P1) OEN
3. Irrational Numbers in Modern (that is, post-mediae'val) Math-
ematics. After the geometrical theory of proportion of the Greeks, we now turn to the arithmetic aspect which becomes important for the develop- ment of mathematics in the modern era. Its history can be traced back to the practical calculation of approximate values, which had been prac- ticed since very early times by mathematicians interested in astronomy and civil engineering. After the Babylonians, we need especially to remember ARCHIMEDES who, in his determination of the circumference of a circle, succeeded in showing that lay between 34 and and PTOLEMY (circa 150 AD) the great astronomer of the Ancient and Mediaeval world, who chose the sexagesimal fraction 3;8,30 as a mean between 34 = 3;8,34 and
= 3;8,27. The process of nesting of intervals is applied here.
While Greek mathematics was showing little interest in arithmetical cal- culations, which were kept very much in the background compared with geometrical constructions and proofs of propositions by logical inference, the development of the number concept gained a decisive impetus from the influence of Indo-Arabic algebra. Thus, for example, the Arab mathe- matician ABU KAMIL (circa 850—930) was able to work with expressions involving square roots, using such rules as, among others:
(TROPFKE [23], p. 135). One begins to operate with new expressions with- out realizing that they are a new type of number. This process received a further impetus through the discovery, in the 16th century, of the formulae for the solution of cubic and biquadratic equations. The reader will find
more on this subject in Chapter 3, §1.
M. STIFEL [22J still wrote, in his Arztlzmetica integra of 1544 "So wie elne unendliche Zahi keine ZahI 1st, 80ist eine irrationale Zahi keine wahre ZahI, weil sie sozusagen unter einem Nebel der Unendlichkeit verborgen 1st." [Just as an infinite number is no number, so an irrational number is not a true number, because it is so to speak concealed under a fog of infinity.]
This "fog of infinity" is already defined rather more precisely by STEVIN (1548—1620) as an infinite sequence of decimal fractions, representing a se- quence of nested intervals, which he develops, for example, in finding succes- sive approximations to the solution of the equation z3 = 300x+33900000.
He writes: "... etprocédant ainsi infiniment, l'on approche infiniment plus près au requis" (and proceeding in this way unendingly, one approaches infinitely closer to the required valuej (S. STEVIN [21], p. 353).
In the Geometrie of 1637 by DESCARTES, the operations of addition, subtraction, multiplication, division and root extraction of line segments are defined in such a way that the result of the operation is again a line seg- ment in each case. Whereas the product of two line segments had hitherto always been interpreted as a rectangle, DESCARTES obtains the product as the fourth proportional in the Intercept, theorem, when the first intercept is taken to be of unit length, so that 1 is to b as a is to a b.
The development of the number concept received a new boost through the infinitesimal calculus in the 17th and 18th century. Here the theory of series, especially from the time of LEIBNIZ and the brothers BERNouLLI, opened up new possibilities for the representation of numbers. In the Arals- metica infinitorum of 1655, by WALLIS (1616—1703), we find, for example,
theinfiniteproduct
Representations of numbers by infinite sums and infinite products were not defined however—as has usually been the case since CAUCHY and WEIERSTRASS—a8 convergent sequences, using the concept of a limit. In- stead, a sum such as
00
k=1 k(k + 1)
was said to differ from I by an "infinitesimal" or "infinitely small" quan- tity. EULER [9] formulated in 1734 a convergence criterion for series in the language of infinitesimals. Apart from the "finite" and "actual" (real) numbers, which found their application as values in measurement, there ap-
peared to be also "infinitesimal" numbers and "ideal" numbers. In the 19th century such terms were banned from mathematics as being too imprecise and "psychologizing" a form of expression, and were felt to be Buperfiuous after the clarification which had been brought about by the introduction of the concept of a limit. It is only with the comparatively recent non- standard analysis (see Chapter 12) that infinitely small numbers have once more come into fashion and achieved full respectability.
4. The Formulation of More Precise Definitions in the Nineteenth
Century. CAUCHY in his Cow-s d'analyse of 1821, formulated the conver- gence criterion called after him and considered it as self-evident as the laws of arithmetic. The completeness of the system of real numbers, the property which CAUCHY is here expressing, had however already been as- sumed before him. Thus, for example, LEIBNIz assumed that a continuous line drawn on a surface, and lying partly within and partly without some portion of that surface, must intersect the boundary of that portion.In 1817, BOLZANO [4] proved the Intermediate value theorem under the assumption of the CAUCHY criterion. However, it should be pointed out that he already had this criterion at his disposal before CAUCHY. Recently,
a BOLZANO manuscript was discovered containing an unpublished draft of a book entitled Grössenlehre (Theory of Quantities) in which he attempted to base the theory of real numbers on firmer foundations by using sequences of intervals.
