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Who said: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
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Home > Category Index for Science Quotations > Category Index C > Category: Continuous

Continuous Quotes (82 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).
From Benoit B. Mandelbrot and Richard Hudson, The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward (2004,2010), 85-86.
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A closer look at the course followed by developing theory reveals for a start that it is by no means as continuous as one might expect, but full of breaks and at least apparently not along the shortest logical path. Certain methods often afforded the most handsome results only the other day, and many might well have thought that the development of science to infinity would consist in no more than their constant application. Instead, on the contrary, they suddenly reveal themselves as exhausted and the attempt is made to find other quite disparate methods. In that event there may develop a struggle between the followers of the old methods and those of the newer ones. The former's point of view will be termed by their opponents as out-dated and outworn, while its holders in turn belittle the innovators as corrupters of true classical science.
In 'On the Development of the Methods of Theoretical Physics in Recent Times', Populäre Schriften, Essay 14. Address (22 Sep 1899) to the Meeting of Natural Scientists at Munich. Collected in Brian McGuinness (ed.), Ludwig Boltzmann: Theoretical Physics and Philosophical Problems, Selected Writings (1974), 79.
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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.
War and Peace (1869), Book 11, Chap. 1.
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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.
From a radio talk, collected in Warren Weaver (ed.), The Scientists Speak (1946)
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As for me ... I would much rather be a perfected ape than a degraded Adam. Yes, if it is shown to me that my humble ancestors were quadrupedal animals, arboreal herbivores, brothers or cousins of those who were also the ancestors of monkeys and apes, far from blushing in shame for my species because of its genealogy and parentage, I will be proud of all that evolution has accomplished, of the continuous improvement which takes us up to the highest order, of the successive triumphs that have made us superior to all of the other species ... the splendid work of progress.
I will conclude in saying: the fixity of species is almost impossible, it contradicts the mode of succession and of the distribution of species in the sequence of extant and extinct creatures. It is therefore extremely likely that species are variable and are subject to evolution. But the causes, the mechanisms of this evolution are still unknown.
'Discussion sur la Machoire Humaine de la Naulette (Belgique)', Bulletin de la Societé d'Anthropologie de Paris, 2nd Series, I (1866), 595. Trans. Erik Trinkaus and Pat Shipman, The Neanderthals: Changing the Image of Mankind (1993), 103-4.
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At the moment I am occupied by an investigation with Kirchoff which does not allow us to sleep. Kirchoff has made a totally unexpected discovery, inasmuch as he has found out the cause for the dark lines in the solar spectrum and can produce these lines artificially intensified both in the solar spectrum and in the continuous spectrum of a flame, their position being identical with that of Fraunhofer’s lines. Hence the path is opened for the determination of the chemical composition of the Sun and the fixed stars.
Letter to H.E. Roscoe (Nov 1859). In The Life and Experiences of Sir Henry Enfield Roscoe (1906), 71.
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At the moment I am occupied by an investigation with Kirchoff which does not allow us to sleep. Kirchoff has made a totally unexpected discovery, inasmuch as he has found out the cause for the dark lines in the solar spectrum and can produce these lines artificially intensified both in the solar spectrum and in the continuous spectrum of a flame, their position being identical with that of Fraunhofer’s lines. Hence the path is opened for the determination of the chemical composition of the Sun and the fixed stars.
Letter to H.E. Roscoe (Nov 1859). In The Life and Experiences of Sir Henry Enfield Roscoe (1906), 81.
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Consciousness is not sharply defined, but fades into sub-consciousness; and beyond that we must postulate something indefinite but yet continuous with our mental nature. This I take it be the world-stuff.
From Gifford Lecture, Edinburgh, (1927), 'Reality', collected in The Nature of the Physical World (1928), 280.
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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.
Second verse of poem, 'I Wandered Lonely as a Cloud', In Poems: Including Lyrical Ballads: In two Volumes (1815), Vol. 1, 328.
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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.”
In article 'Mathematics', Encyclopedia Britannica (1911, 11th ed.), Vol. 17, 880. In the 2006 DVD edition of the encyclopedia, the definition of mathematics is given as “The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects.” [Premiss is a variant form of “premise”. —Webmaster]
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Energy of the tides is continuously being dissipated at a rate whose order of magnitude is a billion horsepower!
In The Tides: Pulse of the Earth (1968), 133.
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Everything does not happen continuously at any one moment in the universe. Neither does everything happen everywhere in it.
