[btilly]

A "renaissance man" now is a dilettante. In Jefferson's day it meant having both broad and deep knowledge.

One could argue that the advances of science have made this impossible in modern times but I disagree.

I believe you're being too fast to dismiss that argument. As is pointed out in http://www.its.caltech.edu/~dg/crunch_art.html, from Jefferson's day to relatively recently, the output in scientific journals increased 10-fold every 50 years.

In Jefferson's day it was truly possible to learn what was known about all fields of science, and keep up with them all. Your knowledge could be broad and deep. But based on the sheer increase of volume, this soon became impossible. Today there are over 100,000 journals devoted to scientific research. Even if you read one page per second, every second, day and night, you would not even be scratching the surface of what is being produced.

Regurgitating the fractured work of others isn't intelligence. Applying this information as a tool to advance oneself and the human good is the goal of education. Applying this information as a tool to advance oneself and the human good is the goal of education.

This is a strong argument, but not necessarily for the point you are trying to make. Your argument leads to the point that we don't want students to just memorize random facts and regurgitate them. It does not lead to the point that it is possible to have both deep and broad knowledge about all fields.

The challenge is that, thanks to the advances of science, today we both have more subjects to learn about (the "broad" is broader) and we know about them in more depth (the "deep" is deeper). This has made the ideal of having both broad and deep knowledge much, much harder. Hard enough that most conclude it is simply impossible.

[_delirium]

I do think breadth and depth have expanded, but I think fundamentally less than the explosion in page count would suggest. There has also been a big decrease in signal-to-noise ratio, with a lot of crap published in journals, and not enough gardening work. IMO a much bigger proportion of scientific resources need to be put into survey articles, "recent advances in X" retrospectives, post-intro-level textbooks and tutorials, and even ideally some cross-field work, matching up equivalent/redundant ideas and harmonizing terminology. In academia at least, the incentives don't encourage that, though: even though a survey article is probably the single quickest way to have significant impact on a field (they're widely read, and you're forging the lens through which many subsequent people will view that field), they're not as well respected for advancement purposes as even very niche original research is.

As it is, tons of stuff keeps getting reinvented just because the state of the literature is so bad that you'll never find it, unless it was invented exactly in your sub-sub-specialty, or you serendipitously found it via a colleague who remarked that what you were doing sounded similar to something he once read.

The decline in scientists writing books also doesn't help. It used to be that prominent scientists would gather up their scattered papers and unify them into a magnum opus laying out their theories, or possibly a few different books, one on each major area they worked on. That sometimes happens, especially in areas like cosmology, but it's much less the norm than it was 100 years ago. Today it's quite common to just publish 200+ papers over your career and not really do any summarization of them, even though there is plenty that could often be done, since it's common for papers to supersede or overlap with previous ones.

Actually, on that front, it'd be a big win if scientists with lots of publications simply provided some sort of brief guide to them. Take the 50 papers on lasers, and provide an annotated bibliography explaining which papers are the important ones, which papers are obsolete or superseded, and which ones might be of particular interest to people working on particular topics. A few people do that, but many don't even separate their list of publications by topic, let alone provide a guide.

Or, shorter: There is a lot of stuff published, but it's terribly indexed and summarized, which I think is a bigger problem than the volume alone.

[btilly]

I strongly agree with this point. See http://bentilly.blogspot.com/2009/11/why-i-left-math.html for how what I saw in math supports your point.

Worse yet, we have not just specialized, we have lost knowledge as well. My favorite example of this is a result I rediscovered. Every mathematician knows that, for instance, sqrt(2)+cube_root(3) must be the root of some polynomial of degree 6. (This one turns out to be x^6 - 6x^4 -6x^3 + 12x^2 - 36x + 1.) I came up with a construction for answering questions like this. I showed it to a number of mathematicians, and none had seen it until I showed an older one who said, "That looks like a very old way to do this. Go to the library, pick up an algebra book from the 1800s, and see if you find it."

I did, it was there, and it turns out that two of the mathematicians I had talked to were in fields that had gotten their start from the very construction I rediscovered! (One was in number theory, dealing with things like algebraic integers. The other was in combinatorics, and did a lot of stuff with symmetric polynomials.)

If you're curious, the observation behind the construction is that any polynomial expression that is symmetric in the roots of a polynomial can be rewritten as a polynomial in the coefficients of that polynomial. So if a1 and a2 are the roots of x^2-2, and b1, b2, b3 are the roots of x^3-3, then (x-a1-b1)(x-a1-b2)(x-a1-b3)(x-a2-b1)(x-a2-b2)(x-a2-b3) is a polynomial in x and the coefficients of x^2-2 and x^3-3, which means that it is an integer polynomial.

All that said, this point doesn't support your argument. It is true that if information were better presented, people could learn more about more subjects than they could otherwise. It is true that trying to do this would have tremendous value. But if you want to learn about multiple subjects then you need to deal with how information actually exists out there in the world, rather than how we'd like it to be organized. The disorganization that you point to is a significant barrier to learning. (This problem does not look like it will improve any time soon.)