Exploring Genius – Terence Tao

Each Exploring Genius article profiles an accomplished and recognised genius, details parts of their life and career, how they’ve influenced society, and what they’re like as people. The previous entry was on Stanley Kubrick.

Genius appears in all fields of human accomplishment so these articles are naturally varied in style, length and approach. Terence Tao works in pioneering-level pure mathematics and I’m about as proficient with mathematics as a salamander, so this entry is coming from a particularly laymen (nay, idiot’s) point of view. It provides a generalised overview of Tao’s life, briefly covers the origins and significance of mathematics for context (which is actually pretty damn interesting), gives rough insight into the significance of his work, explores his giftedness growing up and how it was developed, and ends with an overview of his personality—which is exceptionally kind and humble—and how it all fits together.

Introduction

The term ‘genius’ is more related to accomplishment than ability, and can be equally applied to painting as it can be to theoretical physics. It has very little to do with IQ (though some take having an IQ above 140 to also qualify a person as a genius). There may be a correlation with IQ scores in many cases, but an IQ score is only indicative of isolated aptitudes (such as memory and logical reasoning). Genius-level accomplishment comes from the interplay between cognitive control and creativity; it’s raw intelligence multiplied by open-minded imagination and wonder. Certain fields display a stronger correlation than others, and from what I can tell it appears strongest in mathematics and physics. The Nobel Prize-winning physicist Richard Feynman is notoriously used as an example of the irrelevancy of IQ testing, with a tested score of only 125 and a clearly genius-level intellect, but closer inspection reveals that to be a likely product of the specific test he took, which was heavily language-focused. IQ tests are largely irrelevant, by Feynman isn’t the best example.

The kind of thinking required for mathematics and physics is pure logical reasoning and abstraction, with processing speed, braveness (yep, braveness) and imagination being key bonuses. Terence Tao has a tested IQ score of over 220, and by many accounts demonstrates those attributes better than any mathematician alive today. He’s known as the “Mozart of math” and in the classical sense of the term, he may well be the smartest guy on the planet.

What is Mathematics?

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For a better appreciation of Tao it helps to understand the broader significance of his field, so without deviating too much, here’s a basic rundown:

We don’t exactly know when it ended, but there was a time in human history when we had no concept of counting. We intuitively understood the concepts of ‘more’ and ‘less’—generalised quantity—but couldn’t differentiate anything in abstract terms. Seeing two antelope and recognising them as more than one antelope was one thing, recognising their quantity as an abstract concept equally applicable to fingers and days on a calendar—the concept of the number 2—was a quantum leap in human thought. The first person to achieve this may well be the most important genius in our ancestry. But we have no idea who it was, or how it came about. Anthropologists theorise that counting started as the tallying of single units, seen as vertical lines drawn on a wall, and that symbols were eventually incorporated to represent larger groups of tallies. In ancient Sumerian culture for example, a small clay cone was used to denote ‘1’, a clay sphere ’10’ and a large clay cone was ’60’. Many different systems of symbols were used across the world before the establishment of 0 – 9, which came out of India after 300BC.

The formation of symbols to represent groups of single units created a new dynamic between each symbol, and with each new dynamic came further symbol sub-systems (like algebra) with their own unique interplay, so that complexity grew exponentially from a mathematical big bang—an outward explosion of theory from the use of the first single unit.

The philosopher Bertrand Russell makes the case in The Principles of Mathematics (1903; not to be confused with his Principia Mathematica released in 1928) that mathematics and logic are the same thing (or at least, come from the same place), which becomes easier to comprehend when we consider that numbers are only representative—different systems (such as roman numerals and binary) yield different kinds of patterns, puzzles and insights, but all are bound by logic to the parameters of the system they belong to. Whether or not logic and mathematics are considered the same is a matter of definition, but thinking of logic as being fundamental to math at least helps us understand its nature from a deeper perspective and ponder the question: what exactly is mathematics? Is it something we’ve discovered, or is it something we’ve created?

I think it makes sense to view logic as a core property of the universe, intrinsic to the way everything exists and functions, and that mathematical theory is a form of logical structuring—an interaction of human concepts with the order of the universe. I have nil expertise and may be way off, but it seems like the 0-9 number system could potentially be replaced by something much more complex; it’s just that it works broadly for our population and is complex enough to describe reality to the level we’re capable of being curious.

So is mathematics just a way to describe reality? The physicist Max Tegmark makes the case in his book Our Mathematical Universe that mathematics not only describes reality, but that reality itself is mathematical in nature:

“The idea that everything is, in some sense, mathematical goes back at least to the Pythagoreans of ancient Greece and has spawned centuries of discussion among physicists and philosophers. In the 17th century, Galileo famously stated that our universe is a “grand book” written in the language of mathematics. More recently, the Nobel laureate Eugene Wigner argued in the 1960s that “the unreasonable effectiveness of mathematics in the natural sciences” demanded an explanation.

