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Single Idea 14437

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts ]

Full Idea

Dedekind set up the axiom that the gap in his 'cut' must always be filled …The method of 'postulating' what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others.

Gist of Idea

Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil

Source

report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - Introduction to Mathematical Philosophy VII

Book Ref

Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.71

A Reaction

This remark of Russell's is famous, and much quoted in other contexts, but I have seen the modern comment that it is grossly unfair to Dedekind.

Related Idea

Idea 14436 A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell]

The 11 ideas with the same theme [defining real numbers by cutting the line of rationals]:

10572 A cut between rational numbers creates and defines an irrational number [Dedekind]
18244 I say the irrational is not the cut itself, but a new creation which corresponds to the cut [Dedekind]
14437 Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
18094 Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock]
14436 A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell]
18248 A real number is the class of rationals less than the number [Russell/Whitehead, by Shapiro]
15274 Points are 'continuous' if any 'cut' point participates in both halves of the cut [Harré/Madden]
18093 For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock]
10575 Why should a Dedekind cut correspond to a number? [Fine,K]
18245 Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
15904 The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]