arXiv subjects

Iulian D. Toader

Publications and source records attributed to Iulian D. Toader.

16 recordsLinked to original sources

arXiv2026

Hankel's principle as an anti-Kantian program

Hankel used his principle of the permanence of formal laws (PFL) as a guide for the extension of number systems and as a necessary condition for the legitimacy of their formal theories. He acknowledged that these applications have important limitations, evidenced by the extension to hypercomplex numbers and by what he saw as the unavoidable inconsistency of a formal theory of irrational numbers. Yet, intriguingly enough, he remained fully committed to the PFL. I argue that this was due to his understanding it as an expression of a conservative strategy, inherited from Peacock and Hamilton, which permits the revision of the basic laws of arithmetic if there are reasons for revision that are found, upon deliberation, to outweigh the reasons for their preservation. Then I discuss criticisms by Schubert and Pringsheim, who reformulated the PFL to align it with their own anti-revisionary conservative strategy, at the cost of relinquishing parts of modern mathematics. I conclude by emphasizing the deep philosophical difference between these kinds of conservatism in mathematics.

math.HO↗

arXiv2026

Carnap on Quantum Mechanics

This entry reviews Rudolf Carnap's philosophical views on the quantum mechanics of his time. It also offers some thoughts on how Carnap might have reacted to some recent developments in the foundations of quantum mechanics.

physics.hist-ph↗

arXiv2026

Peacock's Principle as a Conservative Strategy

The view that Peacock's principle of permanence has been invalidated by Hamilton's introduction of non-commutative algebras has always seemed rather odd, in light of Peacock's favorable reception of quaternions and the endorsement of his principle by Hamilton. But the view is not just odd; it is incorrect. In order to show this, I critically analyze Peacock's attempts to reject possible exceptions to his principle, like the factorial function and an infinite series due to Euler. Then I argue that the principle of permanence is best understood as an expression of a conservative strategy, philosophically grounded in Hume's conception of the laws of reasoning, which advocates their preservation to the furthest extent possible, thus allowing exceptions, i.e., violations of these laws. On this reading, non-commutative multiplication does not invalidate Peacock's principle, if the reasons for violating commutativity outweigh the reasons for its preservation. Finally, I show that Hamilton followed a conservative strategy of precisely this sort when he developed his quaternionic calculus.

math.HO↗

arXiv2025

Weyl's Quantifiers

I argue against the predominant view of Weyl's interpretation of the logical signs. Drawing on his correctness-first account of mathematical knowledge, I point out that, according to him, quantified statements generate conditional obligations to act in ways that expand the repository of correct judgments. This clarifies Weyl's reasons for rejecting the law of excluded middle, which have nothing to do with what has been attributed to him by the predominant view. I also offer some preliminary thoughts on how to understand conditional obligations generated by statements with nested quantifiers.

math.HO↗

arXiv2025

Quantum Logic and Meaning

This paper gives a formulation of quantum logic in the abstract algebraic setting laid out by Dunn and Hardegree (2001). On this basis, it provides a comparative analysis of viable quantum logical bivalent semantics and their classical counterparts, thereby showing that the truth-functional status of classical and quantum connectives is not as different as usually thought. Then it points out that bivalent semantics for quantum logic -- compatible with realism about quantum mechanics -- can be maintained, albeit at the price of truth-functionality. Finally, the paper critically addresses Geoffrey Hellman's argument (1980) that this lack of truth-functionality entails a change of meaning between classical and quantum connectives.

physics.hist-ph↗

arXiv2025

Rules and Meaning in Quantum Mechanics

This book concerns the metasemantics of quantum mechanics (QM). Roughly, it pursues an investigation at the intersection of philosophy of physics and philosophy of language, and it offers a critical analysis of rival explanations of the semantic facts of standard QM. Two problems for such explanations are discussed: categoricity and permanence. New results include 1) a reconstruction of Einstein's incompleteness argument, which concludes that a local, separable, and categorical QM cannot exist, 2) a reinterpretation of Bohr's principle of correspondence, grounded in the principle of permanence, 3) a meaning-variance argument for quantum logic, which follows a line of critical reflections initiated by Weyl, and 4) an argument for semantic indeterminacy leveled against inferentialism about QM, inspired by Carnap's work in the philosophy of classical logic.

physics.hist-ph↗

arXiv2025

Bivalent Quantum Indeterminacy

This paper provides a novel metametaphysical approach to quantum indeterminacy. More specifically, it argues that bivalent quantum logic can successfully account for this kind of indeterminacy, given the non-truth-functional character of its disjunction. Furthermore, it suggests that the determinable-based account of quantum indeterminacy illustrates precisely this possibility.

