Terry Tao replies to my comment on his blog. It's about IQ and mathematical ability

I left the following comment on Terry’s blog today. The comment is just a summary of several blog posts from the blogosphere I belong to and is a result of a conversation I was having with a friend (like many blog posts here).

See the original post and comment on Terry’s post “Does one have to be a genius to do maths?”.


The subject of innate vs learned ability in mathematics fascinates me and I really wish there was a simple answer. But there isn’t, such is life. After reading more on the subject, I came to the conclusion that Terry Tao is right in certain things, but the following statement is not something I can defend:

But an exceptional amount of intelligence has almost no bearing on whether one is an exceptional mathematician.

I think we all agree that some kind of innate talent is necessary and just having a lot of innate talent is not sufficient. But can we actually quantify this sentiment?

This is called threshold theory. Once you are above a certain threshold in raw intelligence, other factors start to play a role. Factors such as being studious and creative. Or just being in the right place at the right time. And, of course, deliberate practice makes wonders. This is fairly uncontroversial until we actually get round defining this threshold.

Now what do we actually mean by raw intelligence? No one really knows what it is, but one thing is clear: clever people do well on a lot of seemingly unrelated tasks. And this is highly frustrating for some of us who invest a lot of time in the aforementioned deliberate practice only to be outperformed by “genius amateurs”. The correlations across different fields are referred to as a g-factor. Colloquially, we refer to people who possess “much raw intelligence” as high G people.

So one way to measure G is by distributing an IQ test. The role and value of IQ tests is the fact that people who do well on them tend to do well on other tasks, such as mathematics. Bear in mind, we are talking about a correlation, which is not 1.0. There are and will be people who are not so good at maths yet score high on the IQ test for various reasons.

So what should the threshold be? How high?

If you are aiming for the Fields medal, then, yes, it actually has to be very high. But may be, you just shouldn’t? Why is this a worthy goal? I want to congratulate Terry for pointing this out as he gets this point absolutely right. Repeat after me: your IQ is fixed for you and it doesn’t define your worth as a human being.

Now, going back to the claim that IQ has to be high. Anne Roe wrote in “The Making of a Scientist” eminent scientists in some fields have average IQs around 150 to 160. She measured 64 of them and you can find a summary of her findings here. Please note that Feynman’s low score is addressed there too.

Hence if you are aiming for eminence in some crowded field like mathematics or theoretical physics then spare yourself the hassle and take an IQ test first. It might force you to set more realistic targets. And if you only score 120 this is only evidence that becoming eminent will be unlikely. But you could still do it, against all odds.

So it appears to me that only 1 in 10000 of us even has a chance of becoming eminent and that’s not taking into account other necessary factors. The 150 threshold is quite hard to accept, but I am coming to terms with it.

So what should you do if you have reasonable doubt about being above a certain threshold? It’s complicated but it is ultimately tied to your life goals. If you love the subject, why does it all matter? Oh, of course, you still have to get the funding… Such is our sad reality at the moment. However, we are talking about different threshold now. My guess would be 130, but this is a baseless hunch: please don’t change your entire career plan based on my hunch.

Finally there are other consequences of such a stark realisation. May be spotting kids like Terry Tao is a more efficient way for you to contribute to maths than actually doing it. Again, that is if your life-goal is narrowly defined as “contribute to maths” disregarding the entire process: which is equally unwise as setting out to win the Fields Medal. Another roundabout way to go at achieving a Fields Medal would be to address your shortcomings in innate intelligence by conducting research into neuroscience instead. Or, even, Artificial Intelligence. Ultimately, none of us are actually all that smart compared to what we could have been, even Terry.

It’s all too complicated and I am getting too side-tracked, but thanks for reading all of this.

P.S. read more about Terry’s claim here.

MY COMMENT ENDS. Terry’s reply:

It appears my previous comment may have have been interpreted in a manner differently from what I intended, which was as a statement of (lack of) empirical correlation rather than (lack of) causation. More precisely, the point I was trying to make with the above quote is this: if one considers a population of promising young mathematicians (e.g. an incoming PhD class at an elite mathematics department), they will almost all certainly have some reasonable level of intelligence, and some subset will have particularly exceptional levels of intelligence. A significant fraction of both groups will go on to become professional mathematicians of some decent level of accomplishment, with the fraction likely to (but not necessarily) be a bit higher when restricted to the group with exceptional intelligence. But if one were to try to use “exceptional levels of intelligence” as a predictor as to which members of the population will go on to become exceptionally successful and productive mathematicians, I believe this to be an extremely poor predictor, with the empirical correlation being low or even negative (cf. Berkson’s paradox).

Now, at the level of theoretical causation rather than empirical correlation, I would concede that if one were to take a given mathematician and somehow increase his or her level of intelligence to extraordinary levels, while keeping all other traits (e.g. maturity, work ethic, study habits, persistence, level of rigor and organisation, breadth and retention of knowledge, social skills, etc.) unchanged, then this would likely have a positive effect on his or her ability to be an extraordinarily productive mathematician. However, empirically one finds that mathematicians who did not exhibit precocious levels of intelligence in their youth are likely to be stronger in other areas which will often turn out to be more decisive in the long-term, at least when one restricts to populations that have already reached some level of mathematical achievement (e.g. admission to a top maths PhD program).

For instance, many difficult problems in mathematics require a slow, patient approach in which one methodically digests all the existing techniques in the literature and applies various combinations of them in turn to the problem, until one can isolate the key obstruction that needs to be overcome and the key new insight which, in conjunction with an appropriate combination of existing methods, will resolve the problem. A mathematician who is used to using his or her high levels of intelligence to quickly find original solutions to problems may not have the patience and stamina for such a systematic approach, and may instead inefficiently expend a lot of energy on coming up with creative but inappropriate approaches to the problem, without the benefit of being guided by the accumulated conventional wisdom gained from fully understanding prior approaches to the problem. Of course, the converse situation can also occur, in which an unusually intelligent mathematician comes up with a viable approach missed by all the more methodical people working on the problem, but in my experience this scenario is rarer than is sometimes assumed by outside observers, though it certainly can make for a more interesting story to tell.