PEANO AND RUSSELL Introduction Gottlob Frege is customarily awarded the honorific title ‘the father of modern logic’. Jean van Heijenoort (1992: 242) tells us that ‘Modern logic began in 1879, the year in which Gottlob Frege (1848–1925) published his Begriffsschrift’, while Michael Dum- mett (1993: xiii) regards Frege as the ‘initiator of the modern period in the study of logic’. Yet, the praise so frequently accorded to the originality and quality of Frege’s logic and philosophy has obscured the significance and influence of his contem- porary, Guiseppe Peano. The contribution of the latter has been underestimated; indeed in terms of causal consequence, Peano’s work had a greater impact on the development of logic at a crucial period surrounding the turn of the nineteenth cen- tury. Peano’s influence was felt principally through his impact on Bertrand Russell (indeed, much the same can be said of Frege). In this essay we shall examine this line of influence, looking first at Peano’s logic and then at Russell’s. Guiseppe Peano Guiseppe Peano was born in Piedmont, in 1858, ten years after Frege and fourteen years before Russell. He was educated in the Piedmontese capital, Turin, where he became professor of mathematics, teaching and researching in analysis and logic. As with several other mathematicians, his interest in logic was in part motivated by the need to promote rigour in mathematics. The emphasis he places on rigour can be appreciated through a debate between Peano and his colleague Corrado Segre. Segre had made important contributions to algebraic geometry, and with Peano made Turin a notable centre for geometry. Nonetheless, they had quite different at- 1 titudes towards rigour, proof, and intuition. Segre felt that in order to make progress in a new area of mathematics, it would often be necessary to resort to less rigorous, incomplete procedures (Borga and Palladino 1992: 23). Peano (1891: 66) responded that, ‘a result cannot be considered as obtained if it is not rigourously proved, even if exceptions are not known.’ The difference between the intuitive approach and the rigorous approach is ap- parent in the mathematical understanding of continuity. The intuitive notion of a continuous curve is one that can be divided into parts each of which is contin- uously differentiable. Curves of the latter sort cannot fill a space—however small a space we choose, the curve cannot go through every point in the space. Intu- itively a one-dimensional thing, a continuous curve, cannot fill two-dimensional space. However, Peano shows that some continuous curves can go through every point in a finite space, indeed in infinite space. In a paper published in 1890, Peano gave a formal description of what became known as the ‘Peano curve’, a curve that passes through every point in the unit square. The paper gave no illustration of the curve, precisely because, Peano held, it is thinking in visual terms about geometry and analysis that leads to the use of intuition in mathematics and the breakdown of rigour. The other influence on the Peano curve was Cantor. Peano was intrigued by Can- tor’s proof that the set of points in the unit interval (the real numbers between 0 and 1) has the same cardinality as (i.e. can be put into 1–1 correlation with) the number of points in the unit square. Since the cardinality of the unit interval is equal to that of the whole real line, this raises the possibility that there is a curve that proves Can- tor’s conclusion by going though every point in the unit square. Peano’s curve does exactly that. Peano’s logical symbolism In the year following publication of his space-filling curve, Peano instituted his ‘For- mulario’ project, the aim of which was to articulate all the theorems of mathemat- ics using a new notation that Peano had devised. A little earlier Peano (1888) has 2 published a book on the geometrical calculus, following the approach of Hermann Grassmann’s Ausdehnungslehre. According to Grassmann, geometry does not have physical space as its subject matter. It is rather a purely abstract calculus that may have application to physical space. Conceiving it this way allows for a formal ap- proach to geometrical proof divorced from intuition and thereby allows for flaws in intuition-based reasoning (for example in Euclid) to be avoided. Moritz Pasch, in his Vorlesungen über neuere Geometrie (1882) takes certain terms as primitive and undefined (‘Kernbegriffe’ as Pasch later called them) while certain theorems (‘Kern- sätze’) are taken as unproved, i.e. are the axioms. Derived theorems are to be proved by pure logic alone, independently of any intuition. Pasch’s