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Home > Category Index for Science Quotations > Category Index C > Category: Comprehend

Comprehend Quotes (40 quotes)

And the light shineth in the darkness; and the darkness comprehendeth it not.
Bible
(circa 325 A.D.)
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Clearly, we have compiled a record of serious failures in recent technological encounters with the environment. In each case, the new technology was brought into use before the ultimate hazards were known. We have been quick to reap the benefits and slow to comprehend the costs.
In 'Frail Reeds in a Harsh World', Natural History Journal of the American Museum of Natural History (Feb 1969), 79, No. 2, 44.
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Enough for me the mystery of the eternity of life, and the inkling of the marvellous structure of reality, together with the single-hearted endeavour to comprehend a portion, be it never so tiny, of the reason that manifests itself in nature.
In Alan Harris (ed.), The World As I See It (1934), 242.
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Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case,—which would only indicate some defect in the plan or treatment of the whole,—the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method.
In 'Stücke aus dem Lehrbuche der Arithmetik', Werke, Bd. 2 (1904), 296.
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I think the name atomic theory was an unfortunate one. We talk fluently about atoms as the smallest particles that exist, and chemists regard them as indivisible … To my mind the infinitely small is as incomprehensible as the infinitely great. … we cannot comprehend it, we cannot take it in. And so with the atom. Therefore I think that it would have been better to have taken a different word—say minim—which would have been a safer term than atom.
Address, in 'Report to the Chemical Society's Jubilee', Nature (26 Mar 1891), 43, 493.
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If it is true as Whewell says, that the essence of the triumphs of Science and its progress consists in that it enables us to consider evident and necessary, views which our ancestors held to be unintelligible and were unable to comprehend, then the extension of the number concept to include the irrational, and we will at once add, the imaginary, is the greatest forward step which pure mathematics has ever taken.
In Theorie der Complexen Zahlensysteme (1867), 60. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 281. From the original German, “Wenn es wahr ist, dass, wie Whewell meint, das Wesen der Triumphe der Wissenschaft und ihres Fortschrittes darin besteht, dass wir veranlasst werden, Ansichten, welche unsere Vorfahren für unbegreiflich hielten und unfähig waren zu begreifen, für evident und nothwendig zu halten, so war die Erweiterung des Zahlenbegriffes auf das Irrationale, und wollen wir sogleich hinzufügen, das Imaginäre, der grösste Fortschritt, den die reine Mathematik jemals gemacht hat.”
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If we knew all the laws of Nature, we should need only one fact or the description of one actual phenomenon to infer all the particular results at that point. Now we know only a few laws, and our result is vitiated, not, of course, by any confusion or irregularity in Nature, but by our ignorance of essential elements in the calculation. Our notions of law and harmony are commonly confined to those instances which we detect, but the harmony which results from a far greater number of seemingly conflicting, but really concurring, laws which we have not detected, is still more wonderful. The particular laws are as our points of view, as to the traveler, a mountain outline varies with every step, and it has an infinite number of profiles, though absolutely but one form. Even when cleft or bored through, it is not comprehended in its entireness.
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If you enquire about him [J.J. Sylvester], you will hear his genius universally recognized but his power of teaching will probably be said to be quite deficient. Now there is no man living who is more luminary in his language, to those who have the capacity to comprehend him than Sylvester, provided the hearer is in a lucid interval. But as the barn yard fowl cannot understand the flight of the eagle, so it is the eaglet only who will be nourished by his instruction.
Letter (18 Sep 1875) to Daniel C. Gilman. In Daniel C. Gilman Papers, Ms. 1, Special Collections Division, Milton S. Eisenhower Library, Johns Hopkins University. As quoted in Karen Hunger Parshall, 'America’s First School of Mathematical Research: James Joseph Sylvester at The Johns Hopkins University 1876—1883', Archive for History of Exact Sciences (1988), 38, No. 2, 167.
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In order to comprehend and fully control arithmetical concepts and methods of proof, a high degree of abstraction is necessary, and this condition has at times been charged against arithmetic as a fault. I am of the opinion that all other fields of knowledge require at least an equally high degree of abstraction as mathematics,—provided, that in these fields the foundations are also everywhere examined with the rigour and completeness which is actually necessary.