We humans have gradually discovered many additional recurring shapes and patterns in nature, involving not only motion and gravity, but also electricity, magnetism, light, heat, chemistry, radioactivity and subatomic particles. These patterns are summarized by what we call our laws of physics. Just like the shape of an ellipse, all these laws can be described using mathematical equations.

Equations aren’t the only hints of mathematics that are built into nature: There are also numbers. As opposed to human creations like the page numbers in this book, I’m now talking about numbers that are basic properties of our physical reality.

For example, how many pencils can you arrange so that they’re all perpendicular (at 90 degrees) to each other? The answer is 3, by placing them along the three edges emanating from a corner of your room. Where did that number 3 come sailing in from? We call this number the dimensionality of our space, but why are there three dimensions rather than four or two or 42?”

The example Tegmark gives is a good illustration of the symbolic nature of numbers, showing there to be a fundamental truth of the universe beneath their representation, but whether or not reality is mathematical in nature is mostly redundant to the field; it’s just helpful when trying to understand why it’s all so important, and therefore, the importance of the work being done by someone like Terence Tao. There may be conjecture around the philosophical nature of mathematics but there’s little debate over the benefit. Without it, our cultural and technological evolution wouldn’t have progressed beyond the spear—every scientific and technological advancement involves mathematics to some degree.

The paradigm shift available through understanding mathematics at a deeper level is also about mathematicians. Where once they appeared as number technicians, it now seems talented mathematicians are actually more tuned-in to the universe than anyone else (especially those making that kind of claim). Like a child who develops language early and is therefore at an advantage with interpersonal relationships, the gifted mathematician has an aptitude with the language of the universe, becoming the core force behind the progression of our species within it.

If Tao really is the world’s most gifted mathematician, he’s more than just a guy who solves hard problems: he’s more fluent with universal language than anyone else alive.

The Child Prodigy 

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Terence Tao was born in Adelaide, South Australia in 1975 to Billy and Grace, both Chinese natives who had emigrated to Australia in 1972. They’d met a few years previously at Hong Kong university; Billy there to complete a doctorate in paediatrics while Grace became an honours-roll mathematics and physics graduate. They had three sons within a few years of arriving: Terence (known to his friends as Terry), Nigel and Trevor—their westernised names chosen to reflect the culture of the couple’s new home country. All three brothers would eventually become standout intellectuals, with Nigel scoring a 180 IQ and winning bronze at two international mathematics olympiads, and Trevor becoming a national chess champion at age 14 while winning numerous prizes for his classical music compositions; broad achievements made all the more impressive by the fact he has autism.

Tao’s precocity became evident before the age of two, when his parents noticed him arranging an older child’s letter blocks alphabetically; a skill he’d learnt through watching Sesame Street. Things didn’t slow down: when he was 4 he was able to multiply two-digit numbers by two-digit numbers in his head. It was soon decided that regular schooling wouldn’t be suitable, and so he was placed into accelerated learning, which was eventually monitored by the Davidson Institute (Australia’s centre for the development of gifted children). The institute’s Miraca Gross writes:

“A few months after Terry’s second birthday, the Taos found him using a portable typewriter which stood in Dr. Tao’s office; he had copied a whole page of a children’s book laboriously with one finger! At this stage his parents decided that, although they did not want to ‘push’ their brilliant son, it would be foolish to hold him back. They began to borrow and buy books for him and, indeed, found it hard to keep pace with the boy. They encouraged Terry to read and explore but were careful not to introduce him to highly abstract subjects, believing, rather, that their task was to help him develop basic literacy and numerical skills so that he could learn from books by himself and thus develop at his own rate. “Looking back,” says Dr. Tao, “we are sure that it was this capacity for individual learning which helped Terry to progress so fast without ever becoming bogged down by the inability to find a suitable tutor at a crucial time.” By the age of 3, Terry was displaying the reading, writing and mathematical ability of a 6-year-old.”

Research has shown the likelihood of a child prodigy transitioning into an adult genius to be extremely rare. Genius-level intellect isn’t just about talent; it’s about creativity, inventiveness and open-minded intrigue. Tiger mothers forcing a discipline on a child may eventually produce a fantastically able technician in line with the best of a field, but geniuses are generally made through self-interested goals; at the core of true genius is one defining characteristic: self-propelled passion.

Billy and Grace Tao are exceptional parents. Instead of marshalling their son’s progression forcibly, it was Tao’s own interest and maturity that informed each incremental step in his education. His father explains:

“Firstly we realised that no matter how advanced a child’s intellectual development, he is not ready for formal schooling until he has reached a certain level of maturity, and it is folly to try to expose him to this type of education before he has reached that stage. This experience has made us monitor Terry’s educational progress very carefully. Certainly, he has been radically accelerated, but we have been careful to ensure, at each stage, that he is both ready and eager to move on, and that we are not exposing him to social experiences which could be harmful.