In Teilhard de Chardin and Bernard Wall (trans.), The Phenomenon of Man (1959), 49. Originally published in French as Le Phénomene Humain (1955).
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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.
…...
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First Law
In every animal which has not passed the limit of its development, a more frequent and continuous use of any organ gradually strengthens, develops and enlarges that organ, and gives it a power proportional to the length of time it has been so used; while the permanent disuse of any organ imperceptibly weakens and deteriorates it, and progressively diminishes its functional capacity, until it finally disappears.
Philosophie Zoologique (1809), Vol. 1, 235, trans. Hugh Elliot (1914), 113.
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From packaging materials, through fibers, foams and surface coatings, to continuous extrusions and large-scale moldings, plastics have transformed almost every aspect of life. Without them, much of modern medicine would be impossible and the consumer electronics and computer industries would disappear. Plastic sewage and water pipes alone have made an immeasurable contribution to public health worldwide.
'Plastics—No Need To Apologize', Trends in Polymer Science (Jun 1996), 4, 172. In Paul C. Painter and Michael M. Coleman, Essentials of Polymer Science and Engineering (2008), 21.
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Geology has its peculiar difficulties, from which all other sciences are exempt. Questions in chemistry may be settled in the laboratory by experiment. Mathematical and philosophical questions may be discussed, while the materials for discussion are ready furnished by our own intellectual reflections. Plants, animals and minerals, may be arranged in the museum, and all questions relating to their intrinsic principles may be discussed with facility. But the relative positions, the shades of difference, the peculiar complexions, whether continuous or in subordinate beds, are subjects of enquiry in settling the character of rocks, which can be judged of while they are in situ only.
A Geological and Agricultural Survey of the District Adjoining the Erie Canal (1824), 8.
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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.
In 'On the Principles of Animal Morphology', Proceedings of the Royal Society of Edinburgh (1888), 15, 295. Original as Letter to Mr John Murray, communicated to the Society by Professor Sir William Turner. Page given as in collected volume published 1889.
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I have read somewhere that the resistance offered by a wire ... is affected by the tension of the wire. If this is so, a continuous current of electricity passed through a vibrating wire should meet with a varying resistance, and hence a pulsatory action should be induced in the current ... [corresponding] in amplitude, as well as in rate of movement, to the vibrations of the string ... [Thus] the timbre of a sound [a quality essential to intelligible speech] could be transmitted ... [and] the strength of the current can be increased ad libitum without destroying the relative intensities of the vibrations.
Letter to Gardiner Greene Hubbard (4 May 1875), 3-4. Bell Papers, Library of Congress.
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If a small animal and a lighted candle be placed in a closed flask, so that no air can enter, in a short time the candle will go out, nor will the animal long survive. ... The animal is not suffocated by the smoke of the candle. ... The reason why the animal can live some time after the candle has gone out seems to be that the flame needs a continuous rapid and full supply of nitro-aereal particles. ... For animals, a less aereal spirit is sufficient. ... The movements of the lungs help not a little towards sucking in aereal particles which may remain in said flask and towards transferring them to the blood of the animal.
Remarking (a hundred years before Priestley identified oxygen) that a component of the air is taken into the blood.
Quoted in William Stirling, Some Apostles of Physiology (1902), 45.
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If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.
As quoted, without citation, in Eric Temple Bell, Men of Mathematics (1945), 254.
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If the 'Principle of Relativity' in an extreme sense establishes itself, it seems as if even Time would become discontinuous and be supplied in atoms, as money is doled out in pence or centimes instead of continuously;—in which case our customary existence will turn out to be no more really continuous than the events on a kinematograph screen;—while that great agent of continuity, the Ether of Space, will be relegated to the museum of historical curiosities.
Continuity: The Presidential Address to the British Association (1913), 40-41.
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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 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 Mathematische Annalen (1898), Bd.50, 134-135. As quoted and cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 173.
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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.
…...
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In its earliest development knowledge is self-sown. Impressions force themselves upon men’s senses whether they will or not, and often against their will. The amount of interest in which these impressions awaken is determined by the coarser pains and pleasures which they carry in their train or by mere curiosity; and reason deals with the materials supplied to it as far as that interest carries it, and no further. Such common knowledge is rather brought than sought; and such ratiocination is little more than the working of a blind intellectual instinct. It is only when the mind passes beyond this condition that it begins to evolve science. When simple curiosity passes into the love of knowledge as such, and the gratification of the æsthetic sense of the beauty of completeness and accuracy seems more desirable that the easy indolence of ignorance; when the finding out of the causes of things becomes a source of joy, and he is accounted happy who is successful in the search, common knowledge passes into what our forefathers called natural history, whence there is but a step to that which used to be termed natural philosophy, and now passes by the name of physical science.
