Lynn Thorndike was happy enough (in his "Science & Thought in the XVth Century") to point out that these two major strands were very often at odds with each other: but it should also be noted that the intersection between the two was far from empty. In fact, you might well look at the architects Leon Battista Alberti and Antonio Averlino - both born near Florence near to 1400 - as examples of "Renaissance Men" in the purest sense, in that they exemplified both strands at the same time.
The mystery of the Italian Renaissance (as described by Burckhardt and the generation of gung-ho pro-humanistic historians that followed him) is this: why did it emerge at such a narrow time (circa 1400) and place (Florence)?
For a long time, the dominant view has basically been that this was a random event, just one of those things that happen from time to time. However, some modern writers have begun to speculate whether a particular freak event or a subtle change in diet or eating habits might perhaps been the real "cause" of the Italian Renaissance.
For me, I would be unsurprised if insomnia turned out to be a key: Alberti writes, in his 1441 "On the Tranquillity of the Soul", of "the agitation of his soul"at night, and how he can relieve this by trying to devise amazing machines for lifting and carrying weights. I wonder if an entire generation of Florentines suffered from a kind of intellectual insomnia, perhaps as a result of effectively becoming hyperthyroid from ingesting a particularly iodine-rich salt being brought into the city?
Or might the Florentines have simply become addicted to the sugar confections that had not long before suddenly filled the city's apothecaries and markets? Might the Renaissance have simply been a metabolic balancing act as people tried to compensate for a giant communal sugar rush?
But there's another possibility. If you were looking for a statistical explanation why a particular population produced more geniuses (while the overall bell-curve distribution probably remained intact), there would be two obvious candidates to consider - either (a) the mean IQ got shifted up (i.e. everyone somehow got smarter) , or (b) the variance increased dramatically (i.e. more extreme cases appeared at both ends of the scale).
This set me wondering: as I understand it, one of the problems often put forward with Darwinian evolution is that the natural rate of mutations is too low to support the amount of random change needed. So could it be that stable contexts inhibit mutations (i.e. encourage low adaptation rates), while troubled contexts somehow encourage mutations (i.e. encourage high adaptation rates)?
Interestingly, one medieval obsession presents itself here: that of whether "monsters" (freaks of nature) were signs (de-monstr-ations) of something greater happening in the world. Perhaps "monsters" in the human population were (and possibly still are?) literally a sign that the variance of the population is high.
Thinking about all this, it suddenly then became clear to me why low sperm counts make evolutionary sense: if a body is in significant physical difficulties, it makes no sense for it to try to reproduce offspring that are the same as it, as they would likely experience the same difficulties in the next generation. Instead, perhaps the body deliberately produces crippled, damaged sperm to try to encourage mutations that might be better adapted to the changing physical context. Otherwise, why would the body ever want to deliberately produce poor sperm or eggs? Perhaps the current medical view of "healthy" sperm is somehow clouded by an anti-mutation bias of some sort.
My prediction here is that that there will turn out to be reproductive mechanisms by which (a) healthy eggs repel damaged sperm and (b) damaged eggs discourage healthy sperm: leaving the two pivotal cases of (healthy eggs + healthy sperm) => low mutation rate (low variance, well-adapted to environment), and (damaged eggs + damaged sperm) => high mutation rate (high variance, poorly-adapted to environment).
But there's a timing issue. Whereas sperm production is essentially a "just-in-time" process, women's eggs are produced all in one go, and so form a lagging indicator to environmental adaptation (i.e. egg health in reproduction gives an indication of environmental fit 15-25 years earlier). So, could it be that, when looking for the source of the Renaissance circa 1400, we should instead look for a traumatic event in Florence circa 1380 that significantly affected women's reproductive health, causing a change in the population's IQ variance?
I really don't know (this is a blog, not an article in Nature): but it is tempting to speculate whether it was simply coincidence that the Florentine Renaissance began two generations after the plague had ravaged Florence in 1348 (Boccaccio famously wrote about it). Might children born after the plague have stirred the Florentine gene-pool up in just the right way to set the Renaissance in motion a generation later?