# „If one proves the equality of two numbers a and b by showing first that a \leqq b and then that a \geqq b, it is unfair; one should instead show that they are really equal by disclosing the inner ground for their equality.“

As quoted in Hermann Weyl, "Emmy Noether" (April 26, 1935) in Weyl's Levels of Infinity: Selected Writings on Mathematics and Philosophy (2012) p. 64.

Obtido da Wikiquote. Última atualização 3 de Junho de 2021. História
1882 - 1935

### „If a and b yield C, but C is not equal to a+b, then we have emergence.“

—  Varadaraja V. Raman American physicist 1932

page 313
Truth and Tension in Science and Religion

### „In the algebra of fantasy, A times B doesn't have to equal B times A. But, once established, the equation must hold throughout the story.“

—  Lloyd Alexander American children's writer 1924 - 2007

"The Flat-Heeled Muse", Horn Book Magazine (1 April 1965)

### „b>The first thing is to have the will; the rest is technique.</b“

—  Halldór Laxness, livro Kristnihald undir Jökli (bók)

Kristnihald undir Jökli (Under the Glacier/Christianity at Glacier) (1968)

### „On symbolic use of equalities and proportions. Chapter II.The analytical method accepts as proven the most famous [ as known from Euclid ] symbolic use of equalities and proportions that are found in items such as:1. The whole is equal to the sum of its parts.2. Quantities being equal to the same quantity have equality between themselves. [a = c & b = c => a = b]3. If equal quantities are added to equal quantities the resulting sums are equal.4. If equals are subtracted from equal quantities the remains are equal.5. If equal equal amounts are multiplied by equal amounts the products are equal.6. If equal amounts are divided by equal amounts, the quotients are equal.7. If the quantities are in direct proportion so also are they are in inverse and alternate proportion. [a:b::c:d=>b:a::d:c & a:c::b:d]8. If the quantities in the same proportion are added likewise to amounts in the same proportion, the sums are in proportion. [a:b::c:d => (a+c):(b+d)::c:d]9. If the quantities in the same proportion are subtracted likewise from amounts in the same proportion, the differences are in proportion. [a:b::c:d => (a-c):(b-d)::c:d]10. If proportional quantities are multiplied by proportional quantities the products are in proportion. [a:b::c:d & e:f::g:h => ae:bf::cg:dh]11. If proportional quantities are divided by proportional quantities the quotients are in proportion. [a:b::c:d & e:f::g:h => a/e:b/f::c/g:d/h]12. A common multiplier or divisor does not change an equality nor a proportion. [a:b::ka:kb & a:b::(a/k):(b/k)]13. The product of different parts of the same number is equal to the product of the sum of these parts by the same number. [ka + kb = k(a+b)]14. The result of successive multiplications or divisions of a magnitude by several others is the same regardless of the sequential order of quantities multiplied times or divided into that magnitude.But the masterful symbolic use of equalities and proportions which the analyst may apply any time is the following:15. If we have three or four magnitudes and the product of the extremes is equal to the product means, they are in proportion. [ad=bc => a:b::c:d OR ac=b2 => a:b::b:c]And conversely10. If we have three or four magnitudes and the first is to the second as the second or the third is to the last, the product of the extremes is equal to that of means. [a:b::c:d => ad=bc OR a:b::b:c => ac=b2]We can call a proportion the establishment of an equality [equation] and an equality [equation] the resolution of a proportion.“

—  François Viète French mathematician 1540 - 1603

From Frédéric Louis Ritter's French Tr. Introduction à l'art Analytique (1868) utilizing Google translate with reference to English translation in Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968) Appendix
In artem analyticem Isagoge (1591)

### „Equality gives rise to challenging questions which are not altogether easy to answer… a = a and a = b are obviously statements of differing cognitive value; a = a holds a priori and, according to Kant, is to be labeled analytic, while statements of the form a = b often contain very valuable extensions of our knowledge and cannot always be established a priori. The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even to-day the identification of a small planet or a comet is not always a matter of course. Now if we were to regard equality as a relation between that which the names 'a' and 'b' designate, it would seem that a = b could not differ from a = a (i. e. provided a = b is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing.“

—  Gottlob Frege, Sense and reference

As cited in: M. Fitting, Richard L. Mendelsoh (1999), First-Order Modal Logic, p. 142. They called this Frege's Puzzle.
Über Sinn und Bedeutung, 1892