In 'Die Theorie der algebraischen Zahlkorper', Vorwort, Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 4.
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In the endeavor to clearly comprehend and explain the functions of the combination of forces called “brain,” the physiologist is hindered and troubled by the views of the nature of those cerebral forces which the needs of dogmatic theology have imposed on mankind.
In 'General Conclusions, Anatomy of the Vertebrates (1868, 2011), Vol. 3, Chap 40, 823. Excerpt in Noah Porter (ed.), Half Hours with Modern Scientists (1872), 71.
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Joy in looking and comprehending is nature’s most beautiful gift.
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Let thy speech be short, comprehending much in few words.
Bible
(circa 725 B.C.)
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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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Like almost every subject of human interest, this one [mathematics] is just as easy or as difficult as we choose to make it. A lifetime may be spent by a philosopher in discussing the truth of the simplest axiom. The simplest fact as to our existence may fill us with such wonder that our minds will remain overwhelmed with wonder all the time. A Scotch ploughman makes a working religion out of a system which appalls a mental philosopher. Some boys of ten years of age study the methods of the differential calculus; other much cleverer boys working at mathematics to the age of nineteen have a difficulty in comprehending the fundamental ideas of the calculus.
In Teaching of Mathematics (1902), 19-20.
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Mathematicians attach great importance to the elegance of their methods and their results. This is not pure dilettantism. What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details. But this is exactly what yields great results, in fact the more we see this aggregate clearly and at a single glance, the better we perceive its analogies with other neighboring objects, consequently the more chances we have of divining the possible generalizations. Elegance may produce the feeling of the unforeseen by the unexpected meeting of objects we are not accustomed to bring together; there again it is fruitful, since it thus unveils for us kinships before unrecognized. It is fruitful even when it results only from the contrast between the simplicity of the means and the complexity of the problem set; it makes us then think of the reason for this contrast and very often makes us see that chance is not the reason; that it is to be found in some unexpected law. In a word, the feeling of mathematical elegance is only the satisfaction due to any adaptation of the solution to the needs of our mind, and it is because of this very adaptation that this solution can be for us an instrument. Consequently this esthetic satisfaction is bound up with the economy of thought.
In 'The Future of Mathematics', Monist, 20, 80. Translated from the French by George Bruce Halsted.
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Mere infants of the universe, with no feel for infinity, no sense of place in time and space, we human beings have yet to comprehend the enormity of what we are doing: In a geological second, we are unraveling complexities it took eternity to create.
In Jacques Cousteau and Susan Schiefelbein, The Human, the Orchid, and the Octopus: Exploring and Conserving Our Natural World (2007), 107.
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Nothing has tended more to retard the advancement of science than the disposition in vulgar minds to vilify what they cannot comprehend.
In Maturin Murray Ballou, Treasury of Thought (1894), 459.
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Our imagination is stretched to the utmost, not as in fiction, to imagine things which are not really there, but just to comprehend those things which are there.
In The Character of Physical Law (1965, reprint 2001), 127-128.
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Philosophy [the universe] is written in that great book which ever lies before our eyes ... We cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in the mathematical language ... without whose help it is humanly impossible to comprehend a single word of it, and without which one wanders in vain through a dark labyrinth.
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Religion closes off the central questions of existence by attempting to dissuade us from further enquiry by asserting that we cannot ever hope to comprehend. We are, religion asserts, simply too puny.
Essay collected in John Cornwell (ed.), 'The Limitless Power of Science', Nature's Imagination: The Frontiers of Scientific Vision (1995), 125.
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Religion has been compelled by science to give up one after another of its dogmas—of those assumed cognitions which it could not substantiate. In the mean time, Science substituted for the personalities to which Religion ascribed phenomena certain metaphysical entities; and in doing this it trespassed on the province of religion; since it classed among the things which it comprehended certain forms of the incomprehensible.
In First Principles (1864), 109.
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The chances for favorable serendipity are increased if one studies an animal that is not one of the common laboratory species. Atypical animals, or preparations, force one to use non-standard approaches and non-standard techniques, and even to think nonstandard ideas. My own preference is to seek out species which show some extreme of adaptation. Such organisms often force one to abandon standard methods and standard points of view. Almost inevitably they lead one to ask new questions, and most importantly in trying to comprehend their special and often unusual adaptations one often serendipitously stumbles upon new insights.
