Quite an everyday occurrence

from Huygens and Barrow, Newton and Hooke, Vladimir Arnold, translated by Eric J. F. Primrose (1989):

Hooke was a poor man and began work as an assistant to Boyle (who is now well known thanks to the Boyle-Mariotte law discovered by Hooke). Subsequently Hooke began working in the recently established Royal Society (that is, the English Academy of Sciences) as Curator. The duties of the Curator of the Royal Society were very onerous. According to his contract, at every session of the Society (and they occurred every week except for the summer vacation) he had to demonstrate three or four experiments proving the new laws of nature.

Hooke held the post of Curator for forty years, and all that time he carried out his duties thoroughly. Of course, there was no condition in the contract that all the laws to be demonstrated had to be devised by him. He was allowed to read books, correspond with other scientists, and to be interested in their discoveries. He was only required to verify whether their statements were true and to convince the Royal Society that some law was reliably established. For this it was necessary to prove this law experimentally and demonstrate the appropriate experiment. This was Hooke’s official activity.

[…]

At that time it was easy to carry out fundamental discoveries, and large numbers of them were carried out. Huygens, for example, improved the telescope, looked at Saturn and discovered its ring, and Hooke discovered the red spot on Jupiter. At that time discoveries were not unusual events, they were not registered, not patented, as they are now, they were quite an everyday occurrence. (This was the case not only in the natural sciences. Mathematical discoveries at that time also poured forth as if from a horn of plenty.)

But Hooke never had enough time to dwell on any of his discoveries and develop it in detail, since in the following week he needed to demonstrate new laws. So in the whole manifold of Hooke’s achievements his discoveries appeared somewhat incomplete, and sometimes when he was in a hurry he made assertions that he could not justify accurately and with mathematical rigour.

[…]

Holding the chair at Cambridge, Newton earned considerably more (200 pounds a year), and the farm that he had inherited, which he leased out and where the famous apple tree grew, gave him roughly the same income. Despite the fact that Newton was quite well off, he did not want to spend any money on the publication of the book, so he sent the Principia to the Royal Society, which decided to publish the book at its own expense. But the Society had no money, so the manuscript lay there until Halley (who was the son of a rich soap manufacturer) published it on his own account. Halley took on himself all the trouble of publishing the book, and even read the proofs himself. Newton, in correspondence at this time, called it “Your book”…

An engine of discovery

from the preface to Cell Biology by the Numbers, Ron Milo and Rob Phillips:

One of the great traditions in biology’s more quantitative partner sciences such as chemistry and physics is the value placed on centralized, curated quantitative data. Whether thinking about the astronomical data that describes the motions of planets or the thermal and electrical conductivities of materials, the numbers themselves are a central part of the factual and conceptual backdrop for these fields.  Indeed, often the act of trying to explain why numbers have the values they do ends up being an engine of discovery.

Where the appearance of disorder reigned

Poincaré, “The Future of Mathematics,” 1908, in Science and Method, translated by Francis Maitland:

The importance of a fact is measured by the return it gives—that is, by the amount of thought it enables us to economize.

In physics, the facts which give a large return are those which take their place in a very general law, because they enable us to foresee a very large number of others, and it is exactly the same in mathematics. Suppose I apply myself to a complicated calculation and with much difficulty arrive at a result, I shall have gained nothing by my trouble if it has not enabled me to foresee the results of other analogous calculations, and to direct them with certainty, avoiding the blind groping with which I had to be contented the first time. On the contrary, my time will not have been lost if this very groping has succeeded in revealing to me the profound analogy between the problem just dealt with and a much more extensive class of other problems; if it has shown me at once their resemblances and their differences; if, in a word, it has enabled me to perceive the possibility of a generalization. Then it will not be merely a new result that I have acquired, but a new force.

An algebraical formula which gives us the solution of a type of numerical problems, if we finally replace the letters by numbers, is the simple example which occurs to one’s mind at once. Thanks to the formula, a single algebraical calculation saves us the trouble of a constant repetition of numerical calculations. But this is only a rough example; every one feels that there are analogies which cannot be expressed by a formula, and that they are the most valuable.

If a new result is to have any value, it must unite elements long since known, but till then scattered and seemingly foreign to each other, and suddenly introduce order where the appearance of disorder reigned. Then it enables us to see at a glance each of these elements in the place it occupies in the whole. Not only is the new fact valuable on its own account, but it alone gives a value to the old facts it unites. Our mind is frail as our senses are; it would lose itself in the complexity of the world if that complexity were not harmonious; like the short-sighted, it would only see the details, and would be obliged to forget each of these details before examining the next, because it would be incapable of taking in the whole. The only facts worthy of our attention are those which introduce order into this complexity and so make it accessible to us.

Mathematicians attach a great importance to the elegance of their methods and of their results, and this is not mere dilettantism. What is it that gives us the feeling of elegance in a solution or a demonstration? It is the harmony of the different parts, their symmetry, and their happy adjustment; it is, in a word, all that introduces order, all that gives them unity, that enables us to obtain a clear comprehension of the whole as well as of the parts. But that is also precisely what causes it to give a large return; and in fact the more we see this whole clearly and at a single glance, the better we shall perceive the analogies with other neighbouring objects, and consequently the better chance we shall have of guessing the possible generalizations. Elegance may result from the feeling of surprise caused by the un-looked-for occurrence together of objects not habitually associated. In this, again, it is fruitful, since it thus discloses relations till then unrecognized. It is also fruitful even when it only results from the contrast between the simplicity of the means and the complexity of the problem presented, for it then causes us to reflect on the reason for this contrast, and generally shows us that this reason is not chance, but is to be found in some unsuspected law. Briefly stated, the sentiment of mathematical elegance is nothing but the satisfaction due to some conformity between the solution we wish to discover and the necessities of our mind, and it is on account of this very conformity that the solution can be an instrument for us. This aesthetic satisfaction is consequently connected with the economy of thought. Again the comparison with the Erechtheum occurs to me, but I do not wish to serve it up too often.

