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Today in Science History - Quickie Quiz
Who said: “I seem to have been only like a boy playing on the seashore, ... finding a smoother pebble or a prettier shell ... whilst the great ocean of truth lay all undiscovered before me.”
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Understand Quotes (340 quotes)

...That day in the account of creation, or those days that are numbers according to its recurrence, are beyond the experience and knowledge of us mortal earthbound men. And if we are able to make any effort towards an understanding of those days, we ought not to rush forward with an ill considered opinion, as if no other reasonable and plausible interpretation could be offered.
iv.44
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Every teacher certainly should know something of non-euclidean geometry. Thus, it forms one of the few parts of mathematics which, at least in scattered catch-words, is talked about in wide circles, so that any teacher may be asked about it at any moment. … Imagine a teacher of physics who is unable to say anything about Röntgen rays, or about radium. A teacher of mathematics who could give no answer to questions about non-euclidean geometry would not make a better impression.
On the other hand, I should like to advise emphatically against bringing non-euclidean into regular school instruction (i.e., beyond occasional suggestions, upon inquiry by interested pupils), as enthusiasts are always recommending. Let us be satisfied if the preceding advice is followed and if the pupils learn to really understand euclidean geometry. After all, it is in order for the teacher to know a little more than the average pupil.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 72.
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Ron Hutcheson, a Knight-Ridder reporter: [Mr. President, what are your] personal views [about the theory of] intelligent design?
President George W. Bush: [Laughing. You're] doing a fine job of dragging me back to the past [days as governor of Texas]. ... Then, I said that, first of all, that decision should be made to local school districts, but I felt like both sides ought to be properly taught...”
Hutcheson: Both sides ought to be properly taught?
President: Yes ... so people can understand what the debate is about.
Hutcheson: So the answer accepts the validity of “intelligent design” as an alternative to evolution?
President: I think that part of education is to expose people to different schools of thought, and I'm not suggesting—you're asking me whether or not people ought to be exposed to different ideas, and the answer is yes.
Hutcheson: So we've got to give these groups—...
President: [interrupting] Very interesting question, Hutch. [Laughter from other reporters]
From conversation with reporters at the White House (1 Aug 2005), as quoted by Matthew Cooper in 'Fanning the Controversy Over “Intelligent Design”', Time (3 Aug 2005). The Time writer stated, “The president has gone farther in questioning the widely-taught theories of evolution and natural selection than any president since Ronald Reagan, who advocated teaching creationism in public schools alongside evolution.” Just a few months later, in the nation's first case on that point, on 20 Dec 2005, “a federal judge [John E. Jones] ruled it was unconstitutional for a Pennsylvania school district to present intelligent design as an alternative in high school biology courses, because it is a religious viewpoint,” as reported by Laurie Goodstein in 'Judge Rejects Teaching Intelligent Design', New York Times (21 Dec 2005). Goodstein also wrote “Judge Jones, a Republican appointed by President Bush, concluded that intelligent design was not science,” and that “the evidence in the trial proved that intelligent design was 'creationism relabeled.' The Supreme Court has already ruled that creationism ... cannot be taught as science in a public school.”
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Salviati: …Now you see how easy it is to understand.
Sagredo: So are all truths, once they are discovered; the point is in being able to discover them.
[Commonly seen merged in a paraphrase as: All truths are easy to understand once they are discovered; the point is to discover them.]
Lines of two characters in Dialogue Concerning the Two Chief World Systems (1632), as translated by S. Drake (1967). As quoted and cited in Barnaby Sheppard, The Logic of Infinity (2014), 398 & 440 footnote. Galileo’s work is written as a series of dialogues over four days between Salviati (supporting the Copernican system), Sagredo (a neutral layman) and Simplicio (supporting the Ptolemaic system).
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A good theoretical physicist today might find it useful to have a wide range of physical viewpoints and mathematical expressions of the same theory (for example, of quantum electrodynamics) available to him. This may be asking too much of one man. Then new students should as a class have this. If every individual student follows the same current fashion in expressing and thinking about electrodynamics or field theory, then the variety of hypotheses being generated to understand strong interactions, say, is limited. Perhaps rightly so, for possibly the chance is high that the truth lies in the fashionable direction. But, on the off-chance that it is in another direction—a direction obvious from an unfashionable view of field theory—who will find it?
In his Nobel Prize Lecture (11 Dec 1965), 'The Development of the Space-Time View of Quantum Electrodynamics'. Collected in Stig Lundqvist, Nobel Lectures: Physics, 1963-1970 (1998), 177.
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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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A troubling question for those of us committed to the widest application of intelligence in the study and solution of the problems of men is whether a general understanding of the social sciences will be possible much longer. Many significant areas of these disciplines have already been removed by the advances of the past two decades beyond the reach of anyone who does not know mathematics; and the man of letters is increasingly finding, to his dismay, that the study of mankind proper is passing from his hands to those of technicians and specialists. The aesthetic effect is admittedly bad: we have given up the belletristic “essay on man” for the barbarisms of a technical vocabulary, or at best the forbidding elegance of mathematical syntax.
Opening paragraph of 'The Study of Man: Sociology Learns the Language of Mathematics' in Commentary (1 Sep 1952). Reprinted in James Roy Newman, The World of Mathematics (1956), Vol. 2, 1294.
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A vision of the whole of life!. Could any human undertaking be ... more grandiose? This attempt stands without rival as the most audacious enterprise in which the mind of man has ever engaged ... Here is man, surrounded by the vastness of a universe in which he is only a tiny and perhaps insignificant part—and he wants to understand it.
…...
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Above, far above the prejudices and passions of men soar the laws of nature. Eternal and immutable, they are the expression of the creative power they represent what is, what must be, what otherwise could not be. Man can come to understand the: he is incapable of changing them.
In Cours d’economie Politique (1896-97)
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After an orange cloud—formed as a result of a dust storm over the Sahara and caught up by air currents—reached the Philippines and settled there with rain, I understood that we are all sailing in the same boat.
In Jack Hassard and Julie Weisberg , Environmental Science on the Net: The Global Thinking Project (1999), 40.
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Again and again in reading even his [William Thomson] most abstract writings one is struck by the tenacity with which physical ideas control in him the mathematical form in which he expressed them. An instance of this is afforded by … an example of a mathematical result that is, in his own words, “not instantly obvious from the analytical form of my solution, but which we immediately see must be the case by thinking of the physical meaning of the result.”
As given in Life of Lord Kelvin (1910), Vol. 2, 1136. The ellipsis gives the reference to the quoted footnote, to a passage in his Mathematical and Physical Papers, Vol. 1, 457. [Note: William Thomson, later became Lord Kelvin. —Webmaster]
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Alexander the king of the Macedonians, began like a wretch to learn geometry, that he might know how little the earth was, whereof he had possessed very little. Thus, I say, like a wretch for this, because he was to understand that he did bear a false surname. For who can be great in so small a thing? Those things that were delivered were subtile, and to be learned by diligent attention: not which that mad man could perceive, who sent his thoughts beyond the ocean sea. Teach me, saith he, easy things. To whom his master said: These things be the same, and alike difficult unto all. Think thou that the nature of things saith this. These things whereof thou complainest, they are the same unto all: more easy things can be given unto none; but whosoever will, shall make those things more easy unto himself. How? With uprightness of mind.
In Thomas Lodge (trans.), 'Epistle 91', The Workes of Lucius Annaeus Seneca: Both Morrall and Naturall (1614), 383. Also in Robert Édouard Moritz, Memorabilia Mathematica (1914), 135.
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All knowledge and understanding of the Universe was no more than playing with stones and shells on the seashore of the vast imponderable ocean of truth.
…...
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All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics ... It has survived all tests and there is no reason to believe that there is any flaw in it.... We all know how to use it and how to apply it to problems; and so we have learned to live with the fact that nobody can understand it.
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All of physics is either impossible or trivial. It is impossible until you understand it, and then it becomes trivial.
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All that science can achieve is a perfect knowledge and a perfect understanding of the action of natural and moral forces.
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All the real true knowledge we have of Nature is intirely experimental, insomuch that, how strange soever the assertion seems, we may lay this down as the first fundamental unerring rule in physics, That it is not within the compass of human understanding to assign a purely speculative reason for any one phaenomenon in nature.
In The Procedure, Extent, and Limits of Human Understanding (1728, 1729), 205-206.
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All true science must aim at objective truth, and that means that the human observer must never allow himself to get emotionally mixed up with his subject-matter. His concern is to understand the universe, not to improve it. Detachment is obligatory.
From transcript of BBC radio Reith Lecture (12 Nov 1967), 'A Runaway World', on the bbc.co.uk website.
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An important fact, an ingenious aperçu, occupies a very great number of men, at first only to make acquaintance with it; then to understand it; and afterwards to work it out and carry it further.
In The Maxims and Reflections of Goethe (1906), 189.
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An informed appraisal of life absolutely require(s) a full understanding of life’s arena–the universe. … By deepening our understanding of the true nature of physical reality, we profoundly reconfigure our sense of ourselves and our experience of the universe.
In The Fabric of the Cosmos: Space, Time, and the Texture of Reality (2007), 5.
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And yet I think that the Full House model does teach us to treasure variety for its own sake–for tough reasons of evolutionary theory and nature’s ontology, and not from a lamentable failure of thought that accepts all beliefs on the absurd rationale that disagreement must imply disrespect. Excellence is a range of differences, not a spot. Each location on the range can be occupied by an excellent or an inadequate representative– and we must struggle for excellence at each of these varied locations. In a society driven, of ten unconsciously, to impose a uniform mediocrity upon a former richness of excellence–where McDonald’s drives out the local diner, and the mega-Stop & Shop eliminates the corner Mom and Pop–an understanding and defense of full ranges as natural reality might help to stem the tide and preserve the rich raw material of any evolving system: variation itself.
…...
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Any fool can know. The point is to understand.
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Any one, if he will only observe, can find some little thing he does not understand as a starter for an investigation.
From Address (22 May 1914) to the graduating class of the Friends’ School, Washington, D.C. Printed in 'Discovery and Invention', The National Geographic Magazine (1914), 25, 650.
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Anyone who sits on top of the largest hydrogen-oxygen fueled system in the world; knowing they’re going to light the bottom, and doesn’t get a little worried, does not fully understand the situation.
Response to question whether he was worried about embarking on the first space shuttle flight. As quoted on the nmspacemuseum.org website of the New Mexico Museum of Space History.
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Anyone who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains.
From Recherches sur les Principes Mathématiques de la Théorie des Richesses (1838), as translated by Nathaniel T. Bacon in 'Preface', Researches Into Mathematical Principles of the Theory of Wealth (1897), 4. From the original French, “Quiconque connaît la notation algébrique, lit d'un clin-d'œil dans une équation le résultat auquel on parvient péniblement par des règles de fausse position, dans l'arithmétique de Banque.”
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As language-using organisms, we participate in the evolution of the Universe most fruitfully through interpretation. We understand the world by drawing pictures, telling stories, conversing. These are our special contributions to existence. It is our immense good fortune and grave responsibility to sing the songs of the Cosmos.
Epigraph, without citation, in Michael Dowd, Thank God for Evolution: How the Marriage of Science and Religion Will Transform Your Life and Our World (2008), 103.
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As was predicted at the beginning of the Human Genome Project, getting the sequence will be the easy part as only technical issues are involved. The hard part will be finding out what it means, because this poses intellectual problems of how to understand the participation of the genes in the functions of living cells.
Loose Ends from Current Biology (1997), 71.
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At first it seems obvious, but the more you think about it the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it.
As quoted, without citation, in Stories about Sets (1968), 84.
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Bell’s theorem is easy to understand but hard to believe.
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Biology today is moving in the direction of chemistry. Much of what is understood in the field is based on the structure of molecules and the properties of molecules in relation to their structure. If you have that basis, then biology isn’t just a collection of disconnected facts.
From interview with Neil A. Campbell, in 'Crossing the Boundaries of Science', BioScience (Dec 1986), 36, No. 11, 737.
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Both social and biosocial factors are necessary to interpret crosscultural studies, with the general proviso that one’s research interest determines which elements, in what combinations, are significant for the provision of understanding.
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But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.
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Cayley was singularly learned in the work of other men, and catholic in his range of knowledge. Yet he did not read a memoir completely through: his custom was to read only so much as would enable him to grasp the meaning of the symbols and understand its scope. The main result would then become to him a subject of investigation: he would establish it (or test it) by algebraic analysis and, not infrequently, develop it so to obtain other results. This faculty of grasping and testing rapidly the work of others, together with his great knowledge, made him an invaluable referee; his services in this capacity were used through a long series of years by a number of societies to which he was almost in the position of standing mathematical advisor.
In Proceedings of London Royal Society (1895), 58, 11-12.
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Chaos theory is a new theory invented by scientists panicked by the thought that the public were beginning to understand the old ones.
John Mitchinson and John Lloyd, If Ignorance Is Bliss, Why Aren't There More Happy People?: Smart Quotes for Dumb Times (2009), 273.
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Commenting on Archimedes, for whom he also had a boundless admiration, Gauss remarked that he could not understand how Archimedes failed to invent the decimal system of numeration or its equivalent (with some base other than 10). … This oversight Gauss regarded as the greatest calamity in the history of science.
In Men of Mathematics (1937), 256.
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Common sense … is to the judgment what genius is to the understanding.
In Igerne and Other Writings of Arthur Handly Marks (1897), 349.
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Considering the difficulties represented by the lack of water, by extremes of temperature, by the full force of gravity unmitigated by the buoyancy of water, it must be understood that the spread to land of life forms that evolved to meet the conditions of the ocean represented the greatest single victory won by life over the inanimate environment.
(1965). In Isaac Asimov’s Book of Science and Nature Quotations (1988), 194.
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Dirichlet was not satisfied to study Gauss’ Disquisitiones arithmetical once or several times, but continued throughout life to keep in close touch with the wealth of deep mathematical thoughts which it contains by perusing it again and again. For this reason the book was never placed on the shelf but had an abiding place on the table at which he worked. … Dirichlet was the first one, who not only fully understood this work, but made it also accessible to others.
In Dirichlet, Werke, Bd. 2, 315. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 159.
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Does it mean, if you don’t understand something, and the community of physicists don’t understand it, that means God did it? Is that how you want to play this game? Because if it is, here’s a list of the things in the past that the physicists—at the time—didn’t understand … [but now we do understand.] If that’s how you want to invoke your evidence for God, then God is an ever-receding pocket of scientific ignorance, that’s getting smaller and smaller and smaller, as time moves on. So just be ready for that to happen, if that’s how you want to come at the problem. That’s simply the “God of the Gaps” argument that’s been around for ever.
From interview, The Science Studio video series of The Science Network website, episode 'The Moon, the Tides and why Neil DeGrasse Tyson is Colbert’s God' (20 Jan 2011), time 26:58-27:55.
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Dreams are true interpreters of our inclinations; but there is art required to categorize and understand them.
In The Works of Michael de Montaigne (1849), 536.
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Each thing in the world has names or unnamed relations to everything else. Relations are infinite in number and kind. To be is to be related. It is evident that the understanding of relations is a major concern of all men and women. Are relations a concern of mathematics? They are so much its concern that mathematics is sometimes defined to be the science of relations.
