Quotes about calculus (92 quotes) | Quotes of famous people

“I have completely forgotten the symbolic calculus.”

Emmy Noether (1882–1935) German mathematician

Original: (de) Ich habe das symbolische Rechnen mit Stumpf und Stil verlernt.

Habilitation curriculum vitae (1919) submitted to the Göttingen Faculty, as quoted by Peter Roquette, "Emmy Noether and Hermann Weyl" (Jan. 28, 2008) extended manuscript of a talk presented at the Hermann Weyl conference in Bielefeld, September 10, 2006.

Quotes

“I have completely forgotten the symbolic calculus.”

Emmy Noether (1882–1935) German mathematician

Letter to , "Emmy Noether and Hermann Weyl" (Jan. 28, 2008) extended manuscript of a talk presented at the Hermann Weyl conference in Bielefeld, September 10, 2006.

“It is upon the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place among the acknowledged forms of Mathematical Analysis,”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1840s, The Mathematical Analysis of Logic, 1847, p. iii
Context: That to the existing forms of Analysis a quantitative interpretation is assigned, is the result of the circumstances by which those forms were determined, and is not to be construed into a universal condition of Analysis. It is upon the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place among the acknowledged forms of Mathematical Analysis, regardless that in its object and in its instruments it must at present stand alone.

Mathematics

“During my controlled near-death experiences, I’ve met Sir Isaac Newton, who died back in 1727 as often as I’ve met Saint Peter. They both hang out at the Heaven end of the blue tunnel of the Afterlife. Saint Peter is there because it’s his job. Sir Isaac is there because of his insatiable curiosity about what the blue tunnel is, how the blue tunnel works.
It isn’t enough for Newton that during his eighty-five years on Earth he invented calculus, codified and quantified the laws of gravity, motion and optics, and designed the first reflecting telescope. He can’t forgive himself for having left it to Darwin to come up with the theory of evolution, to Pasteur to come up with the germ theory, and to Albert Einstein to come up with relativity. “I must have been deaf, dumb, and blind not to have come up with those myself,” he said to me. “What could have been more obvious?””

Kurt Vonnegut book God Bless You, Dr. Kevorkian

God Bless You, Dr. Kevorkian (1999)

Death , Law , Forgiveness

“The calculus of utility aims at supplying the ordinary wants of man at the least cost of labour.”

William Stanley Jevons The Theory of Political Economy

Source: The Theory of Political Economy (1871), Chapter I, Introduction, p. 53.

“The Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Source: 1910s, Mysticism and Logic and Other Essays http://archive.org/stream/mysticism00russuoft/mysticism00russuoft_djvu.txt (1918), Ch. 5: Mathematics and the Metaphysicians

“The principle that the sum of the squares of the differences between the observed and computed quantities must be a minimum may, in the following manner, be considered independently of the calculus of probabilities.”

Carl Friedrich Gauss (1777–1855) German mathematician and physical scientist

Theoria motus corporum coelestium in sectionibus conicis solem ambientum (1809) Tr. Charles Henry Davis as Theory of the Motion of the Heavenly Bodies moving about the Sun in Conic Sections http://books.google.com/books?id=cspWAAAAMAAJ& (1857)
Context: The principle that the sum of the squares of the differences between the observed and computed quantities must be a minimum may, in the following manner, be considered independently of the calculus of probabilities. When the number of unknown quantities is equal to the number of the observed quantities depending on them, the former may be so determined as exactly to satisfy the latter. But when the number of the former is less than that of the latter, an absolutely exact agreement cannot be obtained, unless the observations possess absolute accuracy. In this case care must be taken to establish the best possible agreement, or to diminish as far as practicable the differences. This idea, however, from its nature, involves something vague. For, although a system of values for the unknown quantities which makes all the differences respectively less than another system, is without doubt to be preferred to the latter, still the choice between two systems, one of which presents a better agreement in some observations, the other in others, is left in a measure to our judgment, and innumerable different principles can be proposed by which the former condition is satisfied. Denoting the differences between observation and calculation by A, A’, A’’, etc., the first condition will be satisfied not only if AA + A’ A’ + A’’ A’’ + etc., is a minimum (which is our principle) but also if A4 + A’4 + A’’4 + etc., or A6 + A’6 + A’’6 + etc., or in general, if the sum of any of the powers with an even exponent becomes a minimum. But of all these principles ours is the most simple; by the others we should be led into the most complicated calculations.

“Everyone has their weak spot. The one thing that despite your best efforts, will always bring you to your knees, regardless of how strong you are otherwise. For some people, it's love. Others, money or alcohol. Mine was even worse: calculus.”

Sarah Dessen Lock and Key

Source: Lock and Key

Love , Money , People , Alcohol

“Well, I live a double life. Tonight, I'll escort you to dinner. Then I have to rush home and finish my calculus homework.'
'You're not joking, are you?'
'I never joke about calculus homework.”

Rick Riordan book The Sword of Summer

Source: The Sword of Summer

Life , Home

“The calculus is to mathematics no more than what experiment is to physics, and all the truths produced solely by the calculus can be treated as truths of experiment. The sciences must proceed to first causes, above all mathematics where one cannot assume, as in physics, principles that are unknown to us. For there is in mathematics, so to speak, only what we have placed there… If, however, mathematics always has some essential obscurity that one cannot dissipate, it will lie, uniquely, I think, in the direction of the infinite; it is in that direction that mathematics touches on physics, on the innermost nature of bodies about which we know little.”

Bernard Le Bovier de Fontenelle (1657–1757) French writer, satirist and philosopher of enlightenment

Elements de la géométrie de l'infini (1727) as quoted by Amir R. Alexander, Geometrical Landscapes: The Voyages of Discovery and the Transformation of Mathematical Practice (2002) citing Michael S. Mahoney, "Infinitesimals and Transcendent Relations: The Mathematics of Motion in the Late Seventeenth Century" in Reappraisals of the Scientific Revolution, ed. David C. Lindberg, Robert S. Westman (1990)

Truth , Lie , Mathematics , Nature

“The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. …The conic sections, invented in an attempt to solve the problem of doubling the alter of an oracle, ended by becoming the orbits followed by the planets… The imaginary magnitudes invented by Cardan and Bombelli describe… the characteristic features of alternating currents. The absolute differential calculus, which originated as a fantasy of Reimann, became the mathematical model for the theory of Relativity. And the matrices which were a complete abstraction in the days of Cayley and Sylvester appear admirably adapted to the… quantum of the atom.”