In this final state of knowledge the phenomena of nature are regarded as one continuous series of causes and effects; and the ultimate object of science is to trace out that series, from the term which is nearest to us, to that which is at the farthest limit accessible to our means of investigation.
The course of nature as it is, as it has been, and as it will be, is the object of scientific inquiry; whatever lies beyond, above, or below this is outside science. But the philosopher need not despair at the limitation on his field of labor; in relation to the human mind Nature is boundless; and, though nowhere inaccessible, she is everywhere unfathomable.
The Crayfish: an Introduction to the Study of Zoölogy (1880), 2-3. Excerpted in Popular Science (Apr 1880), 16, 789-790.
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In relativity, movement is continuous, causally determinate and well defined; while in quantum mechanics it is discontinuous, not causally determinate and not well defined.
In Wholeness and the Implicate Order (1980), xv.
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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.
In Lectures on Mathematics (1911), 24.
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It is clear, then, that though there may be countless instances of the perishing of unmoved movers, and though many things that move themselves perish and are succeeded by others that come into being, and though one thing that is unmoved moves one thing while another moves another, nevertheless there is something that comprehends them all, and that as something apart from each one of them, and this it is that is the cause of the fact that some things are and others are not and of the continuous process of change; and this causes the motion of the other movers, while they are the causes of the motion of other things. Motion, then, being eternal, the first mover, if there is but one, will be eternal also; if there are more than one, there will be a plurality of such eternal movers.
Aristotle
Physics, 258b, 32-259a, 8. In Jonathan Barnes (ed.), The Complete Works of Aristotle (1984), Vol. 1, 432.
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It is difficult to conceive a grander mass of vegetation:—the straight shafts of the timber-trees shooting aloft, some naked and clean, with grey, pale, or brown bark; others literally clothed for yards with a continuous garment of epiphytes, one mass of blossoms, especially the white Orchids Caelogynes, which bloom in a profuse manner, whitening their trunks like snow. More bulky trunks were masses of interlacing climbers, Araliaceae, Leguminosae, Vines, and Menispermeae, Hydrangea, and Peppers, enclosing a hollow, once filled by the now strangled supporting tree, which has long ago decayed away. From the sides and summit of these, supple branches hung forth, either leafy or naked; the latter resembling cables flung from one tree to another, swinging in the breeze, their rocking motion increased by the weight of great bunches of ferns or Orchids, which were perched aloft in the loops. Perpetual moisture nourishes this dripping forest: and pendulous mosses and lichens are met with in profusion.
Himalayan Journals (1854), vol. 1, 110-1.
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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.
In The Genetical Theory of Natural Selection (1930), 37. Reprinted as A Complete Variorum Edition (2003), 37.
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It is not only a decided preference for synthesis and a complete denial of general methods which characterizes the ancient mathematics as against our newer Science [modern mathematics]: besides this extemal formal difference there is another real, more deeply seated, contrast, which arises from the different attitudes which the two assumed relative to the use of the concept of variability. For while the ancients, on account of considerations which had been transmitted to them from the Philosophie school of the Eleatics, never employed the concept of motion, the spatial expression for variability, in their rigorous system, and made incidental use of it only in the treatment of phonoromically generated curves, modern geometry dates from the instant that Descartes left the purely algebraic treatment of equations and proceeded to investigate the variations which an algebraic expression undergoes when one of its variables assumes a continuous succession of values.
In 'Untersuchungen über die unendlich oft oszillierenden und unstetigen Functionen', Ostwald’s Klassiker der exacten Wissenschaften (1905), No. 153, 44-45. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 115. From the original German, “Nicht allein entschiedene Vorliebe für die Synthese und gänzliche Verleugnung allgemeiner Methoden charakterisiert die antike Mathematik gegenüber unserer neueren Wissenschaft; es gibt neben diesem mehr äußeren, formalen, noch einen tiefliegenden realen Gegensatz, welcher aus der verschiedenen Stellung entspringt, in welche sich beide zu der wissenschaftlichen Verwendung des Begriffes der Veränderlichkeit gesetzt haben. Denn während die Alten den Begriff der Bewegung, des räumlichen Ausdruckes der Veränderlichkeit, aus Bedenken, die aus der philosophischen Schule der Eleaten auf sie übergegangen waren, in ihrem strengen Systeme niemals und auch in der Behandlung phoronomisch erzeugter Kurven nur vorübergehend verwenden, so datiert die neuere Mathematik von dem Augenblicke, als Descartes von der rein algebraischen Behandlung der Gleichungen dazu fortschritt, die Größenveränderungen zu untersuchen, welche ein algebraischer Ausdruck erleidet, indem eine in ihm allgemein bezeichnete Größe eine stetige Folge von Werten durchläuft.”