In Mole Philosophy and Other Essays (1927), 94-95.
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Each worldview was a cultural product, but evolution is true and separate creation is not ... Worldviews are social constructions, and they channel the search for facts. But facts are found and knowledge progresses, however fitfully. Fact and theory are intertwined, and all great scientists understand the interaction.
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Egad, I think the interpreter is the hardest to be understood of the two!
In 'The Critic: Or, A Tragedy Rehearsed', Act 1, Scene 2, as collected in Thomas Moore (ed.), The Works of the Late Right Honourable Richard Brinsley Sheridan (1833), 181.
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Error is often nourished by good sense. … The meaning is, that the powers of the understanding are frequently employed to defend favourite errors; and that a man of sense frequently fortifies himself in his prejudices, or in false opinions which he received without examination, by such arguments as would not have occurred to a fool.
In Maxims, Characters, and Reflections, Critical, Satyrical and Moral (2nd ed., 1757), 9. The meaning is given as a footnote.
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Euclid and Archimedes are allowed to be knowing, and to have demonstrated what they say: and yet whosoever shall read over their writings without perceiving the connection of their proofs, and seeing what they show, though he may understand all their words, yet he is not the more knowing. He may believe, indeed, but does not know what they say, and so is not advanced one jot in mathematical knowledge by all his reading of those approved mathematicians.
In Conduct of the Understanding, sect. 24.
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Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members. On the other hand, the mathematics of variable magnitudes—function theory or analysis—considers mathematical forms in their genesis. By writing the equation of the parabola, we express its law of generation, the law according to which the variable point moves. The path, produced before the eyes of the student by a point moving in accordance to this law, is the parabola.
If, then, Euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body, modern mathematics deals, as it were, with the living body, with growing and changing forms, and thus furnishes an insight, not only into nature as she is and appears, but also into nature as she generates and creates,—reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming. Thus modern mathematics bears the same relation to Euclidean mathematics that physiology or biology … bears to anatomy.
In Die Mathematik die Fackelträgerin einer neuen Zeit (1889), 38. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 112-113.
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Every day we are interacting with the economy, whether we want to or not, and whether we know it or not. To have a level of control over our lives, we need to understand the connections between money and events and ourselves.
As quoted in brochure for PBS TV financial news series, Adam Smith’s Money World.
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Every proposition which we can understand must be composed wholly of constituents with which we are acquainted.
From 'Knowledge by Acquaintance', in Mysticism and Logic: And Other Essays (1918), 219.
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Everyone working in science, no matter their politics, has a stake in cleaning up the mess revealed by the East Anglia emails. Science is on the credibility bubble. If it pops, centuries of what we understand to be the role of science go with it.
Newspaper
In D. Henninger, 'Climategate: Science is Dying', Wall Street Journal (Dec 2009), A21.
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Everything that the human race has done and thought is concerned with the satisfaction of deeply felt needs and the assuagement of pain. One has to keep this constantly in mind if one wishes to understand spiritual movements and their development. Feeling and longing are the motive force behind all human endeavor and human creation, in however exalted a guise the latter may present themselves to us.
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Everywhere you look in science, the harder it becomes to understand the universe without God.
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Evolution is the conviction that organisms developed their current forms by an extended history of continual transformation, and that ties of genealogy bind all living things into one nexus. Panselectionism is a denial of history, for perfection covers the tracks of time. A perfect wing may have evolved to its current state, but it may have been created just as we find it. We simply cannot tell if perfection be our only evidence. As Darwin himself understood so well, the primary proofs of evolution are oddities and imperfections that must record pathways of historical descent–the panda’s thumb and the flamingo’s smile of my book titles (chosen to illustrate this paramount principle of history).
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Facts are to the mind the same thing as food to the body. On the due digestion of facts depends the strength and wisdom of the one, just as vigor and health depend on the other. The wisest in council, the ablest in debate, and the most agreeable in the commerce of life is that man who has assimilated to his understanding the greatest number of facts.
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First, as concerns the success of teaching mathematics. No instruction in the high schools is as difficult as that of mathematics, since the large majority of students are at first decidedly disinclined to be harnessed into the rigid framework of logical conclusions. The interest of young people is won much more easily, if sense-objects are made the starting point and the transition to abstract formulation is brought about gradually. For this reason it is psychologically quite correct to follow this course.
Not less to be recommended is this course if we inquire into the essential purpose of mathematical instruction. Formerly it was too exclusively held that this purpose is to sharpen the understanding. Surely another important end is to implant in the student the conviction that correct thinking based on true premises secures mastery over the outer world. To accomplish this the outer world must receive its share of attention from the very beginning.
Doubtless this is true but there is a danger which needs pointing out. It is as in the case of language teaching where the modern tendency is to secure in addition to grammar also an understanding of the authors. The danger lies in grammar being completely set aside leaving the subject without its indispensable solid basis. Just so in Teaching of Mathematics it is possible to accumulate interesting applications to such an extent as to stunt the essential logical development. This should in no wise be permitted, for thus the kernel of the whole matter is lost. Therefore: We do want throughout a quickening of mathematical instruction by the introduction of applications, but we do not want that the pendulum, which in former decades may have inclined too much toward the abstract side, should now swing to the other extreme; we would rather pursue the proper middle course.
In Ueber den Mathematischen Unterricht an den hoheren Schulen; Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 11, 131.
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For man being the minister and interpreter of nature, acts and understands so far as he has observed of the order, the works and mind of nature, and can proceed no further; for no power is able to loose or break the chain of causes, nor is nature to be conquered but by submission: whence those twin intentions, human knowledge and human power, are really coincident; and the greatest hindrance to works is the ignorance of causes.
In The Great lnstauration.
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For, in mathematics or symbolic logic, reason can crank out the answer from the symboled equations—even a calculating machine can often do so—but it cannot alone set up the equations. Imagination resides in the words which define and connect the symbols—subtract them from the most aridly rigorous mathematical treatise and all meaning vanishes. Was it Eddington who said that we once thought if we understood 1 we understood 2, for 1 and 1 are 2, but we have since found we must learn a good deal more about “and”?
In 'The Biological Basis of Imagination', American Thought: 1947 (1947), 81.
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Formal thought, consciously recognized as such, is the means of all exact knowledge; and a correct understanding of the main formal sciences, Logic and Mathematics, is the proper and only safe foundation for a scientific education.
In Number and its Algebra (1896), 134.
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Freeman’s gift? It’s cosmic. He is able to see more interconnections between more things than almost anybody. He sees the interrelationships, whether it’s in some microscopic physical process or in a big complicated machine like Orion. He has been, from the time he was in his teens, capable of understanding essentially anything that he’s interested in. He’s the most intelligent person I know.
As quoted in Kenneth Brower, 'The Danger of Cosmic Genius', The Atlantic (Dec 2010). Webmaster note: The Orion Project was a study of the possibility of nuclear powered propulsion of spacecraft.
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Fullness of knowledge always means some understanding of the depths of our ignorance; and that is always conducive to humility and reverence.
In 'What I Believe: Living Philosophies II', The Forum (Oct 1929), 82, No. 4, 199.
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Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it is the truth.
In a lecture, after establishing the relation eπ/2 = ii in a lecture, “which evidently had a strong hold on his imagination. He dropped his chalk and rubber, put his hands in his pockets, and after contemplating the formula a few minutes turned to his class and said [this quote] very slowly and impressively.” As quoted in W. E. Byerly (writing as a Professor Emeritus at Harvard University, but a former student of Peirce), 'Benjamin Peirce: II. Reminiscences', The American Mathematical Monthly (Jan 1925), 32, No. 1, 6.
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Geometry may sometimes appear to take the lead of analysis, but in fact precedes it only as a servant goes before his master to clear the path and light him on his way. The interval between the two is as wide as between empiricism and science, as between the understanding and the reason, or as between the finite and the infinite.
From 'Astronomical Prolusions', Philosophical Magazine (Jan 1866), 31, No. 206, 54, collected in Collected Mathematical Papers of James Joseph Sylvester (1908), Vol. 2, 521.
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God is able to do more than man can understand.
Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 156
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Good scholars struggle to understand the world in an integral way (pedants bite off tiny bits and worry them to death). These visions of reality ... demand our respect, for they are an intellectual’s only birthright. They are often entirely wrong and always flawed in serious ways, but they must be understood honorably and not subjected to mayhem by the excision of patches.
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Great spirits have always found violent opposition from mediocrities. The latter cannot understand it when a man does not thoughtlessly submit to hereditary prejudices but honestly and courageously uses his intelligence and fulfills the duty to express the results of his thoughts in clear form.
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Guide to understanding a net.addict’s day: Slow day: didn’t have much to do, so spent three hours on usenet. Busy day: managed to work in three hours of usenet. Bad day: barely squeezed in three hours of usenet.
Anonymous
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He is a learned man that understands one subject, a very learned man that understands two.
In Hialmer Day Gould, New Practical Spelling (1905), 14.
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He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.
Quoted, without citation, in Max Dehn, 'Mathematics, 300 B.C.-200 B.C.', The American Mathematical Monthly (Jan 1944), 51, No. 1, 31.
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He [Lord Bacon] appears to have been utterly ignorant of the discoveries which had just been made by Kepler’s calculations … he does not say a word about Napier’s Logarithms, which had been published only nine years before and reprinted more than once in the interval. He complained that no considerable advance had been made in Geometry beyond Euclid, without taking any notice of what had been done by Archimedes and Apollonius. He saw the importance of determining accurately the specific gravities of different substances, and himself attempted to form a table of them by a rude process of his own, without knowing of the more scientific though still imperfect methods previously employed by Archimedes, Ghetaldus and Porta. He speaks of the εὕρηκα of Archimedes in a manner which implies that he did not clearly appreciate either the problem to be solved or the principles upon which the solution depended. In reviewing the progress of Mechanics, he makes no mention either of Archimedes, or Stevinus, Galileo, Guldinus, or Ghetaldus. He makes no allusion to the theory of Equilibrium. He observes that a ball of one pound weight will fall nearly as fast through the air as a ball of two, without alluding to the theory of acceleration of falling bodies, which had been made known by Galileo more than thirty years before. He proposed an inquiry with regard to the lever,—namely, whether in a balance with arms of different length but equal weight the distance from the fulcrum has any effect upon the inclination—though the theory of the lever was as well understood in his own time as it is now. … He speaks of the poles of the earth as fixed, in a manner which seems to imply that he was not acquainted with the precession of the equinoxes; and in another place, of the north pole being above and the south pole below, as a reason why in our hemisphere the north winds predominate over the south.
From Spedding’s 'Preface' to De Interpretations Naturae Proœmium, in The Works of Francis Bacon (1857), Vol. 3, 511-512. [Note: the Greek word “εὕρηκα” is “Eureka” —Webmaster.]
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Human evolution is nothing else but the natural continuation, at a collective level, of the perennial and cumulative process of “psychogenetic” arrangement of matter which we call life. … The whole history of mankind has been nothing else (and henceforth it will never be anything else) but an explosive outburst of ever-growing cerebration. … Life, if fully understood, is not a freak in the universe—nor man a freak in life. On the contrary, life physically culminates in man, just as energy physically culminates in life.
(1952). As quoted in Stephen Jay Gould, Hen's Teeth and Horse's Toes: Further Reflections in Natural History (1984, 1994), 246.
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I am not insensible to natural beauty, but my emotional joys center on the improbable yet sometimes wondrous works of that tiny and accidental evolutionary twig called Homo sapiens. And I find, among these works, nothing more noble than the history of our struggle to understand nature—a majestic entity of such vast spatial and temporal scope that she cannot care much for a little mammalian afterthought with a curious evolutionary invention, even if that invention has, for the first time in so me four billion years of life on earth, produced recursion as a creature reflects back upon its own production and evolution. Thus, I love nature primarily for the puzzles and intellectual delights that she offers to the first organ capable of such curious contemplation.
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I am of the decided opinion, that mathematical instruction must have for its first aim a deep penetration and complete command of abstract mathematical theory together with a clear insight into the structure of the system, and doubt not that the instruction which accomplishes this is valuable and interesting even if it neglects practical applications. If the instruction sharpens the understanding, if it arouses the scientific interest, whether mathematical or philosophical, if finally it calls into life an esthetic feeling for the beauty of a scientific edifice, the instruction will take on an ethical value as well, provided that with the interest it awakens also the impulse toward scientific activity. I contend, therefore, that even without reference to its applications mathematics in the high schools has a value equal to that of the other subjects of instruction.
In 'Ueber das Lehrziel im mathemalischen Unterricht der höheren Realanstalten', Jahresbericht der Deutschen Mathematiker Vereinigung, 2, 192. (The Annual Report of the German Mathematical Association. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 73.
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I am satisfied with the mystery of life’s eternity and with a knowledge, a sense, of the marvelous structure of existence–as well as the humble attempt to understand even a tiny portion of the Reason that manifests itself in nature.
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I can never satisfy myself until I can make a mechanical model of a thing. If I can make a mechanical model, I can understand it. As long as I cannot make a mechanical model all the way through I cannot understand.
From stenographic report by A.S. Hathaway of the Lecture 20 Kelvin presented at Johns Hopkins University, Baltimore, on 'Molecular Dynamics and the Wave Theory of Light' (1884), 270-271. (Hathaway was a Mathematics fellow there.) This remark is not included in the first typeset publication—a revised version, printed twenty years later, in 1904, as Lord Kelvin’s Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light. The original notes were reproduced by the “papyrograph” process. They are excerpted in Pierre Maurice Marie Duhem, Essays in the History and Philosophy of Science (1996), 55.
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I can see him [Sylvester] now, with his white beard and few locks of gray hair, his forehead wrinkled o’er with thoughts, writing rapidly his figures and formulae on the board, sometimes explaining as he wrote, while we, his listeners, caught the reflected sounds from the board. But stop, something is not right, he pauses, his hand goes to his forehead to help his thought, he goes over the work again, emphasizes the leading points, and finally discovers his difficulty. Perhaps it is some error in his figures, perhaps an oversight in the reasoning. Sometimes, however, the difficulty is not elucidated, and then there is not much to the rest of the lecture. But at the next lecture we would hear of some new discovery that was the outcome of that difficulty, and of some article for the Journal, which he had begun. If a text-book had been taken up at the beginning, with the intention of following it, that text-book was most likely doomed to oblivion for the rest of the term, or until the class had been made listeners to every new thought and principle that had sprung from the laboratory of his mind, in consequence of that first difficulty. Other difficulties would soon appear, so that no text-book could last more than half of the term. In this way his class listened to almost all of the work that subsequently appeared in the Journal. It seemed to be the quality of his mind that he must adhere to one subject. He would think about it, talk about it to his class, and finally write about it for the Journal. The merest accident might start him, but once started, every moment, every thought was given to it, and, as much as possible, he read what others had done in the same direction; but this last seemed to be his real point; he could not read without finding difficulties in the way of understanding the author. Thus, often his own work reproduced what had been done by others, and he did not find it out until too late.