In 'Scientific innovation and creativity: a zoologist’s point of view', American Zoologist (1982), 22, 234.
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The Excellence of Modern Geometry is in nothing more evident, than in those full and adequate Solutions it gives to Problems; representing all possible Cases in one view, and in one general Theorem many times comprehending whole Sciences; which deduced at length into Propositions, and demonstrated after the manner of the Ancients, might well become the subjects of large Treatises: For whatsoever Theorem solves the most complicated Problem of the kind, does with a due Reduction reach all the subordinate Cases.
In 'An Instance of the Excellence of Modern Algebra, etc', Philosophical Transactions, 1694, 960.
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The human mind is not capable of grasping the Universe. We are like a little child entering a huge library. The walls are covered to the ceilings with books in many different tongues. The child knows that someone must have written these books. It does not know who or how. It does not understand the languages in which they are written. But the child notes a definite plan in the arrangement of the books—a mysterious order which it does not comprehend, but only dimly suspects.
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The mathematical method is the essence of mathematics. He who fully comprehends the method is a mathematician.
In Schriften (1901), Zweiter Teil, 190.
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The maxim of science is simply that of common sense—simple cases first; begin with seeing how the main force acts when there is as little as possible to impede it, and when you thoroughly comprehend that, add to it in succession the separate effects of each of the incumbering and interfering agencies.
Collected in The Works of Walter Bagehot (1889), Vol. 5, 319-320.
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The mind comprehends a thing the more correctly the closer the thing approaches toward pure quantity as its origin.
Letter to Mästlin (19 Apr 1597). In Gerald James Holton, Thematic Origins of Scientific Thought: Kepler to Einstein (1985), 74.
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The overwhelming astonishment, the queerest structure we know about so far in the whole universe, the greatest of all cosmological scientific puzzles, confounding all our efforts to comprehend it, is the earth. We are only now beginning to appreciate how strange and splendid it is, how it catches the breath, the loveliest object afloat around the sun, enclosed in its own blue bubble of atmosphere, manufacturing and breathing its own oxygen, fixing its own nitrogen from the air into its own soil, generating its own weather at the surface of its rain forests, constructing its own carapace from living parts: chalk cliffs, coral reefs, old fossils from earlier forms of life now covered by layers of new life meshed together around the globe, Troy upon Troy.
In Late Night Thoughts on Listening to Mahler’s Ninth Symphony (1984), 22-23.
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The Pacific. You don’t comprehend it by looking at a globe, but when you’re traveling at four miles a second and it still takes you twenty-five minutes to cross it, you know it’s big.
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There are then two kinds of intellect: the one able to penetrate acutely and deeply into the conclusions of given premises, and this is the precise intellect; the other able to comprehend a great number of premises without confusing them, and this is the mathematical intellect. The one has force and exactness, the other comprehension. Now the one quality can exist without the other; the intellect can be strong and narrow, and can also be comprehensive and weak.
In Pascal’s Pensées (1958), 3.
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There remains something subtle, intangible and inexplicable. Veneration for this force beyond anything that we can comprehend is my religion.
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Those who are accustomed to judge by feeling do not understand the process of reasoning, because they want to comprehend at a glance and are not used to seeking for first principles. Those, on the other hand, who are accustomed to reason from first principles do not understand matters of feeling at all, because they look for first principles and are unable to comprehend at a glance.
In Pensées (1670), Section 7, No. 33. As translated in W.H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms (1966), 351. Also translated as “Those who are accustomed to judge by feeling do not understand the process of reasoning, for they would understand at first sight, and are not used to seek for principles. And others, on the contrary, who are accustomed to reason from principles, do not at all understand matters of feeling, seeking principles, and being unable to see at a glance,” in Blaise Pascal and W.F. Trotter (trans.), 'Thoughts', No. 3, collected in Charles W. Eliot (ed.), The Harvard Classics (1910), Vol. 48, 9. From the original French, “Ceux qui sont accoutumés à juger par le sentiment ne comprennent rien aux choses de raisonnement, car ils veulent d’abord pénétrer d’une vue et ne sont point accoutumés à chercher les principes. Et les autres, au contraire, qui sont accoutumés à raisonner par principes, ne comprennent rien aux choses de sentiment, y cherchant des principes et ne pouvant voir d’une vue,” in Ernest Havet (ed.), Pensées de Pascal (1892), 224.
