Emmy Noether quote: “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…”

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

Varadaraja V. Raman (1932) American physicist

page 313
Truth and Tension in Science and Religion

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

Josh Duffy (1978) Subject of the documentary THE MAYOR

Film Quotes

“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)

“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)

“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

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

Halldór Laxness book Kristnihald undir Jökli (bók)

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

“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)

“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 (1958) American journalist and movie critic

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

“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 book A Theory of Justice

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

“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.”

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