I had a valuable conversation with Michael A Woodley yesterday, in which he gave me a new and very significant insight into the nature of genius in its social context.
(There are a particularly large number of mathematicians in this short-lived category; but there have been many short-lived examples among geniuses of all types.)
Furthermore, Michael already knew that many geniuses have either zero children - or few/ zero children who survive.
(Shakespeare had zero direct descendants, so far as known. Same for Beethoven and Schubert. Same for Pascal and Turing. Same for Newton and Maxwell.)
Therefore, there is a high probability of reproductive extinction for geniuses - and this implies that geniuses do not need to live beyond early adulthood and do not need to reproduce.
I believe that this insight of Woodley's is the missing key; when added to the insights of Eysenck, he has essentially solved the problem of genius, as I understand it.
One thing that must be emphasized is that each individual genius potentially has a disproportionately massive impact on their society as a whole.
For instance, it is plausible that the unknown Byzantine inventor of Greek Fire quite possibly ensured the survival of Constantinople for several hundred years beyond what would otherwise have happened.
This is totally different from the way that genetics works!
So a single genius may affect - may massively-enhance - the likelihood of survival and growth of a whole human society - even when their genetic impact on that society is near-zero or actually zero.
Which means that the problem in explaining the emergence of a society where there is a high concentration and number of geniuses (e.g. Western Europe from the late Middle Ages and until recently) reduces to the simple problem of first producing geniuses and secondly of society being such as to appreciate and use the products of genius.
In other words, geniuses do not need to live long or have children, they simply need to survive long enough to make their contribution. And having made their contribution they do not personally need to be rewarded for it - because it is likely that they will not live very long anyway.
Why should geniuses so often be short lived (and sterile)?
Plausibly, because that the genius has a brain of exceptionally high complexity (in relevant ways), which means (from the predictions of general complex systems theory) that in the first place this kind of super-brain is harder to grow and therefore more likely to have developmental faults; and in the second place that an exceptionally complex brain is harder to maintain and therefore more likely to malfunction fatally (in many possible ways).
(The minority of long lived and fertile geniuses would include those few who have had the good fortune to develop an extra highly complex brain without any significant or potentially fatal faults; and therefore whose brain is more robust and easier to maintain over the long-term.)
So, what is needed to produce a society with a high concentration of geniuses is:
1. Enough potential geniuses are born
2. Enough of these potential geniuses survive to early adulthood (at least)
These two factors are enough to ensure that geniuses will produce - because the innate tendency to practice genius is a property of geniuses: it takes very little in the way of encouragement, and indeed geniuses will 'do their thing' even in the face of considerable dis-couragement.
The final step is that:
3. Society must value and use the products of genius.
Not all societies do this - or, more often, they only use certain restricted products of genius - such as decorative arts, improvements in statecraft or military technologies.
But, to reiterate, the society need not value or reward the genius himself, indeed the genius can safely be neglected, defrauded of his credit, or even killed (after making his discovery) since he will probably be dead soon anyway.
So long as the genius has the minimal psychological appreciation/ positive valuation - perhaps from just one or two or a few persons - then he will do his thing anyway.
So what kind of society produces geniuses?
Probably all societies produce sporadic geniuses, but only a few produce a high enough concentration of geniuses to make a difference to the society as a whole.
Although even one single genius who produces a particular invention may make a vast difference.
But the genetic pre-requisites of genius are already known: a society with a sufficiently high average general intelligence ('g') and also an average personality of a moderately high level of Psychoticism
('Psychoticism' embodies both creativity and the personality traits necessary for a genius to produce a distinctive contribution - see http://iqpersonalitygenius.blogspot.co.uk/2013/06/eysencks-personality-trait-of.html )
This will ensure a sufficient production of geniuses - then these must survive to adulthood in sufficient number.
I think this implies that the society must have a sufficiently large proportion of 'middle class' people - in other words those whose economic role involves a high level of technical or cognitive skills - and where possession of technical or cognitive skills gives a high probability of a higher than replacement-level of survival for your children.
In other words, what is needed is a sufficiently large and successful middle class.
But what kind of society tends to value the products of genius?
My guess is that it needs to be a society where the middle class of skill-workers has a substantial role in serving itself - because where the skill workers serve the upper classes, then only the types of genius which serve upper class interests will be used - mostly innovations in statecraft and war, and to a lesser extent in upper class luxury goods such as the arts and crafts and architecture, cooking and clothes and the like.
But when a middle class person can sell their skills to other middle class persons, then conditions are ripe for the recognition and use of a wide range of products of genius - including those which increase the functionality of society in many ways - innovations in economics, industrial organization, agriculture, transport, power machinery, communication and so on.
But a society of mandarins, a society in which the skilled middle class survive by serving the ruling class, a society in which the middle class of predominantly officials and bureaucrats, is NOT, on the whole, a society which will be likely to recognize or use the innovations of genius.
All that is somewhat speculative.
But Woodley's insight has simplified, and thereby solved, the basic problem of how genius is produced - by clarifying that genius as such is not, cannot coherently be, regarded as a direct product of natural selection: genius does not happen because it brings reproductive success - quite the reverse.
Genius is a product of heredity - but is itself (on average) a genetic dead-end.
And, having had enough geniuses born and survive to adulthood - the crucial factor is that society as a whole recognizes and uses the innovations which individual (and often short-lived) geniuses produce.