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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts

[defining real numbers by cutting the line of rationals]

11 ideas
A cut between rational numbers creates and defines an irrational number [Dedekind]
     Full Idea: Whenever we have to do a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut.
     From: Richard Dedekind (Continuity and Irrational Numbers [1872], §4)
     A reaction: Fine quotes this to show that the Dedekind Cut creates the irrational numbers, rather than hitting them. A consequence is that the irrational numbers depend on the rational numbers, and so can never be identical with any of them. See Idea 10573.
I say the irrational is not the cut itself, but a new creation which corresponds to the cut [Dedekind]
     Full Idea: Of my theory of irrationals you say that the irrational number is nothing else than the cut itself, whereas I prefer to create something new (different from the cut), which corresponds to the cut. We have the right to claim such a creative power.
     From: Richard Dedekind (Letter to Weber [1888], 1888 Jan), quoted by Stewart Shapiro - Philosophy of Mathematics 5.4
     A reaction: Clearly a cut will not locate a unique irrational number, so something more needs to be done. Shapiro remarks here that for Dedekind numbers are objects.
Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
     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.
     From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - Introduction to Mathematical Philosophy VII
     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.
Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock]
     Full Idea: One view, favoured by Dedekind, is that the cut postulates a real number for each cut in the rationals; it does not identify real numbers with cuts. ....A view favoured by later logicists is simply to identify a real number with a cut.
     From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by David Bostock - Philosophy of Mathematics 4.4
     A reaction: Dedekind is the patriarch of structuralism about mathematics, so he has little interest in the existenc of 'objects'.
A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell]
     Full Idea: There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. This division of a series into two classes is called a 'Dedekind Cut'.
     From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII)
A real number is the class of rationals less than the number [Russell/Whitehead, by Shapiro]
     Full Idea: For Russell the real number 2 is the class of rationals less than 2 (i.e. 2/1). ...Notice that on this definition, real numbers are classes of rational numbers.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Thinking About Mathematics 5.2
Points are 'continuous' if any 'cut' point participates in both halves of the cut [Harré/Madden]
     Full Idea: Divide points into left and right set. They're 'continuous' if that point is either last member of left set, and greatest lower bound of right (so no least member), or least upper bound of left set (so no last member) and first member of right set.
     From: Harré,R./Madden,E.H. (Causal Powers [1975], 6.IV)
     A reaction: The best attempt I have yet encountered to explain a Dedekind Cut for the layperson. I gather modern mathematicians no longer rely on this way of defining the real numbers.
For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock]
     Full Idea: As Eudoxus claimed, two distinct real numbers cannot both make the same cut in the rationals, for any two real numbers must be separated by a rational number. He did not say, though, that for every such cut there is a real number that makes it.
     From: David Bostock (Philosophy of Mathematics [2009], 4.4)
     A reaction: This is in Bostock's discussion of Dedekind's cuts. It seems that every cut is guaranteed to produce a real. Fine challenges the later assumption.
Why should a Dedekind cut correspond to a number? [Fine,K]
     Full Idea: By what right can Dedekind suppose that there is a number corresponding to any pair of irrationals that constitute an irrational cut?
     From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
     Full Idea: A Dedekind Cut is a division of rationals into two set (A1,A2) where every member of A1 is less than every member of A2. If n is the largest A1 or the smallest A2, the cut is produced by n. Some cuts aren't produced by rationals.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.4)
The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]
     Full Idea: Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio.
     From: Shaughan Lavine (Understanding the Infinite [1994], II.6)
     A reaction: Thus there is the problem of whether the contents of the gap are one unique thing, or many.