A notable example of this is in his theory of cyclotomic functions, which he had reproduced in several foreign journals, only to find that he had been greatly anticipated by foreign authors. It was manifest, one of the critics said, that the learned professor had not read Rummer’s elementary results in the theory of ideal primes. Yet Professor Smith’s report on the theory of numbers, which contained a full synopsis of Kummer’s theory, was Professor Sylvester’s constant companion.
This weakness of Professor Sylvester, in not being able to read what others had done, is perhaps a concomitant of his peculiar genius. Other minds could pass over little difficulties and not be troubled by them, and so go on to a final understanding of the results of the author. But not so with him. A difficulty, however small, worried him, and he was sure to have difficulties until the subject had been worked over in his own way, to correspond with his own mode of thought. To read the work of others, meant therefore to him an almost independent development of it. Like the man whose pleasure in life is to pioneer the way for society into the forests, his rugged mind could derive satisfaction only in hewing out its own paths; and only when his efforts brought him into the uncleared fields of mathematics did he find his place in the Universe.
In Florian Cajori, Teaching and History of Mathematics in the United States (1890), 266-267.
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I can understand your aversion to the use of the term ‘religion’ to describe an emotional and psychological attitude which shows itself most clearly in Spinoza ... I have not found a better expression than ‘religious’ for the trust in the rational nature of reality that is, at least to a certain extent, accessible to human reason.
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I cannot separate land and sea: to me they interfinger like a pattern in a moss agate, positive and negative shapes irrevocably interlocked. My knowledge of this peninsula depends on that understanding: of underwater canyons that are continuations of the land, of the shell fossils far inland that measure continuations of the sea in eons past.
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I cannot tell you the efforts to which I was condemned to understand something of the diagrams of Descriptive Geometry, which I detest.
Epigraph, without citation, in E.T. Bell, Men of Mathematics (1937, 1965), 181.
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I can’t explain it, but spiritually it makes sense— though I don’t understand how it does make sense.
Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 120
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I can’t think of any definition of the words mathematician or scientist that would apply to me. I think of myself as a journalist who knows just enough about mathematics to be able to take low-level math and make it clear and interesting to nonmathematicians. Let me say that I think not knowing too much about a subject is an asset for a journalist, not a liability. The great secret of my column is that I know so little about mathematics that I have to work hard to understand the subject myself. Maybe I can explain things more clearly than a professional mathematician can.
In Scot Morris, 'Interview: Martin Gardner', Omni, 4, No. 4 (Jan 1982), 68.
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I can’t understand why men make all this fuss about Everest—it’s only a mountain.
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I decided to study science and, on arrival at Cambridge, became extremely excited and interested in biochemistry when I first heard about it…. It seemed to me that here was a way to really understand living matter and to develop a more scientific basis to many medical problems.
From biographical sketch in Wilhelm Odelberg (ed.) Les Prix Nobel. The Nobel Prizes 1980, (1981).
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I didn’t arrive at my understanding of the fundamental laws of the universe through my rational mind.
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I do believe that a scientist is a freelance personality. We’re driven by an impulse which is one of curiosity, which is one of the basic instincts that a man has. So we are … driven … not by success, but by a sort of passion, namely the desire of understanding better, to possess, if you like, a bigger part of the truth. I do believe that science, for me, is very close to art.
From 'Asking Nature', collected in Lewis Wolpert and Alison Richards (eds.), Passionate Minds: The Inner World of Scientists (1997), 197.
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I do not think it is possible really to understand the successes of science without understanding how hard it is—how easy it is to be led astray, how difficult it is to know at any time what is the next thing to be done.
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I don’t pretend to understand the universe–it’s much bigger than I am.
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I don’t understand why people insist on pitting concepts of evolution and creation against each other. Why can’t they see that spiritualism and science are one? That bodies evolve and souls evolve and the universe is a fluid package that marries them both in a wonderful package called a human being.
The Art of Racing in the Rain. Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 43
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I have a friendly feeling towards pigs generally, and consider them the most intelligent of beasts, not excepting the elephant and the anthropoid ape—the dog is not to be mentioned in this connection. I also like his disposition and attitude towards all other creatures, especially man. He is not suspicious, or shrinkingly submissive, like horses, cattle, and sheep; nor an impudent devil-may-care like the goat; nor hostile like the goose; nor condescending like the cat; nor a flattering parasite like the dog. He views us from a totally different, a sort of democratic, standpoint as fellow-citizens and brothers, and takes it for granted, or grunted, that we understand his language, and without servility or insolence he has a natural, pleasant, camerados-all or hail-fellow-well-met air with us.
In The Book of a Naturalist (1919), 295-296.
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I have always assumed, and I now assume, that he [Robert Oppenheimer] is loyal to the United States. I believe this, and I shall believe it until I see very conclusive proof to the opposite. … [But] I thoroughly disagreed with him in numerous issues and his actions frankly appeared to me confused and complicated. To this extent I feel that I would like to see the vital interests of this country in hands which I understand better, and therefore trust more.
After Teller paid tribute to Oppenheimer’s talents, especially his “very outstanding achievement” as the wartime organizer and director of Los Alamos, Teller continued his testimony to the Gray board hearings (28 Apr 1954) in the Atomic Energy Commission building, “In the Matter of J. Robert Oppenheimer.” The subject quotes were excerpted from Teller’s answers to their questions. As given in Robert Coughlan, 'Dr. Edward Teller’s Magnificent Obsession', Life (6 Sep 1954), 72-74.
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I have always hated machinery, and the only machine I ever understood was a wheelbarrow, and that but imperfectly.
In 'The Queen of mathematics', The World of Mathematics (1956), 515.
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I have always tried to fit knowledge that I acquired into my understanding of the world. … When something comes along that I don’t understand, that I can’t fit in, that bothers me, I think about it, mull over it, and perhaps ultimately do some work with it. That’s perhaps the reason that I’ve been able to make discoveries in molecular biology.
From interview with Neil A. Campbell, in 'Crossing the Boundaries of Science', BioScience (Dec 1986), 36, No. 11, 739.
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I have before mentioned mathematics, wherein algebra gives new helps and views to the understanding. If I propose these it is not to make every man a thorough mathematician or deep algebraist; but yet I think the study of them is of infinite use even to grown men; first by experimentally convincing them, that to make anyone reason well, it is not enough to have parts wherewith he is satisfied, and that serve him well enough in his ordinary course. A man in those studies will see, that however good he may think his understanding, yet in many things, and those very visible, it may fail him. This would take off that presumption that most men have of themselves in this part; and they would not be so apt to think their minds wanted no helps to enlarge them, that there could be nothing added to the acuteness and penetration of their understanding.
In The Conduct of the Understanding, Sect. 7.
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I have deeply regretted that I did not proceed far enough [as a Cambridge undergraduate] at least to understand something of the great leading principles of mathematics; for men thus endowed seem to have an extra sense.
In Charles Darwin and Francis Darwin (ed.), 'Autobiography', The Life and Letters of Charles Darwin (1887, 1896), Vol. 1, 40.
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I have never really had dreams to fulfil…. You just want to go on looking at these ecosystems and trying to understand them and they are all fascinating. To achieve a dream suggests snatching a prize from the top of a tree and running off with it, and that’s the end of it. It isn’t like that. … What you are trying to achieve is understanding and you don’t do that just by chasing dreams.
From interview with Michael Bond, 'It’s a Wonderful Life', New Scientist (14 Dec 2002), 176, No. 2373, 52.
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I have often noticed that when people come to understand a mathematical proposition in some other way than that of the ordinary demonstration, they promptly say, “Oh, I see. That’s how it must be.” This is a sign that they explain it to themselves from within their own system.
Lichtenberg: A Doctrine of Scattered Occasions: Reconstructed From: Reconstructed From His Aphorisms and Reflections (1959), 291.
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I heard … xenon was a good anesthesia. … I thought, “How can xenon, which doesn’t form any chemical compounds, serve as a general anesthetic? … I lay awake at night for a few minutes before going to sleep, and during the next couple of weeks each night I would think, “…how do anesthetic agents work?" Then I forgot to do it after a while, but I’d trained my unconscious mind to keep this question alive and to call [it] to my consciousness whenever a new idea turned up…. So seven years went by. [One day I] put my feet up on the desk and started reading my mail, and here was a letter from George Jeffrey … an x-ray crystallographer, on his determination of the structure of a hydrate crystal. Immediately I sat up, took my feet off the desk, and said, “I understand anesthesia!” … I spent a year [and] determined the structure of chloroform hydrate, and then I wrote my paper published in June of 1961.
Interview with George B. Kauffman and Laurie M. Kauffman, in 'Linus Pauling: Reflections', American Scientist (Nov-Dec 1994), 82, No. 6, 522-523.
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I read them. Not to grade them. No, I read them to see how I am doing. Where am I failing? What don’t they understand? Why do they give wrong answers? Why do they have some point of view that I don’t think is right? Where am I failing? Where do I need to build up.
In The Essential Deming.
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I remember my first look at the great treatise of Maxwell’s when I was a young man… I saw that it was great, greater and greatest, with prodigious possibilities in its power… I was determined to master the book and set to work. I was very ignorant. I had no knowledge of mathematical analysis (having learned only school algebra and trigonometry which I had largely forgotten) and thus my work was laid out for me. It took me several years before I could understand as much as I possibly could. Then I set Maxwell aside and followed my own course. And I progressed much more quickly… It will be understood that I preach the gospel according to my interpretation of Maxwell.
From translations of a letter (24 Feb 1918), cited in Paul J. Nahin, Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age (2002), 24. Nahin footnotes that the words are not verbatim, but are the result of two translations. Heaviside's original letter in English was quoted, translated in to French by J. Bethenode, for the obituary he wrote, "Oliver Heaviside", in Annales des Posies Telegraphs (1925), 14, 521-538. The quote was retranslated back to English in Nadin's book. Bethenode footnoted that he made the original translation "as literally as possible in order not to change the meaning." Nadin assures that the retranslation was done likewise. Heaviside studyied Maxwell's two-volume Treatise on Electricity and Magnetism.
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I respect Kirkpatrick both for his sponges and for his numinous nummulosphere. It is easy to dismiss a crazy theory with laughter that debars any attempt to understand a man’s motivation–and the nummulosphere is a crazy theory. I find that few men of imagination are not worth my attention. Their ideas may be wrong, even foolish, but their methods often repay a close study ... The different drummer often beats a fruitful tempo.
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I returned and saw under the sun that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all.
Bible
Ecclesiastes 9:11. As given in the King James Version.
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I strive that in public dissection the students do as much as possible so that if even the least trained of them must dissect a cadaver before a group of spectators, he will be able to perform it accurately with his own hands; and by comparing their studies one with another they will properly understand, this part of medicine.
In De Humani Corporis Fabrica Libri Septem [Seven Books on the Structure of the Human Body] (1543), 547. Quoted and trans. in Charles Donald O'Malley, Andreas Vesalius of Brussels, 1514-1564 (1964), 144.
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I think I can safely say that nobody understands quantum mechanics.
'Probability and Uncertainty—the Quantum Mechanical View of Nature', the sixth of his Messenger Lectures (1964), Cornell University. Collected in The Character of Physical Law (1967), 129. Often misquoted in various ways, for example, “If people say they understand quantum mechanics they’re lying,” in Ari Ben-Menahem, Historical Encyclopedia of Natural and Mathematical Sciences (2009), 5699.
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I think that intelligence does not emerge from a handful of very beautiful principles—like physics. It emerges from perhaps a hundred fundamentally different kinds of mechanisms that have to interact just right. So, even if it took only four years to understand them, it might take four hundred years to unscramble the whole thing.
As quoted from an interview with author Jeremy Bernstein, in Science Observed: Essays Out of My Mind (1982), 44.
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I try to make a point not to talk about things I don’t understand—at least the things I do not understand at all.
As quoted in Robert Coughlan, 'Dr. Edward Teller’s Magnificent Obsession', Life (6 Sep 1954), 74.
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I want to argue that the ‘sudden’ appearance of species in the fossil record and our failure to note subsequent evolutionary change within them is the proper prediction of evolutionary theory as we understand it ... Evolutionary ‘sequences’ are not rungs on a ladder, but our retrospective reconstruction of a circuitous path running like a labyrinth, branch to branch, from the base of the bush to a lineage now surviving at its top.
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I wanted to be a scientist from my earliest school days. The crystallizing moment came when I first caught on that stars are mighty suns, and how staggeringly far away they must be to appear to us as mere points of light. I’m not sure I even knew the word science then, but I was gripped by the prospect of understanding how things work, of helping to uncover deep mysteries, of exploring new worlds.
In 'With Science on Our Side', Washington Post (9 Jan 1994).
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I would have my son mind and understand business, read little history, study the mathematics and cosmography; these are good, with subordination to the things of God. … These fit for public services for which man is born.
In Letters and Speeches of Oliver Cromwell (1899), Vol. 1, 371.
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I would picture myself as a virus, or as a cancer cell, for example, and try to sense what it would be like to be either. I would also imagine myself as the immune system, and I would try to reconstruct what I would do as an immune system engaged in combating a virus or cancer cell. When I had played through a series of such scenarios on a particular problem and had acquired new insights, I would design laboratory experiments accordingly… Based upon the results of the experiment, I would then know what question to ask next… When I observed phenomena in the laboratory that I did not understand, I would also ask questions as if interrogating myself: “Why would I do that if I were a virus or a cancer cell, or the immune system?” Before long, this internal dialogue became second nature to me; I found that my mind worked this way all the time.
In Anatomy of Reality: Merging of Intuition and Reason (1983), 7, footnote b, as quoted and cited in Roger Frantz, Two Minds: Intuition and Analysis in the History of Economic Thought (2006), 7.
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If I choose to impose individual blame for all past social ills, there will be no one left to like in some of the most fascinating periods of our history. For example ... if I place every Victorian anti-Semite beyond the pale of my attention, my compass of available music and literature will be pitifully small. Though I hold no shred of sympathy for active persecution, I cannot excoriate individuals who acquiesced passively in a standard societal judgment. Rail instead against the judgment, and try to understand what motivates men of decent will.
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If I do not understand a thing, I keep it before me and I wait.
Gull’s restatement of a Newton quote, as given by the Editor in Preface to Sir William Withey Gull and Theodore Dyke Acland (ed.), A Collection of the Published Writings of William Withey Gull (1896), xvi.
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If in physics there’s something you don’t understand, you can always hide behind the uncharted depths of nature. You can always blame God. You didn’t make it so complex yourself. But if your program doesn’t work, there is no one to hide behind. You cannot hide behind an obstinate nature. If it doesn’t work, you’ve messed up.
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If it were always necessary to reduce everything to intuitive knowledge, demonstration would often be insufferably prolix. This is why mathematicians have had the cleverness to divide the difficulties and to demonstrate separately the intervening propositions. And there is art also in this; for as the mediate truths (which are called lemmas, since they appear to be a digression) may be assigned in many ways, it is well, in order to aid the understanding and memory, to choose of them those which greatly shorten the process, and appear memorable and worthy in themselves of being demonstrated. But there is another obstacle, viz.: that it is not easy to demonstrate all the axioms, and to reduce demonstrations wholly to intuitive knowledge. And if we had chosen to wait for that, perhaps we should not yet have the science of geometry.
In Gottfried Wilhelm Leibnitz and Alfred Gideon Langley (trans.), New Essays Concerning Human Understanding (1896), 413-414.
