“If a, c are two different numbers, there are infinitely many different numbers lying between a, c.”

p, 125
Stetigkeit und irrationale Zahlen (1872)

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Richard Dedekind13

German mathematician 1831–1916

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“Suppose we a certain Number of things exposed, different each from other, as a, b, c, d, e, &c.; The question is, how many ways the order of these may be varied? as, for instance, how many changes may be Rung upon a certain Number of Bells; or, how many ways (by way of Anagram) a certain Number of (different) Letters may be differently ordered?
Alt.1,21) If the thing exposed be but One, as a, it is certain, that the order can be but one. That is 1.
2) If Two be exposed, as a, b, it is also manifest, that they may be taken in a double order, as ab, ba, and no more. That is 1 x 2 = 2. Alt.3
3) If Three be exposed; as a, b, c: Then, beginning with a, the other two b, c, may (by art. 2,) be disposed according to Two different orders, as bc, cb; whence arise Two Changes (or varieties of order) beginning with a as abc, acb: And, in like manner it may be shewed, that there be as many beginning with b; because the other two, a, c, may be so varied, as bac, bca. And again as many beginning with c as cab, cba. And therefore, in all, Three times Two. That is 1 x 2, x 3 = 6.
Alt.34) If Four be exposed as a, b, c, d; Then, beginning with a, the other Three may (by art. preceeding) be disposed six several ways. And (by the same reason) as many beginning with b, and as many beginning with c, and as many beginning with d. And therefore, in all, Four times six, or 24. That is, the Number answering to the case next foregoing, so many times taken as is the Number of things here exposed. That is 1 x 2 x 3, x 4 = 6 x 4 = 24.
5) And in like manner it may be shewed, that this Number 24 Multiplied by 5, that is 120 = 24 x 5 = 1 x 2 x 3 x 4 x 5, is the number of alternations (or changes of order) of Five things exposed. (Or, the Number of Changes on Five Bells.) For each of these five being put in the first place, the other four will (by art. preceeding) admit of 24 varieties, that is, in all, five times 24. And in like manner, this Number 120 Multiplied by 6, shews the Number of Alternations of 6 things exposed; and so onward, by continual Multiplication by the conse quent Numbers 7, 8, 9, &c.;
6) That is, how many so ever of Numbers, in their natural Consecution, beginning from 1, being continually Multiplied, give us the Number of Alternations (or Change of order) of which so many things are capable as is the last of the Numbers so Multiplied. As for instance, the Number of Changes in Ringing Five Bells, is 1 x 2 x 3 x 4 x 5 = 120. In Six Bells, 1 x 2 x 3 x 4 x 5 x 6 = 120 x 6 = 720. In Seven Bells, 720 x 7 = 5040. In Eight Bells, 5040 x 8 = 40320, And so onward, as far as we please.”

John Wallis (1616–1703) English mathematician

Source: A Discourse of Combinations, Alterations, and Aliquot Parts (1685), Ch.II Of Alternations, or the different Change of Order, in any Number of Things proposed.

“Man differs from woman in size, bodily strength, hairyness, &c., as well as in mind, in the same manner as do the two sexes of many mammals.”

Charles Darwin book The Descent of Man, and Selection in Relation to Sex

volume I, chapter I: "The Evidence of the Descent of Man from some Lower Form", pages 13-14 http://darwin-online.org.uk/content/frameset?pageseq=26&itemID=F937.1&viewtype=image The Descent of Man (1871)

“Number one is "Do you have curiosity?" Number two is "Does it translate to imagination?" But number three is "Did it translate to action?" That’s the difference between someone with an idea and someone who is an entrepreneur.”

Steve Blank (1953) American businessman

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John Wallis (1616–1703) English mathematician

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.
Source: 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.

“There are few reasons for telling the truth, but for lying the number is infinite.”

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“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 (1616–1703) English mathematician

Source: 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.

“I've been briefed on every contingency you can possibly imagine. Many contingencies. A lot of—a lot of positive. Different numbers. All different numbers. Very large numbers. And some small numbers too, by the way.”

Donald J. Trump (1946) 45th President of the United States of America

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“[Perl] combines all the worst aspects of C and Lisp: a billion different sublanguages in one monolithic executable. It combines the power of C with the readability of PostScript.”

Jamie Zawinski (1968) American programmer

http://groups.google.com/groups?selm=33F4D777.7BF84EA3%40netscape.com
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“Number is different from quantity.”

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“There is a limit to the number of (and I'm going to use this word in an entirely neutral sense) aliens who can be brought into a nation, particularly as close-knit and concentrated a nation as Britain is, without breaking the bounds of that society and setting up intolerable frictions and stresses as damaging to one side as to the other. Now, this is a question of number. But the relationship between number and difference is clearly important, because the more different they are – and colour is a signal, an outward signal of differences (not significant in itself, but it signalizes other differences that one can't deny) – the greater the difference, the smaller the numbers that can at any one time be accepted without breaking, or being thought to break (which comes to the same thing if we're talking about psychology), the framework of a nation and a society. So it's numbers.”

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“If a, c are two different numbers, there are infinitely many different numbers lying between a, c.”

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Richard Dedekind13

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