As the world’s population gets larger, should we expect more Newtons, Mozarts and Tiger Woods or should we expect fewer? By this, I mean should the gap between the best in a field and the second best, be larger than the gap between the second best and the rest? If you presume that the aptitude level of people for some enterprise is normally distributed then the answer is no. The probability to be between n standard deviations and n+1 standard deviations above the mean decreases exponentially. As you go out onto the tail, the probability of someone being a standard deviation better than you is decreasing very quickly. Thus we should see a lot of people bunched near the top.
So why is it that we have Tiger Woods? Tiger is dominating golf like no one since perhaps Bobby Jones in the twenties and Ben Hogan in the early fifties. He was more dominant over a stretch of five years than Jack Nicklaus ever was although his dominance declined slightly last year. There are many more golfers now then there were fifty years ago so if anything we should see a much more competitive environment. Except for Woods and perhaps Vijay Singh, this is true as there is very little separating the rest. Dominance is not just limited to golf. Ten years ago we had Michael Jordan in basketball and Wayne Gretzky in hockey. Right now we have Barry Bonds in baseball and Lance Armstrong in cycling. These people are dominating or have dominated their sports as much or more than anyone else before.
Part of the reason is surely that a normal distribution is a bad approximation for the tail. Anyone in finance certainly knows this. The central limit theorem does not converge uniformly over the entire range, so even though the bulk is well described by a normal distribution, the tail can still experience low number statistics. Thus as the population increases, more of the tail gets sampled and that could actually increase the probability of rare events not decrease it as would be expected. So my prediction is that we will see more dominant athletes in the future.
Now what about physics and music? By this argument we should see more anomalies in these areas as well. However, it is not clear who is the Newton of our day. Ed Witten maybe? I think the reason is that there are fewer objective measures of mastery in these areas as compared to sports. A lot of what leads to great impact in physics is the choice of problem. In this case, there is a greater element of luck. One could be way out on the tail in terms of sheer brilliance (if that can even be measured) but not have a great impact if they worked in a dead end field. In this case, physics may suffer from low number statistics in the bulk and the tail so there will be no correlation of dominance with population size.
Music has a similar problem. Who in the 20th century compared to Mozart? Diatonic Western classical music reached its acme with Mahler. After that, composers either became atonal or “popular”. The 20th century was the century of performers. Perhaps if Rubenstein, Heifetz, or Yo Yo Ma was born in the 18th century they would have been great composers. The playing field changed in the 20th century so the question of dominance became more difficult to answer. In 200 years, we may look back at the 20th century and consider a Jazz or Rock musician to be the Mozart of the day.