### Grattan-Guinness’s *Search for mathematical roots*

I. Grattan-Guinness.

*The search for mathematical roots, 1870–1940. Logics, set theories and the foundations of mathematics from Cantor through Russell to Gödel.*Princeton & Oxford: Princeton University Press, 2000.A richly documented study that takes issue with received views of the origins and content of logicism. GG begins with Lagrange’s “algebraization” of the study of differential equations; this, he argues, was one of the inspirations for Boole’s and De Morgan’s algebraic logic, which in turn fed into the algebraic logics of Peirce and Schröder near the end of the 19th century. I would regard that portion of the history as belonging to the history of the extension and generalization of algebraic notions, a history that also includes the emergence of “abstract” groups, vector algebra, and the beginnings of commutative algebra. (Whitehead’s

*Universal algebra*(1898) belongs here too.)What philosophers think of as “foundations”—the logics of Frege and Russell—was a largely separate line of development whose sources included the “arithmetization” of analysis, Cantor’s

*Mengenlehre*(with Zermelo’s work on the axioms of set theory), and Peano’s efforts to create a symbolic language for mathematics. Whitehead and Russell’s*Principia*(1910–1913), not Frege’s earlier work, is the centerpiece of GG’s history. GG distinguishes the historical Frege from Frege´, the father (with Russell) of analytic philosophy. Frege´, whose influence since World War II has been enormous, is an invention, “an analytic philosopher of language writing in English about meaning and its meaning(s), and putting forward some attendant philosophy of mathematics”. The historical Frege, “a mathematician who wrote in German, in a markedly Platonic spirit, principally on the foundations of arithmetic and on a formal calculus in which it could be expressed”, has only a modest place in the history of logicism, though (as GG argues) he was read more widely than Russell implies in an appendix to the*Principles of mathematics*in 1903.Indeed the whole book can be read as a revision of common conceptions (or rather misconceptions) about logicism, its motives, and its reception. In response, I think, to the predominantly Anglocentric historiography of logicism prevalent among analytic philosophers, GG emphasizes the European-wide enterprise of which the

*Principia*was part. (The Vienna Circle here counts as Anglophone, because Anglo-American philosophers read the works of its members mostly in English, which in any case many of them adopted as their language of publication after emigrating to the US and the UK.) Hence the space devoted to Schröder, Peano, and later to the Polish logicians. Among the revisions of the usual story, I would note the rather limited role played by Gödel’s results in the waning of logicism in the 30s.What disappoints me in

*The search for roots*is that GG, who admittedly has a lot of ground to cover, does not push the conceptual treatment very far. Instead he mostly concerns himself with influences, groups, and doctrinal differences.GG takes note several times, for example, of uncertainty concerning the distinction between an individual and what we would now call its “unit class”. Before set theorists arrived at what is now the standard Zermelo-Fraenkel formulation, the semantics or model theory of formal logics was often based not on the element-set relation but on various part-whole relations, one difference being that in a part-whole model there is no distinction between an individual and the whole of which it is the sole part.

Something interesting is going on here: one would like to know more. The distinction between

*x*and {*x*} is, after all, not especially intuitive: by what magic do the { } create a new entity? There is, of course, a difference in definition once you have arrived at a set theory in the style of ZF: the unit class of*x*is {*y*: I*x**y*} where I*x*is the predicate ‘( ) =*x*’. This class has*x*as its unique member, whereas*x*itself cannot have*x*as a member according to the “axiom of foundation”. Now that axiom is usually mentioned in connection with blocking the Russell paradox: how nice that it*also*distinguishes an individual from its unit class! Some set theories do without the axiom; some even identify individuals with their unit-classes. The axiom is, in other words, not inevitable. Because of it the element-class relation in standard-issue set theory is less intuitive than the part-whole relation (witness Russell’s troubles with “class as one” and “class as many”), and ontologically suspect. Was it understood as an unavoidable cost of avoiding the paradoxes?Another example: there seems to have been, notationally at least, a longstanding asymmetry between the universal and existential quantifiers. The symbol ∀ was introduced, according to GG, by Gentzen in his dissertation, completed under Weyl in 1933. The symbol ∃, on the other hand, was introduced a generation earlier by Peano in the second edition of the

*Formulario*in 1897, where ‘∃*a*’ means that*a*is not equal to the empty set, which in turn is defined as the minimal class (i.e. the class*a*such that for all classes*b*,*b*⊇*a*). There were, of course, notations to signify universal quantification. Some systems did so by attaching a subscripted variable-name to the implication sign ⊃ (a notation that in some systems coexisted with the*prefixed*∃). Others did so by placing a variable-name in parentheses at the beginning of a formula (so that ‘(*x*) .*x*⊇ ∅’ means that every set contains the empty set).Again something interesting seems to be going on. As far as I know, the treatment of universal and particular sentences in Aristotelian logic was not asymmetrical; ‘all’ and ‘some’ are of course distinct in meaning, but syntactically they were denoted in the same way. Why then are they not treated analogously in Peano’s logic or in the logic of the

*Principia*? I remember being struck by this long ago. The first formal logic I learned used Polish notation, and treated universal and existential quantification on a par. When I encountered Peano-style notation, it took me a while to get used to treating the*absence*of a symbol—that is, ‘(*x*)’ in contrast to ‘(∃*x*)’—as significant. The asymmetry is perhaps even more striking when one realizes that many authors treated universal and existential quantification as infinite conjunction and disjunction; the well-known duality between 'and' and 'or' must have suggested that 'all' and 'some' should be on the same footing semantically.GG’s style is at times eccentric. It reminds me of those old-school professors, now mostly gone, whose mostly impersonal style did not preclude the occasional sharp-tongued aside. Some of his translations remind me of William of Moerbeke’s versions of Aristotle: word for word, and syntax be hanged. There are several dull patches—surveys of minor authors with the feel of only slightly digested index-card notes. The book could have used a good copy-editor. Nevertheless it will be indispensable to anyone who wants to understand the history of “foundations”. In its coverage of the relevant sources, published and unpublished, it is unique.

May 26, 2004 in Books · History of Philosophy · Philosophy of Mathematics