This chapter traces the philosophical and mathematical origins of formalist ideas. On the philosophical side, the key element is a certain freedom the mathematician is taken to have—namely, the freedom to “create” methods of reasoning that are symbolical rather than contentual in character, but which nonetheless serve as efficient and reliable means for deriving contentual conclusions. Mathematically, such methods came to prominence with the rapid development of algebra and the algebraic or “analytical” approach to geometry in the 17th century. Formalism received its most mature formulation in the foundational writings of David Hilbert, who maintained that, modulo a proof of their consistency, the mathematician is free to stipulate that the concepts she introduces have exactly the properties she provides for them in the axioms by which she introduces them. Such items as imaginary and complex numbers are, I argue, examples of concepts (or expressions) introduced in this way. Kurt Gödel’s incompleteness theorems pose a serious problem for formalism but do not rule out the possible success of a carefully formulated version of it.
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