Generalize Quotes (19 quotes)
Man muss immer generalisieren
One must always generalize.
One must always generalize.
Another characteristic of mathematical thought is that it can have no success where it cannot generalize.
Before you generalize, formalize, and axiomatize there must be mathematical substance.
But it is just this characteristic of simplicity in the laws of nature hitherto discovered which it would be fallacious to generalize, for it is obvious that simplicity has been a part cause of their discovery, and can, therefore, give no ground for the supposition that other undiscovered laws are equally simple.
Every science begins by accumulating observations, and presently generalizes these empirically; but only when it reaches the stage at which its empirical generalizations are included in a rational generalization does it become developed science.
Having discovered … by observation and comparison that certain objects agree in certain respects, we generalise the qualities in which they coincide,—that is, from a certain number of individual instances we infer a general law; we perform an act of Induction. This induction is erroneously viewed as analytic; it is purely a synthetic process.
How can a modern anthropologist embark upon a generalization with any hope of arriving at a satisfactory conclusion? By thinking of the organizational ideas that are present in any society as a mathematical pattern.
In a word, to get the law from experiment, it is necessary to generalize; this is a necessity imposed upon the most circumspect observer.
Induction is the process of generalizing from our known and limited experience, and framing wider rules for the future than we have been able to test fully. At its simplest, then, an induction is a habit or an adaptation—the habit of expecting tomorrow’s weather to be like today’s, the adaptation to the unwritten conventions of community life.
It is not Cayley’s way to analyze concepts into their ultimate elements. … But he is master of the empirical utilization of the material: in the way he combines it to form a single abstract concept which he generalizes and then subjects to computative tests, in the way the newly acquired data are made to yield at a single stroke the general comprehensive idea to the subsequent numerical verification of which years of labor are devoted. Cayley is thus the natural philosopher among mathematicians.
Man is made for science; he reasons from effects to causes, and from causes to effects; but he does not always reason without error. In reasoning, therefore, from appearances which are particular, care must be taken how we generalize; we should be cautious not to attribute to nature, laws which may perhaps be only of our own invention.
Mathematics is a science of Observation, dealing with reals, precisely as all other sciences deal with reals. It would be easy to show that its Method is the same: that, like other sciences, having observed or discovered properties, which it classifies, generalises, co-ordinates and subordinates, it proceeds to extend discoveries by means of Hypothesis, Induction, Experiment and Deduction.
No generalizing beyond the data, no theory. No theory, no insight. And if no insight, why do research.
One-story intellects, two-story intellects, three-story intellects with skylights. All fact-collectors, who have no aim beyond their facts, are one-story men. Two-story men compare, reason, generalize, using the labors of the fact-collectors as well as their own. Three-story men idealize, imagine, predict; their best illumination comes from above, through the skylight. There are minds with large ground-floors, that can store an infinite amount of knowledge; some librarians, for instance, who know enough of books to help other people, without being able to make much other use of their knowledge, have intellects of this class. Your great working lawyer has two spacious stories; his mind is clear, because his mental floors are large, and he has room to arrange his thoughts so that lie can get at them,—facts below, principles above, and all in ordered series; poets are often narrow below, incapable of clear statement, and with small power of consecutive reasoning, but full of light, if sometimes rather bare of furniture, in the attics.
Random search for data on ... off-chance is hardly scientific. A questionnaire on 'Intellectual Immoralities' was circulated by a well-known institution. 'Intellectual Immorality No. 4' read: 'Generalizing beyond one's data'. [Wilder Dwight] Bancroft asked whether it would not be more correct to word question no. 4 'Not generalizing beyond one's data.'
The book [Future of an Illusion] testifies to the fact that the genius of experimental science is not necessarily joined with the genius of logic or generalizing power.
There are, at present, fundamental problems in theoretical physics … the solution of which … will presumably require a more drastic revision of our fundmental concepts than any that have gone before. Quite likely, these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. The theoretical worker in the future will, therefore, have to proceed in a more direct way. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities.
At age 28.
At age 28.
Thinking is merely the comparing of ideas, discerning relations of likeness and of difference between ideas, and drawing inferences. It is seizing general truths on the basis of clearly apprehended particulars. It is but generalizing and particularizing. Who will deny that a child can deal profitably with sequences of ideas like: How many marbles are 2 marbles and 3 marbles? 2 pencils and 3 pencils? 2 balls and 3 balls? 2 children and 3 children? 2 inches and 3 inches? 2 feet and 3 feet? 2 and 3? Who has not seen the countenance of some little learner light up at the end of such a series of questions with the exclamation, “Why it’s always that way. Isn’t it?” This is the glow of pleasure that the generalizing step always affords him who takes the step himself. This is the genuine life-giving joy which comes from feeling that one can successfully take this step. The reality of such a discovery is as great, and the lasting effect upon the mind of him that makes it is as sure as was that by which the great Newton hit upon the generalization of the law of gravitation. It is through these thrills of discovery that love to learn and intellectual pleasure are begotten and fostered. Good arithmetic teaching abounds in such opportunities.
To generalize is to be an idiot. To particularize is the alone distinction of merit. General knowledges are those knowledges that idiots possess.