Untouched by the parasite idea

T. S. Eliot, “In Memory,” The Little Review, Henry James number, August 1918:

The “influence” of James hardly matters: to be influenced by a writer is to have a chance inspiration from him; or to take what one wants; or to see things one has overlooked; there will always be a few intelligent people to understand James, and to be understood by a few intelligent people is all the influence a man requires.

• • •

James’s critical genius comes out most tellingly in his mastery over, his baffling escape from, Ideas; a mastery and an escape which are perhaps the last test of a superior intelligence. He had a mind so fine that no idea could violate it. Englishmen, with their uncritical admiration (in the present age) for France, like to refer to France as the Home of Ideas; a phrase which, if we could twist it into truth, or at least a compliment, ought to mean that in France ideas are very severely looked after; not allowed to stray, but preserved for the inspection of civic pride in a Jardin des Plantes, and frugally dispatched on occasions of public necessity. England, on the other hand, if it is not the Home of Ideas, has at least become infested with them in about the space of time within which Australia has been overrun by rabbits. In England ideas run wild and pasture on the emotions; instead of thinking with our feelings (a very different thing) we corrupt our feelings with ideas; we produce the political, the emotional idea, evading sensation and thought. George Meredith (the disciple of Carlyle) was fertile in ideas; his epigrams are a facile substitute for observation and inference. Mr. Chesterton’s brain swarms with ideas; I see no evidence that it thinks. James in his novels is like the best French critics in maintaining a point of view, a view-point untouched by the parasite idea. He is the most intelligent man of his generation.

Not results, but powers

George Eliot on Thomas Carlyle, 1855:

It has been well said that the highest aim in education is analogous to the highest aim in mathematics, namely, to obtain not results but powers, not particular solutions, but the means by which endless solutions may be wrought. He is the most effective educator who aims less at perfecting specific acquirements than at producing that mental condition which renders acquirements easy, and leads to their useful application; who does not seek to make his pupils moral by enjoining particular courses of action, but by bringing into activity the feelings and sympathies that must issue in noble action. On the same ground it may be said that the most effective writer is not he who announces a particular discovery, who convinced men of a particular conclusion, who demonstrates that this measure is right and that measure wrong; but he who rouses in others that activities that must issue in discovery, who awakes men from their indifference to right and wrong, who nerves their energies to seek for the truth and live up to it at whatever cost. The influence of such a writer is dynamic. He does not teach men how to use sword and musket, but he inspires their souls with courage and sends a strong will into their muscles. He does not, perhaps, enrich your stock of data, but he clears away the film from your eyes that you may search for data to some purpose. He does not, perhaps, convince you, but he strikes you, undeceives you, animates you. You are not directly fed by his books, but you are braced as by a walk up to an alpine summit, and yet subdued to calm and reverence as by the sublime things to be seen from that summit.

Freed from these irrelevancies

Arthur Eddington, The Internal Constitution of the Stars, 1926:

I conceive that the chief aim of the physicist in discussing a theoretical problem is to obtain ‘insight’—to see which of the numerous factors are particularly concerned in any effect and how they work together to give it. For this purpose a legitimate approximation is not just an unavoidable evil; it is a discernment that certain factors—certain complications of the problem—do not contribute appreciably to the result. We satisfy ourselves that they may be left aside; and the mechanism stands out more clearly freed from these irrelevancies. This discernment is only a continuation of a task begun by the physicist before the mathematical premises of the problem could even be stated; for in any natural problem the actual conditions are of extreme complexity and the first step is to select those which have an essential influence on the result—in short, to get hold of the right end of the stick.

Lawrence Krauss, Fear of Physics, 1993:

A physicist, an engineer, and a psychologist are called in as consultants to a dairy farm whose production has been below par. Each is given time to inspect the details of the operation before making a report.

The first to be called is the engineer, who states: “The size of the stalls for the cattle should be decreased. Efficiency could be improved if the cows were more closely packed, with a net allotment of 275 cubic feet per cow. Also, the diameter of the milking tubes should be increased by 4 percent to allow for a greater average flow rate during the milking periods.”

