The Dedekind Snap

PROPOSAL
Against standard theory, I argue that the "Dedekind cut" describes a gap. That is, the Dedekind cut is unplaced: it does not fall on a line. To restore the concept of the Dedekind cut it would be better to re-define it as a snap.

BACKGROUND INFORMATION
A Dedekind cut can be understood as an attempt to resolve an old problem in mathematics - how to reconcile the continuous with the discrete or quantifiable. Hippasus was allegedly thrown overboard by his fellow Pythagoreans for finding a magnitude (the irrational number square root of two) that could not be quantified or represented by discrete digits.

Dedekind's cut is a "cut" described in the geometric sense, that is, as a line crossing or "cutting" another line. The cut, according to the favoured definition, is not a gap. Dedekind's cut is made upon a continuous line whose continuity is allegedly described by all the real numbers, these comprising the rationals and the irrationals.

The idea is that Dedekind's cut unites the discrete with the continuous because it is a cut that can be made at any point on a continuous line composed of the rationals and the irrationals.

[There are some problems with that idea that are not my concern here, chief among them being how any line that is expressed in terms of digits, whether or not the digits belong to rational or irrational numbers, can yield the continuous if these digits express the transition or movement from rational to irrational or vice versa.]

THE DEDEKIND GAP
Dedekind's cut is not a cut. It is a gap. The point at which it cuts the number line is a point that is said to belong to each of the two half's of the number line. A claimed position on a line which is described by two points is a fractured line. That is, Dedekind's cut describes two lines, these separated, using the geometric metaphor, by a gap. The positions on a continuous line are not retained by the fractured wholes. Thus, Dedekind's cut is an unplaced gap and does not fall on a line.

THE DEDEKIND SNAP
I propose, therefore, that if we are to retain the Dedekind cut in the form which was intended for it (for better or worse), that we use the metaphor of the "Dedekind snap". Like two radii, or like a twig that snaps but does not separate, the Dedekind cut does not fall on two points erronously conceived as belonging to two halfs of one continuous line. Rather, it falls on one point on the line.
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Relevant Pages

  • Re: The Dedekind Snap
    ... the Dedekind cut is unplaced: it does not fall on a line. ... cut "describes a gap", whatever that means, cannot possibly ... continuous line whose continuity is allegedly described by all the real ... these comprising the rationals and the irrationals. ...
    (sci.logic)
  • Re: Grammar informs topic A.
    ... instead that it does not separate, the Dedekind cut reconciles the ... these comprising the rationals ... ' The rationals are discrete. ... The irrationals are not ...
    (sci.logic)
  • Grammar informs topic A.
    ... instead that it does not separate, the Dedekind cut reconciles the ... these comprising the rationals ... ' The rationals are discrete. ... The irrationals are not ...
    (sci.logic)
  • Re: The Dedekind Snap
    ... continuous line whose continuity is allegedly described by all the real ... these comprising the rationals and the irrationals. ... a Dedekind cut is a nonempty set of rational numbers that is ... If you think Ded's cut is only about the rationals, then, you can't say that the Dedekind cut reconciles the discrete with the continuous. ...
    (sci.logic)
  • Re: Origins of Analysis
    ... > Is there a relation ship then between calculus and the first appearance ... The first modern theory of irrationals is to be found in Dedekind's ... reals as undefined. ... reals are then deduced from Dedekind cut as definition or Dedekind cut ...
    (sci.math)
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