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It would be our worst enemy who would wish us to live only on the glories of the past and die off from the face of the earth in sheer passivity. By continuous achievement alone we can justify our great ancestry. We do not honour our ancestors by the false claim that they are omniscient and had nothing more to learn.
From 'Sir J.C. Bose’s Address', Benares Hindu University 1905-1935 (1936), 423. Collected in J. Lourdusamy, Science and National Consciousness in Bengal: 1870-1930 (2004), 106.
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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.
In Rules for the Direction of the Mind, Philosophy of Descartes. [Torrey] (1892), 64-65.
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Life is a partial, continuous, progressive, multiform and conditionally interactive, self-realization of the potentialities of atomic electron states.
In The Origin of Life (1967).
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Life is the continuous adjustment of internal relations to external relations.
In The Principles of Biology (1862), Vol. 1, 84.
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Life is the twofold internal movement of composition and decomposition at once general and continuous.
In The American Journal of the Medical Sciences (1858), 152.
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Mathematical knowledge is not—as all Cambridge men are surely aware—the result of any special gift. It is merely the development of those conceptions of form and number which every human being possesses; and any person of average intellect can make himself a fair mathematician if he will only pay continuous attention; in plain English, think enough about the subject.
'Science', a lecture delivered at the Royal Institution. The Works of Charles Kingsley (1880), 241.
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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.”
Opening statement in article 'Mathematics', Encyclopedia Britannica (1911, 11th ed.), Vol. 17, 878. Whitehead then indicated this was an inadequate definition, which he then discussed at length and tried to give an improved definition later in the article. See the quote beginning “Definition of Mathematics…” on the Alfred North Whitehead Quotes page on this website.
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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.
In The National Gardener (1952?), 7.
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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.
Ontological Relativity and Other Essays, pp. 126-127, Columbia University Press (1969).
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Nature being capricious and taking pleasure in creating and producing a continuous sucession of lives and forms because she knows that they serve to increase her terrestrial substance, is more ready and swift in her creating than time is in destroying, and therefore she has ordained that many animals shall serve as food one for the other; and as this does not satisfy her desire she sends forth frequently certain noisome and pestilential vapours and continual plagues upon the vast accumulations and herds of animals and especially upon human beings who increase very rapidly because other animals do not feed upon them.
'Philosophy', in The Notebooks of Leonardo da Vinci, trans. E. MacCurdy (1938), Vol. 1 80.
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Nature proceeds little by little from things lifeless to animal life in such a way that it is impossible to determine the exact line of demarcation, nor on which side thereof an intermediate form should lie. Thus, next after lifeless things comes the plant, and of plants one will differ from another as to its amount of apparent vitality; and, in a word, the whole genus of plants, whilst it is devoid of life as compared with an animal, is endowed with life as compared with other corporeal entities. Indeed, as we just remarked, there is observed in plants a continuous scale of ascent towards the animal. So, in the sea, there are certain objects concerning which one would be at a loss to determine whether they be animal or vegetable. For instance, certain of these objects are fairly rooted, and in several cases perish if detached.
Aristotle
History of Animals, 588b, 4-14. In Jonathan Barnes (ed.) The Complete Works of Aristotle (1984), Vol. 1, 922.
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Nature, displayed in its full extent, presents us with an immense tableau, in which all the order of beings are each represented by a chain which sustains a continuous series of objects, so close and so similar that their difference would be difficult to define. This chain is not a simple thread which is only extended in length, it is a large web or rather a network, which, from interval to interval, casts branches to the side in order to unite with the networks of another order.
'Les Oiseaux Qui Ne Peuvent Voler', Histoire Naturelle des Oiseaux (1770), Vol. I, 394. Trans. Phillip R. Sloan.
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Nature, the parent of all things, designed the human backbone to be like a keel or foundation. It is because we have a backbone that we can walk upright and stand erect. But this was not the only purpose for which Nature provided it; here, as elsewhere, she displayed great skill in turning the construction of a single member to a variety of different uses.