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If physical science is dangerous, as I have said, it is dangerous because it necessarily ignores the idea of moral evil; but literature is open to the more grievous imputation of recognizing and understanding it too well.
In 'Duties of the Church Towards Knowledge', The Idea of a University Defined and Illustrated (1852, 1873), Discourse 9, 229.
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If the brain were simple enough for us to understand it, we would be too simple to understand it.
Ken Hill
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If we are correct in understanding how evolution actually works, and provided we can survive the complications of war, environmental degradation, and possible contact with interstellar planetary travelers, we will look exactly the same as we do now. We won’t change at all. The species is now so widely dispersed that it is not going to evolve, except by gradualism.
In Pamela Weintraub, The Omni Interviews (1984), 75.
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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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If you want to achieve conservation, the first thing you have to do is persuade people that the natural world is precious, beautiful, worth saving and complex. If people don’t understand that and don’t believe that in their hearts, conservation doesn't stand a chance. That’s the first step, and that is what I do.
From interview with Michael Bond, 'It’s a Wonderful Life', New Scientist (14 Dec 2002), 176, No. 2373, 48.
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Imagine life as a game in which you are juggling five balls in the air. You name them - work, family, health, friends, and spirit - and you’re keeping all of these in the air. You will soon understand that work is a rubber ball. If you drop it, it will bounce back. But the other four balls - family, health, friends, and spirit are made of glass. If you drop one of these, they will be irrevocably scuffed, marked, nicked, damaged, or even shattered. They will never be the same. You must understand that and strive for balance in your life.
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Imagine that … the world is something like a great chess game being played by the gods, and we are observers of the game. … If we watch long enough, we may eventually catch on to a few of the rules…. However, we might not be able to understand why a particular move is made in the game, merely because it is too complicated and our minds are limited…. We must limit ourselves to the more basic question of the rules of the game.
If we know the rules, we consider that we “understand” the world.
In 'Basic Physics', The Feynman Lectures on Physics (1964, 2013), Vol. 1, 2-1.
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In all disciplines in which there is systematic knowledge of things with principles, causes, or elements, it arises from a grasp of those: we think we have knowledge of a thing when we have found its primary causes and principles, and followed it back to its elements. Clearly, then, systematic knowledge of nature must start with an attempt to settle questions about principles.
Aristotle
In Physics Book 1, Chap 1, as translated in J.L. Ackrill, A New Aristotle Reader (1988), 81.
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In all spheres of science, art, skill, and handicraft it is never doubted that, in order to master them, a considerable amount of trouble must be spent in learning and in being trained. As regards philosophy, on the contrary, there seems still an assumption prevalent that, though every one with eyes and fingers is not on that account in a position to make shoes if he only has leather and a last, yet everybody understands how to philosophize straight away, and pass judgment on philosophy, simply because he possesses the criterion for doing so in his natural reason.
From Phänomenologie des Geistes (1807) as translated by J.B. Baillie in 'Preface', The Phenomenology of Mind (1910), Vol. 1, 67.
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In biology, nothing is clear, everything is too complicated, everything is a mess, and just when you think you understand something, you peel off a layer and find deeper complications beneath. Nature is anything but simple.
The Hot Zone
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In mathematics, … and in natural philosophy since mathematics was applied to it, we see the noblest instance of the force of the human mind, and of the sublime heights to which it may rise by cultivation. An acquaintance with such sciences naturally leads us to think well of our faculties, and to indulge sanguine expectations concerning the improvement of other parts of knowledge. To this I may add, that, as mathematical and physical truths are perfectly uninteresting in their consequences, the understanding readily yields its assent to the evidence which is presented to it; and in this way may be expected to acquire the habit of trusting to its own conclusions, which will contribute to fortify it against the weaknesses of scepticism, in the more interesting inquiries after moral truth in which it may afterwards engage.
In Elements of the Philosophy of the Human Mind (1827), Vol. 3, Chap. 1, Sec. 3, 182.
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In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it’s the exact opposite.
As quoted, without citation, in Robert Jungk and James Cleugh (trans.), Brighter Than a Thousand Suns: A Personal History of the Atomic Scientists (1958), 22. The writing of poetry by Robert Oppenheimer was being remarked upon by Dirac, as he was talking with him.
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In the end, we conserve only what we love. We will love only what we understand. We will understand only what we are taught.
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In the light of knowledge attained, the happy achievement seems almost a matter of course, and any intelligent student can grasp it without too much trouble. But the years of anxious searching in the dark, with their intense longing, their alternations of confidence and exhaustion, and the final emergence into the light—only those who have experienced it can understand that.
Quoted in Banesh Hoffmann and Helen Dukas, Albert Einstein: Creator and Rebel (1972), 124.
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In the medical field [scientific ignorance] could lead to horrendous results. People who don’t understand the difference between a controlled experiment and claims by some quack may die as a result of not taking medical science seriously. One of the most damaging examples of pseudoscience is false memory syndrome. I’m on the board of a foundation exposing this problem.
As quoted by Lawrence Toppman, 'Mastermind', The Charlotte Observer (20 Jun 1993), 6E. As quoted and cited in Dana Richards, 'Martin Gardner: A “Documentary”', collected in Elwyn R. Berlekamp and Tom Rodgers (ed.) The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner (1999), 11.
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In the next twenty centuries … humanity may begin to understand its most baffling mystery—where are we going? The earth is, in fact, traveling many thousands of miles per hour in the direction of the constellation Hercules—to some unknown destination in the cosmos. Man must understand his universe in order to understand his destiny. Mystery, however, is a very necessary ingredient in our lives. Mystery creates wonder and wonder is the basis for man’s desire to understand. Who knows what mysteries will be solved in our lifetime, and what new riddles will become the challenge of the new generation? Science has not mastered prophesy. We predict too much for the next year yet far too little for the next ten. Responding to challenges is one of democracy’s great strengths. Our successes in space can be used in the next decade in the solution of many of our planet’s problems.
In a speech to a Joint Meeting of the Two Houses of Congress to Receive the Apollo 11 Astronauts (16 Sep 1969), in the Congressional Record.
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Increasingly, our leaders must deal with dangers that threaten the entire world, where an understanding of those dangers and the possible solutions depend on a good grasp of science. The ozone layer, the greenhouse effect, acid rain, questions of diet and of heredity--all require scientific literacy. Can Americans choose the proper leaders and support the proper programs if they are scientifically illiterate?
articles.latimes.com/1989-03-31/news/vw-543_1_scientific-literacy
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Indeed, if one understands by algebra the application of arithmetic operations to composite magnitudes of all kinds, whether they be rational or irrational number or space magnitudes, then the learned Brahmins of Hindostan are the true inventors of algebra.
In Geschichte der Mathematik im Altertum und im Mittelalter (1874), 195. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 284. From the original German, “Ja, wenn man unter Algebra die Anwendung arithmetischer Operationen auf zusammengesetzte Grössen aller Art, mögen sie rationale oder irrationale Zahl- oder Raumgrössen sein, versteht, so sind die gelehrten Brahmanen Hindustans die wahren Erfinder der Algebra.”
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Infinities and indivisibles transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness; Imagine what they are when combined.
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It feels unacceptable to many people even to think of having a cosmology based on science. … They see fanciful origin stories as spicing up the culture. … Aspects of many origin stories can enrich our understanding of the scientific picture, but they cannot take its place.
As co-author with Nancy Ellen Abrams, in The View from the Center of the Universe: Discovering Our Extraordinary Place in the Cosmos (2006), 85.
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It has always seemed to me extreme presumptuousness on the part of those who want to make human ability the measure of what nature can and knows how to do, since, when one comes down to it, there is not one effect in nature, no matter how small, that even the most speculative minds can fully understand.
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It is a great pity Aristotle had not understood mathematics as well as Mr. Newton, and made use of it in his natural philosophy with good success: his example had then authorized the accommodating of it to material things.
In Second Reply to the Bishop of Worcester.
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It is an irony of fate that I myself have been the recipient of excessive admiration and reverence from my fellow-beings, through no fault, and no merit, of my own. The cause of this may well be the desire, unattainable for many, to understand the few ideas to which I have with my feeble powers attained through ceaseless struggle. I am quite aware that for any organisation to reach its goals, one man must do the thinking and directing and generally bear the responsibility. But the led must not be coerced, they must be able to choose their leader.
In 'What I Believe', Forum and Century (1930), 84, 193-194.
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It is exceptional that one should be able to acquire the understanding of a process without having previously acquired a deep familiarity with running it, with using it, before one has assimilated it in an instinctive and empirical way. Thus any discussion of the nature of intellectual effort in any field is difficult, unless it presupposes an easy, routine familiarity with that field. In mathematics this limitation becomes very severe.
In 'The Mathematician', Works of the Mind (1947), 1, No. 1. Collected in James Roy Newman (ed.), The World of Mathematics (1956), Vol. 4, 2053.
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It is from this absolute indifference and tranquility of the mind, that mathematical speculations derive some of their most considerable advantages; because there is nothing to interest the imagination; because the judgment sits free and unbiased to examine the point. All proportions, every arrangement of quantity, is alike to the understanding, because the same truths result to it from all; from greater from lesser, from equality and inequality.
In On the Sublime and Beautiful, Part 3, sect. 2.
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It is not the fruits of scientific research that elevate man and enrich his nature but the urge to understand, the intellectual work, creative or receptive.
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It is the individual only who is timeless. Societies, cultures, and civilizations - past and present - are often incomprehensible to outsiders, but the individual’s hunger, anxieties, dreams, and preoccupations have remained unchanged through the millennia. Thus, we are up against the paradox that the individual who is more complex, unpredictable, and mysterious than any communal entity is the one nearest to our understanding; so near that even the interval of millennia cannot weaken our feeling of kinshiIf in some manner the voice of an individual reaches us from the remotest distance of time, it is a timeless voice speaking about ourselves.
In Reflections on the Human Condition (1973), 97.
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It is the intact and functioning organism on which natural selection operates. Organisms are therefore the central element of concern to the biologist who aspires to a broad and integrated understanding of biology.
From 'Interspecific comparison as a tool for ecological physiologists', collected in M.E. Feder, A.F. Bennett, W.W. Burggren, and R.B. Huey, (eds.), New Directions in Ecological Physiology (1987), 15.
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It is the man of science, eager to have his every opinion regenerated, his every idea rationalised, by drinking at the fountain of fact, and devoting all the energies of his life to the cult of truth, not as he understands it, but as he does not understand it, that ought properly to be called a philosopher. To an earlier age knowledge was power—merely that and nothing more—to us it is life and the summum bonum.
As quoted in Sir Richard Gregory, Discovery: Or, The Spirit and Service of Science (1916), 24.
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It is therefore easy to see why the churches have always fought science and persecuted its devotees. On the other hand, I maintain that the cosmic religious feeling is the strongest and noblest motive for scientific research. Only those who realize the immense efforts and, above all, the devotion without which pioneer work in theoretical science cannot be achieved are able to grasp the strength of the emotion out of which alone such work, remote as it is from the immediate realities of life, can issue. What a deep conviction of the rationality of the universe and what a yearning to understand, were it but a feeble reflection of the mind revealed in this world, Kepler and Newton must have had to enable them to spend years of solitary labor in disentangling the principles of celestial mechanics! Those whose acquaintance with scientific research is derived chiefly from its practical results easily develop a completely false notion of the mentality of the men who, surrounded by a skeptical world, have shown the way to kindred spirits scattered wide through the world and through the centuries. Only one who has devoted his life to similar ends can have a vivid realization of what has inspired these men and given them the strength to remain true to their purpose in spite of countless failures. It is cosmic religious feeling that gives a man such strength. A contemporary has said, not unjustly, that in this materialistic age of ours the serious scientific workers are the only profoundly religious people.
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It might be thought … that evolutionary arguments would play a large part in guiding biological research, but this is far from the case. It is difficult enough to study what is happening now. To figure out exactly what happened in evolution is even more difficult. Thus evolutionary achievements can be used as hints to suggest possible lines of research, but it is highly dangerous to trust them too much. It is all too easy to make mistaken inferences unless the process involved is already very well understood.
In What Mad Pursuit: A Personal View of Scientific Discovery (1988), 138-139.
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It must happen that in some cases the author is not understood, or is very imperfectly understood; and the question is what is to be done. After giving a reasonable amount of attention to the passage, let the student pass on, reserving the obscurity for future efforts. … The natural tendency of solitary students, I believe, is not to hurry away prematurely from a hard passage, but to hang far too long over it; the just pride that does not like to acknowledge defeat, and the strong will that cannot endure to be thwarted, both urge to a continuance of effort even when success seems hopeless. It is only by experience we gain the conviction that when the mind is thoroughly fatigued it has neither the power to continue with advantage its course in .an assigned direction, nor elasticity to strike out a new path; but that, on the other hand, after being withdrawn for a time from the pursuit, it may return and gain the desired end.
In 'Private Study of Mathematics', Conflict of Studies and other Essays (1873), 68.
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It must, however, be confessed that this species of scepticism, when more moderate, may be understood in a very reasonable sense, and is a necessary preparative to the study of philosophy by preserving a proper impartiality in our judgments and weaning our mind from all those prejudices which we may have imbibed from education or rash opinion.
From 'An Inquiry Concerning Human Understanding', Sec. 7, Pt. 1, collected in Essays and Treatises on Various Subjects: With a Brief Sketch of the (1849), 85.
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It took me so long to understand what I was writing about, that I knew how to write about it so most readers would understand it.
As quoted in Alex Bellos, 'Martin Gardner Obituary', The Guardian (27 May 2010)
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It usually develops that after much laborious and frustrating effort the investigator of environmental physiology succeeds in proving that the animal in question can actually exist where it lives. It is always somewhat discouraging for an investigator to realize that his efforts can be made to appear so trite, but this statement does not belittle the ecological physiologist. If his data assist the understanding of the ways in which an animal manages to live where it does, he makes an important contribution to the study of distribution, for the present is necessarily a key to the past.”
From 'The role of physiology in the distribution of terrestrial vertebrates', collected in C.L. Hubbs (ed.), Zoogeography: Publ. 51 (1958), 84.
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It was his [Leibnitz’s] love of method and order, and the conviction that such order and harmony existed in the real world, and that our success in understanding it depended upon the degree and order which we could attain in our own thoughts, that originally was probably nothing more than a habit which by degrees grew into a formal rule. This habit was acquired by early occupation with legal and mathematical questions. We have seen how the theory of combinations and arrangements of elements had a special interest for him. We also saw how mathematical calculations served him as a type and model of clear and orderly reasoning, and how he tried to introduce method and system into logical discussions, by reducing to a small number of terms the multitude of compound notions he had to deal with. This tendency increased in strength, and even in those early years he elaborated the idea of a general arithmetic, with a universal language of symbols, or a characteristic which would be applicable to all reasoning processes, and reduce philosophical investigations to that simplicity and certainty which the use of algebraic symbols had introduced into mathematics.