With WEIER.STRASS consideration of the foundations of the real number system entered into the basic mathematical curriculum. All that has come
down to us of this, however, are some notes written by his pupils and which were in part criticized by WEIERSTRASS. The central idea of the concept of a real number as visualized by WEIERSTRASS [24:) is expressed in terms of the principle of nesting of intervals. He also uses this to prove his well- known Limit-point Theorem (see, also DUGAC [8]). A systematic definition of real numbers in terms of nested intervals was given by BACHMANN [1]
in 1892.
Another method of defining real numbers was introduced by CANTOR in his theory of fundamental sequences (see 2). Shortly before, MERAY (1835—
1911) had used (though CANTOR was not aware of this) this approach to the definition of irrational numbers by regarding them as "fictive" limits of convergent sequences and, harking back to the discovery in classical times, calling them "nombres incommensurables."
Finally, DEDEKIND (1831—1916) in his famous book Stetigkeit und Ir- rat:onalzahlen [7) published in 1872 took up the theory of proportion of EUDOXUS and presented it in a modernized form with exemplary clarity.
DEDEKIND'S definition expresses our geometrical intuition of the contin- uum, which has been so deeply rooted since the days of classical antiq- uity. This intuition tells us that the points of a straight line are defined by "the bisection of a line into two parts" (DEDEKIND) by "the common
boundary between two parts, which together constitute the whole" (LEIB- NIZ) or by the "extremities of two parts which touch" (AaIsTo'FLE) see
§1). The question of whether EUDOXUS and EUCLID with their theory of proportion had satisfactorily settled the matter of defining the irrational numbers led to some controversy in connection with the work published by DEDEKIND in 1872. Thus LIPsciuTz wrote to DEDEKIND in 1876: "...Ich
kann nur sagen, daB (ich) die von Euclid V, 5 aufgestellte Definition .. .für genauso befriedigend halte, als lhre Definition. Aus diesem Grunde würde ich wünschen, daB namentlich die Behauptung wegfiele, daB solche Sätze wie
=
bisher nicht wirk)ich bewiesen seien." (1 can only say that I personally find the definition in Euclid V, 5 just as satisfactory as yours. For this reason I would have liked to have seen omitted, in particu- lar, the statement that such propositions as = have never yet really been proved.) Characteristic is LWScHITz's remark: "Was Sie an der Vollständigkeit des Gebietes erwähnen, die aus thren Principien abgeleitet wird, so failt dieselbe in der Sache mit der Grundeigenschaft einer Linie zusammen, ohne die kein Mensch sich eine Linie vorstellen kann." (What you say in regard to the completeness of the domain, deduced from your principles, in point, of fact merely coincides with the basic property of a line, without which no one can possibly imagine a line.)While LIPSCHITZ thus expresses an attitude recalling that of the math- of the previous century who were frequently content to rely on an intuitive understanding of the foundations of their science, DEDEKIND
stands at the start of an era heralding a new methodical approach. He
is concerned—as were CANTOR, FREGE, PEANO and others—to formulate explicitly and precisely the concepts on which mathematics is founded. And so DEDEKIND writes to LIPSCHITZ with particular reference to the concept of completeness: ".. .Aber Euklid schweigt vollständig über diesen, für die Arithmetik wichtigsten Punkt, und deshaib kann ich lhrer Ansicht nicht zustimmen, daB bei Euklid die vollstandigen Grundlagen für die Theorie der irrationalen Zahien zu linden seien." ["...But Euclid is completely silent on this, the most important point for arithmetic, and therefore I cannot share your opinion that a complete theory of irrational numbers is to be found in Euclid.")
The real number concept became a problem area once more in the dis- cussions of the nineteen twenties between IIILBERT and BROUWER on the foundations of mathematics, after RUSSELL had derived contradictions from the so-called "naive" set theory of CANTOR and FREGE, and after it was found that even the new axiomatized versions of set theory could not be proved to be consistent, and, as GÔDEL showed, were inherently incapable of being proved consistent by finite methods. Within mathematical logic these considerations led to an interesting discussion, which continues up to the present day, of more limited concepts such as, for example, computable numbers, and constructive real numbers (see, BISHOP [3], HERMES [12],
LORENZEN [18)).