Modern Logic: From Frege to Gödel: Peano

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MODERN LOGIC: FROM FREGE TO GÖDEL: PEANO

Giuseppe Peano (1858–1932), professor of infinitesimal analysis at Turin and a prolific writer on a wide range of mathematical topics, contributed to the early development of both logicism and the formalism to which it is partly opposed. His first book, published under the name of a former teacher, Angelo Genocchi, was devoted to the calculus and featured a careful, systematic treatment of the subject that contrasted favorably with customary texts in rejecting loosely phrased definitions and theorems and in substituting rigorous proof for appeals to intuition. Peano was particularly insistent that the acceptability of a mathematical proposition should depend not on its intuitive plausibility but on its derivability from stated premises and definitions, and he devised a remarkable illustration of the way in which what appears evident to intuition may nonetheless be contradicted by formally incontrovertible considerations. This is his well-known space-filling curve, introduced in 1890 in the paper "Sur une Courbe, qui remplit toute une aire plaine" (Mathematische Annalen 36 [1890]: 157–160). About ten years earlier Camille Jordan had defined a curve as a continuous and single-valued image of the unit segment. This definition accords well enough with our intuitive conception of a curve, but Peano showed that a curve in conformity with this definition could in fact pass through every point in a square based on the unit segment and so would appear as a uniformly shaded surface if plotted on a graph.

Convinced that the development of mathematics must proceed independently of intuitive considerations, Peano embarked upon a program of refounding the various branches of mathematics. Not only geometry and analysis, where we are particularly inclined to make an appeal to what can be grasped pictorially, but even elementary number theory was to be purified of commonsense preconceptions. The entities of a mathematical theory (numbers, points, and so forth) would have to enter into the theory not as idealizations of objects given to intuition but as postulated or defined entities, having only those properties which are explicitly listed or which can be grounded on the initial definitions. To ensure the exclusion of misleading intuitive associations, Peano devised a new symbolic language in which to formalize definitions and other postulates. Principles of reasoning employed within mathematics, as well as conceptions forming the substance of mathematical theories, are transcribed into the new notation. It is at this point that mathematical logic enters into Peano's work, and although he did not carry the development of his system very far, the basic ideas and notation were taken over by Whitehead and Russell as a starting point for the system of logic presented in great detail in Principia Mathematica.

Also important for subsequent developments was Peano's presentation of arithmetic. It is based on a set of postulates known as the Peano axioms, although, as has been noted, Richard Dedekind had published them earlier. The axioms were intended to free the concept of number from dependence on intuition. The essentials of Peano's treatment are embodied in these five axioms:

(1) 0 is a number.
(2) The successor of any number is a number.
(3) No two numbers have the same successor.
(4) 0 is not the successor of any number.
(5) Any class which contains 0 and which contains the successor of n whenever it contains n includes the class of numbers.

The Peano axioms are commonly taken as a basis for the arithmetic of the natural numbers, supplemented by recursive definitions of such arithmetical operations as addition, multiplication, and exponentiation. Peano himself made considerable use of recursive definition, an analogue, for definitions, of the axiom of mathematical induction given by (5), which allows us to calculate the value of a function ƒ(n ) step by step, given an explicit definition of ƒ(0) along with a definition of ƒ(n ′) in terms of ƒ(n )—here "n " means "the successor of n." Thus, for addition Peano provided the two recursion equations a + 0 = a and a + n ′ = (a + n )′. Rewriting the second of these as a + (n + 1) = (a + n ) + 1, we can see that we have here a particular case of the associative law for addition, x + (y + z ) = (x + y ) + z, which can in fact be derived from the recursion equations by means of axiom (5). Multiplication is defined in similar fashion by means of the equations a · 0 = 0 and a · b ′ = a · b + a, and once more familiar arithmetical laws can be extracted by means of induction.

With the assistance of a number of colleagues, including Cesare Burali-Forti, Peano succeeded in reformulating much of existing mathematical theory in accordance with his criteria of rigor and precision, the results of these investigations appearing in the journal Rivista di Matematica (later also Revue de mathématiques and Revista de mathematica ) from 1891 to 1906 and in Peano's Formulaire de mathématiques (5 vols., Turin 1892–1908). The detailed coverage of algebra, arithmetic, set theory, geometry, and other branches of mathematics argues convincingly for Peano's approach, but it is questionable whether it vindicates a formalist philosophy of mathematics, since further metamathematical investigation, notably by Thoralf Skolem, has shown that if Peano's axioms are embedded in an axiomatization of set theory, they do not serve to characterize the natural numbers to the exclusion of other progressions. At the same time, it should be noted that Peano was not himself concerned with advancing either a formalist or a logicist philosophy; his approach was determined by a desire for technical improvements in the presentation of mathematics.