Continuous Quotes (38 quotes)

*Natura non facit saltum*or, Nature does not make leaps… If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).

A modern branch of mathematics, having achieved the art of dealing with the infinitely small, can now yield solutions in other more complex problems of motion, which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion, admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when dealing with separate elements of motion instead of examining continuous motion. In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable human wills, is continuous. To understand the laws of this continuous movement is the aim of history. … Only by taking an infinitesimally small unit for observation (the differential of history, that is, the individual tendencies of man) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.

Applied science, purposeful and determined, and pure science, playful and freely curious, continuously support and stimulate each other. The great nation of the future will be the one which protects the freedom of pure science as much as it encourages applied science.

Continuous as the stars that shine

And twinkle on the milky way,

They stretch’d in never-ending line

Along the margin of a bay:

Ten thousand saw I at a glance

Tossing their heads in sprightly dance.

And twinkle on the milky way,

They stretch’d in never-ending line

Along the margin of a bay:

Ten thousand saw I at a glance

Tossing their heads in sprightly dance.

Definition of Mathematics.—It has now become apparent that the traditional field of mathematics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a discussion as to the mere application of a word easily degenerates into the most fruitless logomachy. It is open to any one to use any word in any sense. But on the assumption that “mathematics” is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ “mathematics” in the general sense of the “science concerned with the logical deduction of consequences from the general premisses of all reasoning.”

Energy of the tides is continuously being dissipated at a rate whose order of magnitude is a billion horsepower!

Everything does not happen continuously at any one moment in the universe. Neither does everything happen everywhere in it.

Evolution is a theory of organic change, but it does not imply, as many people assume, that ceaseless flux is the irreducible state of nature and that structure is but a temporary incarnation of the moment. Change is more often a rapid transition between stable states than a continuous transformation at slow and steady rates. We live in a world of structure and legitimate distinction. Species are the units of nature’s morphology.

Heredity is the general expression of the periodicity of organic life. All generations belong to a continuous succession of waves, in which every single one resembles its predecessors and its followers.

If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.

If we survey the mathematical works of Sylvester, we recognize indeed a considerable abundance, but in contradistinction to Cayley—not a versatility toward separate fields, but, with few exceptions—a confinement to arithmetic-algebraic branches. …

The concept of

The concept of

*Function*of a continuous variable, the fundamental concept of modern mathematics, plays no role, is indeed scarcely mentioned in the entire work of Sylvester—Sylvester was combinatorist [combinatoriker].
In Darwin’s theory, you just have to substitute ‘mutations’ for his ‘slight accidental variations’ (just as quantum theory substitutes ‘quantum jump’ for ‘continuous transfer of energy’). In all other respects little change was necessary in Darwin’s theory.

In relativity, movement is continuous, causally determinate and well defined; while in quantum mechanics it is discontinuous, not causally determinate and not well defined.

It has been the final aim of Lie from the beginning to make progress in the theory of differential equations; as subsidiary to this may be regarded both his geometrical developments and the theory of continuous groups.

It is easy without any very profound logical analysis to perceive the difference between a succession of favorable deviations from the laws of chance, and on the other hand, the continuous and cumulative action of these laws. It is on the latter that the principle of Natural Selection relies.

Let us now declare the means whereby our understanding can rise to knowledge without fear of error. There are two such means: intuition and deduction. By intuition I mean not the varying testimony of the senses, nor the deductive judgment of imagination naturally extravagant, but the conception of an attentive mind so distinct and so clear that no doubt remains to it with regard to that which it comprehends; or, what amounts to the same thing, the self-evidencing conception of a sound and attentive mind, a conception which springs from the light of reason alone, and is more certain, because more simple, than deduction itself. …

It may perhaps be asked why to intuition we add this other mode of knowing, by deduction, that is to say, the process which, from something of which we have certain knowledge, draws consequences which necessarily follow therefrom. But we are obliged to admit this second step; for there are a great many things which, without being evident of themselves, nevertheless bear the marks of certainty if only they are deduced from true and incontestable principles by a continuous and uninterrupted movement of thought, with distinct intuition of each thing; just as we know that the last link of a long chain holds to the first, although we can not take in with one glance of the eye the intermediate links, provided that, after having run over them in succession, we can recall them all, each as being joined to its fellows, from the first up to the last. Thus we distinguish intuition from deduction, inasmuch as in the latter case there is conceived a certain progress or succession, while it is not so in the former; … whence it follows that primary propositions, derived immediately from principles, may be said to be known, according to the way we view them, now by intuition, now by deduction; although the principles themselves can be known only by intuition, the remote consequences only by deduction.