Tobias Dantzig (1884–1956) American mathematician

Number: The Language of Science (1930)

Mathematics , Problem , Appearance

“The first period embraces the time between the first records of empirical determinations of the ratio of the circumference to the diameter of a circle until the invention of the Differential and Integral Calculus, in the middle of the seventeenth century. This period, in which the ideal of an exact construction was never entirely lost sight of, and was occasionally supposed to have been attained, was the geometrical period, in which the main activity consisted in the approximate determination of π by the calculation of the sides or the areas of regular polygons in- and circum-scribed to the circle. The theoretical groundwork of the method was the Greek method of Exhaustions. In the earlier part of the period the work of approximation was much hampered by the backward condition of arithmetic due to the fact that our present system of numerical notation had not yet been invented; but the closeness of the approximations obtained in spite of this great obstacle are truly surprising. In the later part of this first period methods were devised by which the approximations to the value of π were obtained which required only a fraction of the labour involved in the earlier calculations. At the end of the period the method was developed to so high a degree of perfection that no further advance could be hoped for on the lines laid down by the Greek Mathematicians; for further progress more powerful methods were required.”

E. W. Hobson (1856–1933) British mathematician

Source: Squaring the Circle (1913), pp. 10-11

Hope , Time , Parting , Perfection

“A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformation of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs.”

Vannevar Bush book As We May Think

As We May Think (1945)

“If you put it as “complex nervous systems” it sounds pretty deflationary. What's so special about a complex nervous system? But of course, that complex nervous system allows you to do calculus. It allows you to do astrophysics… to write poetry… to fall in love. Put under that description, when asked “What’s so special about humans…?”, I’m at a loss to know how to answer that question. If you don’t see why we’d be special… because we can do poetry [and] think philosophical thoughts [and] we can think about the morality of our behavior, I’m not sure what kind of answer could possibly satisfy you at that point.
…I could pose the same kinds of questions of you… So God says, “You are guys are really, really special.” How does his saying it make us special? “But you see, he gave us a soul.””

Shelly Kagan (1956) American philosopher

How does our having a soul make us special? Whatever answer you give, you could always say… “What’s so special about that?”
Debate: Is God Necessary for Morality? (2011)

Love , God , Guy , Behavior

“Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic 31 geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence.”

James Joseph Sylvester (1814–1897) English mathematician

James Joseph Sylvester. "A Plea for the Mathematician, Nature," Vol. 1, p. 238; Collected Mathematical Papers, Vol. 2 (1908), pp. 655, 656.

Law , Mathematics , Idea , Effort

“Donald Trump's political rivals look at the price exacted by a Muslim like Syed Farouk and his bride in the aggregate. To a politician, 14 lives in 322 million is a small price to pay for 'our freedoms.' Fourteen dead in San Bernardino is not a steep price to pay for unfettered immigration from Islamic countries, peddled politically as 'our values,' 'our tolerance,' 'our greatness.' This callous calculus is second nature to neoconservative politicians like Lindsey Graham or Darth Vader Cheney. Not to Trump.”

Ilana Mercer South African writer

"No Wonder the Pols Think Businessman Trump’s Crazy; He Understands Scarcity," http://www.unz.com/imercer/no-wonder-the-pols-think-businessman-trumps-crazy-he-understands-scarcity/ The Unz Review, December 11, 2015.
2010s, 2015

Nature , Freedom , Dead , Country

“As professor in the Polytechnic School [autumn of 1858] in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt, more keenly than ever before, the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

School , Feeling , Question , Autumn

“The calculus of probabilities, when confined within just limits, ought to interest, in an equal degree, the mathematician, the experimentalist, and the statesman.”

François Arago (1786–1853) French mathematician, physicist, astronomer and politician

Laplace, p. 364.
Biographies of Distinguished Scientific Men (1859)

“The chief object of the present work is, as its title indicates, to furnish to the student examples by which to illustrate the processes of the Differential and Integral Calculus. In this respect it will be seen to agree with Professor Peacock's Collection of Examples; and indeed if a second edition of that excellent work had been published I should not have undertaken the task of making this compilation. But as Professor Peacock informed me that he had not leisure to superintend the publication of a second edition of his "Examples" which had been long out of print, I thought that I should do a service to students by preparing a work on a similar plan, but with such modifications as seemed called for by the increased cultivation of Analysis in this University.”

Duncan Gregory (1813–1844) British mathematician

p. iii http://books.google.com/books?id=h7JT-QDuAHoC&pg=PR3; Lead paragraph of the Preface; Highlighted section cited in: Patricia R. Allaire and Robert E. Bradley. " Symbolical algebra as a foundation for calculus: DF Gregory's contribution http://poncelet.math.nthu.edu.tw/disk5/js/history/gregory.pdf." Historia Mathematica 29.4 (2002): p. 408
Examples of the processes of the differential and integral calculus, (1841)

Universe

“Calculus is the mathematics of change. …Change is characteristic of the world.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

World , Mathematics , Change

“The need for a fundamentally different approach to the study of physical processes at the molecular level motivated the development of relevant statistical methods, which turned out to be applicable not only to the study of molecular processes (statistical mechanics), but to a host of other areas such as the actuarial profession, design of large telephone exchanges, and the like. In statistical methods, specific manifestations of microscopic entities (molecules, individual telephone sites, etc.) are replaced with their statistical averages, which are connected with appropriate macroscopic variables. The role played in Newtonian mechanics by the calculus, which involves no uncertainty, is replaced in statistical mechanics by probability theory, a theory whose very purpose is to capture uncertainty of a certain type.”

George Klir (1932–2016) American computer scientist

Source: Fuzzy sets and fuzzy logic (1995), p. 1-2.

Motivation , Study

“All gone. Zelazny was one of the first times I looked at something I had had familiarity with to find the spot where the memory should have been empty, replaced by a scrawled "Moved South for the Fishing" sign. Calculus was another loss. It was quite upsetting to reach for a skill and find nothing.”

James Nicoll (1961) Canadian fiction reviewer

[c2nsoc$aqk$1@panix3.panix.com, 2004]
2000s

Fish , Memory , Loss , Time

“The theory which follows is entirely based on a calculus of pleasure and pain; and the object of economics is to maximize happiness by purchasing pleasure, as it were, at the lowest cost of pain.”

William Stanley Jevons The Theory of Political Economy

Source: The Theory of Political Economy (1871), Chapter I, Introduction, p. 51.

Pain , Pleasure , Happiness

“It is known that the mathematics prescribed for the high school [Gymnasien] is essentially Euclidean, while it is modern mathematics, the theory of functions and the infinitesimal calculus, which has secured for us an insight into the mechanism and laws of nature. Euclidean mathematics is indeed, a prerequisite for the theory of functions, but just as one, though he has learned the inflections of Latin nouns and verbs, will not thereby be enabled to read a Latin author much less to appreciate the beauties of a Horace, so Euclidean mathematics, that is the mathematics of the high school, is unable to unlock nature and her laws. 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 113 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. But it is exactly in this respect that our view of nature is so far above that of the ancients; that we no longer look on nature as a quiescent complete whole, which compels admiration by its sublimity and wealth of forms, but that we conceive of her as a vigorous growing organism, unfolding according to definite, as delicate as far-reaching, laws; that we are able to lay hold of the permanent amidst the transitory, of law amidst fleeting phenomena, and to be able to give these their simplest and truest expression through the mathematical formulas”

Christian Heinrich von Dillmann (1829–1899) German educationist

Source: Die Mathematik die Fackelträgerin einer neuen Zeit (Stuttgart, 1889), p. 37.

