Quotes about integer

A collection of quotes on the topic of integer, number, position, quantity.

Quotes about integer

“By the time I was a student in high school I was reading the classic Men of Mathematics by E. T. Bell and I remember succeeding in proving the classic Fermat theorem about an integer multiplied by itself p times where p is a prime.”

John Nash (1928–2015) American mathematician and Nobel Prize laureate

Autobiographical essay (1994)

Men, School, Mathematics, Time

“The number of syllables in the English names of finite integers tends to increase as the integers grow larger, and must gradually increase indefinitely, since only a finite number of names can be made with a given finite number of syllables. Hence the names of some integers must consist of at least nineteen syllables, and among these there must be a least. Hence "the least integer not nameable in fewer than nineteen syllables" must denote a definite integer; in fact, it denotes 111, 777. But "the least integer not nameable in fewer than nineteen syllables" is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables, which is a contradiction. This contradiction was suggested to us by Mr. G. G. Berry of the Bodleian Library.”

Bertrand Russell Principia Mathematica

Principia Mathematica, written with Alfred North Whitehead, (1910), vol. I, Introduction, ch. II: The Theory of Logical Types. This is a statement of the Berry paradox.
1910s

“He could remember the idiosyncrasies of numbers in an almost uncanny way. It was Littlewood who said that every positive integer was one of Ramanujan's personal friends. I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."”

G. H. Hardy (1877–1947) British mathematician

Source: Ramanujan (1940), Ch. I : The Indian mathematician Ramanujan.

Hope, Way, Personality

“Planck’s quantization assumption applied to the matter that emits and absorbs radiation, not to radiation itself. As George Gamow later remarked, Planck thought that radiation was like butter; butter itself comes in any quantity, but it can be bought and sold only in multiples of one quarter pound. It was Albert Einstein (1879–1955) who in 1905 proposed that the energy of radiation of frequency ν was itself an integer multiple of hν.”

Steven Weinberg (1933) American theoretical physicist

Source: Lectures on Quantum Mechanics (2012, 2nd ed. 2015), Ch. 1: Historical Introduction

“Is a woman a thinking unit at all, or a fraction always wanting its integer?”

Thomas Hardy (1840–1928) English novelist and poet

Thinking

“Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers).”

Henry Burchard Fine (1858–1928) American academic

Source: The Number-System of Algebra, (1890), p. 86; Reported in Moritz (1914, 282)

Sense

“Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity there is no such jump, and because jump is missing in the world of quantity it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate.”

Gregory Bateson (1904–1980) English anthropologist, social scientist, linguist, visual anthropologist, semiotician and cyberneticist

Bateson (1978) " Number is Different from Quantity http://www.oikos.org/batesnumber.htm". In: CoEvolution Quarterly, Spring 1978, pp. 44-46

World, Water

“The concept of 'number' in its most elementary sense as the signless integer appears to be an immediate abstraction from quantitative reality subjected to processes of counting and measurement. Vulgar fractions arise from division of a quantity into equal parts. But in what sense is zero a number? Are there negative numbers? Are there numbers corresponding to incommensurable ratios? Each question requires for its solution a fresh exercise of that kind of creative imagination which we call mathematical abstraction.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Exercises, Imagination, Mathematics, Question

“God made the integers, all the rest is the work of man.”

Leopold Kronecker (1823–1891) German mathematician who worked on number theory and algebra (1823–1891)

Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.
Quoted in "Philosophies of Mathematics" - Page 13 - by Alexander George, Daniel J. Velleman - Philosophy - 2002

God

“It was Mr. Littlewood (I believe) who remarked that "every positive integer was one of his personal friends."”

John Edensor Littlewood (1885–1977) English Mathematician

(about Ramanujan) p. lvii of [Hardy, G. H., G. H. Hardy, Obituary Notices: Srinivasa Ramanujan, Proceedings of the London Mathematical Society, 19, xl-lviii, 1921, http://www.numbertheory.org/obituaries/LMS/ramanujan/index.html, 2008-05-26]

Personality

“I read in the proof-sheets of Hardy on Ramanujan: 'As someone said, each of the positive integers was one of his personal friends.' My reaction was, 'I wonder who said that; I wish I had.' In the next proof-sheets I read (what now stands): 'It was Littlewood who said…' (What had happened was that Hardy had received the remark in silence and with a poker face, and I wrote it off as a dud….)”

John Edensor Littlewood (1885–1977) English Mathematician

"Cross-purposes, Unconscious Assumptions, Howlers, Misprints, etc.", p. 61.
Littlewood's Miscellany (1986)

Silence, Reading, Personality

“Cadets are people. Behind the gray suits, beneath the Pom-pom and Shako and above the miraculously polished shoes, blood flows through veins and arteries, hearts thump in a regular pattern, stomachs digest food, and kidneys collect waste. Each cadet is unique, a functioning unit of his own, a distinct and separate integer from anyone else. Part of the irony of military schools stems from the fact that everyone in these schools is expected to act precisely the same way, register the same feelings, and respond in the same prescribed manner. The school erects a rigid structure of rules from which there can be no deviation. The path has already been carved through the forest and all the student must do is follow it, glancing neither to the right nor left, and making goddamn sure he participates in no exploration into the uncharted territory around him. A flaw exists in this system. If every person is, indeed, different from every other person, then he will respond to rules, regulations, people, situations, orders, commands, and entreaties in a way entirely depending on his own individual experiences. Te cadet who is spawned in a family that stresses discipline will probably have less difficulty in adjusting than the one who comes from a broken home, or whose father is an alcoholic, or whose home is shattered by cruel arguments between the parents. Yet no rule encompasses enough flexibility to offer a break to a boy who is the product of one of these homes.”