It may perhaps be asked why to intuition we add this other mode of knowing, by deduction, that is to say, the process which, from something of which we have certain knowledge, draws consequences which necessarily follow therefrom. But we are obliged to admit this second step; for there are a great many things which, without being evident of themselves, nevertheless bear the marks of certainty if only they are deduced from true and incontestable principles by a continuous and uninterrupted movement of thought, with distinct intuition of each thing; just as we know that the last link of a long chain holds to the first, although we can not take in with one glance of the eye the intermediate links, provided that, after having run over them in succession, we can recall them all, each as being joined to its fellows, from the first up to the last. Thus we distinguish intuition from deduction, inasmuch as in the latter case there is conceived a certain progress or succession, while it is not so in the former; … whence it follows that primary propositions, derived immediately from principles, may be said to be known, according to the way we view them, now by intuition, now by deduction; although the principles themselves can be known only by intuition, the remote consequences only by deduction.

Life is the continuous adjustment of internal relations to external relations.

Life is the twofold internal movement of composition and decomposition at once general and continuous.

MATHEMATICS … the general term for the various applications of mathematical thought, the traditional field of which is number and quantity. It has been usual to define mathematics as “the science of discrete and continuous magnitude.”

My life has been a continuous fulfillment of dreams. It appears that everything I saw and did has a new, and perhaps, more significant meaning, every time I see it. The earth is good. It is a privilege to live thereon.

My position is a naturalistic one; I see philosophy not as an a priori propaedeutic or groundwork for science, but as continuous with science. I see philosophy and science as in the same boat—a boat which, to revert to Neurath’s figure as I so often do, we can rebuild only at sea while staying afloat in it. There is no external vantage point, no first philosophy.

Now, we propose in the first place to show, that this law of organic progress is the law of all progress. Whether it be in the development of the Earth, in the development in Life upon its surface, in the development of Society, of Government, of Manufactures, of Commerce, of Language, Literature, Science, Art, this same evolution of the simple into the complex, through a process of continuous differentiation, holds throughout. From the earliest traceable cosmical changes down to the latest results of civilization, we shall find that the transformation of the homogeneous into the heterogeneous is that in which Progress essentially consists.

Our mind, by virtue of a certain finite, limited capability, is by no means capable of putting a question to Nature that permits a continuous series of answers. The observations, the individual results of measurements, are the answers of Nature to our discontinuous questioning.

Science is a speculative enterprise. The validity of a new idea and the significance of a new experimental finding are to be measured by the consequences—consequences in terms of other ideas and other experiments. Thus conceived, science is not a quest for certainty; it is rather a quest which is successful only to the degree that it is continuous.

Sometime in my early teens, I started feeling an inner urgency, ups and downs of excitement and frustration, caused by such unlikely occupations as reading Granville’s course of calculus ... I found this book in the attic of a friend’s apartment. Among other standard stuff, it contained the notorious epsilon-delta definition of continuous functions. After struggling with this definition for some time (it was the hot Crimean summer, and I was sitting in the shadow of a dusty apple tree), I got so angry that I dug a shallow grave for the book between the roots, buried it there, and left in disdain. Rain started in an hour. I ran back to the tree and exhumed the poor thing. Thus, I discovered that I loved it, regardless.

The basic ideas and simplest facts of set-theoretic topology are needed in the most diverse areas of mathematics; the concepts of topological and metric spaces, of compactness, the properties of continuous functions and the like are often indispensable.

The difference between an ordinary mind and the mind of Newton consists principally in this, that the one is capable of a more continuous attention than the other,—that a Newton is able, without fatigue, to connect inference with inference in one long series towards a determinate end; while the man of inferior capacity is soon obliged to break or let full the thread which lie had begun to spin.

The fact that Science walks forward on two feet, namely theory and experiment, is nowhere better illustrated than in the two fields for slight contributions to which you have done me the great honour of awarding the the Nobel Prize in Physics for the year 1923. Sometimes it is one foot that is put forward first, sometimes the other, but continuous progress is only made by the use of both—by theorizing and then testing, or by finding new relations in the process of experimenting and then bringing the theoretical foot up and pushing it on beyond, and so on in unending alterations.

The following theorem can be found in the work of Mr. Cauchy: If the various terms of the series

sin

is discontinuous at each value (2

*u*_{0}+*u*_{1}+*u*_{2}+... are continuous functions,… then the sum*s*of the series is also a continuous function of*x*. But it seems to me that this theorem admits exceptions. For example the seriessin

*x*- (1/2)sin 2*x*+ (1/3)sin 3*x*- …is discontinuous at each value (2

*m*+ 1)π of*x*,…
The handling of our forests as a continuous, renewable resource means permanent employment and stability to our country life. The forests are also needed for mitigating extreme climatic fluctuations, holding the soil on the slopes, retaining the moisture in the ground, and controlling the equable flow of water in our streams.