School , Law , Beauty , Mathematics

“Even though principles of rationality seem as often violated as followed, we still cling to the notion that human thought should be rational, logical, and orderly. Much of law is based upon the concept of rational thought and behavior. Much of economic theory is based upon the model of the rational human who attempts to optimize personal benefit, utility, or comfort. Many scientists who study artificial intelligence use the mathematics of formal logic—the predicate calculus—as their major tool to simulate thought. […] Human thought is not like logic; it is fundamentally different in kind and spirit. The difference is neither worse nor better. But it is the difference that leads to creative discovery and to great robustness of behavior.”

Donald A. Norman book The Design of Everyday Things

Source: The Design of Everyday Things (1988, 2002), Ch. 5, pp. 114–115.

Law , Behavior , Intelligence , Mathematics

“James Buchanan … produced several seminal contributions to the literature prior to the appearance of The Calculus, and a continual stream of books and articles afterward. I view Buchanan’s Nobel Prize as a reward for this great body of work, and not just limited to The Calculus. Pursuing this line of reasoning, Gordon Tullock’s prodigious research output following the publication of The Calculus also justifies a Nobel Prize. Indeed, Tullock’s (1967c) seminal analysis of rent seeking, and his subsequent work on this topic might be regarded as sufficient achievements for the awarding of a prize.”

Dennis Mueller (1940) American economist

James Buchanan, Gordon Tullock, and The Calculus (2012)

Books , Literature , Appearance

“The success of mathematical physics led the social scientist to be jealous of its power without quite understanding the intellectual attitudes that had contributed to this power. The use of mathematical formulae had accompanied the development of the natural sciences and become the mode in the social sciences. Just as primitive peoples adopt the Western modes of denationalized clothing and of parliamentatism out of a vague feeling that these magic rites and vestments will at once put them abreast of modern culture and technique, so the economists have developed the habit of dressing up their rather imprecise ideas in the language of the infinitesimal calculus.”

Norbert Wiener (1894–1964) American mathematician

Source: Ex-Prodigy: My Childhood and Youth (1964), p. 89; partly cited in: Herman E. Daly. Steady-State Economics: Second Edition With New Essays. 1977/1991 p. 4

People , Feeling , Mathematics , Idea

“The truth is that the mathematical sciences are growing in complete security and harmony. The ideas of Dedekind, Poincare, and Hilbert have been systematically developed with great success, without any conflict in the results. It is only from the philosophical point of view that objections have been raised. They bear on certain ways of reasoning peculiar to analysis and set theory. These modes of reasoning were first systematically applied in giving a rigorous form to the methods of the calculus. [According to them, ] the objects of a theory are viewed as elements of a totality such that one can reason as follows: For each property expressible using the notions of the theory, it is [an] objectively determinate [fact] whether there is or there is not an element of the totality which possesses this property. Similarly, it follows from this point of view that either all the elements of a set possess a given property, or there is at least one element which does not possess it”

Paul Bernays (1888–1977) Swiss mathematician

Paul Bernays, Platonism in mathematics http://sites.google.com/site/ancientaroma2/book_platonism.pdf (1935)

Truth , Way , Mathematics , Idea

“To make psychology into experimental epistemology is to attempt to understand the embodiment of mind. Here we are confronted by what seem to be three questions, although they may ultimately be only one. The three exist as categorically disperate desiderata.
The first is at the logical level: We lack an adequate, appropriate calculus for triadic relations.
The second is the psychological level: We do not know how we generate hypotheses that are natural and simple.
The third is that the physiological level: we have no circuit theory for the reticular formation that marshals our abductions.
Logically, the problem is far from simple. To be exact, no proposed theory of relations yields a calculus to handle our problem. When I was growing up, only the Aristotelian logic of classes was ever taught, and that badly. The Organon itself contains only a clumsy description of the apagoge - perhaps from the notes of some students who have not understand his master…”

Warren S. McCulloch (1898–1969) American neuroscientist

Source: Embodiments of Mind, (1965), p. 389, Chapter " What's in the brain that ink may character http://vordenker.de/ggphilosophy/mcculloch_whats-in-the-brain.pdf"

Understanding , Question , Nature , Problem

“In this chapter I shall collect those Theorems in the Differential Calculus which, depending only on the laws of combination of the symbols of differentiation, and not on the functions which are operated on by these symbols, may be proved by the method of the separation of the symbols : but as the principles of this method have not as yet found a place in the elementary works on the Calculus, I shall first state? briefly the theory on which it is founded.”

Duncan Gregory (1813–1844) British mathematician

Source: Examples of the processes of the differential and integral calculus, (1841), p. 237; Lead paragraph of Ch. XV, On General Theorems in the Differential Calculus,; Cited in: James Gasser (2000) A Boole Anthology: Recent and Classical Studies in the Logic of George Boole,, p. 52

Law , Dependency

“On a novel plan, I have combined the historical progress with the scientific developement of the subject; and endeavoured to lay down and inculcate the principles of the Calculus, whilst I traced its gradual and successive improvements…. there is little doubt, the student's curiosity and attention will be more excited and sustained, when he finds history blended with science, and the demonstration of formulae accompanied with the object and the causes of their invention, than by a mere analytical exposition of the principles of the subject. He will have an opportunity of observing how a calculus, from simple beginnings, by easy steps, and seemingly the slightest improvements, is advanced to perfection; his curiosity too, may be stimulated to an examination of the works of the contemporaries of Newton; works once read and celebrated: yet the writings of the Bernoullis are not antiquated from loss of beauty, nor deserve neglect…”

Robert Woodhouse (1773–1827) English mathematician

Preface p. iv-v
A Treatise on Isoperimetrical Problems, and the Calculus of Variations (1810)

Beauty , History , Science , Loss

“The Rigveda stated that the earth was a …globe suspended freely in space. The Vedic texts disclosed that the Sun held the earth and heavenly bodies in its orbit. The Shatapatha Brahmana, a treatise of untold antiquity, recognized and explained the fact that the earth was spherical.. Aryabhata explained the daily rising and setting of planets and stars in terms of the earth’s constant revolutionary motion. The Surya Siddhantha said that the earth, owing to its gravitational force draw all things to itself. In physics, the thinker Kanada, explained light and heat as different aspects of the same element, thus anticipating Clarke Maxwell's Electro-magnetic Theory, which unified different forms of radiant energy. Sankaracharya, in his Advaita thought expanded the concept of unity of matter and energy. Vacaspati recognized light as composed of minute particles emitted by substances, anticipating Newton’s Corpuscular Theory of Light and the later discovery of the Photon. In Botany, Sankara Mishra and Kanada have discussed the circulation of sap in the Plant and the Santiparva of Mahabharata has clearly stated that the plants develop on the strength of nutrients made through interaction of sunlight and materials obtained from the air and ground. Bhaskarcharya's concept of Differential Calculus preceded Newton by many centuries. His study of time identified Truti: The 3400th part of a second as the unit of time.”

