Over at Gregory Cochran’s blog, I’ve been having a very minor argument with the host. (Well, he posted a response or two, then I suspect ignored me, but that’s something of an argument.)

He argues that there is unlikely to have been a substantial drop in IQ since about the Victorian Era, and has offered multiple reasons why that’s so. I generally find his reasoning compelling; however, the argument that I find less than convincing is that such a drop would have crippled the progress of higher mathematics and this has not taken place.

I see no reason to assert that a decline or drop of performance in the field of mathematics has occurred. But neither can I find reason to assert that it has not.

When we can identify a continuing trend in the development of human knowledge – such as Moore’s Law – we usually can’t figure out why it’s stable and clear enough for us to pick up on. Not knowing what causes such trends, we can’t predict when they’ll stop. In the case of Moore’s, it has been noted that people predicted the trend would continue only for a decade – and have continued to predict that for the past thirty and more.

It is pretty clear that there’s a ‘low-hanging fruit’ effect in the sciences. The basics of most sciences were within reach of amateur investigators, and the cutting edges frequently require technology that no individual could afford to own, and/or knowledge leading up to the edge that requires years of study to acquire. Sometimes, in highly abstract and rich fields such as mathematics, a particularly gifted and inspired neophyte can find and strike out in a shocking new direction. But each time that happens, the number of unexploited new directions is reduced. So all else being equal, it should be harder and harder to make new discoveries the more that’s discovered.

It is also clear that our understanding of what’s easy and what’s hard changes with time. At one point algebra was advanced mathematics studied by the luminaries in the field, and now bright elementary students are expected to be able to pick up the rudiments. This would tend to shorten the time it takes to find new things and reduce the difficulty to do so.

Human populations have exploded – from about one billion people in 1800 to seven billion today. All else being equal (which is certainly not the case, but the details are both controversial and obscure) there should be seven times as many people of any skill level, talent, or genius as there were roughly two hundred years ago. The degree to which having more people to work on problems affects the time associated with progress is complicated, but I think it’s clear that it should have a net hastening effect.

Flynn Effect, anyone? We don’t know precisely what it means to actual accomplishment, or what’s responsible for it.

We can name more and more conflicting trends in intellectual innovation. But our ability to model and understand those trends isn’t impressive. For most, we can only make rough approximations as to their effects, and as for what happens when they begin to interact? Forget it.

The exciting developments in the field of mathematics are beyond both my intelligence and my education. But I’ve been told that mathematical proofs are becoming so complex that it can’t be ascertained whether they’re logically valid or not. Instead, teams of mathematicians have to give their opinions. Does this represent increased or decreased progress in the field? I have no idea, and no one else seems to have a clear grasp either.

Cochran’s argument is an appeal to normality, asserting that the status quo is appropriate and correct. But that has a great deal to do with our initial assumptions and very little to do with rational conviction. Can we determine, from first principles, what the level of achievement in mathematics would be if there were unquestionably no decrease in mathematical talent in the population at large? Can we even determine whether or not the nature of progress in the field has actually changed? The answer in both cases is ‘no’. Perhaps the level of talent is the same or even greater than it used to be, and we’ve pushed the field so far into the potential for human cognition that it’s truly becoming harder to grasp without tremendous genius and years of focused effort. Or maybe the potential is being lost. Or maybe it’s our educational system, or the way in which research is conducted or… any one of a million other things. And that’s presuming that the nature of generated proofs has actually changed!