The basic dynamical quantities of classical mechanics, such as position, linear momentum, angular momentum and energy, obtain their fundamental epistomological content by means of their intimate relationship to the symmetries of the space-time manifold which is the arena of physics. The program of canonical quantization can be understood as a two stage process. The first stage is Bohr's Correspondence Principle, whereby the basic dynamical quantities of the quantum theory are required to retain precisely the same relationship to the symmetries of (...) the space-time manifold as do their classical counterparts, thereby preserving their epistemological, as well as measurement-theoretic, significance. Having so identified the basic dynamical variables, functions of these may now be used to identify the subtler symmetries of the proper canonical group. The second and determining stage of the quantization program requires the establishment of a correspondence between some of these subtler symmetries of the classical theory and related symmetries of the quantum theory, the relationship being determined by a common algebraic form for their defining functions. (shrink)

A real x is -Kurtz random if it is in no closed null set . We show that there is a cone of -Kurtz random hyperdegrees. We characterize lowness for -Kurtz randomness as being -dominated and -semi-traceable.

This book discusses the problem of mathematical knowledge, and its broader philosophical ramifications. It argues that the problem of explaining the (defeasible) justification of our mathematical beliefs (‘the justificatory challenge’), arises insofar as disagreement over axioms bottoms out in disagreement over intuitions. And it argues that the problem of explaining their reliability (‘the reliability challenge’), arises to the extent that we could have easily had different beliefs. The book shows that mathematical facts are not, in general, empirically accessible, contra Quine, (...) and that they cannot be dispensed with, contra Field. However, it argues that they might be so plentiful that our knowledge of them is intelligible. The book concludes with a complementary ‘pluralism’ about modality, logic and normative theory, highlighting its revisionary implications. Metaphysically, pluralism engenders a kind of perspectivalism and indeterminacy. Methodologically, it vindicates Carnap’s pragmatism, transposed to the key of realism. (shrink)

Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...) vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows (2015), Pruss (2020), and Scambler (2021), aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas some of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that in light of its theoretical and explanatory advantages, Countabilism is more likely true than not. (shrink)

We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well-ordered families of sets and the axiom of choice for sets are both true, but the axiom of choice is false.

1. Frege’s mistake Frege is justifiably considered the most important thinker in the development of our contemporary “modern” logic. One corollary to this historical role of Frege’s is that his mistakes are found in a magnified form in the subsequent development of logic. This paper examines one such mistake and its later history. Diagnosing this history also reveals ways of overcoming some of the limitations that Frege’s mistake has unwittingly imposed on current forms of modern logic. Frege’s mistake concerns the (...) semantics (meaning) of quantifiers. The mistake is to assume that this semantics is exhausted by the quantifiers’ (quantified variables’) ranging over a class of values. These values are the members of the domain (universe of discourse) of the language to which the quantifiers belong. The entire job description of the quantifiers is to indicate whither or not at least one member of the domain has a certain (possible complex) predicate (existential quantifier) and to indicate whether all of them have one (universal quantifier). In other words, quantifiers are higher order predicates indicating whether or not a given lower-order predicate is nonempty or exceptionless. This is in fact precisely how Frege proposes to treat quantifiers in his logical theory. (See Frege 1984, pp. 153-154, pp. 26-27 of the original.) This is obviously part of the semantical task of quantifiers. However, it is not the only one. Quantifiers have another function in language. There is a task that any language must be capable of fulfilling if it is to serve as a language of science and for that matter as a language suitable for innumerable purposes in everyday life. This task is to.. (shrink)

This paper raises the question under what circumstances a plurality forms a set, parallel to the Special Composition Question for mereology. The range of answers that have been proposed in the literature are surveyed and criticised. I argue that there is good reason to reject both the view that pluralities never form sets and the view that pluralities always form sets. Instead, we need to affirm restricted set formation. Casting doubt on the availability of any informative principle which will settle (...) which pluralities form sets, the paper concludes by affirming a naturalistic approach to the philosophy of set theory. (shrink)

It is argued that mathematical statements are "a posteriori synthetic" statements of a very special sort, To be called "structure-Analytic" statements. They follow logically from the axioms defining the mathematical structure they are describing--Provided that these axioms are "consistent". Yet, Consistency of these axioms is an empirical claim: it may be "empirically verifiable" by existence of a finite model, Or may have the nature of an "empirically falsifiable hypothesis" that no contradiction can be derived from the axioms.