A mental attitude such as this is always highly favorable for mathematical as well as for philosophical investigations. Wherever progress depends upon precision and clearness of thought, and wherever such can be gained by reducing a variety of investigations to a general method, by bringing a multitude of notions under a common term or symbol, it proves inestimable. It necessarily imports the special qualities of number—viz., their continuity, infinity and infinite divisibility—like mathematical quantities—and destroys the notion that irreconcilable contrasts exist in nature, or gaps which cannot be bridged over. Thus, in his letter to Arnaud, Leibnitz expresses it as his opinion that geometry, or the philosophy of space, forms a step to the philosophy of motion—i.e., of corporeal things—and the philosophy of motion a step to the philosophy of mind.
In Leibnitz (1884), 44-45. [The first sentence is reworded to better introduce the quotation. —Webmaster]
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It would be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said, “Yes, it is true, but you won’t be able to understand the proof.”
As quoted in John Horgan, 'The Death of Proof', Scientific American (Oct 1993), 269, No. 4, 103.
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It would not be difficult to come to an agreement as to what we understand by science. Science is the century-old endeavor to bring together by means of systematic thought the perceptible phenomena of this world into as thoroughgoing an association as possible. To put it boldly, it is the attempt at the posterior reconstruction of existence by the process of conceptualization. But when asking myself what religion is I cannot think of the answer so easily. And even after finding an answer which may satisfy me at this particular moment, I still remain convinced that I can never under any circumstances bring together, even to a slight extent, the thoughts of all those who have given this question serious consideration.
…...
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It would seem at first sight as if the rapid expansion of the region of mathematics must be a source of danger to its future progress. Not only does the area widen but the subjects of study increase rapidly in number, and the work of the mathematician tends to become more and more specialized. It is, of course, merely a brilliant exaggeration to say that no mathematician is able to understand the work of any other mathematician, but it is certainly true that it is daily becoming more and more difficult for a mathematician to keep himself acquainted, even in a general way, with the progress of any of the branches of mathematics except those which form the field of his own labours. I believe, however, that the increasing extent of the territory of mathematics will always be counteracted by increased facilities in the means of communication. Additional knowledge opens to us new principles and methods which may conduct us with the greatest ease to results which previously were most difficult of access; and improvements in notation may exercise the most powerful effects both in the simplification and accessibility of a subject. It rests with the worker in mathematics not only to explore new truths, but to devise the language by which they may be discovered and expressed; and the genius of a great mathematician displays itself no less in the notation he invents for deciphering his subject than in the results attained. … I have great faith in the power of well-chosen notation to simplify complicated theories and to bring remote ones near and I think it is safe to predict that the increased knowledge of principles and the resulting improvements in the symbolic language of mathematics will always enable us to grapple satisfactorily with the difficulties arising from the mere extent of the subject.
In Presidential Address British Association for the Advancement of Science, Section A., (1890), Nature, 42, 466.
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It’s a vacuous answer … To say that “God made the world” is simply a more or less sophisticated way of saying that we don't understand how the universe originated. A god, in so far as it is anything, is an admission of ignorance.
From Speech, Annual Meeting of the British Association for the Advancement of Science. As quoted in 'Professor Says Science Rules Out Belief in God', The Telegraph (11 Sep 1996). As cited in John C. Weaver and John David Weaver, Christianity and Science (1973, 1984), 22.
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It’s among the intelligentsia … that we often find the glib compulsion to explain everything and to understand nothing.
The Rape of the Mind World 56
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It’s human nature to stretch, to go, to see, to understand. Exploration is not a choice, really; it’s an imperative.
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It’s very good jam, said the Queen.
“Well, I don’t want any to-day, at any rate.”
“You couldn’t have it if you did want it,” the Queen said.
“The rule is jam tomorrow and jam yesterday but never jam to-day.”
“It must come sometimes to “jam to-day,” Alice objected.
“No it can’t,” said the Queen.
“It’s jam every other day; to-day isn’t any other day, you know.”
“I don’t understand you,” said Alice. “It’s dreadfully confusing.”
From Through the Looking Glass. In Alice’s Adventures in Wonderland And, Through the Looking Glass (1898), 149.
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Knowing Pains
I studied parts
of a flower
to understand
its flowering.
I learned much
about my limits.
I had forgotten
Earth and climate.
Poem, in Starting Points: Poems (1971), 38. As quoted in Arthur Lerner, 'Poetry Therapy', The American Journal of Nursing (Aug 1973), 73, No. 8, 1338.
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Learning acquired in youth arrests the evil of old age; and if you understand that old age has wisdom for its food, you will so conduct yourself in youth that your old age will not lack for nourishment.
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Let us first understand the facts, and then we may seek the cause.
Aristotle
Cited as De Part., I., l. 639 in George Henry Lewes, Aristotle: A Chapter from the History of Science including analyses of Aristotle's scientific writings (1864),
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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 too complicated to permit a complete understanding through the study of whole organisms. Only by simplifying a biological problem—breaking it down into a multitude of individual problems—can you get the answers.
From interview with Neil A. Campbell, in 'Crossing the Boundaries of Science', BioScience (Dec 1986), 36, No. 11, 738.
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Little can be understood of even the simplest phenomena of nature without some knowledge of mathematics, and the attempt to penetrate deeper into the mysteries of nature compels simultaneous development of the mathematical processes.
In Teaching of Mathematics in the Elementary and the Secondary School (1906), 16.
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Logic has borrowed the rules of geometry without understanding its power. … I am far from placing logicians by the side of geometers who teach the true way to guide the reason. … The method of avoiding error is sought by every one. The logicians profess to lead the way, the geometers alone reach it, and aside from their science there is no true demonstration.
From De l’Art de Persuader, (1657). Pensées de Pascal (1842), Part 1, Article 3, 41-42. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 202. From the original French, “La logique a peut-être emprunté les règles de la géométrie sans en comprendre la force … je serai bien éloigné de les mettre en parallèle avec les géomètres, qui apprennent la véritable méthode de conduire la raison. … La méthode de ne point errer est recherchée de tout le monde. Les logiciens font profession d'y conduire, les géomètres seuls y arrivent; et, hors de leur science …, il n'y a point de véritables démonstrations ….”
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Look deep into nature, and then you will understand everything better.
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Man’s law changes with his understanding of man. Only the laws of the spirit remain always the same.
Crow
Proverb. Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 19
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Mathematical language is not only the simplest and most easily understood of any, but the shortest also.
In Works of Henry, Lord Brougham: Vol. 7: Rhetorical and literary Dissertions and Addresses (1856), 317, footnote.
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Mathematicians are only dealing with the structure of reasoning, and they do not really care what they are talking about. They do not even need to know what they are talking about … But the physicist has meaning to all his phrases. … In physics, you have to have an understanding of the connection of words with the real world.
In The Character of Physical Law (1965), 55.
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Mathematicians seem to have no difficulty in creating new concepts faster than the old ones become well understood.
Acceptance Speech for the Kyoto Prize (1991), 'A scientist by choice'. On kyotoprize.org website.
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Mathematics is often considered a difficult and mysterious science, because of the numerous symbols which it employs. Of course, nothing is more incomprehensible than a symbolism which we do not understand. … But this is not because they are difficult in themselves. On the contrary they have invariably been introduced to make things easy. … [T]he symbolism is invariably an immense simplification. It … represents an analysis of the ideas of the subject and an almost pictorial representation of their relations to each other.
In Introduction to Mathematics (1911), 59-60.
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Mathematics make the mind attentive to the objects which it considers. This they do by entertaining it with a great variety of truths, which are delightful and evident, but not obvious. Truth is the same thing to the understanding as music to the ear and beauty to the eye. The pursuit of it does really as much gratify a natural faculty implanted in us by our wise Creator as the pleasing of our senses: only in the former case, as the object and faculty are more spiritual, the delight is more pure, free from regret, turpitude, lassitude, and intemperance that commonly attend sensual pleasures.
In An Essay on the Usefulness of Mathematical Learning (1701), 3-4.
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Mathematics, the science of the ideal, becomes the means of investigating, understanding and making known the world of the real. The complex is expressed in terms of the simple. From one point of view mathematics may be defined as the science of successive substitutions of simpler concepts for more complex.
In A Scrap-book of Elementary Mathematics (1908), 215.
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Mathematics, too, is a language, and as concerns its structure and content it is the most perfect language which exists, superior to any vernacular; indeed, since it is understood by every people, mathematics may be called the language of languages. Through it, as it were, nature herself speaks; through it the Creator of the world has spoken, and through it the Preserver of the world continues to speak.
In Die Mathematik die Fackeltragerin einer neutn Zeit (1889), 5. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 194. From the original German, “Denn auch die Mathematik ist eine Sprache, und zwar nach Bau und Inhalt die vollkommenste Sprache, die es giebt, höher als jede Volkssprache; ja, weil alle Völker sie verstehen, kann sie die Sprache der Sprachen heißen. In ihr spricht sozusagen die Natur selbst, in ihr hat der Schöpfer der Welt geredet und in ihr redet noch immer ihr Erhalter.”
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Medicine is not only a science; it is also an art. It does not consist of compounding pills and plasters; it deals with the very processes of life, which must be understood before they may be guided.
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Men are of three different capacities; one understands intuitively, another only understands so far as it is explained; and a third understands neither of himself nor by explanation; the first is excellent, the second commendable, and the third altogether useless.
Collected, without citation, in Day's Collacon: an Encyclopaedia of Prose Quotations (1884), 87.
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Men of science, fit to teach, hardly exist. There is no demand for such men. The sciences make up life; they are important to life. The highly educated man fails to understand the simplest things of science, and has no peculiar aptitude for grasping them. I find the grown-up mind coming back to me with the same questions over and over again.
Giving Evidence (18 Nov 1862) to the Public Schools Commission. As quoted in John L. Lewis, 125 Years: The Physical Society & The Institute of Physics (1999), 168.
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Most impediments to scientific understanding are conceptual locks, not factual lacks. Most difficult to dislodge are those biases that escape our scrutiny because they seem so obviously, even ineluctably, just. We know ourselves best and tend to view other creatures as mirrors of our own constitution and social arrangements. (Aristotle, and nearly two millennia of successors, designated the large bee that leads the swarm as a king.)
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My grandfather pioneered exploration of what he called “our water planet,” then my father sought to understand the human connection, and now, as part of the third generation, I’m dedicated to not only raising awareness but also to empowering people to take action.
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Mystery creates wonder and wonder is the basis of man’s desire to understand.
In a speech to Congress (16 Sep 1969). This is part of a longer quote given on the Neil Armstrong Quotes page of this website, beginning “In the next twenty centuries….”
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Natural science is one of man’s weapons in his fight for freedom. For the purpose of attaining freedom in the world of nature, man must use natural science to understand, conquer, and change nature and thus attain freedom from nature.
In Speech (5 Feb 1940) to the Natural Science Research Society for the Border Regions, The Thoughts of Chairman Mao Tse-tung (1967) 127.
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Nature never “fails.” Nature complies with its own laws. Nature is the law. When Man lacks understanding of Nature’s laws and a Man-contrived structure buckles unexpectedly, it does not fail. It only demonstrates that Man did not understand Nature’s laws and behaviors. Nothing failed. Man’s knowledge or estimating was inadequate.
In "How Little I Know", in Saturday Review (12 Nov 1966), 152. Excerpted in Buckminster Fuller and Answar Dil, Humans in Universe (1983), 31.
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No one believes the results of the computational modeler except the modeler, for only he understands the premises. No one doubts the experimenter’s results except the experimenter, for only he knows his mistakes.
Anonymous
See a similar idea expressed by W.I.B. Beveredge, beginning “No one believes an hypothesis…” on the W.I.B. Beveredge Quotes web page on this site.
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No one can know and understand everything. Even individual scientists are ignorant about most of the body of scientific knowledge, and it is not simply that biologists do not understand quantum mechanics.
From review, 'Billions and Billions of Demons', of the book, The Demon-Haunted World: Science as a Candle in the Dark, by Carl Sagan, in New York Review of Books (9 Jan 1997).
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No one really understood music unless he was a scientist, her father had declared, and not just a scientist, either, oh, no, only the real ones, the theoreticians, whose language mathematics. She had not understood mathematics until he had explained to her that it was the symbolic language of relationships. “And relationships,” he had told her, “contained the essential meaning of life.”
In The Goddess Abides, (1972), 20.
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No theory of physics that deals only with physics will ever explain physics. I believe that as we go on trying to understand the universe, we are at the same time trying to understand man.
In The Intellectual Digest (June 1973), as quoted and cited in Mark Chandos, 'Philosophical Essay: Story Theory", Kosmoautikon: Exodus From Sapiens (2015).
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Notable enough, however, are the controversies over the series 1 – 1 + 1 – 1 + 1 – … whose sum was given by Leibniz as 1/2, although others disagree. … Understanding of this question is to be sought in the word “sum”; this idea, if thus conceived—namely, the sum of a series is said to be that quantity to which it is brought closer as more terms of the series are taken—has relevance only for convergent series, and we should in general give up the idea of sum for divergent series.
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Nothing can be believed unless it is first understood; and that for any one to preach to others that which either he has not understood nor they have understood is absurd.
From Historia Calamitatum, Chap. 9. As translated in Arthur Penrhyn Stanley, 'The Word Amen' reprinted from The Independent in Friends' Intelligencer (1872), Vol. 28, 575.
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Nothing in this world is to be feared … only understood.
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Nothing without understanding would ever be more beauteous than with understanding.
Plato
From 'Timaeus', St. III, 30b, in A E Taylor, Plato: Timaeus and Critias (2013), 27.
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Numbers are intellectual witnesses that belong only to mankind.
From the French original, “Le nombre est un témoin intellectuel qui n’appartient qu’à l’homme”, in Oeuvres illustrées de Balzac (1852), Vol. 3, Louis Lambert, 28. As translated in Robert Collison (ed.), Dictionary of Foreign Quotations (1980), 223.
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Obviously, what our age has in common with the age of the Reformation is the fallout of disintegrating values. What needs explaining is the presence of a receptive audience. More significant than the fact that poets write abstrusely, painters paint abstractly, and composers compose unintelligible music is that people should admire what they cannot understand; indeed, admire that which has no meaning or principle.
In Reflections on the Human Condition (1973), 62.
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Once you have mastered time, you will understand how true it is that most people overestimate what they can accomplish in a year-and underestimate what they can achieve in a decade!
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One can argue that mathematics is a human activity deeply rooted in reality, and permanently returning to reality. From counting on one’s fingers to moon-landing to Google, we are doing mathematics in order to understand, create, and handle things, … Mathematicians are thus more or less responsible actors of human history, like Archimedes helping to defend Syracuse (and to save a local tyrant), Alan Turing cryptanalyzing Marshal Rommel’s intercepted military dispatches to Berlin, or John von Neumann suggesting high altitude detonation as an efficient tactic of bombing.
In 'Mathematical Knowledge: Internal, Social and Cultural Aspects', Mathematics As Metaphor: Selected Essays (2007), 3.
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One may understand the cosmos, but never the ego; the self is more distant than any star.
In Orthodoxy (1908), 96.
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One would have to have been brought up in the “spirit of militarism” to understand the difference between Hiroshima and Nagasaki on the one hand, and Auschwitz and Belsen on the other. The usual reasoning is the following: the former case is one of warfare, the latter of cold-blooded slaughter. But the plain truth is that the people involved are in both instances nonparticipants, defenseless old people, women, and children, whose annihilation is supposed to achieve some political or military objective.… I am certain that the human race is doomed, unless its instinctive detestation of atrocities gains the upper hand over the artificially constructed judgment of reason.
