Any conception which is definitely and completely determined by means of a finite number of specifications, say by assigning a finite number of elements, is a mathematical conception. Mathematics has for its function to develop the consequences involved in the definition of a group of mathematical conceptions. Interdependence and mutual logical consistency among the members of the group are postulated, otherwise the group would either have to be treated as several distinct groups, or would lie beyond the sphere of mathematics.

Biological disciplines tend to guide research into certain channels. One consequence is that disciplines are apt to become parochial, or at least to develop blind spots, for example, to treat some questions as “interesting” and to dismiss others as “uninteresting.” As a consequence, readily accessible but unworked areas of genuine biological interest often lie in plain sight but untouched within one discipline while being heavily worked in another. For example, historically insect physiologists have paid relatively little attention to the behavioral and physiological control of body temperature and its energetic and ecological consequences, whereas many students of the comparative physiology of terrestrial vertebrates have been virtually fixated on that topic. For the past 10 years, several of my students and I have exploited this situation by taking the standard questions and techniques from comparative vertebrate physiology and applying them to insects. It is surprising that this pattern of innovation is not more deliberately employed.

Cauchy is mad, and there is no way of being on good terms with him, although at present he is the only man who knows how mathematics should be treated. What he does is excellent, but very confused…

Differences between individuals are the raw materials for evolutionary change and for the evolution of adaptations, yet of course most physiologists treat these differences as noise that is to be filtered out. From the standpoint of physiological ecology, the traditional emphasis of physiologists on central tendencies rather than on variance has some unhappy consequences. Variation is not just noise; it is also the stuff of evolution and a central attribute of living systems. The physiological differences between individuals in the same species or population, and also the patterns of variation in different groups, must not be ignored.

Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members. On the other hand, the mathematics of variable magnitudes—function theory or analysis—considers mathematical forms in their genesis. By writing the equation of the parabola, we express its law of generation, the law according to which the variable point moves. The path, produced before the eyes of the student by a point moving in accordance to this law, is the parabola.
If, then, Euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body, modern mathematics deals, as it were, with the living body, with growing and changing forms, and thus furnishes an insight, not only into nature as she is and appears, but also into nature as she generates and creates,—reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming. Thus modern mathematics bears the same relation to Euclidean mathematics that physiology or biology … bears to anatomy.

I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail.

I tell [medical students] that they are the luckiest persons on earth to be in medical school, and to forget all this worry about H.M.O.’s and keep your eye on helping the patient. It’s the best time ever to be a doctor because you can heal and treat conditions that were untreatable even a couple of years ago.

I’ve enjoyed life, but it would have been nice to treat my wife to dinner once in a while.

If we would indicate an idea … striving to remove the barriers which prejudice and limited views of every kind have erected among men, and to treat all mankind, without reference to religion, nation, or color, as one fraternity, one great community, fitted for the attainment of one object, the unrestrained development of the physical powers. This is the ultimate and highest aim of society.

In a large proportion of cases treated by physicians the disease is cured by nature, not by them. In a lesser, but not a small proportion, the disease is cured by nature in spite of them.

In abstract mathematical theorems, the approximation to absolute truth is perfect. … In physical science, on the contrary, we treat of the least quantities which are perceptible.

In the secondary schools mathematics should be a part of general culture and not contributory to technical training of any kind; it should cultivate space intuition, logical thinking, the power to rephrase in clear language thoughts recognized as correct, and ethical and esthetic effects; so treated, mathematics is a quite indispensable factor of general education in so far as the latter shows its traces in the comprehension of the development of civilization and the ability to participate in the further tasks of civilization.

It is difficult to know how to treat the errors of the age. If a man oppose them, he stands alone; if he surrender to them, they bring him neither joy nor credit.

It is not a case we are treating; it is a living, palpitating, alas, too often suffering fellow creature.

It took us only a few days to understand why we in the United States used so much energy; oil and gas were as cheap as dirt or water, and so they were treated like dirt or water.

