Reference For The Historical Context of Dedekind Cuts and Why the need

[MidgetDwarf]

I was wondering if anyone knew of sources regarding the history of Dedekind Cuts and its use to construct the Reals. Ie., the mathematical thinking leading to this idea, why it is important, and how it allowed to put Analysis on a rigorous foundation.

 

[member 587159]

I cannot give you a reference for the historical side of the aspect. I can try to answer the rest of the question though.

One often defines the real numbers as a totally ordered field with the property that every set that is bounded above has a supremum (LUB). This definition is ok, but two important questions must be answered:

- Does such a field actually exist?
- Is such a field unambiguously defined? I.e. is such a field unique?

Given the rational numbers, it is easy to construct a totally ordered field satisfying (LUB). There are two main constructions: one defines a real number as a cut, which is a specific kind of subset of the rational numbers, or one defines a real numbers as the Cauchy-completion of the rational numbers. Following either construction, we end up with a totally ordered field satisfying (LUB). We thus have found a model of the real numbers and the first question has a positive answer.

Now, the question is: Is such a field in some sense unique? And in fact, this turns out to be true: given two totally ordered fields satisfying (LUB), there is a unique field isomorphism between the two fields. This means that we can forget our specific construction, which was only important to show something satisfying our axioms exists.

 

[fresh_42]

Dieudonné writes that he had been urged to set arithmetics on a rigor fundament during the preparation of a lecture about differential equations in Zürich for the winter semester 1858/59. His publication dates on 1872: Continuity and irrational numbers.

 

[pbuk]

You can read about Dedekind's motivation in (an English translation of) his own words thanks to Project Gutenberg: https://www.gutenberg.org/files/21016/21016-pdf.pdf.

Dedekinds essay on what we now call set theory and cardinality is included in that link and is also worth a read, although the translator's choice of "The nature and meaning of numbers" as the title I find less inspiring than the original Was sind und was sollen die Zahlen? ("What are numbers and what should they be?"). This essay first published in 1888 not only consolidates Dedekind's work on irrationals but lays essentially the same foundation for natural numbers as Peano did a year later. I have also in a search just now discovered a text From Kant to Hilbert, volume 2 of which seems to cover this history plus a lot more and is available at a reasonable price from Amazon in the UK. I am looking forward to its arrival on Saturday!

[Edited re. inclusion and translation of Was sind und was sollen die Zahlen?]

 

[mathwonk]

In Euclid, real numbers are represented as lengths of segments, or at least one deals there with the notion of lengths of segments, and the ratios of such lengths. Basic geometric results deal with ratios of lengths of sides of triangles, and the condition of similarity of triangles is fundamental. In particular similarity can be defined as equality of angles, but the immediate corollary one wants is proportionality of side lengths. Since numbers are not available yet in Euclid he deals with pairs of segments and their ratios. However there is the problem of defining just what one means by the ratio of two segments. Hence arises the problem of defining carefully real numbers, since a real number is nothing but the ratio of two finite line segments. (If you have read Galileo, he actually deals with pairs of line segments, since he still has no algebraic notation for real numbers.)

Now Euclid has the notions of dividing a segment into a finite number of equal parts, hence he can make sense of the statement that the ratio of two segments is rational, i.e. some finite subdivision of one equals some finite subdivision of the other. But he knew that there exist pairs of incommensurable segments, i.e. such that no segments can be used to measure, i.e. subdivide, both of them. So in order to prove the theorem that two triangles with equal corresponding angles must have proportional sides, he had to devise a way to say that the ratios of all three pairs of sides are equal even when that ratio is not rational.

His solution was the method, attributed to Eudoxus, of infinite approximation by rational ratios, described in Definition 5, Book V, of the Elements. It essentially says that two ratios are equal if and only if every rational ratio which is less than one of them is also less than the other. For us it is only a small step to saying that this means a real number is determined by all rational numbers which are smaller than it is. Equivalently, choosing a point on the line, determines a separation of (the rational points of) that line into two parts, those points on one side and those on the other side of that separation.

The converse notion, that a real number is a separation of the real line, and that this separation is determined by the corresponding separation of the rational points, is exactly our modern notion of a real number as a Dedekind cut. So the Dedekind cut harks back all the way to essentially our first book of mathematics. Formulating it as a definition of real numbers, by Dedekind, may have required waiting for Georg Cantor to enunciate set theory, and the idea that mathematical objects should be defined as sets. At least Dedekind was said to have been an early appreciator of Cantor's ideas. For us it is hard to read Euclid and not think that Dedekind cuts are already there.

In my own view, this definition of real numbers is important more for this close link to Euclid, since cuts are very clumsy and hard to work with. (The details are worked out in Rudin and in Spivak's calculus book I believe.) A much more beautiful definition to me, is that of convergent sequences of rationals, modulo null sequences, i.e. the metric space completion of the rationals using the absolute value metric. But one way or the other, one must deal with real numbers as approximated by rationals. Infinite decimals is also a nice concrete approach that I had success teaching to a small bright special high school class long ago, some members of which became professional mathematicians.