Shankar Dayal Sharma (1918–1999) Indian politician

He has rightly brought out the rationality and application of Sanskrit literature in diverse fields
Source: Aruna Goel Good Governance and Ancient Sanskrit Literature http://books.google.co.in/books?id=El_VADF13pUC&pg=PA16, Deep and Deep Publications, 1 January 2003, p. 16-17

Star , Study , Light , Time

“…nor have I found occasion to depart from the plan… the rejection of the whole doctrine of series in the establishment of the fundamental parts both of the Differential and Integral Calculus. The method of Lagrange… had taken deep root in elementary works; it was the sacrifice of the clear and indubitable principle of limits to a phantom, the idea that an algebra without limits was purer than one in which that notion was introduced. But, independently of the idea of limits being absolutely necessary even to the proper conception of a convergent series, it must have been obvious enough to Lagrange himself, that all application of the science to concrete magnitude, even in his own system, required the theory of limits.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

The Differential and Integral Calculus (1836)

Idea , Science , Parting

“What is at issue here is not: What are the facts? but rather: How are the facts to be described in order to sanction one mode of explaining them rather than another? Some historians will insist that history cannot become a science until it finds the technical terminology adequate to the correct characterization of its objects of study, in the way that physics did in the calculus and chemistry did in the periodic tables. Such is the recommendation of Marxists, Positivists, Cliometricians, and so on. Others will continue to insist that the integrity of historiography depends on its use of ordinary language, its avoidance of jargon. These latter suppose that ordinary language is a safeguard against ideological deformations of the ‘facts.’ What they fail to recognize is that ordinary language itself has its own forms of terminological determinism, represented by the figures of speech without which discourse itself is impossible.”

Hayden White (1928–2018) American historian

"The fictions of factual representation"

Way , History , Study , Science

“The change began with John Stuart Mill and the Utopians. When Mill pointed out that economics had no ultimate solution to the problem of distribution, that society might do with the fruits of its toil as it saw fit, he introduced into the mechanical calculus of the market a conflicting calculus of moral judgment.”

Robert L. Heilbroner book The Worldly Philosophers

Source: The Worldly Philosophers (1953), Chapter XI, Beyond the Economic Revolution, p. 307

Problem , Change

“In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius.”

Carl Friedrich Gauss (1777–1855) German mathematician and physical scientist

As quoted in Gauss, Werke, Bd. 8, page 298
As quoted in Memorabilia Mathematica (or The Philomath's Quotation-Book) (1914) by Robert Edouard Moritz, quotation #1215
As quoted in The First Systems of Weighted Differential and Integral Calculus (1980) by Jane Grossman, Michael Grossman, and Robert Katz, page ii

Nature , Problem

““We live in strange times. I’m fertile right now. I can tell. No more condoms for you, tiger."… [Doob] was already thinking about the videos he was going to make to teach his baby about calculus when he climaxed.”

Neal Stephenson book Seveneves

"Scouts"
Seveneves (2015), Part One

Thinking , Time , Baby

“The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

Way , Time , Dependency

“Israel is the Middle East’s only legitimate democracy, surrounded by cadres, warlords and villains that do not respect democracy or human rights. These bellicose nations jealously regard Israel, envying its success, stability, and might. Israel faces an impossible calculus between defending itself and facing angry outcries or risking its own destruction.”

Stephen Harper (1959) 22nd Prime Minister of Canada

Stephen Harper, as quoted in " We Must Support Democracy in the Middle East http://www.thebarrieexaminer.com/2014/01/31/we-must-support-democracy-in-the-middle-east" (31 January 2014), The Barrie Examiner.
2014

Democracy , Success , Envy

“As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation”

Bernhard Riemann (1826–1866) German mathematician

can be compared with experience
Die partiellen Differentialgleichungen der mathematischen Physik (1882) as quoted by Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book https://books.google.com/books?id=G0wtAAAAYAAJ (1914) p. 239

Law , Nature , Learning , Time

“Rate of change is this mathematics known as Calculus. … Now I hope you understand this, because I've never been able to make head nor tail of it. It must be some sort of a Black Magic operation, started out by the Luce cult — some immoral people who are operating up in New York City, Rockefeller Plaza — been thoroughly condemned by the whole society. Anyway, their rate-of-change theory — I've never seen any use for that mathematics, by the way — I love that mathematics, because it — I asked an engineer, one time, who was in his 6th year of engineering, if he'd ever used Calculus, and he told me yeah, once, once I did, he said. When did you use it? And he said I used it once. Let me see, what did you use it on? Oh yeah. Something on the rate-of-change of steam particles in boilers. And then we went out and tested it and found the answer was wrong.”

L. Ron Hubbard (1911–1986) American science fiction author, philosopher, cult leader, and the founder of the Church of Scientology

Philadelphia Doctorate Course, Tape #58 (November 1952) http://www.rr.cistron.nl/xenu/quotes.htm.

Love , People , Hope , Way

“Science is turning into a monastery for the Order of Capitulant Friars. Logical calculus is supposed to supersede man as moralist. We submit to the blackmail of the "superior knowledge" that has the temerity to assert that nuclear war can be, by derivation, a good thing, because this follows from simple arithmetic.”

Stanisław Lem book His Master's Voice

Source: His Master's Voice (1968), Ch. 9

War , Knowledge , Science

“PLEASURE and pain are undoubtedly the ultimate objects of the calculus of economics. To satisfy our wants to the utmost with the least effort - to procure the greatest amount of what is desirable at the expense of the least that is undesirable - in other words, to maximize pleasure, is the problem of economics.”

William Stanley Jevons The Theory of Political Economy

Source: The Theory of Political Economy (1871), Chapter III, Theory of Utility, p. 61.

