Friday, 9 March 2012

What Do We Mean By “Mathematical Genius”?

“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.


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: 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”:

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):

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”:

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:]

9 One manifestation of this is in “The Flynn Effect”:

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