### „The typical white intellectual considers himself superior to ordinary white people for two contradictory reasons: a] he constantly proclaims belief in human equality, but they don't; b] he has a high IQ, but they don't.“

—  Steve Sailer American journalist and movie critic 1958

How to Help the Left Half of the Bell Curve http://www.isteve.com/How_to_Help_the_Left_Half_of_the_Bell_Curve.htm, VDARE.com, July to September 2000

### „You can only aim at equality by giving some people the right to take things from others. And what ultimately happens when you aim for equality is that A and B decide what C should do for D, except that they take a bit of commission off on the way.“

—  Milton Friedman American economist, statistician, and writer 1912 - 2006

### „Equality gives rise to challenging questions which are not altogether easy to answer… a = a and a = b are obviously statements of differing cognitive value; a = a holds a priori and, according to Kant, is to be labeled analytic, while statements of the form a = b often contain very valuable extensions of our knowledge and cannot always be established a priori.“

—  Gottlob Frege, Sense and reference

The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even to-day the identification of a small planet or a comet is not always a matter of course. Now if we were to regard equality as a relation between that which the names 'a' and 'b' designate, it would seem that a = b could not differ from a = a (i.e. provided a = b is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing.
As cited in: M. Fitting, Richard L. Mendelsoh (1999), First-Order Modal Logic, p. 142. They called this Frege's Puzzle.
Über Sinn und Bedeutung, 1892

### „The first statement of the two principles reads as follows. First: each person is to have an equal right to the most extensive basic liberty compatible with a similar liberty for others. Second: social and economic inequalities are to be arranged so that they are both(a)reasonably expected to be to everyone's advantage, and (b) attached to positions and offices open to all.“

—  John Rawls, livro A Theory of Justice

Fonte: A Theory of Justice (1971; 1975; 1999), Chapter II, Section 11, pg. 60

### „Prior to 1952, when M. Ventris first published his decipherment of the 'Linear B' tablets from Crete and Greece, showing that they were Greek, everyone assumed that Hebrew was recorded in writing before Greek. But now… we are reading Linear B Greek texts, written before the birth of Abraham“

—  Cyrus H. Gordon American linguist 1908 - 2001

let alone before the date of any known Hebrew text
Introduction
Adventures in the Nearest East (1957)

### „In the arithmetic of love, one plus one equals everything, and two minus one equals nothing.“

—  Mignon McLaughlin American journalist 1913 - 1983

The Complete Neurotic's Notebook (1981), Love

### „One plus one does not equal two.“

—  Howard Bloom American publicist and author 1943

Brace Yourself: The Five Heresies
The God Problem: How a Godless Cosmos Creates (2012)

### „In February 1917, it was found that a static solution with a positive curvature—the solution A—was not possible without the λ. In fact the curvature is proportional to λ (in solution A, λ is equal to the curvature; in B… it is three times the curvature). Thus, at the time when we had only the two static solutions A and B, and thought that these were the only possible ones, here was a plausible physical interpretation of the meaning of λ: it was the curvature of the world, and the square root of its reciprocal, the radius of curvature, could be conceived as providing a natural unit of length.“

—  Willem de Sitter Dutch cosmologist 1872 - 1934

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

### „The discovery of Hippocrates amounted to the discovery of the fact that from the relation(1)\frac{a}{x} = \frac{x}{y} = \frac{y}{b}it follows that(\frac{a}{x})^3 = [\frac{a}{x} \cdot \frac{x}{y} \cdot \frac{y}{b} =] \frac{a}{b}and if a = 2b, [then (\frac{a}{x})^3 = 2, and]a^3 = 2x^3.The equations (1) are equivalent [by reducing to common denominators or cross multiplication] to the three equations(2)x^2 = ay, y^2 = bx, xy = ab[or equivalently…y = \frac{x^2}{a}, x = \frac{y^2}{b}, y = \frac{ab}{x} ]Doubling the Cubethe 2 solutions of Menaechmusand the solutions of Menaechmus described by Eutocius amount to the determination of a point as the intersection of the curves represented in a rectangular system of Cartesian coordinates by any two of the equations (2).Let AO, BO be straight lines placed so as to form a right angle at O, and of length a, b respectively. Produce BO to x and AO to y.The first solution now consists in drawing a parabola, with vertex O and axis Ox, such that its parameter is equal to BO or b, and a hyperbola with Ox, Oy as asymptotes such that the rectangle under the distances of any point on the curve from Ox, Oy respectively is equal to the rectangle under AO, BO i. e. to ab. If P be the point of intersection of the parabola and hyperbola, and PN, PM be drawn perpendicular to Ox, Oy, i. e. if PN, PM be denoted by y, x, the coordinates of the point P, we shall have\begin{cases}y^2 = b. ON = b. PM = bx\\ and\\ xy = PM. PN = ab\end{cases}whence\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.In the second solution of Menaechmus we are to draw the parabola described in the first solution and also the parabola whose vertex is O, axis Oy and parameter equal to a.“