Pain , Pleasure , Problem , Effort

“Understanding the methods of calculus is vital to the creative use of mathematics… Without this mastery the average scientist or engineer, or any other user of mathematics, will be perpetually stunted in development, and will at best be able to follow only what the textbooks say; with mastery, new things can be done, even in old, well-established fields.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

Mathematics , Understanding

“Brook Taylor… in his Methodus Incrementorum Directa et Inversa (1715), sought to clarify the ideas of the calculus but limited himself to algebraic functions and algebraic differential equations. …Taylor's exposition, based on what we would call finite differences, failed to obtain many backers because it was arithmetical in nature when the British were trying to tie the calculus to geometry or to the physical notion of velocity.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 427

Idea , Nature

“The calculus is probably the most useful single branch of mathematics. …I have found the ability to do simple calculus, easily and reliably, was the most valuable part of mathematics I ever learned.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

Mathematics , Learning , Parting

“The predecessors of Newton and Leibnitz knew perfectly well how to determine tangents and areas, but they had to approach each problem from first principles. The great contribution of Newton and Leibnitz was precisely to make the procedures for finding tangents, areas, etc. into a calculus, that is, a systematic way of calculating—a collection of algorithms, to use the currently fashionable word.”

Ralph P. Boas Jr. (1912–1992) American mathematician

[Calculus as an Experimental Science, 78, 6, 1971, 664–667, The American Mathematical Monthly, 10.2307/2316582]

Way , Fashion , Problem

“As has become the custom, a wearisome one I admit, this invitation has not been without controversy. Though this unfortunate, this controversy has added little value in the calculus of my decision to be here.”

Clarence Thomas (1948) Associate Justice of the Supreme Court of the United States

1990s, I Am a Man, a Black Man, an American (1998)

Decision

“Continuous distributions are basic to the theory of probability and statistics, and the calculus is necessary to handle them with any ease.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. … Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information.”

George Pólya book How to Solve It

Source: How to Solve It (1945), p. 219

Mathematics

“Calculus systematically evades a great deal of numerical calculation.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“I never failed in mathematics. Before I was fifteen I had mastered differential and integral calculus.”

Albert Einstein (1879–1955) German-born physicist and founder of the theory of relativity

Response to being shown a "Ripley's Believe It or Not!" column with the headline "Greatest Living Mathematician Failed in Mathematics" in 1935. Quoted in Einstein: His Life and Universe by Walter Isaacson (2007), p. 16 http://books.google.com/books?id=cdxWNE7NY6QC&lpg=PP1&pg=PA16#v=onepage&q&f=false
1930s

Mathematics

“The uniformity and repeatability of print created the “political arithmetic” of the seventeenth century and the “hedonistic calculus” of the eighteenth.”

Marshall McLuhan (1911–1980) Canadian educator, philosopher, and scholar-- a professor of English literature, a literary critic, and a …

Source: 1960s, The Gutenberg Galaxy (1962), p. 237

“So intimate is the union between Mathematics and Physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create for each successive class phenomena a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. Sometimes the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armory of the mathematician the weapons which he needed ready made to his hand. But much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma.”

Henry John Stephen Smith (1826–1883) mathematician

As quoted in The Century: A Popular Quarterly (1874) ed. Richard Watson Gilder, Vol. 7, pp. 508-509, https://books.google.com/books?id=ceYGAQAAIAAJ&pg=PA508 "Relations of Mathematics to Physics". Earlier quote without citation in Nature, Volume 8 (1873), page 450.
Also quoted partially in Michael Grossman and Robert Katz, Calculus http://babel.hathitrust.org/cgi/mb?a=listis;c=216746186|Non-Newtonian (1972) p. iv. ISBN 0912938013.

Mathematics , Question , Nature , Knowledge

“I have throughout introduced the Integral Calculus in connexion with the Differential Calculus…. Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction? The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold…”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

Advertisement, pp.3-4
The Differential and Integral Calculus (1836)

Learning

“The student of the Differential Calculus may… be brought to think it possible that the terms and ideas which that science requires may exist in his own mind in the same rude form as that of a straight line in the conceptions of a beginner in geometry. …he must be prepared to stop his course until he can form exact notions, acquire precise ideas, both of resemblance between those things which have appeared most distinct, and of distinction between those which have appeared most alike. To do this… formal definitions would be useless; for he cannot be supposed to have one single notion in that precise form which would make it worth while to attach it to a word. One reason of the great difficulty which is found in treatises on this subject… the tacit assumption that nothing is necessary previously to actually embodying the terms and rules of the science, as if mere statement of definitions could give instantaneous power of using terms rightly. We shall here attempt… a wider degree of verbal explanation than is usual with the view of enabling the student to come to the definitions in some state of previous preparation.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

The Differential and Integral Calculus (1836)

Idea , Thinking , Appearance , Science

“In this work I have attempted to treat economy as a calculus of pleasure and pain, and have sketched out, almost irrespective of previous opinions, the form which the science, as it seems to me, must ultimately take.”

William Stanley Jevons The Theory of Political Economy

Preface To The First Edition, p. 4.
The Theory of Political Economy (1871)

Pain , Pleasure , Science

“The method applied by Newton to the grounding of the Infinitesimal Calculus, and which since the beginning of this century has been recognised by the best mathematicians as the only one that furnishes sure results, is the method of limits. The method consists in this, viz., instead of considering a continuous transition from one value of a quantity to another, from one position to another, or, speaking generally, from one determination of a concept to another, one considers in the first place a transition through a finite number of intervals and then allows the number of these intervals to increase so that the distances of two successive points of division all decrease infinitely.”

Bernhard Riemann (1826–1866) German mathematician

General Relation of the Concept System of Thesis and Antithesis
Gesammelte Mathematische Werke (1876)

Success

“The actual evolution of mathematical theories proceeds by a process of induction strictly analogous to the method of induction employed in building up the physical sciences; observation, comparison, classification, trial, and generalisation are essential in both cases. Not only are special results, obtained independently of one another, frequently seen to be really included in some generalisation, but branches of the subject which have been developed quite independently of one another are sometimes found to have connections which enable them to be synthesised in one single body of doctrine. The essential nature of mathematical thought manifests itself in the discernment of fundamental identity in the mathematical aspects of what are superficially very different domains. A striking example of this species of immanent identity of mathematical form was exhibited by the discovery of that distinguished mathematician... Major MacMahon, that all possible Latin squares are capable of enumeration by the consideration of certain differential operators. Here we have a case in which an enumeration, which appears to be not amenable to direct treatment, can actually be carried out in a simple manner when the underlying identity of the operation is recognised with that involved in certain operations due to differential operators, the calculus of which belongs superficially to a wholly different region of thought from that relating to Latin squares.”

E. W. Hobson (1856–1933) British mathematician

Source: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 290; Cited in: Moritz (1914, 27): The Nature of Mathematics.

