Incidental Quotes (12 quotes)
A superficial knowledge of mathematics may lead to the belief that this subject can be taught incidentally, and that exercises akin to counting the petals of flowers or the legs of a grasshopper are mathematical. Such work ignores the fundamental idea out of which quantitative reasoning growsthe equality of magnitudes. It leaves the pupil unaware of that relativity which is the essence of mathematical science. Numerical statements are frequently required in the study of natural history, but to repeat these as a drill upon numbers will scarcely lend charm to these studies, and certainly will not result in mathematical knowledge.
Before delivering your lectures, the manuscript should be in such a perfect form that, if need be, it could be set in type. Whether you follow the manuscript during the delivery of the lecture is purely incidental. The essential point is that you are thus master of the subject matter.
Before the promulgation of the periodic law the chemical elements were mere fragmentary incidental facts in nature; there was no special reason to expect the discovery of new elements, and the new ones which were discovered from time to time appeared to be possessed of quite novel properties. The law of periodicity first enabled us to perceive undiscovered elements at a distance which formerly were inaccessible to chemical vision, and long ere they were discovered new elements appeared before our eyes possessed of a number of well-defined properties.
Language is a guide to 'social reality.' Though language is not ordinarily thought of as essential interest to the students of social science, it powerfully conditions all our thinking about social problems and processes. Human beings do not live in the objective world alone, nor alone in the world of social activity as ordinarily understood, but are very much at the mercy of the particular language which has become the medium of expression for their society. It is quite an illusion to imagine that one adjusts to reality essentially without the use of language and that language is merely an incidental means of solving specific problems of communication or reflection. The fact of the matter is that the 'real world' is to a large extent unconsciously built up on the language habits of the group. No two languages are ever sufficiently similar to be considered as representing the same social reality. The worlds in which different societies live are distinct worlds, not merely the same world with different labels attached.
One feature which will probably most impress the mathematician accustomed to the rapidity and directness secured by the generality of modern methods is the deliberation with which Archimedes approaches the solution of any one of his main problems. Yet this very characteristic, with its incidental effects, is calculated to excite the more admiration because the method suggests the tactics of some great strategist who foresees everything, eliminates everything not immediately conducive to the execution of his plan, masters every position in its order, and then suddenly (when the very elaboration of the scheme has almost obscured, in the mind of the spectator, its ultimate object) strikes the final blow. Thus we read in Archimedes proposition after proposition the bearing of which is not immediately obvious but which we find infallibly used later on; and we are led by such easy stages that the difficulties of the original problem, as presented at the outset, are scarcely appreciated. As Plutarch says: It is not possible to find in geometry more difficult and troublesome questions, or more simple and lucid explanations. But it is decidedly a rhetorical exaggeration when Plutarch goes on to say that we are deceived by the easiness of the successive steps into the belief that anyone could have discovered them for himself. On the contrary, the studied simplicity and the perfect finish of the treatises involve at the same time an element of mystery. Though each step depends on the preceding ones, we are left in the dark as to how they were suggested to Archimedes. There is, in fact, much truth in a remark by Wallis to the effect that he seems as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results. Wallis adds with equal reason that not only Archimedes but nearly all the ancients so hid away from posterity their method of Analysis (though it is certain that they had one) that more modern mathematicians found it easier to invent a new Analysis than to seek out the old.
Specialists never contribute anything to their specialty; Helmholtz wasnt an eye-specialist, but a German army doctor who invented the ophthalmoscope one Saturday afternoon when there wasnt anything else to do. Incidentally, he rewrote whole chapters of physics, so that the physicists only know him as one of their own. Robert Mayer wasnt a physicist, but another country doctor; and Pasteur, who made bacteriology, was a tanners son or a chemist, as you will.
Such men as Newton and Linnaeus are incidental, but august, teachers of religion.
The aid which we feel impelled to give to the helpless is mainly an incidental result of the instinct of sympathy, which was originally acquired as part of the social instincts, but subsequently rendered, in the manner previously indicated, more tender and more widely diffused. Nor could we check our sympathy, even at the urging of hard reason, without deterioration in the noblest part of our nature.
The responsibility for maintaining the composition of the blood in respect to other constituents devolves largely upon the kidneys. It is no exaggeration to say that the composition of the blood is determined not by what the mouth ingests but by what the kidneys keep; they are the master chemists of our internal environment, which, so to speak, they synthesize in reverse. When, among other duties, they excrete the ashes of our body fires, or remove from the blood the infinite variety of foreign substances which are constantly being absorbed from our indiscriminate gastrointestinal tracts, these excretory operations are incidental to the major task of keeping our internal environment in an ideal, balanced state. Our glands, our muscles, our bones, our tendons, even our brains, are called upon to do only one kind of physiological work, while our kidneys are called upon to perform an innumerable variety of operations. Bones can break, muscles can atrophy, glands can loaf, even the brain can go to sleep, without immediately endangering our survival, but when the kidneys fail to manufacture the proper kind of blood neither bone, muscle, gland nor brain can carry on.
The science hangs like a gathering fog in a valley, a fog which begins nowhere and goes nowhere, an incidental, unmeaning inconvenience to passers-by.
What was at first merely by-the-way may become the very heart of a matter. Flints were long flaked into knives, arrowheads, spears. Incidentally it was found that they struck fire; to-day that is their one use.
Whatever may happen to the latest theory of Dr. Einstein, his treatise represents a mathematical effort of overwhelming proportions. It is the more remarkable since Einstein is primarily a physicist and only incidentally a mathematician. He came to mathematics rather of necessity than by predilection, and yet he has here developed mathematical formulae and calculations springing from a colossal knowledge.