Max Born
In The Born-Einstein Letters: Correspondence Between Albert Einstein and Max Born (1971), 205. Born’s commentary (at age 86) added for the book, printed after letter to Albert Einstein, 8 Nov 1953.
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Only for you, children of doctrine and learning, have we written this work. Examine this book, ponder the meaning we have dispersed in various places and gathered again; what we have concealed in one place we have disclosed in another, that it may be understood by your wisdom.
In De Occulta Philosophia (1531), Vol. 3, 65. As quoted and cited in epigraph, Umberto Eco and William Weaver (trans.), Foucault’s Pendulum (2007), Front matter before title page.
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Our time is distinguished by wonderful achievements in the fields of scientific understanding and the technical application of those insights. Who would not be cheered by this? But let us not forget that human knowledge and skills alone cannot lead humanity to a happy and dignified life. Humanity has every reason to place the proclaimers of high moral standards and values above the discoverers of objective truth. What humanity owes to personalities like Buddha, Moses, and Jesus ranks for me higher than all the achievements of the inquiring constructive mind.
(Sep 1937). In Helen Dukas and Banesh Hoffman (ed.), Albert Einstein, the Human Side by (1979).
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Our ultimate end must be precisely what Dr. Pauling says, peace based on agreement, upon understanding, on universally agreed and enforced law. I think this is a wonderful idea, but peace based on force buys us the necessary time, and in this time we can work for better understanding, for closer collaboration.
From debate (20 Feb 1958) between Linus Pauling and Edward Teller on WQED-TV, San Francisco. Transcript published as Fallout and Disarmament: The Pauling-Teller Debate (1958). Reprinted in 'Fallout and Disarmament: A Debate between Linus Pauling and Edward Teller', Daedalus (Spring 1958), 87, No. 2, 160.
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Our understanding of the causes of biological diversity is still crude. The science addressing it can be generously put at about the level of physics in the late nineteenth century.
In 'Edward O. Wilson: The Biological Diversity Crisis: A Challenge to Science', Issues in Science and Technology (Fall 1985), 2, No. 1, 24.
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Peace cannot be achieved through violence, it can only be attained through understanding.
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Peace cannot be kept by force. It can only be achieved by understanding.
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Peace means creative giving, making a contribution to the betterment of our own people and to the betterment of people throughout the world. And who better understands that than the men and women of American agriculture?
'Remarks to Farm Leaders Participating in the “Salute to Agriculture”' (7 May 1971), Public Papers of the Presidents of the United States: Richard M. Nixon (1972), 627.
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People still do not understand that a live fish is more valuable than a dead one, and that destructive fishing techniques are taking a wrecking ball to biodiversity.
In 'Can We Stop Killing Our Oceans Now, Please?', Huffington Post (14 Aug 2013).
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Philosophy is written in that great book that lies before our gaze—I mean the universe—but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written.
In Francis Crick, The Astonishing Hypothesis: the Scientific Search for the Soul (1995), 203.
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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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Physicists are people, differing from the common run of humanity only in that from time to time they tend to speak a strange language of their own, much of which they understand.
In 'A Newsman Looks at Physicists', Physics Today (May 1948), 1, No. 1, 15.
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Physics is becoming so unbelievably complex that it is taking longer and longer to train a physicist. It is taking so long, in fact, to train a physicist to the place where he understands the nature of physical problems that he is already too old to solve them.
As quoted by Colin Pittendrigh (1971). In George C. Beakley, Ernest G. Chilton, Introduction to Engineering Design and Graphics (1973), 40
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Prayer is not an old woman’s idle amusement. Properly understood and applied, it is the most potent instrument of action.
Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 178
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Psychoanalysis has changed American psychiatry from a diagnostic to a therapeutic science, not because so many patients are cured by the psychoanalytic technique, but because of the new understanding of psychiatric patients it has given us and the new and different concepts of illness and health.
News summaries 29 Apr 56
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Science and religion are not at odds. Science is simply too young to understand.
Dan Brown
Angels & Demons. Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 35
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Science arouses a soaring sense of wonder. But so does pseudoscience. Sparse and poor popularizations of science abandon ecological niches that pseudoscience promptly fills. If it were widely understood that claims to knowledge require adequate evidence before they can be accepted, there would be no room for pseudoscience.
The Demon-Haunted World: Science as a Candle in the Dark (1996), 6.
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Science is what we understand well enough to explain to a computer. Art is everything else we do.
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Science must be taught well, if a student is to understand the coming decades he must live through.
Epigraph in Isaac Asimov’s Book of Science and Nature Quotations (1988), 255.
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Science not only purifies the religious impulse of the dross of its anthropomorphism but also contributes to a religious spiritualization of our understanding of life.
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Scientific findings do not threaten anyone (except to the extent that Homo sapiens may prove incapable of controlling what science makes possible). But what is critical to understand is that our species (or, for that matter, God) is not in the least diminished by the idea that we emerged thanks to the processes of evolution.
In The Monkey in the Mirror: Essays on the Science of What Makes Us Human (2003), 55.
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Should I refuse a good dinner simply because I do not understand the process of digestion?
In Electromagnetic Theory (1892), Vol. 2, 9. With this quote, he illustrates that testing a mathematically complex theory and getting accurate results has significance, even while rigorous logic in the theory is not yet plain. The mathematician may not be satisfied. The physicist meanwhile sees value based on experimental results, but does still stay aware that the theory may not be infallible.
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Since an organism is inseparable from its environment, any person who attempts to understand an organism’s distribution must keep constantly in mind that the item being studied is neither a stuffed skin, a pickled specimen, nor a dot on a map. It is not even the live organism held in the hand, caged in a laboratory, or seen in the field. It is a complex interaction between a self-sustaining physicochemical system and the environment. An obvious corollary is that to know the organism it is necessary to know its environment.
From 'The role of physiology in the distribution of terrestrial vertebrates', collected in C.L. Hubbs (ed.), Zoogeography: Publ. 51 (1958), 83.
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Since the Creator had made the facts of the after-life inaccessible to man, He must not have required that man understand death in order to live fruitfully.
In The Lost World of Thomas Jefferson (1948, 1993), 262.
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Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
Quoted as “Einstein once joked”, in Carl Seelig, Albert Einstein: A Documentary Biography (1956), 28.
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Some of the men stood talking in this room, and at the right of the door a little knot had formed round a small table, the center of which was the mathematics student, who was eagerly talking. He had made the assertion that one could draw through a given point more than one parallel to a straight line; Frau Hagenström had cried out that this was impossible, and he had gone on to prove it so conclusively that his hearers were constrained to behave as though they understood.
In Little Herr Friedemann (1961), 25.
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String theory is revealing the deepest understanding of the Universe we have ever had.
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Telescopes are in some ways like time machines. They reveal galaxies so far away that their light has taken billions of years to reach us. We in astronomy have an advantage in studying the universe, in that we can actually see the past.
We owe our existence to stars, because they make the atoms of which we are formed. So if you are romantic you can say we are literally starstuff. If you’re less romantic you can say we’re the nuclear waste from the fuel that makes stars shine.
We’ve made so many advances in our understanding. A few centuries ago, the pioneer navigators learnt the size and shape of our Earth, and the layout of the continents. We are now just learning the dimensions and ingredients of our entire cosmos, and can at last make some sense of our cosmic habitat.
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The astronomer may speak to you of his understanding of space, but he cannot give you his understanding. … And he who is versed in the science of numbers can tell of the regions of weight and measure, but he cannot conduct you thither.
In Kahlil Gibran: The Collected Works (207), 134.
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The biggest thrill of my life was finding out something that nobody in the world ever knew before. Another gratification is a recognition of the fact that you really do understand a lot of things that go on in the world that most people don’t—like planets moving around the sun.
In interview, Rushworth M. Kidder, 'Grounded in Space Science', Christian Science Monitor (22 Dec 1989).
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The chief malady of man is restless curiosity about things which he cannot understand; and it is not so bad for him to be in error as to be curious to no purpose.
In Pensées. As translated by W.F. Trotter in Blaise Pascal: Thoughts, Letters, and Minor Works (1910), 13.
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The complexity of contemporary biology has led to an extreme specialization, which has inevitably been followed by a breakdown in communication between disciplines. Partly as a result of this, the members of each specialty tend to feel that their own work is fundamental and that the work of other groups, although sometimes technically ingenious, is trivial or at best only peripheral to an understanding of truly basic problems and issues. There is a familiar resolution to this problem but it is sometimes difficulty to accept emotionally. This is the idea that there are a number of levels of biological integration and that each level offers problems and insights that are unique to it; further, that each level finds its explanations of mechanism in the levels below, and its significances in the levels above it.
From 'Interaction of physiology and behavior under natural conditions', collected in R.I. Bowman (ed.), The Galapagos (1966), 39.
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The contributions of physiological knowledge to an understanding of distribution are necessarily inferential. Distribution is a historical phenomenon, and the data ordinarily obtained by students of physiology are essentially instantaneous. However, every organism has a line of ancestors which extends back to the beginning of life on earth and which, during this immensity of time, has invariably been able to avoid, to adapt to, or to compensate for environmental changes.
From 'The role of physiology in the distribution of terrestrial vertebrates', collected in C.L. Hubbs (ed.), Zoogeography: Publ. 51 (1958), 84.
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The custom of eating the lover after consummation of the nuptials, of making a meal of the exhausted pigmy, who is henceforth good for nothing, is not so difficult to understand, since insects can hardly be accused of sentimentality; but to devour him during the act surpasses anything the most morbid mind could imagine. I have seen the thing with my own eyes, and I have not yet recovered from my surprise.
In Jean-Henri Fabre and B. Miall (trans.), Social Life in the Insect World (1912), 84.
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The desire to understand the world and the desire to reform it are the two great engines of progress.
In Marriage and Morals (1929), 301.
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The ecologist cannot remain a voice crying in the wilderness—if he is to be heard and understood.
In M.W. Holdgate and M.J. Woodman (eds.) The Breakdown and Restoration of Ecosystems: Proceedings of the Conference (2012), 478. This is part of a final Conclusions. Hildgate as editor summarized the remarks of Talbot and Nicholson.
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The fact that man produces a concept ‘I’ besides the totality of his mental and emotional experiences or perceptions does not prove that there must be any specific existence behind such a concept. We are succumbing to illusions produced by our self-created language, without reaching a better understanding of anything. Most of so-called philosophy is due to this kind of fallacy.
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The facts once classified, once understood, the judgment based upon them ought to be independent of the individual mind which examines them.
From The Grammar of Science (1892), 7.
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The final results [of work on the theory of relativity] appear almost simple; any intelligent undergraduate can understand them without much trouble. But the years of searching in the dark for a truth that one feels, but cannot express; the intense effort and the alternations of confidence and misgiving, until one breaks through to clarity and understanding, are only known to him who has himself experienced them.
Concluding remark of George Gibson lecture at the University of Glasgow, 'The Origins of the General Theory of Relativity', (20 Jun 1933). Published by Glasgow University as The Origins of the General Theory of Relativity: Being the First Lecture on the George A. Gibson Foundation in the University of Glasgow, Delivered on June 20th, 1933 (1933), 11. Also quoted in 'No Hitching Posts' The Atlantic (1936), 157, 251.
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The first question which you will ask and which I must try to answer is this, “What is the use of climbing Mount Everest ?” and my answer must at once be, “It is no use.” There is not the slightest prospect of any gain whatsoever. Oh, we may learn a little about the behavior of the human body at high altitudes, and possibly medical men may turn our observation to some account for the purposes of aviation. But otherwise nothing will come of it. We shall not bring back a single bit of gold or silver, not a gem, nor any coal or iron. We shall not find a single foot of earth that can be planted with crops to raise food. It’s no use. So, if you cannot understand that there is something in man which responds to the challenge of this mountain and goes out to meet it, that the struggle is the struggle of life itself upward and forever upward, then you won’t see why we go. What we get from this adventure is just sheer joy. And joy is, after all, the end of life. We do not live to eat and make money. We eat and make money to be able to enjoy life. That is what life means and what life is for.
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The genuine spirit of Mathesis is devout. No intellectual pursuit more truly leads to profound impressions of the existence and attributes of a Creator, and to a deep sense of our filial relations to him, than the study of these abstract sciences. Who can understand so well how feeble are our conceptions of Almighty Power, as he who has calculated the attraction of the sun and the planets, and weighed in his balance the irresistible force of the lightning? Who can so well understand how confused is our estimate of the Eternal Wisdom, as he who has traced out the secret laws which guide the hosts of heaven, and combine the atoms on earth? Who can so well understand that man is made in the image of his Creator, as he who has sought to frame new laws and conditions to govern imaginary worlds, and found his own thoughts similar to those on which his Creator has acted?
In 'The Imagination in Mathematics', North American Review, 85, 226.
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The hardest thing to understand in the world is the income tax.
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The hardest thing to understand is why we can understand anything at all.
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The human heart feels things the eyes cannot see, and knows what the mind cannot understand.
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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 human race has to be bad at psychology; if it were not, it would understand why it is bad at everything else.
In The Decline and Fall of Science (1976), 3.
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The ill and unfit choice of words wonderfully obstructs the understanding.
From Novum Organum (1620), Book 1, Aphorisms 39, 41-44. Translated as The New Organon: Aphorisms Concerning the Interpretation of Nature and the Kingdom of Man), collected in James Spedding, Robert Ellis and Douglas Heath (eds.), The Works of Francis Bacon (1857), Vol. 4, 55.
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The key to understanding overpopulation is not population density but the numbers of people in an area relative to its resources and the capacity of the environment to sustain human activities; that is, to the area’s carrying capacity. When is an area overpopulated? When its population can’t be maintained without rapidly depleting nonrenewable resources…. By this standard, the entire planet and virtually every nation is already vastly overpopulated.
In The Population Explosion (1990), 39.
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The large collection of problems which our modern Cambridge books supply will be found to be almost an exclusive peculiarity of these books; such collections scarcely exist in foreign treatises on mathematics, nor even in English treatises of an earlier date. This fact shows, I think, that a knowledge of mathematics may be gained without the perpetual working of examples. … Do not trouble yourselves with the examples, make it your main business, I might almost say your exclusive business, to understand the text of your author.
In 'Private Study of Mathematics', Conflict of Studies and other Essays (1873), 74.
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The last word in ignorance is the man who says of an animal or plant: “What good is it?” If the land mechanism as a whole is good, then every part is good, whether we understand it or not.
From 'Conservation' (c.1938), Round River: From the Journals of Aldo Leopold (1953), 141. Collected in The Essential Aldo Leopold: Quotations and Commentaries (1999), 142.
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The layman is delighted to learn that after all, in spite of science being so impossibly difficult to understand, Scientists Are Human!
In What Mad Pursuit (1988), 83.