Mathematics , Nature , Appearance , Science

“Isaac Newton’s Philosophae Naturalis Principia Mathematica abstracted time from events, establishing its tractability to scientific calculation. Conceived as pure, absolute duration, without qualities, it conforms perfectly to its mathematical idealization (as the real number line). Since time is already pure, its reality indistinguishable from its formalization, a pure mathematics of change – the calculus – can be applied to physical reality without obstruction. The calculus can exactly describe things as they occur in themselves, without straying, even infinitesimally, from the rigorous dictates of formal intelligence. In this way natural philosophy becomes modern science.”

Nick Land (1962) British philosopher

"Time in Transition" https://web.archive.org/web/20121113235339/http://www.thatsmags.com/shanghai/article/777/time-in-transition (2011) (original emphasis)

Way , Intelligence , Mathematics , Nature

“Geometric calculus consists in a system of operations analogous to those of algebraic calculus, but in which the entities on which the calculations are carried out, instead of being numbers, are geometric entities which we shall define.”

Giuseppe Peano (1858–1932) Italian mathematician

Geometric Calculus (1895) as translated by Lloyd C. Kannenberg (2000) "The Operations of Deductive Logic'" Ch. 1 "Geometric Formations"

“The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.”

Nicholas Murray Butler (1862–1947) American philosopher, diplomat, and educator

The Meaning of Education and other Essays and Addresses https://books.google.com/books?id=H9cKAAAAIAAJ (1898) p. 45 as quoted by Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book https://books.google.com/books?id=G0wtAAAAYAAJ (1914)

History , Mathematics , Sense , Science

“Looking backward on a period which is finished, we are looking at actually realized returns, costs, etc., as those items are registered in the bookkeeping of business. In such an ex post calculation there is, as we will show later, an exact balance between the invested waiting and the value of gross investment [Phil: he appears to mean savings and investment]. Looking forward there is no such balance except under certain conditions which remain to be ascertained. In the ex ante calculus it is a question not of realized results but of anticipations, calculations, and plans driving the dynamic process forward. Had this distinction been kept in mind, much confusion about “saving and investment” would have been avoided. There is in fact no contradiction at all between the statement of an exact bookkeeping balance ex post and the obvious inference that in a situation in which saving is increasing without a corresponding increase in investment, or perhaps with an adverse movement in investment, there must be a tendency ex ante to disparity.”

Gunnar Myrdal (1898–1987) Swedish economist

Source: Monetary Equilibrium (1939), p. 34

Question , Business , Appearance , Waiting

“The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, An Investigation of the Laws of Thought (1854), p. 1; Ch. 1. Nature And Design Of This Work, lead paragraph

Truth , Law , Mathematics , Nature

“When looking at the two streams of research these men produced, it is tempting to characterize Buchanan as a somewhat romantic optimist holding the belief that the pursuits of selfish individuals can be channeled for the common good, if the right set of constitutional constraints are adopted, and believing that individuals are capable of designing constitutions that achieve this goal. Thus, some 35 years after the publication of The Calculus, we find Buchanan, in joint work with Roger Congleton (1998), accepting the fact that legislatures always will adopt the simple majority rule, and trying to design constitutional constraints that will lead to outcomes benefiting all members of the community under that rule.”

Dennis Mueller (1940) American economist

James Buchanan, Gordon Tullock, and The Calculus (2012)

Men , Communication , Selfishness

“It has always appeared to me that we sacriﬁce many of the advantages and more of the pleasures of studying any science by omitting all reference to the history of its progress: I have therefore occasionally introduced historical notices of those problems which are interesting either from the nature of the questions involved, or from their bearing on the history of the Calculus. …[T]hese digressions may serve to relieve the dryness of a mere collection of Examples.”

Duncan Gregory (1813–1844) British mathematician

p. vi http://books.google.com/books?id=h7JT-QDuAHoC&pg=PR6, as cited in: Patricia R. Allaire and Robert E. Bradley. " Symbolical algebra as a foundation for calculus: DF Gregory's contribution http://poncelet.math.nthu.edu.tw/disk5/js/history/gregory.pdf." Historia Mathematica 29.4 (2002): p. 409.
Examples of the processes of the differential and integral calculus, (1841)

Pleasure , History , Question , Nature

“The science of the age, in short, is physical, chemical, physiological; in all shapes mechanical. Our favourite Mathematics, the highly prized exponent of all these other sciences, has also become more and more mechanical. Excellence in what is called its higher departments depends less on natural genius than on acquired expertness in wielding its machinery. Without undervaluing the wonderful results which a Lagrange or Laplace educes by means of it, we may remark, that their calculus, differential and integral, is little else than a more cunningly-constructed arithmetical mill; where the factors, being put in, are, as it were, ground into the true product, under cover, and without other effort on our part than steady turning of the handle. We have more Mathematics than ever; but less Mathesis.”

Thomas Carlyle (1795–1881) Scottish philosopher, satirical writer, essayist, historian and teacher

1820s, Signs of the Times (1829)

Age , Education , Mathematics , Nature

“Indeed, most bright, intellectually curious college students suffer through Econ 101, are happy to pass, and then wave goodbye to the subject forever. Economics is filed away with calculus and chemistry—rigorous subjects that required a lot of memorization and have little to do with anything that will come later in life. And, of course, a lot of bright students avoid the course in the first place. This is a shame on two levels.
First, many intellectually curious people are missing a subject that is provocative, powerful, and highly relevant to almost every aspect of our lives.…Second, many of our brightest citizens are economically illiterate.”

Charles Wheelan (1966) American politician

Introduction
Naked Economics (rev. and updated ed., 2010)

Life , People , Suffering , Happiness

“A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics.
The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field.
The duality principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side.
Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.”

Rudolf E. Kálmán (1930–2016) Hungarian-born American electrical engineer

Source: New results in linear filtering and prediction theory (1961), p. 95 Article summary; cited in: " Rudolf E. Kálmán http://www-history.mcs.st-andrews.ac.uk/Biographies/Kalman.html", MacTutor History of Mathematics archive, 2010

Error , Optimism , Problem

“The nature of the firmis not simply a minimizer of transaction costs, but a kind of protective enclave from the potentially volatile and sometimes destructive, ravaging speculation of a competitive market. In the market the rational calculus depends upon the fragile price conven- tion which can often depend on ‘whim or sentiment or chance’. Habits and traditions within the firm are necessarily more enduring because they embody skills and information which cannot always or easily be codified or made subject to a rational calculus. what the tlrm achieves is an institutionalization of these rules and routines within a durable organizational structure. In consequence they are given some degree of permanence and guarded to some extent from the mood waves of speculation in the market.”

