
it or not, it must be done—hard work and deep thought are the only way. As the
Lord said to Oliver, “You must study it out.” 8
Step 3.
Get Up and Act on Your Best Approximation
It is not
enough just to find our approximate answer to the problems. We must also act on
that approximate, imperfect answer. This is hard because we know our answer is
not perfect. That might scare you. It often scares me. But we cannot let our
fear of imperfection, our fear of making a mistake, prevent us from acting on
our best approximation.
As Paul
told Timothy, “God hath not given us the spirit of fear; but of power, and of
love, and of a sound mind.” 9
When I
went to graduate school many years ago, I was lucky enough to be admitted to a
school with some famous faculty members. I had done pretty well in my
undergraduate classes, and I thought I was pretty smart; so when it came time
to choose classes and teachers, I chose some of the most famous teachers I
could.
One of
them was especially impressive. He was a Fields medalist—the nearest thing in
mathematics to a Nobel Prize winner. Other students spoke of him with awe, both
because of his brilliance and because of his reputation for criticizing students.
When I heard how he had berated a student for being both lazy and stupid, I
determined that I would never give him cause to criticize me like that. I
decided never to ask him for advice or help until I had exhausted every other
means for solving a problem. The result was that he never criticized me, but I
also never learned much from him. I spent three years in his classes and only
spoke with him for a total of about forty-five minutes.
Another
of his students had no fear at all—he would go to this professor almost every
day to ask questions. And he was criticized regularly, but he never seemed to
care. I thought he was completely crazy, but he went back almost every day, got
his questions answered, and learned a lot.
It took
me several years after graduation to realize that I had wasted the opportunity
of a lifetime—this other student wasn’t so crazy after all. He got many hours
of personal tutoring from one of the world’s greatest mathematical minds, and I
got—well, I got through graduate school safely, without being criticized.
Don’t
Bury Your Talent
The Lord
is pretty clear that He wants something more from us than this. He has given us
many talents and opportunities. He wants us to face our fears and do something
good with all He has given us.
Consider
His parable about the rich nobleman who entrusted his wealth with his servants
while he went traveling abroad.
You know
the story, so I’ll skip to the end. “He which had received the one talent came
and said, Lord . . . I was afraid, and went and hid thy talent in the earth. .
. . His lord answered and said unto him, Thou wicked and slothful servant. . .
. Take therefore the talent from him, and give it unto him which hath ten
talents. . . . And cast ye the unprofitable servant into outer darkness. 10
Now think
a moment about that servant with one talent. This is a very severe punishment.
After all, he took good care of that money. Not one cent was lost. Yet the
master not only chastised him and took his talent, he cast him into outer
darkness.
The Lord
doesn’t care about not
messing up—not
losing what we have. It isn’t enough to preserve what He
has given us. He wants us to get up and do something with it.
Fear
Causes Failure
Like the
one-talent servant, when we are afraid of failure we are more likely to fail.
When I
was a teenager I worked for two summers as a lifeguard and a swim instructor.
One of the things I learned in that job was that people’s fear of water is
usually their greatest obstacle to successful swimming.
A person
learning to float on his back, for example, when he is afraid, will
instinctively try to sit up, which causes him to sink. To float he must relax
and put his head back and his legs down. Then he can float on his back with
very little effort.
The real
key is learning to trust that a little water splashing in your face does not
mean you are drowning. It may not be what we imagine to be perfect floating,
but it is good enough. Even going completely below the water is not failure—if
you remain relaxed, a gentle hand motion quickly brings you back up to the top.
But as soon as you become nervous, you try to sit up, and you will sink.
Similarly,
when my kids first started ice-skating lessons, they were unable to stand on
their skates without help and clung to the little walkers that were available
at the arena for nonskaters. Even with the little walkers they fell often. They
were surprised when the first thing the instructor did was to take away the
walkers and teach them to fall. They practiced falling over and over. One of my
daughters complained that falling was the one thing she could do without
lessons or help.
But once
they had finished their falling lesson they could skate—almost as if by
magic.And they didn’t even fall much after that. By overcoming their fear of
falling, by embracing the fall, they were able to learn to avoid it and were
able to try new things without fear.
The Plan
of Salvation
The most
important example of all is the plan of salvation. The entire plan depends upon
our engaging in a very dangerous enterprise.
Recall
that Satan’s plan was to guarantee that bad things would not happen, that we
would all be safe, and that we would all return to our Father after our time on
earth. This plan was rejected not only because Satan wanted all the glory for
himself but also, more importantly, because it would not work. It just wouldn’t work.
We cannot
become like our Father in Heaven unless we learn for ourselves to refuse the
evil and choose the good. We must act for ourselves and choose for ourselves.
But, as Lehi tells us, we cannot do this unless we are enticed by two
opposites—unless we have the option to fail. 11
Not only
must we have the option to fail—we will fail. We do fail. Often.
The most
miraculous proof of God’s love for us is the Atonement of Jesus Christ. The
purpose of the Atonement is precisely to allow us to recover from our many
failures. He knows we will fail, despite our best efforts, and He has provided
a way for us to be freed from our sins, to be healed, and to return to Him.
We show
our gratitude for the Atonement when we use it to overcome our fear of failure,
trust in Him, and act on our best approximation.
Aim High
Recently
our academic vice president, Brent W. Webb, reminded us of something Elder
Jeffrey R. Holland taught us way back when he was president of BYU. He said
that we hit what we aim at: “So, not failure but low aim would be the most
severe indictment of a Latter-day Saint fortunate enough to be at BYU.” 12
I should
have listened more carefully to President Holland before I went off to graduate
school. I hit what I aimed at—I graduated successfully without criticism from
my teachers. Technically I did not fail. But, boy, did I aim low.
Fear of
embarrassment, fear of failure, fear of being considered dumb by someone I
thought was smart—fear made me aim low. Don’t make that mistake. Aim high.
Please
don’t misunderstand me. When I say “aim high,” I do not mean aim to be
successful in your career. I do not mean aim to become rich or famous or
powerful.
Those
might be things that happen along the way for a few of us. But if they are your
target, you are aiming way too low. Section 121 in the Doctrine and Covenants
tells us that if we aim at these things—if our hearts are set upon the things
of this world and if we aspire to the honors of men—we may be called, but we
will not be chosen. 13
No—when I
say “aim high,” I mean we must aim to develop our talents and use our
opportunities the best we can to build His kingdom, bless His children, spread
His gospel, care for the needy, heal the sick, discover truth, teach that
truth, and bring ourselves and our families back to live with Him.
We will make errors along the way. Aim high anyway. “Not failure but low
aim would be the most severe indictment.”
Step 4.
Do It Again
Finally,
an essential step after we try and fail is to repeat the cycle and improve on
each attempt.
Some of
the most powerful methods for solving hard mathematical problems are what we
call iterative methods. You start with an approximate answer—sometimes
just a random guess—but you use that guess to generate a new, slightly better
answer. Then you take that new answer and apply the method again and again
until you get as close as you need to the correct answer. There certainly are
situations where these iterative methods don’t work, but in many settings they
are both the fastest and most robust ways to solve problems.
Iteration
is a powerful tool in our lives as well. We repeat these three steps over and
over again.
Let me
tell you about my son Spencer, who likes to run. His first official race, when
he was eight years old, was a 3K in Kiwanis Park. He did not do nearly as well
as he had hoped to do. He really had to push hard to stay with the leaders,
and, by the end, he just didn’t have the strength to keep up. He was a little
disappointed. The next two days he was really sore. He was forced to admit he
wasn’t yet as fit or as good at running as he wanted to be. But during that
week he pushed himself harder in our daily run than he had in the past—he
worked hard to prepare for his next attempt.
At the
next race, one week later, he was a little worried he wouldn’t do well. But he
got up and ran the race despite his fears. This time he still had to push hard
to keep up, and he still didn’t stay with the leaders for the whole race, but
he was able to stay with them for longer and his time improved a lot. Some of
that improvement was from working harder in our daily workout, and some of it
was from really running hard in the previous race. He was still a little
disappointed that he didn’t do as well as he wanted to.
But he
repeated the process. Again he was sore after the race. Admitting that he
didn’t know as much about how to prepare for a race as he wanted to, he asked his
uncle—an experienced distance runner—for advice on how to train better. Again
he worked hard all week. Again he felt nervous before his next race, but again
he put his fear aside, ran hard, and did better than the week before.
All
through that first season he repeated this process. He had a race almost every
week, and with each race he improved his time—a lot. Each race also motivated
him to try harder in his daily runs and to learn more about running. And not
only the preparation but also the races themselves helped make him stronger for
his subsequent races. By the end of the season he had cut almost three minutes
off his 3K time and had become one of the race leaders that others tried to
keep up with.
Spencer
was successful that running season because he followed the iterative method for
pursuing perfection. Each week he admitted and accepted imperfection, he worked
hard to improve, he braved his fears and made another attempt, and then he
repeated the process over and over again.
This same
process, this iterative method, will bring each of us closer and closer to
perfection. We will not actually reach that goal in this life, but we will be
better than before. We will get better and better.
Conclusion
Let me
conclude with the chorus of Leonard Cohen’s song “Anthem.” Cohen may not have
originally meant this verse exactly the way I interpret it, but for me it
captures very well the idea I am trying to express today:
Ring the
bells that still can ring
Forget your perfect offering
There is a crack in everything
That’s how the light gets in.
Our bells
are cracked. But let us ring those bells that still can ring. Stop worrying
about your failure to achieve perfection—perfection is not possible in this
life. Instead, embrace the light and healing power of Christ that come in
through our cracks and imperfections.
Ring the
bells that still can ring
Forget your perfect offering
There is a crack in everything
That’s how the light gets in.
In the
name of Jesus Christ, amen.
Notes
1. See
Marcus Wohlsen, “The Astronomical Math Behind UPS’ New Tool to Deliver Packages
Faster,” Wired, 13 June 2013;
wired.com/business/2013/06/ups-astronomical-math.
2. Yes, I
spelled it right—it is not the same as the search engine.
3. See
“Julia Robinson: Functional Equations in Arithmetic,” Association for Women in
Mathematics, 2005; awm-math.org/noetherbrochure/Robinson82.html.
4.
Jeffrey R. Holland, “Lord, I Believe,” Ensign, May 2013,
94.
5.
D&C 9:7–8.
6.
Brigham Young, in JD 8:137 (29 July 1860); quoted in Jeffrey R.
Holland, “A School in Zion,” BYU annual university conference address, 22
August 1988; http://president.byu.edu/documents/holland.htm.
7. Eliot
Butler, “Everybody Is Ignorant, Only on Different Subjects,” BYU forum address,
14 September 1976; in BYU
Studies 17, no. 3
(spring 1977): 282; emphasis in original (Eliot Butler’s title is a quotation
from Will Rogers).
8.
D&C 9:8; emphasis added.
9. 2
Timothy 1:7.
10.
Matthew 25:24–26, 28, 30.
11. See 2
Nephi 2:11, 15.
12.
Holland, “A School in Zion”; quoted in Brent W. Webb, “Where There Is No
Vision, the People Perish,” BYU annual university conference address, 23 August
2011; http://speeches.byu.edu/?act=viewitem&id=1990.
13. See
D&C 121:34–35.