The next to report is the psychologist, who proposes: “The inside of the barn should be painted green. This is a more mellow color than brown and should help induce greater milk flow. Also, more trees should be planted in the fields to add diversity to the scenery for the cattle during grazing, to reduce boredom.”

Finally, the physicist is called upon. He asks for a blackboard and then draws a circle. He begins: “Assume the cow is a sphere. . . .”

This old joke, if not very funny, does illustrate how—at least metaphorically—physicists picture the world. The set of tools physicists have to describe nature is limited. Most of the modern theories you read about began life as simple models by physicists who didn’t know how else to start to solve a problem. These simple little models are usually based on even simpler little models, and so on, because the class of things that we do know how to solve exactly can be counted on the fingers of one, maybe two, hands. For the most part, physicists follow the same guidelines that have helped keep Hollywood movie producers rich: If it works, exploit it. If it still works, copy it.

I like the cow joke because it provides an allegory for thinking simply about the world, and it allows me to jump right in to an idea that doesn’t get written about too much, but that is essential for the everyday workings of science: Before doing anything else, abstract out all irrelevant details!

There are two operative words here: abstract and irrelevant. (Getting rid of irrelevant details is the first step in building any model of the world, and we do it subconsciously from the moment we are born). Doing it consciously is another matter. Overcoming the natural desire not to throw out unnecessary information is probably the hardest and most important part of learning physics. In addition, what may be irrelevant in a given situation is not universal but depends in most cases on what interests you. This leads us to the second operative word: abstraction. Of all the abstract thinking required in physics, probably the most challenging lies in choosing how to approach a problem. The mere description of movement along a straight line—the first major development in modern physics—required enough abstraction that it largely eluded some pretty impressive intellects until Galileo.

In need of clarification

There is no way to run out of ideas in need of clarification.

—Fields medalist Bill Thurston, in reply to “What’s a mathematician to do?” on MathOverflow, 2010

Rigidity and continuity

John von Neumann, “The General and Logical Theory of Automata,” 1948:

There exists today a very elaborate system of formal logic, and, specifically, of logic as applied to mathematics. This is a discipline with many good sides, but also with certain serious weaknesses. This is not the occasion to enlarge upon the good sides, which I have certainly no intention to belittle. About the inadequacies, however, this may be said: Everybody who has worked in formal logic will confirm that it is one of the technically most refractory parts of mathematics. The reason for this is that it deals with rigid, all-or-none concepts, and has very little contact with the continuous concept of the real or of the complex number, that is, with mathematical analysis. Yet analysis is the technically most successful and best-elaborated part of mathematics. Thus formal logic is, by the nature of its approach, cut off from the best cultivated portions of mathematics, and forced onto the most difficult part of the mathematical terrain, into combinatorics.

The theory of automata, of the digital, all-or-none type, as discussed up to now, is certainly a chapter in formal logic. It would, therefore, seem that it will have to share this unattractive property of formal logic. It will have to be, from the mathematical point of view, combinatorial rather than analytical.

A total devotion to hedgehoggism

Richard Powers in conversation with Jeffrey Williams, Cultural Logic, spring 1999:

RP: I was the kind of kid who really didn’t make great distinctions between different fields and who took huge amounts of pleasure in being able to solve problems in very different intellectual disciplines. If anything, I would say my problem-solving abilities in math and science were always a good deal stronger than my verbal skills. I always thought that I would end up becoming one kind of scientist or another. It wasn’t always physics. For a while it was oceanography. For a while it was paleontology.

JW: Unusual for a novelist . . .

RP: Well, I’m not sure what the usual novelist trajectory is! But my orientation was definitely empirical, a real bias toward the “non-subjective” disciplines. I guess the difficulty for me growing up was this constant sensation that every decision to commit myself more deeply to any of these fields meant closing several doors. Specializing involved almost perpetual leave-taking from other pursuits that I loved and that gave me great pleasure. I really resisted the process, as long as I could. I just wanted to arrive somewhere where I could be the last generalist and do that in good faith. I thought for a long time that physics might be that place.