Geoffrey Hodgson (1946) British economist

Source: Economics and Institutions, 1988, p. 208

Mood , Nature , Chance , Dependency

“The minimum principle that unified the knowledge of light, gravitation, and electricity of Hamilton's time no longer suffices to relate these fundamental branches of physics. Within fifty years of its creation, the belief that Hamilton's principle would outlive all other physical laws of physics was shattered. Minimum principles have since been created for separate branches of physics… but these are not only restricted… but seem to be contrived…
A single minimum principle, a universal law governing all processes in nature, is still the direction in which the search for simplicity is headed, with the price of simplicity now raised from a mastery of differential equations to a mastery of the calculus of variations.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 442.

Law , Search , Nature , Knowledge

“The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.”

John Von Neumann (1903–1957) Hungarian-American mathematician and polymath

As quoted in Bigeometric Calculus: A System with a Scale-Free Derivative (1983) by Michael Grossman, and in Single Variable Calculus (1994) by James Stewart.

Mathematics , Thinking

“As a public teacher of mere striplings, I am often amazed by the facility and absence of resistance with which the principles of the infinitesimal calculus are accepted and assimilated by the present race of learners. When I was young, a boy of sixteen or seventeen who knew his infinitesimal calculus would have been almost pointed at in the streets as a prodigy, like Dante, as a man who had seen hell. Now-a-days, our Woolwich cadets at the same age, talk with glee of tangents and asymptotes and points of contrary flexure and discuss questions of double maxima and minima, or ballistic pendulums, or motion in a resisting medium, under the familiar and ignoble name of sums.”

James Joseph Sylvester (1814–1897) English mathematician

J. J. Sylvester. "Additional Notes to Prof. Sylvester's Exeter British Association Address", Collected Mathematical Papers, Vol. 2 (1908), pp. 717–718 https://babel.hathitrust.org/cgi/pt?id=miun.aas8085.0002.001;view=1up;seq=732

Age , Boy , Hell , Question

“Even the simplest calculation in the purest mathematics can have terrible consequences. Without the invention of the infinitesimal calculus most of our technology would have been impossible. Should we say therefore that calculus is bad?”

Stanislaw Ulam (1909–1984) Polish-American mathematician

Source: Adventures of a Mathematician - Third Edition (1991), Chapter 11, The 'Super', p. 222

Mathematics

“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.”

Antonin Artaud (1896–1948) French-Occitanian poet, playwright, actor and theatre director

Van Gogh, the Man Suicided by Society (1947)

Understanding

“At the time the book of Marquis de l'Hôpital had appeared, and almost all mathematicians began to turn to the new geometry of the infinite [that is, the new infinitesimal calculus], until then little known. The surprising universality of the methods, the elegant brevity of the proofs, the neatness and speed of the most difficult solutions, a singular and unexpected novelty, all attracted the mind and there was in the mathematical world a well marked revolution [une révolution bien marquée.”

Bernard Le Bovier de Fontenelle (1657–1757) French writer, satirist and philosopher of enlightenment

1792) as quoted by I. Bernard Cohen, Revolution in Science (1985

Books , World , Mathematics , Universe

“…I don't believe in the idea that there are a few peculiar people capable of understanding math and the rest of the world is normal. Math is a human discovery, and it's no more complicated than humans can understand. I had a calculus book once that said, "What one fool can do, another fool can." What we've been able to work out about nature may look abstract and threatening to someone who hasn't studied it, but it was fools who did it, and in the next generation, all the fools will understand it. There's a tendency to pomposity in all this, to make it all deep and profound…”

Richard Feynman book The Pleasure of Finding Things Out

From Omni interview, "The Smartest Man in the World" (1979) or from the book p. 194.
The Pleasure of Finding Things Out (1999)

People , Books , World , Fools

“"One sees, from this Essay, that the theory of probabilities is basically just common sense reduced to calculus; it makes one appreciate with exactness that which accurate minds feel with a sort of instinct, often without being able to account for it."”

Pierre-Simon Laplace (1749–1827) French mathematician and astronomer

"On voit, par cet Essai, que la théorie des probabilités n'est, au fond, que le bon sens réduit au calcul; elle fait apprécier avec exactitude ce que les esprits justes sentent par une sorte d'instinct, sans qu'ils puissent souvent s'en rendre compte."
From the Introduction to Théorie Analytique des Probabilités http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-88764, second and later editions; also published separately as Essai philosophique sur les Probabilités (1814). Œuvres complètes de Laplace, tome VII, p. cliii, Paris: Gauthier-Villars, 1878-1912.
Also reported as: "The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which ofttimes they are unable to account."
Or as: "Probability theory is nothing but common sense reduced to calculation."

Feeling , Sense

“\frac {dy}{dx} = \frac {\omega^2x}{g}…The first derivative, the result of the differentiation of y with respect to x, was written by Leibniz in the form
\frac {dy}{dx}…Leibniz's notation …is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation… need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative \frac {dy}{dx} was considered as the ratio of two "infinitely small quanitites", of the infinitesimals dy and dx. …it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure… it brought mathematics into disrepute… some of the best minds… such as… Berkeley, complained that calculus is incomprehensible. …\frac {dy}{dx} is the limit of a ratio of dy to dx… Once we have realized this sufficiently clearly, we may, under certain circumstances, treat \frac {dy}{dx} so as if it were a ratio… and multiply by dx to achieve the separation of variables. We get
{dy} = \frac {\omega^2x}{g}xdx”

George Pólya (1887–1985) Hungarian mathematician

Mathematical Methods in Science (1977)

Mathematics , Time

“To history we shall adhere no farther, than is sufficient to preserve an unbroken series of methods gradually becoming more exact and extensive; the series beginning with the first rude, though perfectly just, method of James Bernoulli, and ending with Lagrange's exquisite and refined Calculus of Variations.”

Robert Woodhouse (1773–1827) English mathematician

p, 125
A Treatise on Isoperimetrical Problems, and the Calculus of Variations (1810)

History

“Suppose then I want to give myself a little training in the art of reasoning; suppose I want to get out of the region of conjecture and probability, free myself from the difficult task of weighing evidence, and putting instances together to arrive at general propositions, and simply desire to know how to deal with my general propositions when I get them, and how to deduce right inferences from them; it is clear that I shall obtain this sort of discipline best in those departments of thought in which the first principles are unquestionably true. For in all 59 our thinking, if we come to erroneous conclusions, we come to them either by accepting false premises to start with—in which case our reasoning, however good, will not save us from error; or by reasoning badly, in which case the data we start from may be perfectly sound, and yet our conclusions may be false. But in the mathematical or pure sciences,—geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves,—we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline. When Plato wrote over the portal of his school. “Let no one ignorant of geometry enter here,” he did not mean that questions relating to lines and surfaces would be discussed by his disciples. On the contrary, the topics to which he directed their attention were some of the deepest problems,—social, political, moral,—on which the mind could exercise itself. Plato and his followers tried to think out together conclusions respecting the being, the duty, and the destiny of man, and the relation in which he stood to the gods and to the unseen world. What had geometry to do with these things? Simply this: That a man whose mind has not undergone a rigorous training in systematic thinking, and in the art of drawing legitimate inferences from premises, was unfitted to enter on the discussion of these high topics; and that the sort of logical discipline which he needed was most likely to be obtained from geometry—the only mathematical science which in Plato’s time had been formulated and reduced to a system. And we in this country [England] have long acted on the same principle. Our future lawyers, clergy, and statesmen are expected at the University to learn a good deal about curves, and angles, and numbers and proportions; not because these subjects have the smallest relation to the needs of their lives, but because in the very act of learning them they are likely to acquire that habit of steadfast and accurate thinking, which is indispensable to success in all the pursuits of life.”