Secondly, we have become aware that it is not enough for a school to have a fine reputation and even a principal who is perceptive and supportive of gifted education. The teacher who actually works with the gifted student must be a very flexible type of person who can facilitate and guide the gifted child’s development and who will herself model creative thinking and the love of intellectual activity.

Also, and possibly most importantly, we learned that education cannot be the responsibility of the school alone. Probably for most children, but certain for the highly gifted, the educational program should be designed by the teachers and parents working together, sharing their knowledge of the child’s intellectual growth, his social and emotional development, his relationships with family and friends, his particular needs and interests… that is, all the aspects of his cognitive and affective development. This did not happen during Terry’s first school experience but I am convinced that the subsequent success of his academic program from the age of 5 onwards has been largely due to the quality of the relationships my wife and I have had with his teachers and mentors.”

Contrasting this approach to other accelerated prodigies, the Taos seem to have viewed their son as his own person rather than as an extension of themselves. They cultivated an environment of deep caring and unconditional support around the interests of their children, allowing the spark of internal genius to ignite without the repressive force of projected self-expectation. The Davidson Institute’s Marica Gross continues:

“In November of 1983, at the age of 8 years 3 months, Terry informally took the South Australian Matriculation (university entrance) examination in Mathematics 1 and 2 and passed with scores of 90% and 85%, respectively. In February the following year, on the advice of both his primary and secondary teachers, who felt he was emotionally, as well as academically ready, the Taos agreed that he should begin to attend high school full time. He was based in Grade 8 so that he could be with friends with whom he had undertaken some Grade 7 work the year before, and at this level he took English, French, general studies, art, and physical education. Continuing his integration pattern, however, he also studied Grade 12 physics, Grade 11 chemistry, and Grade 10 geography. He also began studying first-year university mathematics, initially by himself and then, after a few months, with help from a professor of mathematics at the nearby Flinders University of South Australia. In September that year he began to attend tutorials in first-year physics at the university, and 2 months later he passed university entrance physics with a score in the upper 90s. In the same month, finding that he had some time on his hands after the matriculation and internal exams, he started Latin at high school.”

Though Tao’s education was governed by his parents and teachers, the trajectory was entirely driven by himself and was aided dramatically by an attention to his emotional and social maturity. In many respects he was actually held back. He was moved into high school at aged 10, but as noted above, he’d nearly aced university entrance exams two years previously (in Australia high school goes up to grade 12). He spent two thirds of his time with grade 11 and 12 students and the remainder attending 1st and 2nd year university maths and physics classes. This was all down to his parents, who felt strongly about not doing anything simply for appearances sake, and only taking steps when it was in their son’s best interests:

“There is no need for him to rush ahead now. If he were to enter full-time now, just for the sake of being the youngest child to graduate, or indeed for the sake of doing anything ‘first,’ that would simply be a stunt. Much more important is the opportunity to consolidate his education, to build a broader base.

If Terry entered university now he would certainly be able to handle the work but he would have little time to indulge in original exploration. Attending part-time, as he is now, he can progress at a more leisurely rate and more emphasis can be placed on creativity, original thinking, and broader knowledge. Later, when he does enter full time, he will have much more time for research or anything else he finds interesting. He may be a few years older when he graduates but he will be much better prepared for the more rigorous graduate and post-doctoral work.”

Sitting among students nearly twice his age, the young Terry Tao became known for his humble and friendly nature, and by all accounts, was universally liked by teachers, mentors and peers alike. This may be his nature, but being as precocious as he was, his personality was undoubtedly benefited by the unwillingness of his parents to treat him any differently to his brothers (and other children of a similar age in ‘regular’ families). Modesty was a virtue in the Tao household; show-boating and arrogance made as much sense as a clown at a librarian convention. He didn’t care about winning prizes or being the best at anything; he just really loved doing maths, and received the perfect balance of encouragement and structure to reach his full potential without ever feeling superior. He knew he was different, but had no value placed on that difference: everyone else was viewed as a human equal. When the 10 year-old Tao was offered a prize for scoring the highest mark ever on the American SAT for a child of his age, he chose a chocolate bar, and when it was handed to him, broke it in half and shared it with his father!