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The majority of mathematical truths now possessed by us presuppose the intellectual toil of many centuries. A mathematician, therefore, who wishes today to acquire a thorough understanding of modern research in this department, must think over again in quickened tempo the mathematical labors of several centuries. This constant dependence of new truths on old ones stamps mathematics as a science of uncommon exclusiveness and renders it generally impossible to lay open to uninitiated readers a speedy path to the apprehension of the higher mathematical truths. For this reason, too, the theories and results of mathematics are rarely adapted for popular presentation … This same inaccessibility of mathematics, although it secures for it a lofty and aristocratic place among the sciences, also renders it odious to those who have never learned it, and who dread the great labor involved in acquiring an understanding of the questions of modern mathematics. Neither in the languages nor in the natural sciences are the investigations and results so closely interdependent as to make it impossible to acquaint the uninitiated student with single branches or with particular results of these sciences, without causing him to go through a long course of preliminary study.
In Mathematical Essays and Recreations (1898), 32.
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The mediocre mind is incapable of understanding the man who refuses to bow blindly to conventional prejudices and chooses instead to express his opinions courageously and honestly.
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The microwave oven is the consolation prize in our struggle to understand physics.
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The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.
In 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences', Communications in Pure and Applied Mathematics (Feb 1960), 13, No. 1, 14. Collected in Eugene Paul Wigner, A.S. Wightman (ed.), Jagdish Mehra (ed.), The Collected Works of Eugene Paul Wigner (1955), Vol. 6, 537.
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The more a science advances, the more will it be possible to understand immediately results which formerly could be demonstrated only by means of lengthy intermediate considerations: a mathematical subject cannot be considered as finally completed until this end has been attained.
In Formensystem binärer Formen (1875), 2.
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The more you understand the significance of evolution, the more you are pushed away from the agnostic position and towards atheism. Complex, statistically improbable things are by their nature more difficult to explain than simple, statistically probable things.
From edited version of a speech, at the Edinburgh International Science Festival (15 Apr 1992), as reprinted from the Independent newspaper in Alec Fisher, The Logic of Real Arguments (2004), 84.
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The most important thing for us to recall may be, that the crucial quality of science is to encourage, not discourage, the testing of assumptions. That is the only ethic that will eventually start us on our way to a new and much deeper level of understanding.
Concluding sentences of Preface, Quasars, Redshifts and Controversies (1987).
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The motive for the study of mathematics is insight into the nature of the universe. Stars and strata, heat and electricity, the laws and processes of becoming and being, incorporate mathematical truths. If language imitates the voice of the Creator, revealing His heart, mathematics discloses His intellect, repeating the story of how things came into being. And Value of Mathematics, appealing as it does to our energy and to our honor, to our desire to know the truth and thereby to live as of right in the household of God, is that it establishes us in larger and larger certainties. As literature develops emotion, understanding, and sympathy, so mathematics develops observation, imagination, and reason.
In A Theory of Motives, Ideals and Values in Education (1907), 406.
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The opinion appears to be gaining ground that this very general conception of functionality, born on mathematical ground, is destined to supersede the narrower notion of causation, traditional in connection with the natural sciences. As an abstract formulation of the idea of determination in its most general sense, the notion of functionality includes and transcends the more special notion of causation as a one-sided determination of future phenomena by means of present conditions; it can be used to express the fact of the subsumption under a general law of past, present, and future alike, in a sequence of phenomena. From this point of view the remark of Huxley that Mathematics “knows nothing of causation” could only be taken to express the whole truth, if by the term “causation” is understood “efficient causation.” The latter notion has, however, in recent times been to an increasing extent regarded as just as irrelevant in the natural sciences as it is in Mathematics; the idea of thorough-going determinancy, in accordance with formal law, being thought to be alone significant in either domain.
In Presidential Address British Association for the Advancement of Science, Sheffield, Section A, Nature (1 Sep 1910), 84, 290.
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The owner of the means of production is in a position to purchase the labor power of the worker. By using the means of production, the worker produces new goods which become the property of the capitalist. The essential point about this process is the relation between what the worker produces and what he is paid, both measured in terms of real value. In so far as the labor contract is free what the worker receives is determined not by the real value of the goods he produces, but by his minimum needs and by the capitalists’ requirements for labor power in relation to the number of workers competing for jobs. It is important to understand that even in theory the payment of the worker is not determined by the value of his product.
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The path isn’t a straight line; it’s a spiral. You continually come back to things you thought you understood and see deeper truths.
Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 246
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The problems of the infinite have challenged man’s mind and have fired his imagination as no other single problem in the history of thought. The infinite appears both strange and familiar, at times beyond our grasp, at times easy and natural to understand. In conquering it, man broke the fetters that bound him to earth. All his faculties were required for this conquest—his reasoning powers, his poetic fancy, his desire to know.
With co-author James R Newman, in 'Beyond the Google', Mathematics and the Imagination (1940), 35.
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The rainbow, “the bridge of the gods,” proved to be the bridge to our understanding of light—much more important.
Epigraph in Isaac Asimov’s Book of Science and Nature Quotations (1988), 189.
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The reason of ordinary man is the reason of knowledge. The reason of normal man is the reason of understanding. Knowledge is temporary, can be changed. Understanding is permanent—unchangeable.
In On Love & Psychological Exercises: With Some Aphorisms & Other Essays (1998), 52.
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The sciences are said, and they are truly said, to have a mutual connection, that any one of them may be the better understood, for an insight into the rest.
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The scientific world-picture vouchsafes a very complete understanding of all that happens–it makes it just a little too understandable. It allows you to imagine the total display as that of a mechanical clockwork which, for all that science knows, could go on just the same as it does, without there being consciousness, will, endeavor, pain and delight and responsibility connected with it–though they actually are. And the reason for this disconcerting situation is just this: that for the purpose of constructing the picture of the external world, we have used the greatly simplifying device of cutting our own personality out, removing it; hence it is gone, it has evaporated, it is ostensibly not needed.
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The student should read his author with the most sustained attention, in order to discover the meaning of every sentence. If the book is well written, it will endure and repay his close attention: the text ought to be fairly intelligible, even without illustrative examples. Often, far too often, a reader hurries over the text without any sincere and vigorous effort to understand it; and rushes to some example to clear up what ought not to have been obscure, if it had been adequately considered. The habit of scrupulously investigating the text seems to me important on several grounds. The close scrutiny of language is a very valuable exercise both for studious and practical life. In the higher departments of mathematics the habit is indispensable: in the long investigations which occur there it would be impossible to interpose illustrative examples at every stage, the student must therefore encounter and master, sentence by sentence, an extensive and complicated argument.
In 'Private Study of Mathematics', Conflict of Studies and other Essays (1873), 67.
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The theoretical understanding of the world, which is the aim of philosophy, is not a matter of great practical importance to animals, or to savages, or even to most civilized men.
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The tool which serves as intermediary between theory and practice, between thought and observation, is mathematics; it is mathematics which builds the linking bridges and gives the ever more reliable forms. From this it has come about that our entire contemporary culture, inasmuch as it is based on the intellectual penetration and the exploitation of nature, has its foundations in mathematics. Already Galileo said: one can understand nature only when one has learned the language and the signs in which it speaks to us; but this language is mathematics and these signs are mathematical figures.
Radio broadcast (8 Sep 1930). As quoted in Michael Fitzgerald and Ioan James, The Mind of the Mathematician (2007), 6-7.
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The understanding of atomic physics is child’s play, compared with the understanding of child’s play.
As quoted in Alan L. Mackay, A Dictionary of Scientific Quotations (1991), 144. Need primary source; can you help?
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The Universe forces those who live in it to understand it. Those creatures who find everyday experience a muddled jumble of events with no predictability, no regularity, are in grave peril. The Universe belongs to those who, at least to some degree, have figured it out.
In Broca’s Brain: Reflections on the Romance of Science (1979, 1980), 19.
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The vigorous branching of life’s tree, and not the accumulating valor of mythical marches to progress, lies behind the persistence and expansion of organic diversity in our tough and constantly stressful world. And if we do not grasp the fundamental nature of branching as the key to life’s passage across the geological stage, we will never understand evolution aright.
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The whole thing is so much bigger than I am, and I can't understand it, so I just trust myself to it; and forget about it.
Reply to a “deeply depressed” friend’s question about his belief in a Creator of the universe. As quoted in Gale E. Christianson, Edwin Hubble: Mariner of the Nebulae (1996), 183.
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The ‘mad idea’ which will lie at the basis of a future fundamental physical theory will come from a realization that physical meaning has some mathematical form not previously associated with reality. From this point of view the problem of the ‘mad idea’ is the problem of choosing, not of generating, the right idea. One should not understand that too literally. In the 1960s it was said (in a certain connection) that the most important discovery of recent years in physics was the complex numbers. The author [Yuri Manin] has something like that in mind.
Mathematics and Physics (1981), Foreward. Reprinted in Mathematics as Metaphor: Selected Essays of Yuri I. Manin (2007), 90.
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There are no deep theorems—only theorems that we have not understood very well.
In 'Reflections on Bishops Philosophy of Mathematics', Constructive Mathematics: Proceedings of the New Mexico State University Conference Held at Las Cruces, New Mexico, August 11-15, 1980 (1981), 137.
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There is no inductive method which could lead to the fundamental concepts of physics. Failure to understand this fact constituted the basic philosophical error of so many investigators of the nineteenth century.
Opening of section 4, 'The Theory of Relativity', in Physics and Reality (1936), collected in Essays in Physics (1950), 34.
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There is not so contemptible a Plant or Animal that does not confound the most enlarged Understanding. Though the familiar use of Things, take off our Wonder; yet it cures not our Ignorance.
In An Essay Concerning Humane Understanding (1690), 211.
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There is something sublime in the secrecy in which the really great deeds of the mathematician are done. No popular applause follows the act; neither contemporary nor succeeding generations of the people understand it. The geometer must be tried by his peers, and those who truly deserve the title of geometer or analyst have usually been unable to find so many as twelve living peers to form a jury. Archimedes so far outstripped his competitors in the race, that more than a thousand years elapsed before any man appeared, able to sit in judgment on his work, and to say how far he had really gone. And in judging of those men whose names are worthy of being mentioned in connection with his,—Galileo, Descartes, Leibnitz, Newton, and the mathematicians created by Leibnitz and Newton’s calculus,—we are forced to depend upon their testimony of one another. They are too far above our reach for us to judge of them.
In 'Imagination in Mathematics', North American Review, 86, 223.
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There was no point in telling your bosses everything; they were busy men, they didn’t want explanations. There was no point in burdening them. What they wanted was little stories that they felt they could understand, and then they’d go away and stop worrying.
In Terry Pratchett, Ian Stewart and Jack Cohen, The Science of Discworld (2014), 17.
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Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all The second time you go through it, you think you understand it, except for one or two points. The third time you go through it, you know you don't understand it, but by that time you are so used to the subject, it doesn't bother you anymore.
Quoted, without citation, in Stanley W. Angrist and Loren G. Hepler, Order and Chaos: Laws of Energy and Entropy (1967), 215. The authors identify it as “perhaps apocryphal.” The quote is used as epigraph, dated as 1950 in Anton Z. Capri, Quips, Quotes, and Quanta: An Anecdotal History of Physics (2011), 50. The quote is introduced as “When asked why he did not write on that field he replied somewhat as follows,” by Keith J. Laidler in Physical Chemistry with Biological Applications (1978), 145.
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This discovery, indeed, is almost of that kind which I call serendipity, a very expressive word, which as I have nothing better to tell you, I shall endeavour to explain to you: you will understand it better by the derivation than by the definition. I once read a silly fairy tale, called The Three Princes of Serendip: as their highnesses travelled, they were always making discoveries, by accidents and sagacity, of things which they were not in quest of: for instance, one of them discovered that a mule blind of the right eye had travelled the same road lately, because the grass was eaten only on the left side, where it was worse than on the right—now do you understand serendipity?
Letter to Sir Horace Mann (28 Jan 1754), in W. S. Lewis, Warren Hunting Smith and George L. Lam (eds.), Horace Walpole's Correspondence with Sir Horace Mann (1960), Vol. 20, 407-408.
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This interpretation of the atomic number [as the number of orbital electrons] may be said to signify an important step toward the solution of the boldest dreams of natural science, namely to build up an understanding of the regularities of nature upon the consideration of pure number.
Atomic Theory and the Description of Nature (1934), 103-104Cited in Gerald James Holton, Thematic Origins of Scientific Thought: Kepler to Einstein (1985), 74.
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This science, Geometry, is one of indispensable use and constant reference, for every student of the laws of nature; for the relations of space and number are the alphabet in which those laws are written. But besides the interest and importance of this kind which geometry possesses, it has a great and peculiar value for all who wish to understand the foundations of human knowledge, and the methods by which it is acquired. For the student of geometry acquires, with a degree of insight and clearness which the unmathematical reader can but feebly imagine, a conviction that there are necessary truths, many of them of a very complex and striking character; and that a few of the most simple and self-evident truths which it is possible for the mind of man to apprehend, may, by systematic deduction, lead to the most remote and unexpected results.
In The Philosophy of the Inductive Sciences Part 1, Bk. 2, chap. 4, sect. 8 (1868).
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Those intervening ideas, which serve to show the agreement of any two others, are called proofs; and where the agreement or disagreement is by this means plainly and clearly perceived, it is called demonstration; it being shown to the understanding, and the mind made to see that it is so. A quickness in the mind to find out these intermediate ideas, (that shall discover the agreement or disagreement of any other) and to apply them right, is, I suppose, that which is called sagacity.
In An Essay concerning Human Understanding, Bk. 6, chaps. 2, 3.
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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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Those who are not shocked when they first come across quantum mechanics cannot possibly have understood it.
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Thus a statement may be pseudoscientific even if it is eminently ‘plausible’ and everybody believes in it, and it may he scientifically valuable even if it is unbelievable and nobody believes in it. A theory may even be of supreme scientific value even if no one understands it, let alone believes it.
In Radio Lecture (30 Jun 1973) broadcast by the Open University, collected in Imre Lakatos, John Worrall (ed.) and Gregory Currie (ed.), 'Introduction: Science and Pseudoscience', The Methodology of Scientific Research Programmes (1978, 1980), Vol. 1, 1.
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Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the logic piano imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.
From 'Les Mathématiques et la Logique', Science et Méthode (1908, 1920), Livre 2, Chap. 3, Sec. 2, 157. English as in Henri Poincaré and George Bruce Halsted (trans.), 'Mathematics and Logic', Science and Method collected in The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method (1913), 451. From the French, “Ainsi, c’est bien entendu, pour démontrer un théorème, il n’est pas nécessaire ni même utile de savoir ce qu’il veut dire. On pourrait remplacer le géomètre par le piano à raisonner imaginé par Stanley Jevons; ou, si l’on aime mieux, on pourrait imaginer une machine où l’on introduirait les axiomes par un bout pendant qu’on recueillerait les théorèmes à l’autre bout, comme cette machine légendaire de Chicago où les porcs entrent vivants et d’où ils sortent transformés en jambons et en saucisses. Pas plus que ces machines, le mathématicien n’a besoin de comprendre ce qu’il fait”.
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Time’s arrow of ‘just history’ marks each moment of time with a distinctive brand. But we cannot, in our quest to understand history, be satisfied only with a mark to recognize each moment and a guide to order events in temporal sequence. Uniqueness is the essence of history, but we also crave some underlying generality, some principles of order transcending the distinction of moments–lest we be driven mad by Borges’s vision of a new picture every two thousand pages in a book without end. We also need, in short, the immanence of time’s cycle.
