Varadaraja V. Raman quote: “If a and b yield C, but C is not equal to a+b, then we have emergence.”

“I don't believe that A B & C = D. It equals Z.”

Josh Duffy (1978) Subject of the documentary THE MAYOR

Film Quotes

“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 conversely
10. 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 (1540–1603) French mathematician

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)

“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 (1912–2006) American economist, statistician, and writer

“Milton Friedman vs Free Lunch Advocate” https://www.youtube.com/watch?v=9Qe7fLL25AQ (1980s)

“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.”

Emmy Noether (1882–1935) German mathematician

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.

“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 (1924–2007) American children's writer

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

“There is a familiar trio of reactions by scientists to a purportedly radical hypothesis: (a) "You must be out of your mind!", (b) "What else is new? Everybody knows that!", and, later — if the hypothesis is still standing — (c) "Hmm. You *might* be on to something!" Sometimes these phases take years to unfold, one after another, but I have seen all three emerge in near synchrony in the course of a half-hour's heated discussion following a conference paper.”

Daniel Dennett book Darwin's Dangerous Idea

Darwin's Dangerous Idea (1995)

“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

“A policeman is a charlatan who offers, in return for obedience, to protect him (a) from his superiors, (b) from his equals, and (c) from himself. This last service, under democracy, is commonly the most esteemed of them all. In the United States, at least theoretically, it is the only thing that keeps ice-wagon drivers, Y.M.C.A. secretaries, insurance collectors and other such human camels from smoking opium, ruining themselves in the night clubs, and going to Palm Beach with Follies girls”

H.L. Mencken (1880–1956) American journalist and writer

1920s, Notes on Democracy (1926)
Context: What the common man longs for in this world, before and above all his other longings, is the simplest and most ignominious sort of peace: the peace of a trusty in a well-managed penitentiary. He is willing to sacrifice everything else to it. He puts it above his dignity and he puts it above his pride. Above all, he puts it above his liberty. The fact, perhaps, explains his veneration for policemen, in all the forms they take–his belief that there is a mysterious sanctity in law, however absurd it may be in fact.
A policeman is a charlatan who offers, in return for obedience, to protect him (a) from his superiors, (b) from his equals, and (c) from himself. This last service, under democracy, is commonly the most esteemed of them all. In the United States, at least theoretically, it is the only thing that keeps ice-wagon drivers, Y. M. C. A. secretaries, insurance collectors and other such human camels from smoking opium, ruining themselves in the night clubs, and going to Palm Beach with Follies girls... Under the pressure of fanaticism, and with the mob complacently applauding the show, democratic law tends more and more to be grounded upon the maxim that every citizen is, by nature, a traitor, a libertine, and a scoundrel. In order to dissuade him from his evil-doing the police power is extended until it surpasses anything ever heard of in the oriental monarchies of antiquity.

“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 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 Cube
the 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 (1861–1940) British civil servant and academic

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

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

Help us to complete the source, original and additional information

Varadaraja V. Raman 23

Related quotes

Related topics