I don't know about 'intelligence,' a term that borders on the vagaries of philosophical terminology, but if you mean cognitive ability, a concept best defined via Spearman's 'g', then all the evidence - countless kinship studies including studies of identical twins, and increasingly genomic data - indicates that it's nearly as hereditary as height in middle-class populations in the developed world.
Nowadays 'g' can be established using a combination of basic mental tests with physiological measures: you give people tests of reaction time, working memory, processing speed (how fast they can solve simple problems) and also measure physiological aspects of their cognition like brain size corrected for body mass, brain glucose metabolism, nerve cell conduction velocity etc.
You then use a statistical technique called factor analysis to distill all of those variables into a single one - a "one ring to rule them all" sort of thing - traditionally called Spearman's 'g' after the Englishman Charles Spearman who first developed the whole methodology. Spearman's 'g' is basically a measure of the correlation between ALL mental tests and abilities so the higher your 'g' the higher you score on all tests of mental ability, in other words, it's a measure of mental horsepower.
Perhaps the most convincing way to show someone that 'g' is real and that it is significantly neurobiological in nature is to show them very extreme individuals on the measure.
Consider the case of a man who was basically the equivalent of being 8 ft tall on mental horsepower - John von Neumann.
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By the age of 6 he was conversant in Hungarian, English, German, French, Italian and was able to read both classical Greek and Latin. Here is a typical anecdote of him at this age:
"The Neumann family sometimes entertained guests with demonstrations of Johnny's ability to memorise phone books. A guest would select a page and column of the phone book at random. Young Johnny read the column over a few times, then handed the book back to the guest. He could answer any question put to him (who has number such and such?) or recite names, addresses, and numbers in order."
By the age of 8 he was already familiar with calculus (a branch of maths that is today first introduced to above-average 17 year olds) and was reading 46 volume books on history.
At age 15 he started studying what is now serious undergraduate standard maths - analysis - with the renowned Hungarian mathematician Gabor Szego. Apparently on their first meeting, Szego was so shocked by the teenager's mathematical talent that he was brought to tears.
By 19 he had already made mathematical history by introducing the set-theoretic construction of the ordinals (unfortunately von Nuemann's father forced him to complete undergraduate studies in engineering/chemistry so this was probably later than it would have been otherwise).
He would go on to father game theory, axiomatize quantum mechanics (and construct a new system of logic for quantum mechanics), play a historical role in the birth of computer science and the hydrogen nuclear bomb, and make stunningly profound contributions to many branches of mathematics and physics.
His mental horsepower was such that it habitually shocked world-class mathematicians and scientists in the 20th century:
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Paul Halmos states that "von Neumann's speed was awe-inspiring."
Edward Teller admitted that he "never could keep up with him." Teller also said "von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us..."
Lothar Wolfgang Nordheim described von Neumann as the "fastest mind I ever met," and Jacob Bronowski wrote "He was the cleverest man I ever knew, without exception. He was a genius."
George Polya said "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."
Herman Goldstine wrote: "One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how A Tale of Two Cities started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes."
"I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man", said physics Nobel Laureate Hans Bethe
His super-human memory seems to have remained intact even in his 50s as apparently on his deathbed, he read Goethe's novel Faust word for word from memory to his brother.
On the other hand, cognitive ability and originality are two different things. Von Neumann may have had very, very few competitors on mental horsepower but he was outclassed in originality by quite a few people. As Wigner put it:
“I have known a great many intelligent people in my life. I knew Max Planck (Nobel Prize 1918), von Laue (Nobel Prize 1914) and Heisenberg (Nobel Prize 1932). Paul Dirac (Nobel Prize 1933) was my brother in law; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. But none of them had a mind as quick and acute as John von Neumann. I have often remarked this in the presence of those men and no one ever disputed me.…
But Einstein’s understanding was deeper even than von Neumann’s. His mind was both more penetrating and more original than von Neumann’s. And that is a very remarkable statement. Einstein took an extraordinary pleasure in invention. Two of his greatest inventions are the Special and General Theories of Relativity; and for all of von Neumann’s brilliance, he never produced anything as original."
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