—  Thomas Little Heath British civil servant and academic 1861 - 1940

The point P where the two parabolas intersect is given by<center>$\begin{cases}y^2 = bx\\x^2 = ay\end{cases}$</center>whence, as before,<center>$\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.$</center>
Apollonius of Perga (1896)

### „One hundred years after the Declaration that 'all men are created equal,' there began to gather in Newport a colony of the rich, determined to show that some Americans were conspicuously more equal than others.“

—  Alistair Cooke British journalist and broadcaster 1908 - 2004

### „And if any number of equals to a first sine be multiplied together producing a second, just so many equals to the Logarithm of the first added together produce the Logarithm of the second.“

—  John Napier Scottish mathematician 1550 - 1617

Appendix, The relations of Logarithms & their natural numbers to each other
The Construction of the Wonderful Canon of Logarithms (1889)

### „If two equally qualified persons apply for a job at a workplace with few immigrants, the one called Mohammed should get the job.“

—  Mona Sahlin Swedish politician 1957

Mona Sahlin in an interview with the Swedish newspaper Göteborgs-Posten, October 22, 2000.

### „In the beginning of 1917, two solutions of the field equations for a homogeneous isotropic universe had been found, which I… call the solutions "A" and "B."… at that time only static solutions were looked for. It was thought that the universe must be a stable structure… In one of these solutions (B) the average density was zero, it was empty; the other one (A) had a finite density…. In B, to get the real universe, we should have to put in a few galactic systems, in A we should have to condense the evenly distributed matter into galactic systems. The universe A… has an average density, but no expansion. It is therefore called the static universe. B, on the other hand… expands, and it could only parade in the garb of a static universe because there is nothing in it to show the expansion. B is therefore called the empty universe. Thus we had two approximations : the static universe with matter and without expansion, and the empty one without matter and with expansion. The actual universe… has both matter and expansion… In 1917… the actual value of the density was still entirely unknown, and the expansion had not yet been discovered.“

—  Willem de Sitter Dutch cosmologist 1872 - 1934

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

### „If two right lines cut one another, they will form the angles at the vertex equal.“

—  Proclus Greek philosopher 412 - 485

...
This... is what the the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented according to Eudemus by Thales...
Proposition XV. Thereom VIII.

### „Let as many Numbers, as you please, be proposed to be Combined: Suppose Five, which we will call a b c d e. Put, in so many Lines, Numbers, in duple proportion, beginning with 1. The Sum (31) is the Number of Sumptions, or Elections; wherein, one or more of them, may several ways be taken. Hence subduct (5) the Number of the Numbers proposed; because each of them may once be taken singly. And the Remainder (26) shews how many ways they may be taken in Combination; (namely, Two or more at once.) And, consequently, how many Products may be had by the Multiplication of any two or more of them so taken. But the same Sum (31) without such Subduction, shews how many Aliquot Parts there are in the greatest of those Products, (that is, in the Number made by the continual Multiplication of all the Numbers proposed,) a b c d e. For every one of those Sumptions, are Aliquot Parts of a b c d e, except the last, (which is the whole,) and instead thereof, 1 is also an Aliquot Part; which makes the number of Aliquot Parts, the same with the Number of Sumptions. Only here is to be understood, (which the Rule should have intimated;) that, all the Numbers proposed, are to be Prime Numbers, and each distinct from the other. For if any of them be Compound Numbers, or any Two of them be the same, the Rule for Aliquot Parts will not hold.“

—  John Wallis English mathematician 1616 - 1703

Fonte: A Discourse of Combinations, Alterations, and Aliquot Parts (1685), Ch.I Of the variety of Elections, or Choice, in taking or leaving One or more, out of a certain Number of things proposed.