physics.hist-ph↗

arXiv2024

Distribution can be Dropped: Reply to Rumfitt

Most believe that there are no empirical grounds that make the adoption of quantum logic necessary. Ian Rumfitt has further argued that this adoption is not possible, either, for the proof that distribution fails in quantum mechanics is rule-circular or unsound. I respond to Rumfitt, by showing that neither is the case: rule-circularity disappears when an appropriate semantics is considered, and soundness is restored by slightly modifying standard quantum mechanics. Thus, albeit this is indeed not necessary, it is however possible for a quantum logician to rationally adjudicate against classical logic.

physics.hist-ph↗

arXiv2024

Permanence as a Principle of Practice

The paper discusses Peano's argument for preserving familiar notations. The argument reinforces the principle of permanence, articulated in the early 19th century by Peacock, then adjusted by Hankel and adopted by many others. Typically regarded as a principle of theoretical rationality, permanence was understood by Peano, following Mach, and against Schubert, as a principle of practical rationality. The paper considers how permanence, thus understood, was used in justifying Burali-Forti and Marcolongo's notation for vectorial calculus, and in rejecting Frege's logical notation, and closes by considering Hahn's revival of Peano's argument against Pringsheim' reading of permanence as a logically necessary principle.

math.HO↗

arXiv2024

Why Did Weyl Think that Emmy Noether Made Algebra the Eldorado of Axiomatics?

The paper attempts to clarify Weyl's metaphorical description of Emmy Noether's algebra as the Eldorado of axiomatics. It discusses Weyl's early view on axiomatics, which is part of his criticism of Dedekind and Hilbert, as motivated by Weyl's acquiescence to a phenomenological epistemology of correctness, then it describes Noether's work in algebra, emphasizing in particular its ancestral relation to Dedekind's and Hilbert's works, as well as her mathematical methods, characterized by non-elementary reasoning, i.e., reasoning detached from mathematical objects. The paper turns then to Weyl's remarks on Noether's work, and argues against assimilating her use of the axiomatic method in algebra to his late view on axiomatics, on the ground of the latter's resistance to Noether's principle of detachment.

math.HO↗

arXiv2024

Quantum Mechanics as a Carnapian Language

The paper discusses Carnap's claim that a proper philosophical analysis of quantum mechanics, including a determination of whether its logic has to be revised, requires a rational reconstruction of the theory. Several articulations of the notion of rational reconstruction are recalled, followed by a brief analysis of two standard criticisms of Carnap's claim. The paper suggests that adopting inferentialism overcomes both criticisms, and then considers the possibility of formulating quantum mechanics as a Carnapian language with an inferentialist semantics.

physics.hist-ph↗

arXiv2023

Is Bohr's Correspondence Principle just Hankel's Principle of Permanence?

No, but the paper argues that Bohr understood his correspondence principle, or at least an aspect of that principle expressed by the notion of rational generalization, as grounded in Hankel's principle of permanence, adapted to new historical and theoretical contexts. This is shown to illuminate some otherwise obscure aspects of Bohr's approach to quantum theory, as well as a seemingly strange criticism against this approach, due to Feyerabend and Bohm.

physics.hist-ph↗

arXiv2023

An Alleged Tension between Non-classical Logics and Applied Classical Mathematics

Timothy Williamson has recently argued that the applicability of classical mathematics in the natural and social sciences raises a problem for the endorsement, in non-mathematical domains, of a wide range of non-classical logics. We first reconstruct his argument and present its restriction to the case of quantum logic (QL). Then we show that there is no problematic tension between the applicability of classical mathematical models to quantum phenomena and the endorsement of QL in the reasoning about the latter. Once we identify the premise in Williamson's argument that turns out to be false when restricted to QL, we argue that the same premise fails for a wider variety of non-classical logics. In the end, we use our discussion to draw some general lessons concerning the relationship between applied logic and applied mathematics.

physics.hist-ph↗

arXiv2023

On the Categoricity of Quantum Mechanics

The paper offers an argument against an intuitive reading of the Stone-von Neumann theorem as a categoricity result, thereby pointing out that, against what is usually taken to be the case, this theorem does not entail any model-theoretical difference between the theories that validate it and those that don't.

quant-ph↗

arXiv2023

Einstein Completeness as Categoricity

This paper provides an algebraic reconstruction of Einstein's own argument for the incompleteness of quantum mechanics -- the one that he thought did not make it into the EPR paper -- in order to clarify the assumptions that underlie an understanding of Einstein completeness as categoricity, the sense in which it is a type of descriptive completeness, and some of the various ways in which it has been more often misconstrued.

quant-ph↗

arXiv2023

Spacetime Singularities and Invariance

This paper explains why spacetime singularities do not constitute a breakdown of physical laws, and points out that the difference between the metrics at singularities and those outside of singularities is factual, rather than nomological.

physics.hist-ph↗