Abstract
Formalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism (advocated by Heine), game formalism (advocated by Thomae) and instrumental formalism (advocated by Hilbert). After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention to Hilbert’s instrumental formalism. My primary aim there will be to develop its formalist elements more fully. These are, in the main, (i) its rejection of the axiom-centric focus of traditional model-construction approaches to consistency problems, (ii) its departure from the traditional understanding of the basic nature of proof and (iii) its distinctively descriptive or observational orientation with regard to the consistency problem for arithmetic. More specifically, I will highlight what I see as the salient points of connection between Hilbert’s formalist attitude and his finitist standard for the consistency proof for arithmetic. I will also note what I see as a significant tension between Hilbert’s observational approach to the consistency problem for arithmetic and his expressed hope that his solution of that problem would dispense with certain epistemological concerns regarding arithmetic once and for all.
Highlights
In Goethe’s Faust, Mephistopheles offered the following cynical assessment of the role(s) played by language in theological discourse.“Where concepts fail, a word appears at just the right time.”1 In Goethe’s view, genuine ideas, thoughts or judgements were in short supply in theological discourse and were commonly verbal mirages developed through the rhetorical skills of those involved
À propos the former, Hilbert regarded the threat to consistency posed by the nonaxiomatic elements of a body of axiomatic reasoning as serious. Given his awareness of Brouwer’s views concerning the capacity of the principles of classical logic to sustain paradox, it is perhaps only natural that he (i.e., Hilbert) should have felt a need to pay more careful attention to the role played by choice of logical principles in the emergence of inconsistencies
The manifest use and usefulness of non-contentual “algebraic calculation” (“algebraic calculation with letters” ( [18], 174-175)) as an instrument of mathematical reasoning made this clear. In keeping with this conviction, Hilbert opted for a conception of proof that left room for a type of proof or proof-like enterprise that did not consist in deductive reasoning with judgements having propositional contents
Summary
In Goethe’s Faust, Mephistopheles offered the following cynical assessment of the role(s) played by language in theological discourse. Hilbert’s collaborator, Paul Bernays, famously paraphrased Mephistopheles’ remark at one point by replacing ‘Wort’ with ‘Zeichen’—“Wo Begriffe fehlen, da stellt ein Zeichen zu rechter Zeit sich ein.” (cf [1], 16).2 He observed that, far from being an expression of skepticism concerning the legitimacy of non-contentual uses of language in mathematics, it was a statement of the core “methodological principle (methodische Prinzip)” In the example mentioned above, the calculation in question results in the eventual elimination of the ideal or non-contentual expressions (i.e., expressions which contain radical signs over negative numerals) If this is correct, non-contentual employments of expressions can function as useful instruments in the development of mathematical knowledge (more exactly, the knowledge of contentual mathematical truths). My aim in doing so is to bring the distinctive motives of Hilbert’s formalism, and their mathematical origins, into clearer perspective
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