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To ask what qualities distinguish good from routine scientific research is to address a question that should be of central concern to every scientist. We can make the question more tractable by rephrasing it, “What attributes are shared by the scientific works which have contributed importantly to our understanding of the physical world—in this case the world of living things?” Two of the most widely accepted characteristics of good scientific work are generality of application and originality of conception. . These qualities are easy to point out in the works of others and, of course extremely difficult to achieve in one’s own research. At first hearing novelty and generality appear to be mutually exclusive, but they really are not. They just have different frames of reference. Novelty has a human frame of reference; generality has a biological frame of reference. Consider, for example, Darwinian Natural Selection. It offers a mechanism so widely applicable as to be almost coexistent with reproduction, so universal as to be almost axiomatic, and so innovative that it shook, and continues to shake, man’s perception of causality.
In 'Scientific innovation and creativity: a zoologist’s point of view', American Zoologist (1982), 22, 230.
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To fully understand the mathematical genius of Sophus Lie, one must not turn to books recently published by him in collaboration with Dr. Engel, but to his earlier memoirs, written during the first years of his scientific career. There Lie shows himself the true geometer that he is, while in his later publications, finding that he was but imperfectly understood by the mathematicians accustomed to the analytic point of view, he adopted a very general analytic form of treatment that is not always easy to follow.
In Lectures on Mathematics (1911), 9.
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To have peace with this peculiar life; to accept what we do not understand; to wait calmly for what awaits us, you have to be wiser than I am.
As quoted, without citation, on the mcescher.com website.
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To suppose that so perfect a system as that of Euclid’s Elements was produced by one man, without any preceding model or materials, would be to suppose that Euclid was more than man. We ascribe to him as much as the weakness of human understanding will permit, if we suppose that the inventions in geometry, which had been made in a tract of preceding ages, were by him not only carried much further, but digested into so admirable a system, that his work obscured all that went before it, and made them be forgot and lost.
In Essay on the Powers of the Human Mind (1812), Vol. 2, 368.
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To understand a science it is necessary to know its history.
From Cours de Philosophie Positive (1830), Tome 1, 82, as often seen in English, by an unidentified translator. From the original French, “Je pense même qu’on ne connaît pas complétement une science tant qu’on n’eu sait pas l’histoire”, which translates more literally as, “I even think that one does not completely understand a science completely as long as one did not know the history”. Other translations include, for example, “A science can not be completely understood without a knowledge of how it arose”, in Auguste Comte and Harriet Martineau (trans.), A Course of Positive Philosophy (1832-1842), Vol. 1, 43.
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To understand is to invent.
Quoted in Richard Saul Wurman, Information Anxiety 2 (2001), 240.
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To understand this for sense it is not required that a man should be a geometrician or a logician, but that he should be mad.
In Thomas Hobbes and William Molesworth (ed.), 'Considerations Upon the Answer of Doctor Wallis to the Three Papers of Mr. Hobbes', Leviathan: Or the Matter, Form and Power of a Commonwealth Ecclesiastical and Civil (1845), Vol. 7, 445. What Hobbes refers to as “this” is described as “that the volume generated by revolving the region under 1/x from 1 to infinity has finite volume,” in In Nicholas J. Rose Mathematical Maxims and Minims (1988).
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To understand [our cosmological roots]...is to give voice to the silent stars. Stand under the stars and say what you like to them. Praise them or blame them, question them, pray to them, wish upon them. The universe will not answer. But it will have spoken.
…...
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Understanding the world for a man is reducing it to the human, stamping it with his seal.
…...
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Understanding … must begin by saturating itself with facts and realities. … Besides, we only understand that which is already within us. To understand is to possess the thing understood, first by sympathy and then by intelligence. Instead of first dismembering and dissecting the object to be conceived, we should begin by laying hold of it in its ensemble. The procedure is the same, whether we study a watch or a plant, a work of art or a character.
(7 Apr 1886). In Mary A. Ward (trans.), Amiel’s Journal: The Journal Intime of Henri-Frédéric Amiel (1889), 119.
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We are all travelers who are journeying … not knowing where the next day of our life is going to take us. We have no understanding of the surprises that are in store for us. Steadily we will know, understand and decipher and then it will all start to make sense. Until then keep travelling.
Anonymous
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We are concerned to understand the motivation for the development of pure mathematics, and it will not do simply to point to aesthetic qualities in the subject and leave it at that. It must be remembered that there is far more excitement to be had from creating something than from appreciating it after it has been created. Let there be no mistake about it, the fact that the mathematician is bound down by the rules of logic can no more prevent him from being creative than the properties of paint can prevent the artist. … We must remember that the mathematician not only finds the solutions to his problems, he creates the problems themselves.
In A Signpost to Mathematics (1951), 19. As quoted and cited in William L. Schaaf, 'Memorabilia Mathematica', The Mathematics Teacher (Mar 1957), 50, No. 3, 230. Note that this paper incorrectly attributes “A.H. Head”.
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We are just beginning to understand how molecular reaction systems have found a way to “organize themselves”. We know that processes of this nature ultimately led to the life cycle, and that (for the time being?) Man with his central nervous system, i.e. his memory, his mind, and his soul, stands at the end of this development and feels compelled to understand this development. For this purpose he must penetrate into the smallest units of time and space, which also requires new ideas to make these familiar concepts from physics of service in understanding what has, right into our century, appeared to be beyond the confines of space and time.
Answering “Where Now?” as the conclusion of his Nobel Lecture (11 Dec 1967) on 'Immeasurably Fast Reactions', published in Nobel Lectures, Chemistry 1963-1970 (1972).
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We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context.
Unterrichtsblätter für Mathematik und Naturwissenschaften (1932), 38, 177-188. As translated by Abe Shenitzer, in 'Part I. Topology and Abstract Algebra as Two Roads of Mathematical Comprehension', The American Mathematical Monthly (May 1995), 102, No. 7, 453.
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We believe that interest in nature leads to knowledge,
which is followed by understanding,
and later, appreciation.
Once respect is gained
it is a short step to responsibility,
and ultimately action
to preserve our Earth.
…...
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We can learn a lot from living organisms. An organism is a pretty complicated thing, which can tolerate surgery, which can tolerate injury, which can tolerate all kinds of perturbation provided they are not too great and do not come too suddenly. There’s something we call trauma, however. We don’t really understand what it is—but organisations can suffer from it too.
From interview with Graham Chedd, 'The Lady Gets Her Way', New Scientist (5 Jul 1973), 59, No. 853, 16.
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We cannot doubt the existence of an ultimate reality. It is the universe forever masked. We are a part of it, and the masks figured by us are the universe observing and understanding itself from a human point of view.
…...
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We have no faculties for passing beyond ourselves, yet in ourselves are unfathomed depths, unexplored powers and relations which need fathoming and searching into. As Schopenhauer says, if we would understand nature our course must not only be horizontal, but perpendicular.
In Sir William Withey Gull and Theodore Dyke Acland (ed.), A Collection of the Published Writings of William Withey Gull (1896), li.
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We have reached the end of our journey into the depths of matter. We have sought for firm ground and found none. The deeper we penetrate, the more restless becomes the universe…: all is rushing about and vibrating in a wild dance.
Max Born
In The Restless Universe (2013), Chap. 5, 277.
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We live in a democracy and I do not understand why highly respected scientists from top international branches are not able express themselves!
…...
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We live in a society absolutely dependent on science and technology and yet have cleverly arranged things so that almost no one understands science and technology. That’s a clear prescription for disaster.
From interview with Anne Kalosh in her article 'Bringing Science Down to Earth', in Hemispheres (Oct 1994), 99. Collected and cited in Tom Head (ed.), Conversations with Carl Sagan (2006), 100. This was Sagan’s answer to her question, “How does not understanding science cripple people in their daily lives.”
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We may need simple and heroic legends for that peculiar genre of literature known as the textbook. But historians must also labor to rescue human beings from their legends in science–if only so that we may understand the process of scientific thought aright.
…...
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We see a universe marvelously arranged and obeying certain laws, but only dimly understand these laws. Our limited minds cannot grasp the mysterious force that moves the constellations. I am fascinated by Spinoza’s pantheism, but admire even more his contributions to modern thought because he is the first philosopher to deal with the soul and the body as one, not two separate things.
…...
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We shall never understand each other until we reduce the language to seven words.
In Kahlil Gibran: The Collected Works (207), 187.
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We should be most careful about retreating from the specific challenge of our age. We should be reluctant to turn our back upon the frontier of this epoch… We cannot be indifferent to space, because the grand slow march of our intelligence has brought us, in our generation, to a point from which we can explore and understand and utilize it. To turn back now would be to deny our history, our capabilities.
At a 1979 U.S. Senate hearing. As quoted in House Congressional Record (21 Jun 1991), 13874. Also quoted in James E. Oberg, Mission to Mars: Plans and Concepts for the First Manned Landing (2017), 174.
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We want them to use the education to be leaders in their community with an understanding of ecology and conservation for the wild outdoors far beyond their legislators back home. We expect these people to he a grain of sand on the beach of future leadership.
…...
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We’ve arranged a global civilization in which most critical elements profoundly depend on science and technology. We have also arranged things so that almost no one understands science and technology. This is a prescription for disaster. We might get away with it for a while, but sooner or later this combustible mixture of ignorance and power is going to blow up in our faces.
From The Demon-Haunted World: Science as a Candle in the Dark (1996), 26.
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What a splendid perspective contact with a profoundly different civilization might provide! In a cosmic setting vast and old beyond ordinary human understanding we are a little lonely, and we ponder the ultimate significance, if any, of our tiny but exquisite blue planet, the Earth… In the deepest sense the search for extraterrestrial intelligence is a search for ourselves.
…...
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What else can the human mind hold besides numbers and magnitudes? These alone we apprehend correctly, and if piety permits to say so, our comprehension is in this case of the same kind as God’s, at least insofar as we are able to understand it in this mortal life.
As quoted in Epilogue, The Sleepwalkers: A History of Man’s Changing Vision of the Universe (1959), 524.
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What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school… It is my task to convince you not to turn away because you don’t understand it. You see my physics students don’t understand it… That is because I don’t understand it. Nobody does.
From Lecture, the first in the first series of Alix G. Mauntner Lectures, trascribed and editted by Ralph Leighton, 'Introduction', QED, The Strange Theory of Light and Matter (1985, 1988), 9.
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When faced with a problem you do not understand, do any part of it you do understand and then look at it again.
Anonymous
Webmaster has not yet been able to identify a primary source. Can you help? Although the quote appears in Jon, Michael and Deborah Fripp, Speaking of Science (2000), 32, attributed to Robert Heinlein, it has no citation. Without finding a primary source citation, Webmaster is dubious. The advice is good, and is notable as a quote, but for now, “Anonymous” seems appropriate.
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When I was a child, I spoke as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.
Bible
(circa 325 A.D.)
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When the boy begins to understand that the visible point is preceded by an invisible point, that the shortest distance between two points is conceived as a straight line before it is ever drawn with the pencil on paper, he experiences a feeling of pride, of satisfaction. And justly so, for the fountain of all thought has been opened to him, the difference between the ideal and the real, potentia et actu, has become clear to him; henceforth the philosopher can reveal him nothing new, as a geometrician he has discovered the basis of all thought.
In Sprüche in Reimen. Sprüche in Prosa. Ethisches (1850), Vol. 3, 214. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 67. From the original German, “Wenn der knabe zu begreifen anfängt, daß einem sichtbaren Punkte ein unsichtbarer vorhergehen müsse, daß der nächste Weg zwischen zwei Punkten schon als Linie gedacht werde, ehe sie mit dem Bleistift aufs Papier gezogen wird, so fühlt er einen gewissen Stolz, ein Behagen. Und nicht mit Unrecht; denn ihm ist die Quelle alles Denkens aufgeschlossen, Idee und Verwirklichtes, potentia et actu, ist ihm klargeworden; der Philosoph entdeckt ihm nichts Neues; dem Geometer war von seiner Seite der Grund alles Denkens aufgegangen.” The Latin phrase, “potentia et actu” means “potentiality and actuality”.
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When the child outgrows the narrow circle of family life … then comes the period of the school, whose object is to initiate him into the technicalities of intercommunication with his fellow-men, and to familiarize him with the ideas that underlie his civilization, and which he must use as tools of thought if he would observe and understand the phases of human life around him; for these … are invisible to the human being who has not the aid of elementary ideas with which to see them.
In Psychologic Foundations of Education: An Attempt to Show the Genesis of the Higher Faculties of the Mind (1907), 265.
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When the principles of breeding and of inheritance are better understood, we shall not hear ignorant members of our legislature rejecting with scorn a plan for ascertaining by an easy method whether or not consanguineous marriages are injurious to man.
…...
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When you have made a thorough and reasonably long effort, to understand a thing, and still feel puzzled by it, stop, you will only hurt yourself by going on. Put it aside till the next morning; and if then you can’t make it out, and have no one to explain it to you, put it aside entirely, and go back to that part of the subject which you do understand.
From letter to Edith Rix with hints for studying (about Mar 1885), in Stuart Dodgson Collingwood, The Life and Letters of Lewis Carroll (1898), 240.
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When you live in a complex world, you have to simplify it in order to understand it.
In Terry Pratchett, Ian Stewart and Jack Cohen, The Science of Discworld (2014), 44.
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Whether science is seen as genie or devil, the attitude is wrong. We need to get some sort of perspective, so that people understand science is just one more intellectual tool, one more way of knowing enough things to give society a means of living on Earth.
From interview with Graham Chedd, 'The Lady Gets Her Way', New Scientist (5 Jul 1973), 59, No. 853, 16.
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Why does a man want to be a scientist? There are many goals: fame, position, a thirst for understanding. The first two can be attained without intellectual integrity; the third cannot. … The thirst for knowledge, what Thomas Huxley called the ‘Divine dipsomania’, can only be satisfied by complete intellectual integrity. It seems to me the only one of the three goals that continues to reward the pursuer. He presses on, “knowing that Nature never did betray the heart that loved her”. Here is another kind of love, that has so many faces. Love is neither passion, nor pride, nor pity, nor blind adoration, but it can be any or all of these if they are transfigured by deep and unbiased understanding.
In Cecilia Payne-Gaposchkin: An Autobiography and Other Recollections (1996), 123.
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Why then does science work? The answer is that nobody knows. It is a complete mystery—perhaps the complete mystery&mdashwhy the human mind should be able to understand anything at all about the wider universe. ... Perhaps it is because our brains evolved through the working of natural law that they somehow resonate with natural law. ... But the mystery, really, is not that we are at one with the universe, but that we are so to some degree at odds with it, different from it, and yet can understand something about it. Why is this so?
Coming of Age in the Milky Way (1988), 385. In Vinoth Ramachandra, Subverting Global Myths: Theology and the Public Issues Shaping our World (2008), 185.
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With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which numbers holds sway above the flux. A little of this, but not much, I have achieved.
In 'Prologue: What I Have Lived For', The Autobiography of Bertrand Russell (1969). 3-4.
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