Does Mathematics Have Objects? in what Sense?

ION The topologist Salomon Bochner considers the iteration of abstraction as the distinctive feature of the mathematics since the Scientific Revolution of the 17th century. “In Greek mathematics, whatever its originality and DOES MATHEMATICS HAVE OBJECTS? IN WHAT SENSE? 199 reputation, symbolization . . . did not advance beyond a first stage, namely, beyond the process of idealization, which is a process of abstraction from direct actuality. . . . However . . . full-scale symbolization is much more than mere idealization. It involves, in particular, untrammeled escalation of abstraction, that is, abstraction from abstraction, abstraction from abstraction from abstraction, and so forth; and, all importantly, the general abstract objects thus arising, if viewed as instances of symbols, must be eligible for the exercise of certain productive manipulations and operations, if they are mathematically meaningful. . . . On the face of it, modern mathematics, that is, mathematics of the 16th century and after, began to undertake abstractions from possibility only in the 19th century; but effectively it did so from the outset” (Bochner 1966, 18, 57). In a similar vein Peirce writes: “One extremely important grade of thinking about thought, which my logical analyses have shown to be one of the chief, if not the chief, explanation of the power of mathematical reasoning, is a stock topic of ridicule among the wits. This operation is performed when something, that one has thought about any subject, is itself made a subject of thought” (NEM IV, 49). In this way even the means and conditions of thought become an object of it. A predicative or attributive use of some concept is transformed into a referential use in order to incorporate the entity thus synthesized into new relational structures. The above mentioned example of the introduction of the imaginary numbers provides a case in point. At first after having been introduced to generalize certain algebraic operations, these “numbers” seemed the paradigmatic model of an artificial invention, whilst the subsequent history of complex functiontheory would tend to provide this invention with the characteristics of something indubitably objective. In all necessary reasoning, Peirce continues “the greatest point of art consists in the introduction of suitable abstractions. By this I mean such a transformation of our diagrams that characters of one diagram may appear in another as things. A familiar example is where in analysis we treat operations as themselves the subject of operations” (CP 5.162). Piaget, as we have seen, entertains a similar view of mathematical generalization but misses the fact that perception and observation will necessarily play a role throughout the process. In the discussion of the so-called fundamental theorem of algebra, it is said, that Lagrange had tacitly and implicitly used the intermediate value theorem for continuous functions to give a proof of this theorem. In actual fact Lagrange did nothing but provide algorithms for attaining approximate solutions of algebraic equations. It was Cauchy who read the intermediate value theorem for continuous functions into Lagrange’s argument, trying