We have this notion of physics—especially cosmology, I guess—as representing a fundamental kind of knowledge, and that it’s a great field to be in if you want the aerial view of how things work. In fact, in some ways, almost the opposite may be true. The enormous success of the reductionist program depends upon absolute applications of Occam’s razor on every level. You have to make yourself expert in a field that’s too small even to be called a specialization. The whole overwhelming success of physics as a discipline depends upon dividing and conquering, on separating fields of research into ever smaller domains. And so it became clear to me pretty quickly, to use Isaiah Berlin’s hedgehog and fox distinction, that rather than becoming a fox, I had in fact landed in a place that demanded of me a total devotion to hedgehoggism. I got pretty claustrophobic pretty quickly, and it made me look for other fields where I could preserve that sense of multiplicity, of generalism.

JW: So that induced your turn to literature?

RP: That’s right. Initially, I thought that in the study of literature, I’d really found that aerial view again.

JW: So you thought that you’d be a literary critic and a professor?

RP: Right, or at least that that’s how I would make my living. Since literature seemed to be about everything that there is—about the human condition—I figured that a good literary critic would have to make himself expert at that big picture. It didn’t take me long to realize that the professionalization of literary criticism has taken reductionism as its model, and that it too can lead to learning more and more about less and less until you’re in danger of knowing everything there is to know about nothing.

The hazards of learning

Dewey, from Chapter 4, §3, How We Think, 1910:

Studies are conventionally and conveniently grouped under these heads: (1) Those especially involving the acquisition of skill in performance—the school arts, such as reading, writing, figuring, and music. (2) Those mainly concerned with acquiring knowledge—”informational” studies, such as geography and history. (3) Those in which skill in doing and bulk of information are relatively less important, and appeal to abstract thinking, to “reasoning,” is most marked—”disciplinary” studies, such as arithmetic and formal grammar. Each of these groups of subjects has its own special pitfalls.

(a) In the case of the so-called disciplinary or pre-eminently logical studies, there is danger of the isolation of intellectual activity from the ordinary affairs of life. Teacher and student alike tend to set up a chasm between logical thought as something abstract and remote, and the specific and concrete demands of everyday events. The abstract tends to become so aloof, so far away from application, as to be cut loose from practical and moral bearing. The gullibility of specialized scholars when out of their own lines, their extravagant habits of inference and speech, their ineptness in reaching conclusions in practical matters, their egotistical engrossment in their own subjects, are extreme examples of the bad effects of severing studies completely from their ordinary connections in life.

(b) The danger in those studies where the main emphasis is upon acquisition of skill is just the reverse. The tendency is to take the shortest cuts possible to gain the required end. This makes the subjects mechanical, and thus restrictive of intellectual power. In the mastery of reading, writing, drawing, laboratory technique, etc., the need of economy of time and material, of neatness and accuracy, of promptness and uniformity, is so great that these things tend to become ends in themselves, irrespective of their influence upon general mental attitude. Sheer imitation, dictation of steps to be taken, mechanical drill, may give results most quickly and yet strengthen traits likely to be fatal to reflective power. The pupil is enjoined to do this and that specific thing, with no knowledge of any reason except that by so doing he gets his result most speedily; his mistakes are pointed out and corrected for him; he is kept at pure repetition of certain acts till they become automatic. Later, teachers wonder why the pupil reads with so little expression, and figures with so little intelligent consideration of the terms of his problem. In some educational dogmas and practices, the very idea of training mind seems to be hopelessly confused with that of a drill which hardly touches mind at all—or touches it for the worse—since it is wholly taken up with training skill in external execution. This method reduces the “training” of human beings to the level of animal training. Practical skill, modes of effective technique, can be intelligently, non-mechanically used, only when intelligence has played a part in their acquisition.

(c) Much the same sort of thing is to be said regarding studies where emphasis traditionally falls upon bulk and accuracy of information. The distinction between information and wisdom is old, and yet requires constantly to be redrawn. Information is knowledge which is merely acquired and stored up; wisdom is knowledge operating in the direction of powers to the better living of life. Information, merely as information, implies no special training of intellectual capacity; wisdom is the finest fruit of that training. In school, amassing information always tends to escape from the ideal of wisdom or good judgment. The aim often seems to be—especially in such a subject as geography—to make the pupil what has been called a “cyclopedia of useless information.” “Covering the ground” is the primary necessity; the nurture of mind a bad second. Thinking cannot, of course, go on in a vacuum; suggestions and inferences can occur only upon a basis of information as to matters of fact.

