Date of Award

Fall 12-2026

Document Type

Dissertation

Degree Name

Doctor of Science (D.Sc.)

Department

Mathematics, Philosophy and Physics

First Advisor

Marco Panza

Second Advisor

Andrew Moshier

Abstract

This dissertation addresses the necessity of arithmetic by distinguishing two tasks that are often run together: the derivation task, which seeks to prove that mathematical truths are necessary, and the characterization task, which asks what kind of necessity arithmetic has and what grounds it. Setting the former aside for now, I focus on the latter, arguing that an adequate explanation of arithmetical necessity requires two sources: logical necessity and the necessary possibility that the right kind of structure exists. The second source makes mathematical modality distinctively mathematical, and the dissertation’s positive contribution emerges chiefly in that area. I survey and ultimately cannot accept several existing attempts to ground mathematical necessity: Carnapian linguistic frameworks, set-theoretic and neo-logicist reductions, and recent derivation-task arguments appealing to rigid comprehension or S5. I then develop a quasi-concrete modal-structuralist account. Working within Hellman’s modal structuralism (but departing from it in some respects), I construct, rather than merely posit, a witness for the categorical component of the theory. I use a Fitch-style construction over formula-types and argue that it is necessarily possible that an omega-sequence exists without assuming either Platonic abstracta or an actual infinity of concrete atoms. The construction instead relies on quasi-concrete objects, in Charles Parsons’ sense, that occupy an intermediate ontological position between the abstract and the concrete. The resulting account is nominalist-friendly but not zealous, constructivist without identifying mathematical objects with mental acts, and groundable in modal facts about objects already accessible to us.

DOI

10.36837/chapman.000770

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

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