It is for the same reason that, when a somewhat lengthy calculation has conducted us to some simple and striking result, we are not satisfied until we have shown that we might have foreseen, if not the whole result, at least its most characteristic features. Why is this? What is it that prevents our being contented with a calculation which has taught us apparently all that we wished to know? The reason is that, in analogous cases, the lengthy calculation might not be able to be used again, while this is not true of the reasoning, often semi-intuitive, which might have enabled us to foresee the result. This reasoning being short, we can see all the parts at a single glance, so that we perceive immediately what must be changed to adapt it to all the problems of a similar nature that may be presented. And since it enables us to foresee whether the solution of these problems will be simple, it shows us at least whether the calculation is worth undertaking.

What I have just said is sufficient to show how vain it would be to attempt to replace the mathematician’s free initiative by a mechanical process of any kind. In order to obtain a result having any real value, it is not enough to grind out calculations, or to have a machine for putting things in order: it is not order only, but unexpected order, that has a value. A machine can take hold of the bare fact, but the soul of the fact will always escape it.

Since the middle of last century, mathematicians have become more and more anxious to attain to absolute exactness. They are quite right, and this tendency will become more and more marked. In mathematics, exactness is not everything, but without it there is nothing: a demonstration which lacks exactness is nothing at all. This is a truth that I think no one will dispute, but if it is taken too literally it leads us to the conclusion that before 1820, for instance, there was no such thing as mathematics, and this is clearly an exaggeration. The geometricians of that day were willing to assume what we explain by prolix dissertations. This does not mean that they did not see it at all, but they passed it over too hastily, and, in order to see it clearly, they would have had to take the trouble to state it.

Only, is it always necessary to state it so many times? Those who were the first to pay special attention to exactness have given us reasonings that we may attempt to imitate; but if the demonstrations of the future are to be constructed on this model, mathematical works will become exceedingly long, and if I dread length, it is not only because I am afraid of the congestion of our libraries, but because I fear that as they grow in length our demonstrations will lose that appearance of harmony which plays such a useful part, as I have just explained.

It is economy of thought that we should aim at, and therefore it is not sufficient to give models to be copied. We must enable those that come after us to do without the models, and not to repeat a previous reasoning, but summarize it in a few lines. And this has already been done successfully in certain cases. For instance, there was a whole class of reasonings that resembled each other, and were found everywhere; they were perfectly exact, but they were long. One day some one thought of the term “uniformity of convergence,” and this term alone made them useless; it was no longer necessary to repeat them, since they could now be assumed. Thus the hair-splitters can render us a double service, first by teaching us to do as they do if necessary, but more especially by enabling us as often as possible not to do as they do, and yet make no sacrifice of exactness.

One example has just shown us the importance of terms in mathematics; but I could quote many others. It is hardly possible to believe what economy of thought, as Mach used to say, can be effected by a well-chosen term. I think I have already said somewhere that mathematics is the art of giving the same name to different things. It is enough that these things, though differing in matter, should be similar in form, to permit of their being, so to speak, run in the same mould. When language has been well chosen, one is astonished to find that all demonstrations made for a known object apply immediately to many new objects: nothing requires to be changed, not even the terms, since the names have become the same.

A well-chosen term is very often sufficient to remove the exceptions permitted by the rules as stated in the old phraseology. This accounts for the invention of negative quantities, imaginary quantities, decimals to infinity, and I know not what else. And we must never forget that exceptions are pernicious, because they conceal laws.

This is one of the characteristics by which we recognize facts which give a great return: they are the facts which permit of these happy innovations of language. The bare fact, then, has sometimes no great interest: it may have been noted many times without rendering any great service to science; it only acquires a value when some more careful thinker perceives the connexion it brings out, and symbolizes it by a term.

The most vigorous exercise

C. S. Peirce, §10. Kinds of Reasoning, in Chapter 2, Lessons from the History of Science, Principles of Philosophy:

The methods of reasoning of science have been studied in various ways
and with results which disagree in important particulars. The followers of Laplace treat the subject from the point of view of the theory of probabilities. After corrections due to Boole and others, that method yields substantially the results stated above. Whewell described the reasoning just as it appeared to a man deeply conversant with several branches of science as only a genuine researcher can know them, and adding to that knowledge a full acquaintance with the history of science. These results, as might be expected, are of the highest value, although there are important distinctions and reasons which he overlooked. John Stuart Mill endeavored to explain the reasonings of science by the nominalistic metaphysics of his father. The superficial perspicuity of that kind of metaphysics rendered his logic extremely popular with those who think, but do not think profoundly; who know something of science, but more from the outside than the inside, and who for one reason or another delight in the simplest theories even if they fail to cover the facts.

Mill denies that there was any reasoning in Kepler’s procedure. He says it is merely a description of the facts. He seems to imagine that Kepler had all the places of Mars in space given him by Tycho’s observations; and that all he did was to generalize and so obtain a general expression for them. Even had that been all, it would certainly have been inference. Had Mill had even so much practical acquaintance with astronomy as to have practised discussions of the motions of double stars, he would have seen that. But so to characterize Kepler’s work is to betray total ignorance of it. Mill certainly never read the De Motu [Motibus] Stellae Martis, which is not easy reading. The reason it is not easy is that it calls for the most vigorous exercise of all the powers of reasoning from beginning to end.

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.