Professional Career

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Tao’s work has achieved everything from progressing prime number and infinity theory to advancing MRI scanning technology—rapidly improving the detection rate of tumours and spinal injuries across the globe. Professor of mathematics at Princeton University Charles Fefferman said in an interview:

“Such is Tao’s reputation that mathematicians now compete to interest him in their problems, and he is becoming a kind of Mr Fix-it for frustrated researchers. If you’re stuck on a problem, then one way out is to interest Terence Tao”

The influence of mathematical advancement on society is almost entirely indirect: it usually functions as a basis to the advancement of other sciences, especially physics, so drawing a clear line between Tao and the broader value of his work quickly becomes convoluted by additional theory and speculation. Not to mention, explaining pure mathematics in laymen’s terms is extremely difficult. The concepts being used are comprised of other concepts that themselves require their own multi-conceptual explanations, all of which are already well beyond the learning level of the average person (myself included). What I do understand though, is that mathematics at an advanced level can be a truly beautiful and creative phenomenon, and for many, an emotional one as well.

It’s been said that most people don’t enjoy math because the schooling curriculum gives a vastly incomplete picture of the subject, analogous to an art class only teaching how to paint a single-coloured wall and never showing a Picasso or Rembrandt. For most of us it’s easy to recognise artistic and social talents as we have our own abilities as a point of reference, allowing us to perceive a distance between our own output and that of the great masters. In the case of mathematics it’s usually a case of viewing some kind of alien language. For example, here’s what Tao has been working on most recently:

“I’ve been meaning to return to fluids for some time now, in order to build upon my construction two years ago of a solution to an averaged Navier-Stokes equation that exhibited finite time blowup.

One of the biggest deficiencies with my previous result is the fact that the averaged Navier-Stokes equation does not enjoy any good equation for the vorticity {\omega = \nabla \times u}, in contrast to the true Navier-Stokes equations which, when written in vorticity-stream formulation, become

\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u + \nu \Delta \omega

\displaystyle u = (-\Delta)^{-1} (\nabla \times \omega).

(Throughout this post we will be working in three spatial dimensions {{\bf R}^3}.) So one of my main near-term goals in this area is to exhibit an equation resembling Navier-Stokes as much as possible which enjoys a vorticity equation, and for which there is finite time blowup.

Heuristically, this task should be easier for the Euler equations (i.e. the zero viscosity case {\nu=0} of Navier-Stokes) than the viscous Navier-Stokes equation, as one expects the viscosity to only make it easier for the solution to stay regular. Indeed, morally speaking, the assertion that finite time blowup solutions of Navier-Stokes exist should be roughly equivalent to the assertion that finite time blowup solutions of Euler exist which are “Type I” in the sense that all Navier-Stokes-critical and Navier-Stokes-subcritical norms of this solution go to infinity…”

I don’t know about you, but I almost need a lay-down after reading that.

It’s my goal over the next 12 months to both increase my own base understanding of mathematics and to source mathematicians capable of providing effective metaphors to better illustrate the work they’re doing for the rest of us. I’ll post more specifically on the subject then, and will potentially revisit this section to give it some greater context.

Closing

It’s no accident that Tao became passionate about mathematics, and it’s not just a matter of encouragement. His parents instilled him with a positive and compassionate outlook and supported him, but it was ultimately the conscious absence of his parents that helped him the most. The common sense fact is, if someone is good at anything, they’re much more inclined towards it over other activities, especially without there being any pressure around their achievements. The brain naturally releases higher dopamine levels when the mind perceives self-accomplishment easily relative to a common standard, which in Tao’s case, came very early when he was teaching children twice his age how to count before turning 3. His aptitude then went on to connect his developing interest to higher-concept (more elegant and interesting) mathematics much sooner than most professionals in the field, thereby giving him an enormous hook. The message for parents here is a clear one: for a child’s potential to be reached, their talent needs guidance without any pressure and expectation.

The choices and direction of Tao’s parents were paramount to his development. They worked tirelessly in the background to create new and nurturing environments for him to grow in, and in terms of his personal experience, they were largely invisible. They recognised the importance of balance in the growth of modest self-confidence, a concept equally important to all avenues of his life—whether it be at school, at home or among friends.

Most importantly, Tao’s parents understood his genius. His father sums it up:

“I have seen too many situations where the parents did the wrong thing. A brilliant mind is not just a cluster of neurons crunching numbers but a deep pool of creativity, originality, experience and imagination. This is the difference between genius and people who are just bright. The genius will look at things, try things, do things, totally unexpectedly. It’s higher-order thinking. Genius is beyond talent. It’s something very original, very hard to fathom.”

Terence Tao is more than a mathematical genius; he’s a role model for human conduct, a rare example of supreme talent and supreme humility existing in side-by-side unison. We may not be able to learn much directly from his work, or even understand the first thing about it, but I think most of us can learn from his outlook on the world: no matter who you are or how good at something you are, be humble, let your work speak for itself, and be a good and genuine person without motive.

If you haven’t seen him before, here’s a brief interview he had on the Colbert Report a couple of years ago. Note his demeanour and the speed of his brain compared to his speech. He’s one of a kind.

If you’re interested in learning more about the ‘Navier-Stokes’ equation or checking out more of his work, Tao runs his own WordPress blog here.

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