Brandom's method of analyzing pragmatic relations among different practices and vocabularies through meaning-use diagrams is used to specify how Laruelle's nonphilosophical suspension of the Principle of Sufficient Philosophy may be distinguished from the philosophical auto-critiques of such thinkers as Badiou and Derrida. A superposition of diagrams modeling philosophical sufficiency on the one hand and supplementation through the Other on the other provides a schematic representation of the core duality of what Laruelle calls The-Philosophy. In contrast to this self-implicating and self-reproducing (...) structure, Laruelle's axiomatic method is shown to enable a unilateral usage of the practices and vocabularies of philosophy as mere material that is no longer subject to the presuppositions or ineluctable restitution of philosophy as such. (shrink)

One can add the machinery of relation symbols and terms to a propositional modal logic without adding quantifiers. Ordinarily this is no extension beyond the propositional. But if terms are allowed to be non-rigid, a scoping mechanism (usually written using lambda abstraction) must also be introduced to avoid ambiguity. Since quantifiers are not present, this is not really a first-order logic, but it is not exactly propositional either. For propositional logics such as K, T and D, adding such machinery produces (...) a decidable logic, but adding it to S5 produces an undecidable one. Further, if an equality symbol is in the language, and interpreted by the equality relation, logics from K4 to S5 yield undecidable versions. (Thus transitivity is the villain here.) The proof of undecidability consists in showing that classical first-order logic can be embedded. (shrink)

In this paper, we strengthen Hardy’s [1995] and Ketland’s [2005] arguments on the issues surrounding the self-referential nature of Yablo’s paradox [1993]. We first begin by observing that Priest’s [1997] construction of the binary satisfaction relation in revealing a fixed point relies on impredicative definitions. We then show that Yablo’s paradox is ‘ω-circular’, based on ω-inconsistent theories, by arguing that the paradox is not self-referential in the classical sense but rather admits circularity at the least transfinite countable ordinal. Hence, we (...) both strengthen arguments for the ω-inconsistency of Yablo’s paradox and present a compromise solution of the problem emerged from Yablo’s and Priest’s conflicting theses. (shrink)

In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results. MSC: 03E25, 03E50, 04A25, 04A50.

This unique anthology of new, contributed essays offers a range of perspectives on various aspects of ontic vagueness. It seeks to answer core questions pertaining to onticism, the view that vagueness exists in the world itself. The questions to be addressed include whether vague objects must have vague identity, and whether ontic vagueness has a distinctive logic, one that is not shared by semantic or epistemic vagueness. The essays in this volume explain the motivations behind onticism, such as the plausibility (...) of mereological vagueness and indeterminacy in quantum mechanics and they offer various arguments both for and against ontic vagueness; onticism is also compared with other, competing theories of vagueness such as semanticism, the view that vagueness exists only in our linguistic representation of the world. Gareth Evans’s influential paper of 1978, “Can There Be Vague Objects?” gave a simple but cogent argument against the coherence of ontic vagueness. Onticism was subsequently dismissed by many. However, in recent years, researchers have become aware of the logical gaps in Evans’s argument and this has triggered a new wave of interest in onticism. Onticism is now widely regarded as at least a coherent view. Reflecting this growing consensus, the present anthology for the first time puts together essays that are focused on onticism and its various facets and it fills in the lacuna in the literature on vagueness, a much-discussed subject in contemporary philosophy. (shrink)

Adding higher types to set theory differs from adding inaccessible cardinals, in that higher type arguments apply to all sets rather than just ordinary ones. Levy's reflection axiom is justified, by considering the principle that we can pretend that the universe is a set, together with methods of Gaifman [8]. We reprove some results of Gaifman, and some facts about Levy's reflection axiom, including the fact that adding higher types yields no new theorems about sets. Some remarks on standard models (...) are made. An obvious strengthening of Levy's axiom to higher types is considered, which implies the existence of indescribable cardinals. Other remarks about larger cardinals are made; some questions of Gloede [9] are settled. Finally we argue that the evidence for V = L is strong, and that CH is certainly true. MSC: 03E30, 03E55. (shrink)

Recent mathematical results, obtained by the author, in collaboration with Alexander Stokolos, Olof Svensson, and Tomasz Weiss, in the study of harmonic functions, have prompted the following reflections, intertwined with views on some turning points in the history of mathematics and accompanied by an interpretive key that could perhaps shed some light on other aspects of (the development of) mathematics.

We introduce a new simple first-order framework for theories whose objects are well-orderings (lists). A system ALT (axiomatic list theory) is presented and shown to be equiconsistent with ZFC (Zermelo Fraenkel Set Theory with the Axiom of Choice). The theory sheds new light on the power set axiom and on Gs axiom of constructibility. In list theory there are strong arguments favoring Gs axiom, while a bare analogon of the set theoretic power set axiom looks artificial. In fact, there is (...) a natural and attractive modification of ALT where every object is constructible and countable. In order to substantiate our foundational interest in lists, we also compare sets and lists from the perspective of finite objects, arguing that lists are, from a certain point of view, conceptually simpler than sets. (shrink)

This paper reviews the claims of several main-stream candidates to be the foundations of mathematics, including set theory. The review concludes that at this level of mathematical knowledge it would be very unreasonable to settle with any one of these foundations and that the only reasonable choice is a pluralist one.