It Provides a Path for the Spinal Marrow, Yet is Flexible.
Firstly, she bored a hole through the posterior region of the bodies of all the vertebrae, thus fashioning a suitable pathway for the spinal marrow which would descend through them.
Secondly, she did not make the backbone out of one single bone with no joints. Such a unified construction would have afforded greater stability and a safer seat for the spinal marrow since, not having joints, the column could not have suffered dislocations, displacements, or distortions. If the Creator of the world had paid such attention to resistance to injury and had subordinated the value and importance of all other aims in the fabric of parts of the body to this one, he would certainly have made a single backbone with no joints, as when someone constructing an animal of wood or stone forms the backbone of one single and continuous component. Even if man were destined only to bend and straighten his back, it would not have been appropriate to construct the whole from one single bone. And in fact, since it was necessary that man, by virtue of his backbone, be able to perform a great variety of movements, it was better that it be constructed from many bones, even though as a result of this it was rendered more liable to injury.
From De Humani Corporis Fabrica Libri Septem: (1543), Book I, 57-58, as translated by William Frank Richardson, in 'Nature’s Skill in Creating a Backbone to Hold Us Erect', On The Fabric of the Human Body: Book I: The Bones and Cartilages (1998), 138.
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Nothing is accomplished all at once, and it is one of my great maxims, and one of the most completely verified, that Nature makes no leaps: a maxim which I have called the law of continuity.
[Referring to the gradual nature of all change from an initial state, through a continuous series of intermediate stages, to a final state.]
Letter to de Volder, in Nouveaux Essais, New Essays (1703) translated by A. G. Langley (3rd ed. 1949). In Frederick Copleston, A History of Philosophy (1999), 292.
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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.
'Progress: Its Law and Cause', Westminster Review (1857), 67, 446-7.
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October 9, 1863
Always, however great the height of the balloon, when I have seen the horizon it has roughly appeared to be on the level of the car though of course the dip of the horizon is a very appreciable quantity or the same height as the eye. From this one might infer that, could the earth be seen without a cloud or anything to obscure it, and the boundary line of the plane approximately the same height as the eye, the general appearance would be that of a slight concavity; but I have never seen any part of the surface of the earth other than as a plane.
Towns and cities, when viewed from the balloon are like models in motion. I shall always remember the ascent of 9th October, 1863, when we passed over London about sunset. At the time when we were 7,000 feet high, and directly over London Bridge, the scene around was one that cannot probably be equalled in the world. We were still so low as not to have lost sight of the details of the spectacle which presented itself to our eyes; and with one glance the homes of 3,000,000 people could be seen, and so distinct was the view, that every large building was easily distinguishable. In fact, the whole of London was visible, and some parts most clearly. All round, the suburbs were also very distinct, with their lines of detached villas, imbedded as it were in a mass of shrubs; beyond, the country was like a garden, its fields, well marked, becoming smaller and smaller as the eye wandered farther and farther away.
Again looking down, there was the Thames, throughout its whole length, without the slightest mist, dotted over its winding course with innumerable ships and steamboats, like moving toys. Gravesend was visible, also the mouth of the Thames, and the coast around as far as Norfolk. The southern shore of the mouth of the Thames was not so clear, but the sea beyond was seen for many miles; when at a higher elevation, I looked for the coast of France, but was unable to see it. On looking round, the eye was arrested by the garden-like appearance of the county of Kent, till again London claimed yet more careful attention.
Smoke, thin and blue, was curling from it, and slowly moving away in beautiful curves, from all except one part, south of the Thames, where it was less blue and seemed more dense, till the cause became evident; it was mixed with mist rising from the ground, the southern limit of which was bounded by an even line, doubtless indicating the meeting of the subsoils of gravel and clay. The whole scene was surmounted by a canopy of blue, everywhere free from cloud, except near the horizon, where a band of cumulus and stratus extended all round, forming a fitting boundary to such a glorious view.
As seen from the earth, the sunset this evening was described as fine, the air being clear and the shadows well defined; but, as we rose to view it and its effects, the golden hues increased in intensity; their richness decreased as the distance from the sun increased, both right and left; but still as far as 90º from the sun, rose-coloured clouds extended. The remainder of the circle was completed, for the most part, by pure white cumulus of well-rounded and symmetrical forms.
I have seen London by night. I have crossed it during the day at the height of four miles. I have often admired the splendour of sky scenery, but never have I seen anything which surpassed this spectacle. The roar of the town heard at this elevation was a deep, rich, continuous sound the voice of labour. At four miles above London, all was hushed; no sound reached our ears.
