FREGE'S CONSTRAINT AND THE NATURE OF FREGE'S FOUNDATIONAL PROGRAM release_jp2gwmkn5femje6rw5s2nk32u4

by MARCO PANZA, ANDREA SERENI

Published in The Review of Symbolic Logic by Cambridge University Press (CUP).

2018   Volume 12, p97-143

Abstract

<jats:title>Abstract</jats:title>Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either 'Application Constraint' (<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline1" /><jats:tex-math>$AC$</jats:tex-math></jats:alternatives></jats:inline-formula>) or 'Frege Constraint' (<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline2" /><jats:tex-math>$FC$</jats:tex-math></jats:alternatives></jats:inline-formula>), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline3" /><jats:tex-math>$AC$</jats:tex-math></jats:alternatives></jats:inline-formula> generalizes Frege's views while <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline4" /><jats:tex-math>$FC$</jats:tex-math></jats:alternatives></jats:inline-formula> comes closer to his original conceptions. Different authors diverge on the interpretation of <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline5" /><jats:tex-math>$FC$</jats:tex-math></jats:alternatives></jats:inline-formula> and on whether it applies to definitions of both natural and real numbers. Our aim is to trace the origins of <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline6" /><jats:tex-math>$FC$</jats:tex-math></jats:alternatives></jats:inline-formula> and to explore how different understandings of it can be faithful to Frege's views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline7" /><jats:tex-math>$AC$</jats:tex-math></jats:alternatives></jats:inline-formula> from <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline8" /><jats:tex-math>$FC$</jats:tex-math></jats:alternatives></jats:inline-formula> (§2). We discuss six rationales which may motivate the adoption of different instances of <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline9" /><jats:tex-math>$AC$</jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline10" /><jats:tex-math>$FC$</jats:tex-math></jats:alternatives></jats:inline-formula> (§3). We turn to the possible interpretations of <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline11" /><jats:tex-math>$FC$</jats:tex-math></jats:alternatives></jats:inline-formula> (§4), and advance a Semantic <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline12" /><jats:tex-math>$FC$</jats:tex-math></jats:alternatives></jats:inline-formula> (§4.1), arguing that while it suits Frege's definition of natural numbers (4.1.1), it cannot reasonably be imposed on definitions of real numbers (§4.1.2), for reasons only partly similar to those offered by Crispin Wright (§4.1.3). We then rehearse a recent exchange between Bob Hale and Vadim Batitzky to shed light on Frege's conception of real numbers and magnitudes (§4.2). We argue that an Architectonic version of <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline13" /><jats:tex-math>$FC$</jats:tex-math></jats:alternatives></jats:inline-formula> is indeed faithful to Frege's definition of real numbers, and compatible with his views on natural ones. Finally, we consider how attributing different instances of <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline14" /><jats:tex-math>$FC$</jats:tex-math></jats:alternatives></jats:inline-formula> to Frege and appreciating the role of the Architectonic <jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S1755020318000278_inline15" /><jats:tex-math>$FC$</jats:tex-math></jats:alternatives></jats:inline-formula> can provide a more perspicuous understanding of his foundational program, by questioning common pictures of his logicism (§5).
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