Historical note

(Chapter III, § 5)

(Numbers in brackets refer to the bibliography at the end of this Note.)

The evolution of ideas relating to the notions of integer and cardinal number is inseparable from the history of the theory of sets and mathematical logic for which the reader is referred to the Historical Note following Chapter IV. The purpose of this Note is to indicate briefly some of the salient facts in the history of numeration and "combinatorial analysis".

History and archaeology have revealed to us a large number of "systems of numeration", the prime aim of which is to attach to each individual integer (up to some limit depending on the demands of practical use) a name and a written representation, formed from a restricted number of signs according to more or less regular laws. By far the most common procedure is to decompose the integers into sums of "successive units" $b_1$, $b_2$, ..., $b_n$, ..., each of which is an integral multiple of its predecessor; and although in general $b_n/b_{n-1}$ is taken to be a fixed number $b$ (the "base" of the system, usually 10), there are many known exceptions to this rule. For example, in the Babylonian system, $b_n/b_{n-1}$ is sometimes 10 and sometimes 6 [1], and in the chronological system of the Mayas $b_n/b_{n-1}$ is equal to 20 except for $n = 2$, and $b_2/b_1 = 18$ [2]. As to the corresponding written symbol, it must indicate the number of "units" $b_i$ of each order $i$. In many systems (for example, the Egyptian, the Greek, and the Roman) the successive multiples $k.b_i$ (where $k$ varies from 1 up to $(b_{i+1}/b_i) - 1$) are denoted by symbols which depend on both $k$ and $i$. A first and important step foward was to denote all the numbers $k.b_i$ (for the same value of $k$) by the same sign: this is the principle of "numeration by position", where the index $i$ is indicated by the fact that the symbol representing $k.b_i$ appears "in the $i$th place". The first system of this nature is that of the Babylonians, who, certainly as early as 2000 B.C., denoted by the same sign all the multiples $k.60^{\pm i}$ corresponding to various values of the exponent $i$ ([1], pp. 93-109). The inconvenience of such a system is of course its ambiguity so long as there is nothing to indicate whether or not the units of a certain order are absent, i.e., so long as the system is not completed by the introduction of a “zero”. Nevertheless, the Babylonians managed without such a sign for the greater part of their history, and did not make use of a “zero” except in the last two centuries B.C., and then only inside a number; up to that time, the context alone could clarify the meaning of the symbol under consideration. Only two other systems made systematic use of a “zero”: that of the Mayas (in use, apparently, since the beginning of the Christian era [2]) and our present decimal system, which comes (via the Arabs) from Hindu mathematics, where the use of zero is attested since the first centuries A.D. Moreover, the conception of zero as a number (and not merely as a separating sign) and its introduction into calculations are original contributions of the Hindus [3]. Of course, once the principle of “numeration by position” had been acquired, it was easy to extend it to an arbitrary base. A discussion of the merits of the different “bases” proposed since the 17th century depends on the techniques of numerical computation, and cannot be entered into here. We note only that the operation which lies at the root of these systems, the so-called “Euclidean division”, did not appear before the time of the Greeks, and undoubtedly goes back to the early Pythagoreans, who made it the essential tool in their theoretical arithmetic.

The general problems of enumeration, grouped together under the name of “combinatorial analysis”, seem not to have been attempted before the last centuries of classical antiquity; only the formula $\binom{n}{2}=\frac{1}{2}n(n-1)$ is attested, in the third century A.D. The Hindu mathematician Bhaskara (12th century) knew the general formula for $\binom{n}{p}$. A more systematic study is found in a manuscript of Levi ben Gerson (the beginning of the 13th century): he obtained the inductive formula for the number $V_n^p$ of arrangements of $n$ objects $p$ at a time, and in particular for the number of permutations of $n$ objects, and he stated rules which are equivalent to the relations $\binom{n}{p}=V_n^p/p!$ and $\binom{n}{p}=\binom{n}{n-p}$ ([4], pp. 64-65). But this manuscript seems to have remained unknown to his contemporaries, and the results were only gradually rediscovered by mathematicians in the subsequent centuries. As regards later progress, let us record that Cardan proved that the number of non-empty subsets of a set of $n$ elements is $2^n-1$. Pascal and Fermat, in founding the calculus of probability, rediscovered the expression for $\binom{n}{p}$, and Pascal was the first to observe the relation between these numbers and the binomial theorem, which seems to have been known to the Arabs since the 13th century, to the Chinese in the 14th century, and was rediscovered in the West at the beginning of the 16th century, together with the inductive method of calculation of the coefficients known as "Pascal's triangle" ([4], pp. 35-38). Finally, about 1676, Leibniz obtained (but did not publish) the general formula for "multinomial coefficients", which was rediscovered independently and published 20 years later by de Moivre.

Bibliography

1. O. NEUGEBAUER, Vorlesungen über die Geschichte der antiken Mathematik, Bd. I : Vorgriechische Mathematik, Berlin (Springer), 1934. 2. S. G. MORLEY, The Ancient Maya, Stanford University Press, 1946. 3. B. DATTA and A. N. SINGH, History of Hindu Mathematics, vol. I, Lahore (Motilal Banarsi Das), 1935. 4. J. TROPFKE, Geschichte der Elementar-Mathematik, vol. VI : Analysis, Analytische Geometrie, Berlin-Leipzig (de Gruyter), 1924.