Travels in the Air (1871), 99-100.
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On the whole, at least in the author's experience, the preparation of species-specific antiserum fractions and the differentiation of closely related species with precipitin sera for serum proteins does not succeed so regularly as with agglutinins and lysins for blood cells. This may be due to the fact that in the evolutional scale the proteins undergo continuous variations whereas cell antigens are subject to sudden changes not linked by intermediary stages.
The Specificity of Serological Reactions (1936), 12-3.
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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.
…...
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Physical changes take place continuously, while chemical changes take place discontinuously. Physics deals chiefly with continuous varying quantities, while chemistry deals chiefly with whole numbers.
Treatise on Thermodynamics (1897), trans. Alexander Ogg (1903), 22, footnote.
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Science advances, not by the accumulation of new facts … but by the continuous development of new concepts.
In A Dictionary of Scientific Quotations by Alan L. Mackay (1991).
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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.
In Science and Common Sense (1951), 25-26.
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Science is intimately integrated with the whole social structure and cultural tradition. They mutually support one other—only in certain types of society can science flourish, and conversely without a continuous and healthy development and application of science such a society cannot function properly.
The Social System (1951, 1977), Chap. 8, 111. As a functionalist, Parsons argued that social practices had to be studied in terms of their function in maintaining society.
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Some of Feynman’s ideas about cosmology have a modern ring. A good example is his attitude toward the origin of matter. The idea of continuous matter creation in the steady state cosmology does not seriously offend him (and he notes … that the big bang cosmology has a problem just as bad, to explain where all the matter came from in the beginning). … He emphasizes that the total energy of the universe could really be zero, and that matter creation is possible because the rest energy of the matter is actually canceled by its gravitational potential energy. “It is exciting to think that it costs nothing to create a new particle, …”
In John Preskill and Kip S. Thorne, 'Foreword to Feynman Lectures on Gravitation' (15 May 1995). Feynman delivered his lectures in 1962–63.
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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.
'Mathematics as Profession and vocation', in V. Arnold et al. (eds.), Mathematics: Frontiers and Perspectives (2000), 153. Reprinted in Mathematics as Metaphor: Selected Essays of Yuri I. Manin (2007), 79.
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The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new, but to the continuous evolution of zoologic types which develop ceaselessly and end by becoming unrecognisable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past. One must not think then that the old-fashioned theories have been sterile and vain.
The Value of Science (1905), in The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method(1946), trans. by George Bruce Halsted, 208.
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The alternative to the Big Bang is not, in my opinion, the steady state; it is instead the more general theory of continuous creation. Continuous creation can occur in bursts and episodes. These mini-bangs can produce all the wonderful element-building that Fred Hoyle discovered and contributed to cosmology. This kind of element and galaxy formation can take place within an unbounded, non-expanding universe. It will also satisfy precisely the Friedmann solutions of general relativity. It can account very well for all the facts the Big Bang explains—and also for those devastating, contradictory observations which the Big Bang must, at all costs, pretend are not there
In 'Letters: Wrangling Over the Bang', Science News (27 Jul 1991), 140, No. 4, 51. Also quoted in Roy C. Martin, Astronomy on Trial: A Devastating and Complete Repudiation of the Big Bang Fiasco (1999), Appendix I, 217.
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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.
As co-author with H. Hopf, Topologie I (1935), 23. As quoted, translated and cited in Reinhold Remmert, Theory of Complex Functions (1991), 9-10. From the original German, “Die Grundbegriffe und die einfachsten Tatsachen aus der mengentheoretischen Topologie braucht man in sehr verschiedenen Gebieten der Mathematik; die Begriffe des topologischen und des metrischen Raumes, der Kompaktheit, die Eigenschaften stetiger Abbildungen u. dgl. sind oft unentbehrlich.”
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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.
In Lectures on Metaphysics and Logic (1860), Vol. 1, 178.
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The Earth Speaks, clearly, distinctly, and, in many of the realms of Nature, loudly, to William Jennings Bryan, but he fails to hear a single sound. The earth speaks from the remotest periods in its wonderful life history in the Archaeozoic Age, when it reveals only a few tissues of its primitive plants. Fifty million years ago it begins to speak as “the waters bring forth abundantly the moving creatures that hath life.” In successive eons of time the various kinds of animals leave their remains in the rocks which compose the deeper layers of the earth, and when the rocks are laid bare by wind, frost, and storm we find wondrous lines of ascent invariably following the principles of creative evolution, whereby the simpler and more lowly forms always precede the higher and more specialized forms.