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To overturn orthodoxy is no easier in science than in philosophy, religion, economics, or any of the other disciplines through which we try to comprehend the world and the society in which we live.
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Try and penetrate with our limited means the secrets of nature and you will find that, behind all the discernible concatenations, there remains something subtle, intangible and inexplicable. Veneration for this force beyond anything that we can comprehend is my religion. To that extent I am, in point of fact, religious.
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We have become a people unable to comprehend the technology we invent.
Report, Integrity in the College Curriculum (Feb 1985).
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We have not been seeing our Spaceship Earth as an integrally-designed machine which to be persistently successful must be comprehended and serviced in total.
In Operating Manual for Spaceship Earth (1969), 52.
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What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.
From 'L’Avenir des Mathématiques', Science et Méthode (1908, 1920), Livre Premier, Chap. 2, 25. English as in Henri Poincaré and George Bruce Halsted (trans.), 'The Future of Mathematics', Science and Method collected in The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method (1913), 372. From the French, “Qu’est-ce qui nous donne en effet dans une solution, dans une démonstration, le sentiment de l’élégance? C’est l’harmonie des diverses parties, leur symétrie, leur heureux balancement; c’est en un mot tout ce qui y met de l'ordre, tout ce qui leur donne de l’unité, ce qui nous permet par conséquent d’y voir clair et d’en comprendre l’ensemble en même temps que les détails”.
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[I find it as difficult] to understand a scientist who does not acknowledge the presence of a superior rationality behind the existence of the universe as it is to comprehend a theologian who would deny the advances of science.
Speech, Huntsville Ministerial Association, in Wernher Von Braun and Irene E. Powell-Willhite (ed.), The Voice of Dr. Wernher Von Braun: An Anthology (2007), 89.
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[In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general propositions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity.
In Ueber das Verhältnis der Naturwissenschaften zur Gesammtheit der Wissenschaft, Vorträge und Reden (1896), Bd. 1, 176. Also seen translated as “In mathematics we see the conscious logical activity of our mind in its purest and most perfect form; here is made manifest to us all the labor and the great care with which it progresses, the precision which is necessary to determine exactly the source of the established general theorems, and the difficulty with which we form and comprehend abstract conceptions; but we also learn here to have confidence in the certainty, breadth, and fruitfulness of such intellectual labor”, in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 20. From the original German, “Hier sehen wir die bewusste logische Thätigkeit unseres Geistes in ihrer reinsten und vollendetsten Form; wir können hier die ganze Mühe derselben kennen lernen, die grosse Vorsicht, mit der sie vorschreiten muss, die Genauigkeit, welche nöthig ist, um den Umfang der gewonnenen allgemeinen Sätze genau zu bestimmen, die Schwierigkeit, abstracte Begriffe zu bilden und zu verstehen; aber ebenso auch Vertrauen fassen lernen in die Sicherheit, Tragweite und Fruchtbarkeit solcher Gedankenarbeit.”
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“Wu Li” was more than poetic. It was the best definition of physics that the conference would produce. It caught that certain something, that living quality that we were seeking to express in a book, that thing without which physics becomes sterile. “Wu” can mean either “matter” or “energy.” “Li” is a richly poetic word. It means “universal order” or “universal law.” It also means “organic patterns.” The grain in a panel of wood is Li. The organic pattern on the surface of a leaf is also Li, and so is the texture of a rose petal. In short, Wu Li, the Chinese word for physics, means “patterns of organic energy” (“matter/ energy” [Wu] + “universal order/organic patterns” [Li]). This is remarkable since it reflects a world view which the founders of western science (Galileo and Newton) simply did not comprehend, but toward which virtually every physical theory of import in the twentieth century is pointing!
In The Dancing Wu Li Masters: An Overview of the New Physics (1979), 5.
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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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- 90 -
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- 80 -
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- 70 -
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- 50 -
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- 40 -
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