The distinct existences argument revisited

The aim of this paper is to take a fresh look at a discussion about the distinct existences argument that took place between David Armstrong and Frank Jackson more than 50 years ago. I will try to show that Armstrong’s argument can be successfully defended against Jackson’s objections (albeit at the price of certain concessions concerning Armstrong’s view on the meaning of psychological terms as well as his conception of universals). Focusing on two counterexamples that Jackson put forward against Hume’s principle (which is central to Armstrong’s argument), I will argue that they are either compatible with Hume’s principle, or imply a false claim. I will also look at several other considerations that go against Hume’s principle, such as, for example, Kripke’s origin essentialism and counterexamples from aposteriori necessity.

Essence and Definition by Abstraction

We may define words. We may also define the things for which words stand. Definitions of words may be explicit or implicit, and may seek to report pre-existing synonymies, but they may instead be wholly or partly stipulative. Definition by abstraction seeks to define a term-forming operator by fixing the truth-conditions of identity-statements featuring terms formed by means of that operator. Such definitions are a species of implicit definition. They are typically at least partly stipulative. Definitions of things (real definitions) are typically conceived as statements about the essence of their definienda, and so not stipulative. There thus appears to be a clash between taking Hume's principle as an implicit, at least partly stipulative definition of the number operator and as a real definition of cardinal numbers. This chapter argues that this apparent tension can be resolved, and that resolving it shows how some modal knowledge can be a priori.

Logicism and Logical Consequence

According to Crispin Wright’s neo-logicist reconstruction of Frege’s philosophy of arithmetic, the truths of arithmetic are logical consequences, in the semantic sense, of second-order logic, augmented with an analytic axiom (Hume’s Principle). Neo-logicism thus views arithmetic truths as analytic, being the logical consequences of an analytic axiom. This chapter argues that the semantic relation of second-order logical consequence that is most naturally suited to the practice of arithmetic is proof-theoretically complete, and that given this, Gödel’s incompleteness result shows that there are arithmetical truths which are not derivable in Wright’s proof theory augmented by Hume’s Principle. The chapter thus challenges Wright’s programme of neo-Fregean logicism.

God, Logic, and Quantum Information

Quantum information is discussed as the universal substance of the world. It is interpreted as that generalization of classical information, which includes both finite and transfinite ordinal numbers. On the other hand, any wave function and thus any state of any quantum system is just one value of quantum information. Information and its generalization as quantum information are considered as quantities of elementary choices. Their units are correspondingly a bit and a qubit. The course of time is what generates choices by itself, thus quantum information and any item in the world in final analysis. The course of time generates necessarily choices so: The future is absolutely unorderable in principle while the past is always well-ordered and thus unchangeable. The present as the mediation between them needs the well-ordered theorem equivalent to the axiom of choice. The latter guarantees the choice even among the elements of an infinite set, which is the case of quantum information. The concrete and abstract objects share information as their common base, which is quantum as to the formers and classical as to the latters. The general quantities of matter in physics, mass and energy can be considered as particular cases of quantum information. The link between choice and abstraction in set theory allows of “Hume’s principle” to be interpreted in terms of quantum mechanics as equivalence of “many” and “much” underlying quantum information. Quantum information as the universal substance of the world calls for the unity of physics and mathematics rather than that of the concrete and abstract objects and thus for a form of quantum neo-Pythagoreanism in final analysis.

The Proof of Hume’s Principle

Beginning in Grundgesetze §53, Frege presents proofs of a set of theorems known to encompass the Peano-Dedekind axioms for arithmetic. The initial part of Frege’s deductive development of arithmetic, to theorems (32) and (49), contains fully formal proofs that had merely been sketched out in Grundlagen. Theorems (32) and (49) are significant because they are the right-to-left and left-to-right directions respectively of what we call today “Hume’s Principle” (HP). The core observation that we explore is that in Grundgesetze, Frege does not prove Hume’s Principle, not at least if we take HP to be the principle he introduces, and then rejects, as a definition of number in Grundlagen. In order better to understand why Frege never considers HP as a biconditional principle in Grundgesetze, we explicate the theorems Frege actually proves in that work, clarify their conceptual and logical status within the overall derivation of arithmetic, and ask how the definitional content that Frege intuited in Hume’s Principle is reconstructed by the theorems that Frege does prove.

Second-Order Abstraction Before and After Russell’s Paradox

In this article, I discuss certain aspects of Frege’s paradigms of second-order abstraction principles, Hume’s Principle and Basic Law V, with special emphasis on the latter. I begin by arguing that, contrary to a widespread view, Frege did not express any dissatisfaction with Basic Law V before 1902. In particular, he did not raise any doubt about its assumed logical nature. I then show why Frege nonetheless fails to justify Basic Law V as a primitive logical truth along the lines of the semantic justification that he provides for the other axioms of his system. In subsequent sections, I argue (a) that Frege could not have chosen Hume’s Principle as a logical axiom, neither before 1902 nor after 1902; (b) that even if in the wake of Russell’s Paradox Frege had accepted Hume’s Principle as a logical axiom, such an axiom could not have replaced Basic Law V which was designed to introduce logical objects of a fundamental and irreducible kind and to afford us the right cognitive access to them; (c) that Frege most likely held that the two sides of Basic Law V express different thoughts; (d) that for Basic Law V or for any other Fregean abstraction principle that is laid down as an axiom of a theory, the case in which both real epistemic value and self-evidence are given their due is ruled out. I make a proposal as to how Frege might have escaped this epistemic dilemma.

Ontogenetic origins of human integer representations

Do children learn number words by associating them with perceptual magnitudes? Recent studies argue that approximate numerical magnitudes play a foundational role in the development of integer concepts. Against this, we argue that approximate number representations fail both empirically and in principle to provide the content required of integer concepts. Instead, we suggest that children’s understanding of integer concepts proceeds in two phases. In the first phase, children learn small exact number word meanings by associating words with small sets. In the second phase, children learn the meanings of larger number words by mastering the logic of exact counting algorithms, which implement the successor function and Hume’s principle (that 1-to-1 correspondence guarantees exact equality). In neither phase do approximate number representations play a foundational role.