 

[mathwonk]

In reference to the discussion above, here is a quote from the article linked above of Dedekind himself:
"
In the preceding section attention was called to the fact that every point p of the straight line produces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the converse, i. e., in the following principle:
“If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.”"Then following excerpt from that same article of Dedekind gives exactly Euclid's or Eudoxus' definition of equality of real numbers:

"
Whenever, then, we have to do with a cut (A1,A2) produced by no rational number, we create a new, an irrational number α, which we regard as completely defined by this cut (A1, A2); we shall say that the number α corresponds to this cut, or that it produces this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to essentially different cuts."Ah yes, in the second linked article, Ddekind explicitly says that Euclid's definition 5 in Book V was the origin of his definition of equality of cuts:

"
I cannot quite agree with Tannery when he calls this theory the development of an idea due to J. Bertrand and contained in his Trait ́e d’arithm ́etique, consisting in this that an irrational number is de- fined by the specification of all rational numbers that are less and all those that are greater than the number to be defined. As regards this statement which is repeated by Stolz—apparently without careful investigation—in the preface to the second part of his Vorlesungen u ̈ber allgemeine Arithmetik (Leipzig, 1886), I venture to remark the following: That an irrational number is to be consid- ered as fully defined by the specification just described, this conviction certainly long before the time of Bertrand was the common property of all mathematicians who concerned themselves with the notion of the irrational. Just this manner of determining it is in the mind of every computer who calculates the irrational root of an equation by approximation, and if, as Bertrand does exclusively in his book, (the eighth edition, of the year 1885, lies before me,) one regards the irrational number as the ratio of two measurable quantities, then is this manner of determining it already set forth in the clearest possible way in the celebrated definition which Euclid gives of the equality of two ratios (Elements, V., 5). This same most ancient conviction has been the source of my theory as well as that of Bertrand and many other more or less complete attempts to lay the foundations for the introduction of irrational numbers into arithmetic."

nice link pbuk!

 

[mathwonk]

reading further in the linked articles, one finds this incorrect definition of a finite set, and one wonders if the translator changed the phrase "cannot be transformed so that...does happen," into :"can be transformed so that ... does not happen". I.e. having more faith in Dedekind than in the translator, I have an opinion. Does anyone have the original German article? (Not that 19th century German is that easy to get straight, at least not for me.)

“A system S is said to be finite when it may be so transformed in itself (36) that no proper part (6) of S is transformed in itself; in the contrary case S is called an infinite system.”

OK, the translator seems to have screwed up only in the preface, as the textual definition following is correct. Of course it is the logical negation of the statement needed in the preface, so maybe that logical construction is what stumped the translator:

"
64. Definition.20 A system S is said to be infinite when it is similar to a
proper part of itself (32); in the contrary case S is said to be a finite system."

 

[pbuk]

reading further in the linked articles, one finds this incorrect definition of a finite set, and one wonders if the translator changed the phrase "cannot be transformed so that...does happen," into :"can be transformed so that ... does not happen". I.e. having more faith in Dedekind than in the translator, I have an opinion. Does anyone have the original German article? (Not that 19th century German is that easy to get straight, at least not for me.)

“A system S is said to be finite when it may be so transformed in itself (36) that no proper part (6) of S is transformed in itself; in the contrary case S is called an infinite system.”

The original German (1893 edition in manuscript here but search for the text and you will find it) is "Ein System S heißt endlich, wenn es sich so in sich selbst abbilden läßt (36), daß kein echter Theil (6) von S in sich selbst abgebildet wird; I am entgegengeseßten Falle heißt S ein unendliches System." and this seems to be translated accurately (although my German isn't up to much either).

But I don't see anything wrong with this - in modern language it is saying "A set S is finite if its cardinality is greater than that of any proper subset of S" or equivalently "A set S is finite if there is no bijection between S and any proper subset of S".

And I've just realized that the second part of the link I posted in #4 is "Was sind und was sollen...", I'll edit it.

 

[mathwonk]

Thank you! Although my german is worse than yours I am sure, I am going to disagree with you and say that the original German says roughly that "whenever a finite set is transformed into itself, then no proper part is transformed bijectively into the whole", rather than stating that "a set is finite if it can be transformed into itself so that a proper part does not transform bijectively into the whole." Hence I reaffirm my, admittedly possibly incorrect, claim that the translator is at fault. But notice I am betting that a genius, namely Dedekind, is not making an incorrect statement, which is kind of cheating in my own favor on the bet.

And in regards to your justification of the, I claim incorrect, version, you are making reference to the concept of cardinality, which Dedekind is going to some pains to avoid, as a separate concept he does not wish to invoke.