The individual on his own is stable only so long as he is possessed of self-esteem. The maintenance of self-esteem is a continuous task which taxes all of the individual’s powers and inner resources. We have to prove our worth and justify our existence anew each day. When, for whatever reason, self-esteem is unattainable, the autonomous individual becomes a highly explosive entity. He turns away from an unpromising self and plunges into the pursuit of pride—the explosive substitute for self-esteem. All social disturbances and upheavals have their roots in crises of individual self-esteem, and the great endeavor in which the masses most readily unite is basically a search for pride.

The long-range trend toward federal regulation, which found its beginnings in the Interstate Commerce Act of 1887 and the Sherman Act of 1890, which was quickened by a large number of measures in the Progressive era, and which has found its consummation in our time, was thus at first the response of a predominantly individualistic public to the uncontrolled and starkly original collectivism of big business. In America the growth of the national state and its regulative power has never been accepted with complacency by any large part of the middle-class public, which has not relaxed its suspicion of authority, and which even now gives repeated evidence of its intense dislike of statism. In our time this growth has been possible only under the stress of great national emergencies, domestic or military, and even then only in the face of continuous resistance from a substantial part of the public. In the Progressive era it was possible only because of widespread and urgent fear of business consolidation and private business authority. Since it has become common in recent years for ideologists of the extreme right to portray the growth of statism as the result of a sinister conspiracy of collectivists inspired by foreign ideologies, it is perhaps worth emphasizing that the first important steps toward the modern organization of society were taken by arch-individualists—the tycoons of the Gilded Age—and that the primitive beginning of modern statism was largely the work of men who were trying to save what they could of the eminently native Yankee values of individualism and enterprise.

The object of pure mathematics is those relations which may be conceptually established among any conceived elements whatsoever by assuming them contained in some ordered manifold; the law of order of this manifold must be subject to our choice; the latter is the case in both of the only conceivable kinds of manifolds, in the discrete as well as in the continuous.

The one who stays in my mind as the ideal man of science is, not Huxley or Tyndall, Hooker or Lubbock, still less my friend, philosopher and guide Herbert Spencer, but Francis Galton, whom I used to observe and listen to—I regret to add, without the least reciprocity—with rapt attention. Even to-day. I can conjure up, from memory’s misty deep, that tall figure with its attitude of perfect physical and mental poise; the clean-shaven face, the thin, compressed mouth with its enigmatical smile; the long upper lip and firm chin, and, as if presiding over the whole personality of the man, the prominent dark eyebrows from beneath which gleamed, with penetrating humour, contemplative grey eyes. Fascinating to me was Francis Galton’s all-embracing but apparently impersonal beneficence. But, to a recent and enthusiastic convert to the scientific method, the most relevant of Galton’s many gifts was the unique contribution of three separate and distinct processes of the intellect; a continuous curiosity about, and rapid apprehension of individual facts, whether common or uncommon; the faculty for ingenious trains of reasoning; and, more admirable than either of these, because the talent was wholly beyond my reach, the capacity for correcting and verifying his own hypotheses, by the statistical handling of masses of data, whether collected by himself or supplied by other students of the problem.

The progress of synthesis, or the building up of natural materials from their constituent elements, proceeds apace. Even some of the simpler albuminoids, a class of substances of great importance in the life process, have recently been artificially prepared. ... Innumerable entirely new compounds have been produced in the last century. The artificial dye-stuffs, prepared from materials occurring in coal-tar, make the natural colours blush. Saccharin, which is hundreds of times sweeter than sugar, is a purely artificial substance. New explosives, drugs, alloys, photographic substances, essences, scents, solvents, and detergents are being poured out in a continuous stream.

The second [argument about motion] is the so-called Achilles, and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

*Statement of the Achilles and the Tortoise paradox in the relation of the discrete to the continuous.; perhaps the earliest example of the reductio ad absurdum method of proof.*
— Zeno

The third [argument of motion is] to the effect that the flying arrow is at rest, which result follows from the assumption that time is composed of moments: if this assumption is not granted, the conclusion will not follow.

*Arrow paradox*
— Zeno

World-wide practice of Conservation and the fair and continued access by all nations to the resources they need are the two indispensable foundations of continuous plenty and of permanent peace.