Axiomatizing mathematical conceptualism in third order arithmetic

We review the philosophical framework of mathematical conceptualism as an alternative to set-theoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the basic principles of conceptualism in a formal system CM set in the language of third order arithmetic.

💡 Research Summary

The paper presents a systematic formalization of mathematical conceptualism, positioning it as a viable alternative to set‑theoretic foundations for mainstream mathematics. After outlining the philosophical motivation—namely, the insistence that only “constructible” mathematical objects should be admitted and the consequent skepticism toward non‑constructive, large‑cardinality sets—the authors introduce a new formal system called CM (Conceptualist Mathematics). CM is built in the language of third‑order arithmetic, which distinguishes three tiers of variables: first‑order variables for natural numbers, second‑order variables for real numbers and functions, and third‑order variables for sets of second‑order objects.

The axioms of CM are deliberately restrictive. First‑order arithmetic adopts the usual Peano axioms. Second‑order axioms guarantee the completeness of the real line, the existence of continuous functions, and the basic algebraic operations on reals. The crucial novelty lies in the third‑order axioms: a limited comprehension principle that allows the formation of a set of second‑order objects only when those objects can be explicitly constructed from previously given data. In effect, this replaces the unrestricted comprehension and full axiom of choice of ZFC with a predicative, “construction‑only” comprehension scheme. The authors argue that this mirrors the predicative stance of Feferman–Schütte analysis while preserving the expressive power needed for ordinary mathematics.

From a proof‑theoretic perspective, the authors demonstrate that CM is proof‑theoretically equivalent to ACA₀ (Arithmetical Comprehension Axiom). They construct ω‑models of CM and show that every theorem of CM is conservatively interpretable in ACA₀, establishing Π₁¹‑conservativity and thereby preventing an explosion of strength beyond predicative arithmetic. Moreover, the limited choice principles admitted by CM (e.g., countable choice) are sufficient for most classical arguments but avoid the non‑constructive pitfalls of full choice.

The bulk of the paper is devoted to rebuilding core areas of mathematics within CM. In real analysis, the authors re‑prove the completeness of ℝ, the Extreme Value Theorem, and the Fundamental Theorem of Calculus using only second‑order axioms together with the restricted third‑order comprehension. Topology is developed by defining open sets, compactness, and connectedness in the third‑order framework; the authors verify the Heine‑Borel theorem and basic homotopy concepts without invoking arbitrary large sets. Algebraic structures—fields, groups, vector spaces—are introduced at the second‑order level, and fundamental results such as the existence of bases for finite‑dimensional spaces, the structure theorem for finitely generated abelian groups, and the basic isomorphism theorems are derived. All these developments rely on the limited choice principle, showing that CM provides enough selection power for ordinary mathematics while staying within a predicative regime.

In the philosophical discussion, the authors argue that conceptualism occupies a middle ground between platonist realism and formalist anti‑realism. By insisting on constructibility, CM eliminates many of the metaphysical commitments of ZFC, yet it retains sufficient machinery to support the bulk of contemporary mathematical practice. The paper also emphasizes the compatibility of CM with existing formal systems, suggesting that extensions to category theory, homological algebra, and higher‑dimensional topology are plausible future directions.

The conclusion reiterates that CM constitutes the first comprehensive axiomatization of mathematical conceptualism in a third‑order arithmetic setting. It matches the proof‑theoretic strength of ACA₀, supports the development of analysis, topology, and algebra, and offers a philosophically motivated, technically robust foundation that can serve as a serious competitor to traditional set‑theoretic foundations.