The earth speaks not of a succession of distinct creations but of a continuous ascent, in which, as the millions of years roll by, increasing perfection of structure and beauty of form are found; out of the water-breathing fish arises the air-breathing amphibian; out of the land-living amphibian arises the land-living, air-breathing reptile, these two kinds of creeping things resembling each other closely. The earth speaks loudly and clearly of the ascent of the bird from one kind of reptile and of the mammal from another kind of reptile.
This is not perhaps the way Bryan would have made the animals, but this is the way God made them!
The Earth Speaks to Bryan (1925), 5-6. Osborn wrote this book in response to the Scopes Monkey Trial, where William Jennings Bryan spoke against the theory of evolution. They had previously been engaged in the controversy about the theory for several years. The title refers to a Biblical verse from the Book of Job (12:8), “Speak to the earth and it shall teach thee.”
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The events of the past few years have led to a critical examination of the function of science in society. It used to be believed that the results of scientific investigation would lead to continuous progressive improvements in conditions of life; but first the War and then the economic crisis have shown that science can be used as easily for destructive and wasteful purposes, and voices have been raised demanding the cessation of scientific research as the only means of preserving a tolerable civilization. Scientists themselves, faced with these criticisms, have been forced to consider, effectively for the first time, how the work they are doing is connected around them. This book is an attempt to analyse this connection; to investigate how far scientists, individually and collectively, are responsible for this state of affairs, and to suggest what possible steps could be taken which would lead to a fruitful and not to a destructive utilization of science.
The Social Function of Science (1939), xlii.
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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 Electron and the Light-quant from the Experimental Point of View', Nobel Lecture (23 May 1924). In Nobel Lectures: Physics 1922-1941 (1998), 54.
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The fire at Lipara, Xenophanes says, ceased once for sixteen years, and came back in the seventeenth. And he says that the lavastream from Aetna is neither of the nature of fire, nor is it continuous, but it appears at intervals of many years.
Aristotle
De mirac. oscult. 38; 833 a 16. Quoted in Arthur Fairbanks (ed. And trans.), The First Philosophers of Greece (1898), 79.
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The following theorem can be found in the work of Mr. Cauchy: If the various terms of the series u0 + u1 + u2 +... 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 series
sin x - (1/2)sin 2x + (1/3)sin 3x - …
is discontinuous at each value (2m + 1)π of x,…
In Oeuvres (1826), Vol. 1, 224-225. As quoted and cited in Ernst Hairer and Gerhard Wanner Analysis by Its History (2008), 213.
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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.
From 'A Presidential Statement on Receipt of the Award of the Schlich Forestry Medal' (29 Jan 1935) in Public Papers of the Presidents of the United States: F.D. Roosevelt, 1935, Volume 4 (1938), 65. Roosevelt was awarded the medal by the Society of American Foresters. This quote continues with the line “The forests are the ‘lungs’ of our land….”
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The incessant driving of the pen over paper causes intense fatigue of the hand and the whole arm because of the continuous ... strain on the muscles and tendons.
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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.
In The Passionate State of Mind (1955), 18
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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.
…...
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The meaning of human life and the destiny of man cannot be separable from the meaning and destiny of life in general. 'What is man?' is a special case of 'What is life?' Probably the human species is not intelligent enough to answer either question fully, but even such glimmerings as are within our powers must be precious to us. The extent to which we can hope to understand ourselves and to plan our future depends in some measure on our ability to read the riddles of the past. The present, for all its awesome importance to us who chance to dwell in it, is only a random point in the long flow of time. Terrestrial life is one and continuous in space and time. Any true comprehension of it requires the attempt to view it whole and not in the artificial limits of any one place or epoch. The processes of life can be adequately displayed only in the course of life throughout the long ages of its existence.
The Meaning of Evolution: A Study of the History of Life and of its Significance for Man (1949), 9.
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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.
In Über das System der rein mathematischen Wissenschaften, Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. 1, 36. As quoted and cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 3.
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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.
In My Apprenticeship (1926), 134-135.
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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.
In Matter and Energy (1912), 45-46.
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The proof given by Wright, that non-adaptive differentiation will occur in small populations owing to “drift,” or the chance fixation of some new mutation or recombination, is one of the most important results of mathematical analysis applied to the facts of neo-mendelism. It gives accident as well as adaptation a place in evolution, and at one stroke explains many facts which puzzled earlier selectionists, notably the much greater degree of divergence shown by island than mainland forms, by forms in isolated lakes than in continuous river-systems.
