Our best measures of intelligence are lifetime achievements (as per Tesla, Einstein, etc.) and IQ tests. Among the most popular IQ tests is the WAIS IV which consists of various components that measure several dimensions of cognitive processing:
The test items measuring of each of the four scales (or indices) of WAIS IV are given below:
• verbal comprehension: vocabulary, information, identifying similarities, comprehension
• perceptional reasoning: picture completion, block design, matrix reasoning
• working memory: arithmetic, digit span, letter-number reasoning
• processing speed: digit symbol coding, symbol search
(More details are available in the book, Intelligence, IQ & Perception, or on specific IQ sites.) People who achieve high IQ scores usually obtain higher than average scores on all four components. This means that virtually all highly intelligent people (I.Q. above 120) should be able to successfully complete all math courses up to the end of high school and even most at the undergraduate level.
However, we humans differ significantly in our capacity for abstract thought, and that ability is reflected in our performance on certain sections of IQ tests. For example, physicist Richard Feynman scored only 125 on his IQ test, because he achieved only average scores on the language component, though performing exceptionally in the math. Almost all people who are engaged in research in mathematics or the so-called “hard sciences” are significantly above average in their perceptional reasoning, short-term memory and processing speed, (e.g. John von Neumann). How are these differences observed?
In describing the results of his research during the 1960’s and 70’s, USSR mathematician Vladimir Krutetskii noted innate differences in the cognitive capacities of children, and expressed his observations carefully to sidestep the Soviet prohibition on linking talent to inheritance:
The difference between capable, average, and incapable pupils, as our research permits us to conclude, comes down to the following. In able pupils these associations can be formed “on the spot”; in this sense they are “born,” if one can so express it, already generalized, with a minimal number of exercises. In average pupils these associations are established and reinforced gradually, as a result of a whole series of exercises. They form isolated, concrete associations, related only to a given problem, “on the spot.” Through single-type exercises these associations are gradually transformed into generalized associations. In incapable pupils, even the isolated, concrete associations are formed with difficulty, their generalizations are still more difficult, and sometimes such generalizations do not occur at all.
So almost all highly intelligent people have the ability to do mathematics at an undergraduate level; that is, they can be “good at mathematics.” However, being “great at mathematics” and working on its frontiers requires an exceptional ability to make the associations described by Krutetskii. It also requires personality attributes like intensity, perseverance and, to some extent, obsession. In my current research on the genius mentality, I am discovering that these qualities, as much as innate talent are vital to high success in the mathematical disciplines.