Joshua Girling Fitch (1824–1903) British educationalist

Source: Lectures on Teaching, (1906), pp. 291-292

Life , God , Truth , World

“It's a hard, simple calculus: Run until you can't run anymore. Then run some more. Find a new source of energy and will. Then run even faster.”

Scott Jurek (1973) American ultramarthon runner

Source: Eat and Run (2012), Ch. 1, p. 5

“The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 346
Context: Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits.

“I believe mysticism is a very serious endeavor. One must be equipped for it. One doesn't study calculus before studying arithmetic.”

Elie Wiesel (1928–2016) writer, professor, political activist, Nobel Laureate, and Holocaust survivor

As quoted in "10 Questions for Elie Wiesel" by Jeff Chu in TIME (22 January 2006) http://www.time.com/time/magazine/article/0,9171,1151803,00.html
Context: I believe mysticism is a very serious endeavor. One must be equipped for it. One doesn't study calculus before studying arithmetic. In my tradition, one must wait until one has learned a lot of Bible and Talmud and the Prophets to handle mysticism. This isn't instant coffee. There is no instant mysticism.

Study

“In the first solution of Isoperimetrical problems, the Bernoullis use diagrams and their properties. Euler, in his early essays, does the same; then, as he improves the calculus he gets rid of constructions. In his Treatise”

Robert Woodhouse (1773–1827) English mathematician

A Treatise on Isoperimetrical Problems, and the Calculus of Variations (1810)
Context: There is another point... and that is the method of demonstration by geometrical figures. In the first solution of Isoperimetrical problems, the Bernoullis use diagrams and their properties. Euler, in his early essays, does the same; then, as he improves the calculus he gets rid of constructions. In his Treatise [footnote: Methodus inveniendi, &c. ], he introduces geometrical figures, but almost entirely, for the purpose of illustration: and finally, in the tenth volume of the Novi Comm. Petrop. as Lagrange had done in the Miscellanea Taurinensea, he expounds the calculus, in its most refined state, entirely without the aid of diagrams and their properties. A similar history will belong to every other method of calculation, that has been advanced to any degree of perfection. <!--Preface p. vii-viii

Problem

“Remember, that though life is short, Thought and the influences of what we do or say, are immortal; and that no calculus has yet pretended to ascertain the law of proportion between cause and effect.”

Albert Pike book Morals and Dogma of the Ancient and Accepted Scottish Rite of Freemasonry

Source: Morals and Dogma of the Ancient and Accepted Scottish Rite of Freemasonry (1871), Ch. II : The Fellow-Craft, p. 43
Context: Remember, that though life is short, Thought and the influences of what we do or say, are immortal; and that no calculus has yet pretended to ascertain the law of proportion between cause and effect. The hammer of an English blacksmith, smiting down an insolent official, led to a rebellion which came near being a revolution. The word well spoken, the deed fitly done, even by the feeblest or humblest, cannot help but have their effect. More or less, the effect is inevitable and eternal. The echoes of the greatest deeds may die away like the echoes of a cry among the cliffs, and what has been done seem to the human judgment to have been without result. The unconsidered act of the poorest of men may fire the train that leads to the subterranean mine, and an empire be rent by the explosion.

Life , Law

“Just as no thing or organism exists on its own, it does not act on its own. Furthermore, every organism is a process: thus the organism is not other than its actions. To put it clumsily: it is what it does. More precisely, the organism, including its behavior, is a process which is to be understood only in relation to the larger and longer process of its environment. For what we mean by "understanding" or "comprehension" is seeing how parts fit into a whole, and then realizing that they don't compose the whole, as one assembles a jigsaw puzzle, but that the whole is a pattern, a complex wiggliness, which has no separate parts. Parts are fictions of language, of the calculus of looking at the world through a net which seems to chop it up into bits. Parts exist only for purposes of figuring and describing, and as we figure the world out we become confused if we do not remember this all the time.”

Alan Watts (1915–1973) British philosopher, writer and speaker

Source: The Book on the Taboo Against Knowing Who You Are (1966), p. 73

World , Behavior , Understanding , Time

“Images and pictures… have played a key role in shaping our scientific picture of the world. …Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking.”

John D. Barrow (1952–2020) British scientist

Introduction
Cosmic Imagery: Key Images in the History of Science (2008)

World , Family , Thinking , Success

“Thoughts in time
And out of season.
The hitchhiker stood
By the side of the road,
And leveled his thumb
In the calm the calculus
Of reason… ”

Jim Morrison (1943–1971) lead singer of The Doors

Time , Season

“Our problem is that we do not fully understand Putin’s calculus, just as he does not understand ours. In Putin’s view, the United States, the European Union and NATO have launched an economic and proxy war in Ukraine to weaken Russia and push it into a corner. As Valery Gerasimov, chief of staff of the Russian armed forces, has underscored, this is a hybrid, 21st-century conflict, in which financial sanctions, support for oppositional political movements and propaganda have all been transformed from diplomatic tools to instruments of war. Putin likely believes that any concession or compromise he makes will encourage the West to push further.”

Fiona Hill (presidential advisor) (1965) American policy adviser

How aiding the Ukrainian military could push Putin into a regional war, By Fiona Hill and Clifford Gaddy, The Washington Post https://www.washingtonpost.com/opinions/giving-weapons-to-ukraine-could-goad-putin-into-a-regional-war/2015/02/05/ec2e9680-abf5-11e4-ad71-7b9eba0f87d6_story.html (5 February 2015)

Understanding , War , Support , Problem

“That was the danger of being old and a politician. Habits outlived the situations that created them. Policies remained in place after the situations that inspired them had changed. The calculus of all human power was changing, and the models she used to make sense of it shifted with them, and she had to keep reminding herself that the past was a different place. She didn’t live there anymore.”

Daniel Abraham (1969) speculative fiction writer from the United States

Source: Cibola Burn (2014), Epilogue (p. 576)

Sense , Past , Making love , Change

“I have throughout introduced the Integral Calculus in connexion with the Differential Calculus. …Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction?”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold...
The Differential and Integral Calculus (1836)

Relation ships , Learning