Pat Conroy book The Boo

Source: The Boo (1970), p. 10

People, Alcohol, School, Family

“We started out in the middle ages creating music which had certain desirable physical properties (for example, a major chord sounds "nice" because the frequencies are in integer ratios to each other). And then as society evolved, we created these emotional contexts for certain instruments and progressions. Major-chord arpeggios sound "happy", minor chords sound "sad", chromatic scales can sound "scary", et cetera. In the 20th century, film soundtracks reinforced this point as people associated certain kinds of music with certain visual and emotional experiences. It's a giant feedback loop, really; once you grow up in a given culture, it leaves this musical fingerprint on you which colors your experiences.”

Andrew Sega (1975) musician from America

Andrew Sega Shrine interview, 2011

People, Age, Sadness, Music

“There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are
(1) ab(u) = ba (u),
(2) a(u + v) = a (u) + a (v),
(3) am. an. u = am + n. u.
The first of these laws is called the commutative law, and symbols which are subject to it are called commutative symbols. The second law is called distributive, and the symbols subject to it distributive symbols. The third law is not so much a law of combination of the operation denoted by a, but rather of the operation performed on a, which is indicated by the index affixed to a. It may be conveniently called the law of repetition, since the most obvious and important case of it is that in which m and n are integers, and am therefore indicates the repetition m times of the operation a.”

Duncan Gregory (1813–1844) British mathematician

That these are the laws employed in the demonstration of the principal theorems in Algebra, a slight examination of the processes will easily shew ; but they are not confined to symbols of numbers ; they apply also to the symbol used to denote differentiation. <br class="br"> p. 237 http://books.google.com/books?id=8lQ7AQAAIAAJ&pg=PA237; Highlighted section cited in: George Boole " Mr Boole on a General Method in Analysis http://books.google.com/books?pg=PA225-IA15&id=aGwOAAAAIAAJ&hl," Philosophical Transactions, Vol. 134 (1844), p. 225; Other section (partly) cited in: James Gasser (2000) A Boole Anthology: Recent and Classical Studies in the Logic of George Boole,, p. 52 <br class="br">Examples of the processes of the differential and integral calculus, (1841)

Law, Nature, Time, Dependency

“Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

Law, Question

“I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

Law, Nature, Success, Wealth

“The integers, the rationals, and the irrationals, taken together, make up the continuum of real numbers. It's called a continuum because the numbers are packed together along the real number line with no empty spaces between them.”

Brian Hayes (scientist) (1900) American scientist, columnist and author

Source: Group Theory in the Bedroom (2008), Chapter 11, Identity Crisis, p. 206 (See also: George Cantor)

“It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. …Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4=4x+20 as ᾰτοπος because it would give x=-4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly.”

Thomas Little Heath (1861–1940) British civil servant and academic

Diophantos of Alexandria: A Study in the History of Greek Algebra (1885)

Problem, Sense, Time

“But what, after all, are the integers? Everyone thinks that he or she knows, for example, what the number three is — until he or she tries to define or explain it.”

Carl B. Boyer (1906–1976) American mathematician

As quoted in Everything and More: A Compact History of Infinity (2013) by David Foster Wallace, p. 8

Thinking

“Bohr had… discovered that the frequencies corresponding to very large integers could be calculated accurately from the classical mechanics; they were simply the number of times that an ordinary electron would complete the circuit of its orbit in one second when it was at a very great distance from the nucleus of the atom to which it belonged. This could only mean that when an electron receded to a great distance from the nucleus of its atom, it not only assumed the properties of an ordinary electron, but also behaved as directed by the classical mechanics. Yet the classical mechanics failed completely for the calculation of frequencies corresponding to small orbits.”

James Jeans (1877–1946) British mathematician and astronomer

Physics and Philosophy (1942)

Time

“He gave more elegant rules for the sum of the squares and cubes of an initial segment of the positive integers. The sixth part of the product of three quantities consisting of the number of terms, the number of terms plus one, and twice the number of terms plus one is the sum of the squares. The square of the sum of the series is the sum of the cubes.”

Aryabhata (476–550) Indian mathematician-astronomer

Roger Cooke in: The history of mathematics: a brief course http://books.google.co.in/books?id=z-ruAAAAMAAJ, Wiley, 7 October 1997, p. 207.

Parting

“When the Löwenheim-Skolem theorem is applied to particular formal systems, we obtain as special cases: Every group, field, ordered field, etc., has a countable subsystem of the same type. A more spectacular result follows from applying the theorem to set theory (a system which we shall later formalize): There is a countable collection of sets, such that if restrict the membership relation to these sets alone, they form a model for set theory (more precisely all the true statements of set theory are true in this model). In particular, within this model which we may denote by M, there must be an uncountable set. This paradox, that a countable model can contain an uncountable set, is explained by noting that to say a set is uncountable merely asserts the nonexistence of a one-one mapping of the set with the set of integers. The "uncountable" set in M set actually has only countably many members in M, but there is no one-one correspondence \underline {within} M of this set with the set of integers.”

Paul Cohen (1934–2007) American mathematician

Set theory and the continuum hypothesis, pp. 19–20 https://books.google.com/books?id=Z4NCAwAAQBAJ&pg=PA19 <br class="br">Set Theory and the Continuum Hypothesis (1966)

Relation ships