Evolution: The Modern Synthesis (1942), 199-200.
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The rudest numerical scales, such as that by which the mineralogists distinguish different degrees of hardness, are found useful. The mere counting of pistils and stamens sufficed to bring botany out of total chaos into some kind of form. It is not, however, so much from counting as from measuring, not so much from the conception of number as from that of continuous quantity, that the advantage of mathematical treatment comes. Number, after all, only serves to pin us down to a precision in our thoughts which, however beneficial, can seldom lead to lofty conceptions, and frequently descend to pettiness.
On the Doctrine of Chances, with Later Reflections (1878), 61-2.
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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
Aristotle, Physics, 239b, 14-6. In Jonathan Barnes (ed.), The Complete Works of Aristotle (1984), Vol. 1, 404.
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The so-called “scientific revolution,” popularly associated with the sixteenth and seventeenth centuries, but reaching back in an unmistakably continuous line to a period much earlier still. Since that revolution overturned the authority in science not only of the middle ages but of the ancient world—since it ended not only in the eclipse of scholastic philosophy but in the destruction of Aristotelian physics—it outshines everything since the rise of Christianity and reduces the Renaissance and Reformation to the rank of mere episodes, mere internal displacements, within the system of medieval Christendom … It looms so large as the real origin of the modern world and of the modern mentality that our customary periodisation of European history has become an anachronism and an encumbrance.
The Origins of Modern Science (1949), viii.
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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
Aristotle, Physics, 239b, 30-1. In Jonathan Barnes (ed.), The Complete Works of Aristotle (1984), Vol. 1, 405.
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There is nothing distinctively scientific about the hypothetico-deductive process. It is not even distinctively intellectual. It is merely a scientific context for a much more general stratagem that underlies almost all regulative processes or processes of continuous control, namely feedback, the control of performance by the consequences of the act performed. In the hypothetico-deductive scheme the inferences we draw from a hypothesis are, in a sense, its logical output. If they are true, the hypothesis need not be altered, but correction is obligatory if they are false. The continuous feedback from inference to hypothesis is implicit in Whewell’s account of scientific method; he would not have dissented from the view that scientific behaviour can be classified as appropriately under cybernetics as under logic.
Induction and Intuition in Scientific Thought (1969), 54-5.
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Too early and perverse sexual satisfaction injures not merely the mind, but also the body; inasmuch as it induces neuroses of the sexual apparatus (irritable weakness of the centres governing erection and ejaculation; defective pleasurable feeling in coitus), while, at the same time, it maintains the imagination and libido in continuous excitement.
Psychopathia Sexualis: With Special Reference to Contrary Sexual Instinct: A Medico-Legal Study (1886), trans. Charles Gilbert Chaddock (1892), 189.
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We shall therefore take an appropriately correct view of the origin of our life, if we consider our own embryos to have sprung immediately from those embryos whence our parents were developed, and these from the embryos of their parents, and so on for ever. We should in this way look on the nature of mankind, and perhaps on that of the whole animated creation, as one Continuous System, ever pushing out new branches in all directions, that variously interlace, and that bud into separate lives at every point of interlacement.
'Hereditary Talent and Character', Macmillan's Magazine, 1865, 12, 322.
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Where a cell arises, there a cell must have previously existed (omnis cellula e cellula), just as an animal can spring only from an animal, a plant only from a plant. In this manner, although there are still a few spots in the body where absolute demonstration has not yet been afforded, the principle is nevertheless established, that in the whole series of living things, whether they be entire plants or animal organisms, or essential constituents of the same, an eternal law of continuous development prevails.
In Lecture II 'Physiological Tissues' (17 Feb 1858), as translated by Frank Chance in Cellular Pathology: As Based Upon Physiological and Pathological Histology. Twenty Lectures Delivered in the Pathological Institute of Berlin During the Months of February, March and April, 1858 (1860), 27-28.
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With the sole guidance of our practical knowledge of those physical agents which we see actually used in the continuous workings of nature, and of our knowledge of the respective effects induced by the same workings, we can with reasonable basis surmise what the forces were which acted even in the remotest times.
Quoted in Francesco Rodolico, 'Arduino', In Charles Coulston Gillispie (ed.), Dictionary of Scientific Biography (1970), Vol. 1, 234.
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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.
…...
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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