Just as it will never be successfully challenged that the French language, progressively developing and growing more perfect day by day, has the better claim to serve as a developed court and world language, so no one will venture to estimate lightly the debt which the world owes to mathematicians, in that they treat in their own language matters of the utmost importance, and govern, determine and decide whatever is subject, using the word in the highest sense, to number and measurement.

Many quite nefarious ideologies pass for common sense. For decades of American history, it was common sense in some quarters for white people to own slaves and for women not to vote. … If common sense sometimes preserves the social status quo, and that status quo sometimes treats unjust social hierarchies as natural, it makes good sense on such occasions to find ways of challenging common sense.

Men cannot be treated as units in operations of political arithmetic because they behave like the symbols for zero and the infinite, which dislocate all mathematical operations.

Most students treat knowledge as a liquid to be swallowed rather than as a solid to be chewed, and then wonder why it provides so little nourishment.

Nature has no reverence towards life. Nature treats life as though it were the most valueless thing in the world. … Nature does not act by purposes.

Nevertheless, his [Dostoyevsky’s] personality retained sadistic traits in plenty, which show themselves in his irritability, his love of tormenting, and his intolerance even towards people he loved, and which appear also in the way in which, as an author, he treats his readers. Thus in little things he was a sadist towards others, and in bigger things a sadist towards himself, in fact a masochist—that is to say the mildest, kindliest, most helpful person possible.

No man treats a motorcar as foolishly as he treats another human being. When the car will not go, he does not attribute its annoying behavior to sin; he does not say, “You are a wicked motorcar, and I shall not give you any more petrol until you go.” He attempts to find out what is wrong and to set it right. An analogous way of treating human beings is, however, considered to be contrary to the truths of our holy religion.

Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols.

See in Nature the cylinder, the sphere, the cone.

The flights of the imagination which occur to the pure mathematician are in general so much better described in his formulas than in words, that it is not remarkable to find the subject treated by outsiders as something essentially cold and uninteresting— … the only successful attempt to invest mathematical reasoning with a halo of glory—that made in this section by Prof. Sylvester—is known to a comparative few, …

The ideas which these sciences, Geometry, Theoretical Arithmetic and Algebra involve extend to all objects and changes which we observe in the external world; and hence the consideration of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature, as Astronomy, Optics, and Mechanics. Such sciences are hence often termed Mixed Mathematics, the relations of space and number being, in these branches of knowledge, combined with principles collected from special observation; while Geometry, Algebra, and the like subjects, which involve no result of experience, are called Pure Mathematics.

The Internet treats censorship as a malfunction and routes around it.

The mathematic, then, is an art. As such it has its styles and style periods. It is not, as the layman and the philosopher (who is in this matter a layman too) imagine, substantially unalterable, but subject like every art to unnoticed changes form epoch to epoch. The development of the great arts ought never to be treated without an (assuredly not unprofitable) side-glance at contemporary mathematics.

The ocean is not just blank blue space but rather the habitat for amazing wildlife, and we have to take care how we use it. If we want to keep having the goods and services it provides, we have to treat it more carefully in terms of fishing and dumping.

The only way in which to treat the elements of an exact and rigorous science is to apply to them all the rigor and exactness possible.

The superior doctor prevents sickness; The mediocre doctor attends to impending sickness; The inferior doctor treats actual sickness.

The trees are man’s best friends; but man has treated them as his worst enemies. The history of our race may be said to be the history of warfare upon the tree world. But while man has seemed to be the victor, his victories have brought upon him inevitable disasters.

There is the immense sea of energy ... a multidimensional implicate order, ... the entire universe of matter as we generally observe it is to be treated as a comparatively small pattern of excitation. This excitation pattern is relatively autonomous and gives rise to approximately recurrent, stable separable projections into a three-dimensional explicate order of manifestation, which is more or less equivalent to that of space as we commonly experience it.

Treat people as if they were what they ought to be, and you help them to become what they are capable of being.

Treat the patient, not the X-ray

Treat your password like your toothbrush. Don’t let anybody else use it, and get a new one every six months.

When you treat a disease, first treat the mind.

You treat world history as a mathematician does mathematics, in which nothing but laws and formulae exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.