Instance Quotes (8 quotes)
...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly generality is, in essence, the same as a small and concrete special case.
For a stone, when it is examined, will be found a mountain in miniature. The fineness of Natures work is so great, that, into a single block, a foot or two in diameter, she can compress as many changes of form and structure, on a small scale, as she needs for her mountains on a large one; and, taking moss for forests, and grains of crystal for crags, the surface of a stone, in by far the plurality of instances, is more interesting than the surface of an ordinary hill; more fantastic in form and incomparably richer in colourthe last quality being, in fact, so noble in most stones of good birth (that is to say, fallen from the crystalline mountain ranges).
I am told that the wall paintings which we had the happiness of admiring in all their beauty and freshness [in the chapel she discovered at Abu Simbel] are already much injured. Such is the fate of every Egyptian monument, great or small. The tourist carves it over with names and dates, and in some instances with caricatures. The student of Egyptology, by taking wet paper squeezes sponges away every vestige of the original colour. The Collector buys and carries off everything of value that he can, and the Arab steals it for him. The work of destruction, meanwhile goes on apace. The Museums of Berlin, of Turin, of Florence are rich in spoils which tell their lamentable tale. When science leads the way, is it wonderful that ignorance should follow?
If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.
Such instances of the almost infinite unpredictability of man are known to social scientists, but they are no more affected by them than the asylum inmate is by being told that he is not Napoleon.
The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.
The significance of a fact is relative to [the general body of scientific] knowledge. To say that a fact is significant in science, is to say that it helps to establish or refute some general law; for science, though it starts from observation of the particular, is not concerned essentially with the particular, but with the general. A fact, in science, is not a mere fact, but an instance. In this the scientist differs from the artist, who, if he deigns to notice facts at all, is likely to notice them in all their particularity.
Yet as I cast my eye over the whole course of science I behold instances of false science, even more pretentious and popular than that of Einstein gradually fading into ineptitude under the searchlight; and I have no doubt that there will arise a new generation who will look with a wonder and amazement, deeper than now accompany Einstein, at our galaxy of thinkers, men of science, popular critics, authoritative professors and witty dramatists, who have been satisfied to waive their common sense in view of Einstein's absurdities.