But there is all the difference in the world whether the acquisition of information is treated as an end in itself, or is made an integral portion of the training of thought. The assumption that information which has been accumulated apart from use in the recognition and solution of a problem may later on be freely employed at will by thought is quite false. The skill at the ready command of intelligence is the skill acquired with the aid of intelligence; the only information which, otherwise than by accident, can be put to logical use is that acquired in the course of thinking. Because their knowledge has been achieved in connection with the needs of specific situations, men of little book-learning are often able to put to effective use every ounce of knowledge they possess; while men of vast erudition are often swamped by the mere bulk of their learning, because memory, rather than thinking, has been operative in obtaining it.

The glad news bubbling within him

Mencken, “Education,” in Prejudices: Third Series, 1922:

That ability to impart knowledge, it seems to me, has very little to do with technical method. It may operate at full function without any technical method at all, and contrariwise, the most elaborate of technical methods, whether out of Switzerland, Italy or Gary, Ind., cannot make it operate when it is not actually present.

And what does it consist of?

It consists, first, of a natural talent for dealing with children, for getting into their minds, for putting things in a way that they can comprehend. And it consists, secondly, of a deep belief in the interest and importance of the thing taught, a concern about it amounting to a sort of passion. A man who knows a subject thoroughly, a man so soaked in it that he eats it, sleeps it and dreams it—this man can always teach it with success, no matter how little he knows of technical pedagogy. That is because there is enthusiasm in him, and because enthusiasm is almost as contagious as fear or the barber’s itch.

An enthusiast is willing to go to any trouble to impart the glad news bubbling within him. He thinks that it is important and valuable for to know; given the slightest glow of interest in a pupil to start with, he will fan that glow to a flame. No hollow formalism cripples him and slows him down. He drags his best pupils along as fast as they can go, and he is so full of the thing that he never tires of expounding its elements to the dullest.

This passion, so unordered and yet so potent, explains the capacity for teaching that one frequently observes in scientific men of high attainments in their specialties—for example, Huxley, Ostwald, Karl Ludwig, Virchow, Billroth, Jowett, William G. Sumner, Halsted and Osler—men who knew nothing whatever about the so-called science of pedagogy, and would have derided its alleged principles if they had heard them stated.

Teachers: preachers, propagandists, gurus

Gian-Carlo Rota, “Ten Lessons I Wished I Had Learned Before I Started Teaching Differential Equations,” an address to the Mathematical Association of America at Simmons College on April 24, 1997:

Who cares whether the students become skilled at working out tricky problems? What matters is their getting a feeling for the importance of the subject, their coming out of the course with the conviction of the inevitability of differential equations, and with enhanced faith in the power of mathematics. These objectives are better achieved by stretching the students’ minds to the utmost limits of cultural breadth of which they are capable, and by pitching the material at a level that is just a little higher than they can reach.

We are kidding ourselves if we believe that the purpose of undergraduate teaching is the transmission of information. Information is an accidental feature of an elementary course in differential equations; such information can nowadays be gotten in much better ways than sitting in a classroom. A teacher of undergraduate courses belongs in a class with P.R. men, with entertainers, with propagandists, with preachers, with magicians, with gurus. Such a teacher will be successful if at the end of the course every one of his or her students feels they have taken “a good course,” even though they may not quite be able to pin down anything specific they have learned in the course.

The work of reason produces monsters

Poincaré, “Mathematical definitions and education,” 1906, in Science and Method, translated by Francis Maitland:

Logic sometimes breeds monsters. For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemblance as possible to honest functions that are of some use. No more continuity, or else continuity but no derivatives, etc. More than this, from the point of view of logic, it is these strange functions that are the most general; those that are met without being looked for no longer appear as more than a particular case, and they have only quite a little corner left them.

Cf. Goya’s El sueño de la razón produce monstruoscirca 1799: