“The mathematician’s best work is art, a high perfect art, as daring as
the most secret dreams of imagination, clear and limpid. Mathematical genius
and artistic genius touch one another”
-Gosta Mittag-Leffler
Consider Arthur Benjamin, a self-described “mathemagician” who can perform extremely complex computations in his head, (he has computed the square of a random 5 digit number to a live audience, for example). While such feats are certainly impressive to behold, most of us would feel slight misgivings attaching the word “genius” to Arthur Benjamin based only on his computational abilities1. What about Daniel Tammet, the gifted savant, who “intuitively ‘sees’ results of calculations as synesthetic landscapes without using conscious mental effort and can ‘sense’ whether a number is prime or composite”2? Among other amazing feats that demonstrated his mathematical and linguistic prowess, he has recited pi to 22,500 decimal places from memory3. We would be tempted to label such savants as mathematical genii were it not for this little snag: when we bestow someone the prestigious title of “genius” we usually require him to achieve an insight; a flash of inspiration born of a creativity that the rest of us don’t have access to. This is why someone who can perform ridiculous calculations in his head isn’t normally called a “genius”; on the flipside, this is also why most people have no trouble calling Einstein a genius.
So what makes a mathematical genius?
It would appear that the answer to this question would have something to do
with the ability to look at several disconnected bits of information, and
having the insight to realise that they
somehow all fit together; to see a connection where most of us see nothing. Let
us consider an example4:
In a world where people have never
heard of chairs, suppose they stumbled across a warehouse full of them, in
every conceivable shape, size, colour and make. Most people would be content to
just wander around looking at these “things”, entranced by the colours and
strange shapes. Now suppose a genius walks into the warehouse; (to make this a
little fun) let’s call him “Steve”. As Steve looks at the objects around him,
he starts to notice similarities and differences between them, and the wheels
in his head start turning. In a flash of insight,
he realises that they are all made to sit on. He christens them “chairs”.
Immediately, he is a step ahead of everyone else in the warehouse, because now
he can not only distinguish between a chair and something that is not made to
sit on, but with a little imagination (and some carpentry), he can make basic chairs
of his own! More than that, armed with this newfound definition of his, Steve
can now start classifying chairs
based on several criterions. He may look around and start noticing the
differences between barstools, recliners, armchairs, rocking chairs etc. and
start segregating them in terms of size, say, or weight, or functionality. Pretty
soon, he has moved into quite advanced territory. He can now look at the
construction of these chairs, the materials that they are made of, the
ergonomics; he is limited only by his imagination and ever-burgeoning knowledge.
As he gets deeper into the subject (you would be surprised at how much there is
to know about chairs!5), he gets closer and closer to the final
frontier; the chair as art. In one
sense, Steve has come full circle; the people in the warehouse were pretty
blown away by the funny objects around them, just like a casual observer
looking at a Rembrandt, but to Steve, just like to someone who has spent years
studying art, the chair represents so much more (though I doubt that a chair
has ever stirred such emotions in anyone). It is important to note that anyone
would have eventually figured out what Steve did. Given an infinite amount of time, anyone can
figure out almost anything, but there is nothing commendable, or special, about
that.
Here we can finally say what we mean
by “mathematical genius”. A genius is
someone who can derive connections, definitions
even, from a relatively small sample space. And just like Steve could build his own chairs
once he figured out what they were for, a mathematical genius, from his now
higher vantage point, can look down and provide new examples of objects that
fit his definition that possibly weren’t even in his sample space to begin with;
and it only gets better from there. Note that the mathematical genius often
regards what he does as art; indeed, this is why the best math looks beautiful
even to the untrained eye.
With this definition in hand, the
logical next question is: how does one get to this heightened state? The long
and short of it is that we don’t conclusively know how to make a genius;
indeed, there isn’t even a current scientifically precise definition of the
word6. The sobering truth is that the ages old “practice makes
perfect” method is what we’re left with (for the most part). On the bright side
however, there are principles of smart, focused practice that us 21st
century people must adhere to if we are to scale the lofty heights of genius. David
Shenk, in his bestselling book “The Genius in All of Us”, makes the compelling
argument that we all have the ability to do extraordinary things in any field;
it just takes an incredible amount of something he calls deliberate practice7, and just a little bit of luck.
Oftentimes, the words “mathematical
genius” border on the mystic in our society. We speak of the mental prowess of Gauss,
Euler and Archimedes in almost hushed voices, as though they possessed
intellectual gifts that we could not dream of having. In one sense this is
true; but the insights that science is gaining every day into the working of
our minds, speak of different times to come. Changes in education systems, an
ever-increasing literacy rate8, the spread of information technology
and increased access to learning resources, mean that we are getting smarter
with each passing generation9. It isn’t a stretch of the imagination
to envision a world where ground-breaking discoveries are made every day across
the wide spectrum of science, technology and even philosophy; this is our world
today. If we continue progressing at this rate, there’s no telling where we
might go as a race; what frontiers our minds may yet conquer. For now however,
we must continue to look to the Giants of Mathematics for inspiration, guidance
and strength.
____________________________________
Endnotes
1 I mean to take away nothing from
this gentleman’s achievements. He is a distinguished professor who has won many
awards, and currently teaches Mathematics at Harvey Mudd College [Source:
Wikipedia]. He is also a great entertainer; I recommend watching his Ted talk.
2, 3 The quote, and information about the
record was taken from his Wikipedia page: http://en.wikipedia.org/wiki/Daniel_Tammet.
He is a truly fascinating,
gifted individual.
4 The idea for this example, and the
subsequent definition that came from it, was the product of a long discussion
with a mathematician and good friend, Stephen J. Cooper. He recently wrote the
book “The Mathematical Foundations of the Universe”: http://www.amazon.ca/Mathematical-Foundations-Universe-Topological-Ontology/dp/0773415815
5 I was amazed at the depth of
information on the subject. Check out the Wikipedia page on chairs, and this
link for an article on the history of chairs (yes, people have actually studied this): http://www.randomhistory.com/2008/11/11_chair.html
6 “There is no scientifically precise definition
of genius, and indeed the question of whether the notion itself has any real
meaning has long been a subject of debate.” [Source: Wikipedia]
7 I highly recommend this book. Here
is a well written article about “deliberate practice”: http://artofmanliness.com/2010/11/07/the-secret-of-great-men-deliberate-practice/
8 “The adult literacy rate increased
by about 8 percentage points globally over the past 20 years – an increase of 6
percent for men and 10 percent for women. Progress was strong in Eastern and
Southern Asia, which saw an increase of 15 percent. Western Asia’s increase was
11 percent, while Southeast Asia saw a 7 percent increase in adult literacy
rates since 1990.” [Source: http://www.asianscientist.com/academia/international-literacy-day-september-8-2011-unesco-room-to-read/]
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