Can identity itself be vague? Can there be vague objects? Does a positive answer to either question entail a positive answer to the other? In this paper we answer these questions as follows: No, No, and Yes. First, we discuss Evans’s famous 1978 argument and argue that the main lesson that it imparts is that identity itself cannot be vague. We defend the argument from objections and endorse this conclusion. We acknowledge, however, that the argument does not by itself establish (...) either that there cannot be vague objects or that there cannot be identity statements that are indeterminate for ontic reasons. And we further acknowledge that it does not by itself establish that there cannot be identity statements that are indeterminate in virtue of the existence of vague objects. We then go on to argue that, despite this, one who believes in vague objects cannot endorse Evans’s argument. To establish this we offer supplementary arguments that show that if vague objects exist then identity is vague, and that if identity is vague then vague objects exist. Finally we draw attention to an argument parallel to that of Evans’s, but safer, which can be employed against the putative ontic indeterminacy in identity of vague objects which can be differentiated by identity-free properties. (shrink)

5.5. Every topos is linguistic: the equivalence theorem.

The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...) the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences — many indispensable, some startling — and has come to figure prominently in discussions on the foundations of mathematics. It (or its equivalents) have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it. (shrink)

This paper is divided in five sections. Section 11.1 sketches the history of the distinction between speech act with negative content and negated speech act, and gives a general dynamic interpretation for negated speech act. “Downdate semantics” for AGM contraction is introduced in Section 11.2. Relying on semantically interpreted contraction, Section 11.3 develops the dynamic semantics for constative and directive speech acts, and their external negations. The expressive completeness for the formal variants of natural language utterances, none of which is (...) a retraction, has been proved in Section 11.4. The last section gives a laconic answer to the question posed in the title of the paper. (shrink)

The Boolean many-valued approach to vagueness is similar to the infinite-valued approach embraced by fuzzy logic in the respect in which both approaches seek to solve the problems of vagueness by assigning to the relevant sentences many values between falsity and truth, but while the fuzzy-logic approach postulates linearly-ordered values between 0 and 1, the Boolean approach assigns to sentences values in a many-element complete Boolean algebra. On the modal-precisificational approach represented by Kit Fine, if a sentence is indeterminate in (...) truth value in some world, it is taken to be true in one precisified world accessible from that world and false in another. This paper points to a way to unify these two approaches to vagueness by showing that Fine’s version of the modal-precisificational approach can be combined with the Boolean many-valued approach instead of supervaluationism, one of the most popular approaches to vagueness. (shrink)

In this paper I discuss possible ways of measuring the power of arithmetical theories, and the possiblity of making an explication in Carnap's sense of this concept. Chaitin formulates several suggestions how to construct measures, and these suggestions are reviewed together with some new and old critical arguments. I also briefly review a measure I have designed together with some shortcomings of this measure. The conclusion of the paper is that it is not possible to formulate an explication of the (...) concept. (shrink)

Iterated admissibility embodies a minimal criterion of rationality in interactions. The epistemic characterization of this solution has been actively investigated in recent times: it has been shown that strategies surviving \ rounds of iterated admissibility may be identified as those that are obtained under a condition called rationality and m assumption of rationality in complete lexicographic type structures. On the other hand, it has been shown that its limit condition, with an infinity assumption of rationality ), might not be satisfied (...) by any state in the epistemic structure, if the class of types is complete and the types are continuous. In this paper we analyze the problem in a different framework. We redefine the notion of type as well as the epistemic notion of assumption. These new definitions are sufficient for the characterization of iterated admissibility as the class of strategies that indeed satisfy \. One of the key methodological innovations in our approach involves defining a new notion of generic types and employing these in conjunction with Cohen’s technique of forcing. (shrink)

A nucleus is an operation on the collection of truth values which, like double negation in intuitionistic logic, is monotone, inflationary, idempotent and commutes with conjunction. Any nucleus determines a proof-theoretic translation of intuitionistic logic into itself by applying it to atomic formulas, disjunctions and existentially quantified subformulas, as in the Gödel–Gentzen negative translation. Here we show that there exists a similar translation of intuitionistic logic into itself which is more in the spirit of Kuroda’s negative translation. The key is (...) to apply the nucleus not only to the entire formula and universally quantified subformulas, but to conclusions of implications as well. (shrink)

En este artículo se examinan algunas implicaciones del naturalismo matemático de P. Maddy como una concepción filosófica que permite superar las dificultades del ficcionalismo y el realismo fisicalista en matemáticas. Aparte de esto, la mayor virtud de tal concepción parece ser que resuelve el problema que plantea para la aplicabilidad de la matemática el no asumir la tesis de indispensabilidad de Quine sin comprometerse con su holismo confirmacional. A continuación, sobre la base de dificultades intrínsecas al programa de Maddy, exploramos (...) un camino naturalista mejor motivado, el enfoque cognitivista corporeizado, y sugerimos que él permite explicar de manera adecuada la postulación de ciertos cardinales grandes, en particular, la aceptación de conjuntos no-construibles. (shrink)

Commençant par une revue sommaire des types de questions qu’une fondation des mathématiques devrait poser, cet article présente premièrement une critique des fondements basés sur la théorie des ensembles, puis propose l’idée que plusieurs fondements catégoriques, reliés les uns aux autres, seraient plus avantageux, et finalement indique une méthode pour retrouver la théorie des ensembles à travers une approche catégorique.