Generalization Quotes (17 quotes)
A mathematician who can only generalise is like a monkey who can only climb UP a tree. ... And a mathematician who can only specialise is like a monkey who can only climb DOWN a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise. ... There is, I think, a moral for the teacher. A teacher of traditional mathematics is in danger of becoming a down monkey, and a teacher of modern mathematics an up monkey. The down teacher dishing out one routine problem after another may never get off the ground, never attain any general idea. and the up teacher dishing out one definition after the other may never climb down from his verbiage, may never get down to solid ground, to something of tangible interest for his pupils.
Analogy is a wonderful, useful and most important form of thinking, and biology is saturated with it. Nothing is worse than a horrible mass of undigested facts, and facts are indigestible unless there is some rhyme or reason to them. The physicist, with his facts, seeks reason; the biologist seeks something very much like rhyme, and rhyme is a kind of analogy.... This analogizing, this fine sweeping ability to see likenesses in the midst of differences is the great glory of biology, but biologists don't know it.... They have always been so fascinated and overawed by the superior prestige of exact physical science that they feel they have to imitate it.... In its central content, biology is not accurate thinking, but accurate observation and imaginative thinking, with great sweeping generalizations.
Both religion and natural science require a belief in God for their activities, to the former He is the starting point, and to the latter the goal of every thought process. To the former He is the foundation, to the latter, the crown of the edifice of every generalized world view.
During the first half of the present century we had an Alexander von Humboldt, who was able to scan the scientific knowledge of his time in its details, and to bring it within one vast generalization. At the present juncture, it is obviously very doubtful whether this task could be accomplished in a similar way, even by a mind with gifts so peculiarly suited for the purpose as Humboldt's was, and if all his time and work were devoted to the purpose.
I hope that in due time the chemists will justify their proceedings by some large generalisations deduced from the infinity of results which they have collected. For me I am left hopelessly behind and I will acknowledge to you that through my bad memory organic chemistry is to me a sealed book. Some of those here, Hoffman [August Hofmann] for instance, consider all this however as scaffolding, which will disappear when the structure is built. I hope the structure will be worthy of the labour. I should expect a better and a quicker result from the study of the powers of matter, but then I have a predilection that way and am probably prejudiced in judgment.
I see with much pleasure that you are working on a large work on the integral Calculus [ ... ] The reconciliation of the methods which you are planning to make, serves to clarify them mutually, and what they have in common contains very often their true metaphysics; this is why that metaphysics is almost the last thing that one discovers. The spirit arrives at the results as if by instinct; it is only on reflecting upon the route that it and others have followed that it succeeds in generalising the methods and in discovering its metaphysics.
If it were possible for a metaphysician to be a golfer, he might perhaps occasionally notice that his ball, instead of moving forward in a vertical plane (like the generality of projectiles, such as brickbats and cricket balls), skewed away gradually to the right. If he did notice it, his methods would naturally lead him to content himself with his caddies's remark-'ye heeled that yin,' or 'Ye jist sliced it.' ... But a scientific man is not to be put off with such flimsy verbiage as that. He must know more. What is 'Heeling', what is 'slicing', and why would either operation (if it could be thoroughly carried out) send a ball as if to cover point, thence to long slip, and finally behind back-stop? These, as Falstaff said, are 'questions to be asked.'
In our day grand generalizations have been reached. The theory of the origin of species is but one of them. Another, of still wider grasp and more radical significance, is the doctrine of the Conservation of Energy, the ultimate philosophical issues of which are as yet but dimly seem-that doctrine which 'binds nature fast in fate' to an extent not hitherto recognized, exacting from every antecedent its equivalent consequent, and bringing vital as well as physical phenomena under the dominion of that law of causal connexion which, so far as the human understanding has yet pierced, asserts itself everywhere in nature.
One of the grandest generalizations formulated by modern biological science is that of the continuity of life; the protoplasmic activity within each living body now on earth has continued without cessation from the remote beginnings of life on our planet, and from that period until the present no single organism has ever arisen save in the form of a bit of living protoplasm detached from a pre-existing portion; the eternal flame of life once kindled upon this earth has passed from organism to organism, and is still, going on existing and propagating, incarnated within the myriad animal and plant forms of everyday life.
Pavlov’s data on the two fundamental antagonistic nervous processes—stimulation and inhibition—and his profound generalizations regarding them, in particular, that these processes are parts of a united whole, that they are in a state of constant conflict and constant transition of the one to the other, and his views on the dominant role they play in the formation of the higher nervous activity—all those belong to the most established natural—scientific validation of the Marxist dialectal method. They are in complete accord with the Leninist concepts on the role of the struggle between opposites in the evolution, the motion of matter.
Science quickens and cultivates directly the faculty of observation, which in very many persons lies almost dormant through life, the power of accurate and rapid generalizations, and the mental habit of method and arrangement; it accustoms young persons to trace the sequence of cause and effect; it familiarizes then with a kind of reasoning which interests them, and which they can promptly comprehend; and it is perhaps the best corrective for that indolence which is the vice of half-awakened minds, and which shrinks from any exertion that is not, like an effort of memory, merely mechanical.
The essence of knowledge is generalization. That fire can be produced by rubbing wood in a certain way is a knowledge derived by generalization from individual experiences; the statement means that rubbing wood in this way will always produce fire. The art of discovery is therefore the art of correct generalization. ... The separation of relevant from irrelevant factors is the beginning of knowledge.
The increasing technicality of the terminology employed is also a serious difficulty. It has become necessary to learn an extensive vocabulary before a book in even a limited department of science can be consulted with much profit. This change, of course, has its advantages for the initiated, in securing precision and concisement of statement; but it tends to narrow the field in which an investigator can labour, and it cannot fail to become, in the future, a serious impediment to wide inductive generalisations.
The scientist, if he is to be more than a plodding gatherer of bits of information, needs to exercise an active imagination. The scientists of the past whom we now recognize as great are those who were gifted with transcendental imaginative powers, and the part played by the imaginative faculty of his daily life is as least as important for the scientist as it is for the worker in any other field—much more important than for most. A good scientist thinks logically and accurately when conditions call for logical and accurate thinking—but so does any other good worker when he has a sufficient number of well-founded facts to serve as the basis for the accurate, logical induction of generalizations and the subsequent deduction of consequences.
The trick in discovering evolutionary laws is the same as it is in discovering laws of physics or chemistry—namely, finding the right level of generalization to make prediction possible. We do not try to find a law that says when and where explosions will occur. We content ourselves with saying that certain sorts of compounds are explosive under the right conditions, and we predict that explosions will occur whenever those conditions are realized.
The true method of discovery is like the flight of an aeroplane. It starts from the ground of particular observation; it makes a flight in the thin air of imaginative generalization; and it again lands for renewed observation rendered acute by rational interpretation.
We lay down a fundamental principle of generalization by abstraction: The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features.