approach influenced Peano’s I principii di geometria logicamente esposti (1889). For this programme to be realised fully, Peano held, the geometrical calculus needed the framework of a logical calculus. In 1889 Peano also published his Arithmetices principia, nova methodo exposita. This work (in Latin) contained his famous five postulates for arithmetic—the ‘arith- metices principia’. The phrase ‘nova methodo exposita’ refers to the new symbolism that Peano introduced for logical relations: the now familiar ‘²’ for set/class mem- bership and ‘ ’C for implication (a rotated ‘C’ for ‘consequentia’—in the second vol- ume of the Formulaire Peano replaced ‘ ’C by the now-familiar horseshoe‘ ’). Peano ⊃ has already used several other new symbols that were to become central to the sym- bolism of logic and set-theory, such as the symbols for union and intersection, ‘ ’ [ and ‘ ’, and existential quantification, ‘ ’, and the tilde ‘ ’ for negation (Cajori 1929: \ 9 » 298–302). Peano signified universal generalisation using subscripts with his ‘ ’.C The familiarity of these symbols tends to mask the importance of their introduction. But the idea is familiar that having the appropriate terminology is crucial not just to ex- pressing thoughts clearly but often also to having the relevant thoughts at all. The development of a symbolism for logic was not just a lubricant to the accelerating ve- hicle that was mathematical logic, but was an essential part of its motor, as Bertrand Russell was soon to acknowledge. 3 From Peano to Russell In the last summer of the nineteenth century, the first International Congress of Phi- losophy took place in Paris (it was to be followed immediately by the second Interna- tional Congress of Mathematics). Peano was on the committee for the Congress and one of its leading lights. Also attending was twenty-eight year old Bertrand Russell. Born into the aristocracy, the grandson of a Prime Minister, Russell came from quite a different background from Peano. Nonetheless, their interests converged on the intersection of mathematics, logic, and philosophy. Russell went up to Cambridge in 1890, where he completed both the Mathematical and Moral Sciences (Philoso- phy) Triposes. Like Peano, Russell was much taken by the work of Georg Cantor. He had begin to study Cantor’s work in 1896 (Grattan–Guinness 1978: 129), and had been thinking a great deal when he met Peano in August 1900. In his autobiography, Russell (1967: 144) records the significance of his meeting thus: The Congress was the turning point of my intellectual life, because there I met Peano. I already knew him by name and had seen some of his work, but had not taken the trouble to master his notation. In discus- sions at the Congress I observed that he was always more precise than anyone else, and that he invariably got the better of any argument on which he embarked. As the days went by, I decided that this must be ow- ing to his mathematical logic . It became clear to me that his notation afforded an instrument of logical analysis such as I had been seeking for years . and thus (Russell 1959: 65): I went to [Peano] and said, ‘I wish to read all your works. Have you got copies with you?’ He had, and I immediately read them all. It was they that gave the impetus to my own views on the principles of mathemat- ics. 4 Burali-Forti’s paradox and Cantor’s theorem Russell, who had been intending to stay for both congresses, left Paris in order to return home to study Peano’s work in greater detail. He was, however, already aware of Burali-Forti’s paradox and of Cantor’s paradox. Cesare Burali-Forti was Peano’s assistant and lectured on Peano’s mathematical logic and used Peano’s symbolism in his own work. In 1897 Burali-Forti had published his famous paradox concerning ordinals. In a simple terms the class of all ordinals, ­, is an ordinal or corresponds to an ordinal. But then we can construct a greater ordinal, which is both a member of ­, (because it is an ordinal) and not a member of ­, (because it is greater than ­). Georg Cantor had earlier (1891) published his diagonal proof that the power set of a set is always larger than the set itself (as follows). Let us assume that there is a 1-1 correspondence between S and its power set, P (S). This 1-1 correspondence, f : S P (S) is a bijection; for every member of x ! of S, f (x) P (S), and for every member y of P (S), there is a unique x S such that 2 2 f (x) y (Table 1). Æ S P (S) x f (x) Table 1: f is a 1-1 correlation between S and P (S) The various f (x) are subsets of S. And so some members of S may be correlated by f with sets of which they are themselves members, i.e.