Expertise slows the progress of knowledge, some say. First, it delays arrival at the cutting edge: if you must master everything that came before, you may not begin original research until your 30s, when your brain is a rigid fossil and retirement is already near. Also, it blinds you to new ideas: after years of seeing with accepted principles, reliance on those principles becomes second nature, a dusty, comfortable cow-path in your mind, and new and better ideas—the advances possible by coming at things sideways—become invisible. In The Lever of Riches Joel Mokyr documents one after another innovation created by amateurs just messing around, or discovered by accident by people working in other fields, while the so-called experts got nowhere. Serious money was invested by governments and corporations into the research and development of flying machines, only for all of them to be beaten by two bicycle mechanics from Ohio, working on their own.
On the other hand, there’s the story of convoy in World War II. Britain depended on supplies from America to stay in the war. But German U-boats patrolled the Atlantic. Merchant ships with food and arms were therefore sent across in convoy, guarded by armed escorts. With national survival on the line, losses had to be minimized, but how? Send many small convoys, or fewer large ones? More small convoys, one would guess from the armchair, means more chance one of them meets a U-boat, but fewer losses per meeting, and it’s unclear how those risks add up.
The Admiralty believed that sending many small convoys would minimize losses. Fortunately, the British government did not rely solely on that ancient institution’s ancient wisdom. It recruited scientists to look into such questions and produce more than hunches. Patrick Blackett was famous among them, and in his government service he more or less invented operations research as a field of study.
Blackett’s group first surveyed existing statistics on convoy losses. They found that large convoys were safer: a smaller percentage of their merchant ships were lost. But this did not satisfy them:
Though the statistics seems quite reliable, the scientists felt it necessary to make as sure as was humanly possible that large convoys were in fact safer than smaller ones, before attempting to convince the Admiralty that their long-founded preference for small convoys was mistaken...We felt that if we could find a rational explanation of why large convoys should be safer than small ones, it would strengthen the case for a change of policy.
A rational explanation was found. It relied on three facts:
1. Even though large convoys take up more space, it turned out that the chance a large convoy would be sighted by a U-boat was nearly the same as the chance a small convoy would be.
2. To protect the convoy, its defending escorts would form a circular perimeter around the merchant ships. As one would guess, the chance that a U-boat would penetrate this perimeter depended on the distance between the escort vessels: the denser the escorts—the more of them per mile of perimeter—the stronger the defense.
3. If a U-boat made it past the escort screen, the absolute number of merchant ships it sunk was (on average) the same for large and small convoys—either way, there were more than enough targets.
With this information, Blackett’s group argued as follows. Consider two convoys, one with escort perimeter twice the length of the other. By (2), the escorts provide the same protection if the large convoy has twice the number of escort ships. Now when the perimeter of a circle is doubled, the area inside is quadrupled. Therefore, the larger convoy can protect four times as many merchant ships clustered inside the escort perimeter. By (3), if caught both convoys lose the same number of ships. The larger convoy, then, loses a smaller percentage of its material (indeed 1/4 of the percent lost if the small convoy is caught), even with half the number of escorts per merchant ship. Since by (1) larger convoys are no more likely to be caught, using them minimizes losses.
I learned all this from The Pleasures of Counting by T. W. Körner. He uses Blackett’s achievements as a tool for teaching math and inspiring potential mathematicians. But interspersed with calculate-it-yourself questions are some of Körner’s historical and dare I say philosophical observations. Section 4.4, a concluding meditation on Blackett’s accomplishments, is titled “What Can We Learn?” Blackett’s argument for large convoys can be understood by anyone with some elementary geometry. Yet “Blackett’s circus contained two future Nobel Prizewinners, five Fellows of the Royal Society, and several future professors.” Körner writes,
In view of this, the reader may ask whether the arguments I have shown here are typical. Surely such a high-powered group used high-powered mathematical arguments?
But no, they did not. This is puzzling:
If the results were produced by such simple tools, why was such a high-powered group needed to produce them? I think the answer lies in the difference between ‘knowledge’ and ‘competence.’ To an 8 year old, multiplication may appear a difficult operation. To a 16 year old, algebraic manipulation may seem hard but multiplication is trivial....We lack the confidence and competence to make use of our highest levels of knowledge but the possession of the highest levels gives us the confidence required to make use of the lower levels.
Acting on the limits of one’s knowledge is like “buying a railway ticket in a foreign language,” a “tired and emotional experience.” One’s mind is distracted from what one wants to say, by the need to think about how to say it. But acting on mastery is speaking in your native tongue, the tools are transparent, and
For Blackett’s team, their mathematics was a well known language which left them free to concentrate on the essentials of their problems.
The discouragement of expertise will be no boon to science or knowledge, if it also discourages this.
A final footnote. Most philosophy PhD programs require some knowledge of advanced logic. To earn their doctorate students must demonstrate an ability to perform complicated formal deductions, and to understand, and maybe prove, such things as completeness and compactness. If in the mid-20th century the sophisticated manipulation of formal systems was central to the work of a large percentage of philosophers, this is no longer so. It does remain central to the work of many, but still, open a philosophy journal today, and you’re unlikely to encounter an argument much more complex than “P; and if P, then Q.” Are logic requirements therefore out of date? I don’t think so, and Körner’s moral from Blackett is why.
See also: On Toleration; On Balance.
"Expertise slows the progress of knowledge, some say. First, it delays arrival at the cutting edge: if you must master everything that came before, you may not begin original research until your 30s, when your brain is a rigid fossil and retirement is already near."
30 is before cognitive decline sets in in most people. It's really not that old.
"Also, it blinds you to new ideas: after years of seeing with accepted principles, reliance on those principles becomes second nature, a dusty, comfortable cow-path in your mind, and new and better ideas—the advances possible by coming at things sideways—become invisible."
This is why teaching is important. When students ask questions and look at the material with fresh perspectives, the teacher can use it as an opportunity to sharpen their thinking on the subject. The teacher can know whether the question has been answered before, how it's been answered before, whether a topic has been broached a certain way before, whether the current approach is sufficient.
"In The Lever of Riches Joel Mokyr documents one after another innovation created by amateurs just messing around, or discovered by accident by people working in other fields, while the so-called experts got nowhere. Serious money was invested by governments and corporations into the research and development of flying machines, only for all of them to be beaten by two bicycle mechanics from Ohio, working on their own."
Perhaps, this is why universities are so successful. It brings together the amateur (the student) and the expert. In addition, it exposes experts to the work done it different fields.
I think Körner draws the wrong lesson here.
The success of these sorts of simple mathematical models mostly depends on clear thinking about the real-world variables that need to be incorporated. Yes, you need to be able to handle the necessary math, and you need to be able to handle it fluently, so you probably need to have studied one or two levels of math beyond what the models require. But studying three or four levels beyond what the models require is not going to help. To continue the language analogy: once someone has achieved basic competence in Japanese, studying the Heian-period literature isn't going to make it any easier to buy a train ticket.
And I think this may extend to the question of advanced logic training requirements for philosophy PhDs. I won't pretend to have real expertise (!) on that topic, but I'm skeptical that it offers the benefits you're suggesting, as opposed to being more like requiring accountants to study complex analysis. Furthermore, I'm not sure the overall impact of such requirements on the discipline as a whole are positive, since they may reinforce the unfortunate tendency in Anglophone philosophy to exaggerate the importance of formal(izable) arguments in philosophical thinking.