The present paper shows how one might model Everettian quantum mechanics using hyperfinitely many worlds. A hyperfinite model allows one to consider idealized measurements of observables with continuous-valued spectra where different outcomes are associated with possibly infinitesimal probabilities. One can also prove hyperfinite formulations of Everett’s limiting relative-frequency and randomness properties, theorems he considered central to his formulation of quantum mechanics. Finally, this model provides an intuitive framework in which to consider no-collapse formulations of quantum mechanics more generally.

The definition of a pseudofinite structure can be translated verbatim into continuous logic, but it also gives rise to a stronger notion and to two parallel concepts of pseudocompactness. Our purpose is to investigate the relationship between these four concepts and establish or refute each of them for several basic theories in continuous logic. Pseudofiniteness and pseudocompactness turn out to be equivalent for relational languages with constant symbols, and the four notions coincide with the standard pseudofiniteness in the case of (...) classical structures, but the details appear to be slightly more important here than in the usual translation of definitions from classical logic. We also prove that injective “formula-definable” endofunctions are surjective, and conversely, in strongly pseudofinite omega-saturated structures. (shrink)

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in absolutely ubiquitous structures from classical logic.

The Connes Embedding Problem asks whether every separable II1 factor embeds into an ultrapower of the hyperfinite II1 factor. We show that the CEP is equivalent to the statement that every type II1 tracial von Neumann algebra has a computable universal theory.

We show that any countable model of a model complete theory has an elementary extension with a “pseudofinite-like” quasi-dimension that detects dividing.

Let H be an infinite-dimensional (real or complex) Hilbert space, viewed as a metric structure in its natural signature. We characterize the definable linear operators on H as exactly the "scalar plus compact" operators.

Answering a question of Cifú Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced by the second author.

Answering a question of Cifú Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced by the second author.

The present paper shows how one might model Everettian quantum mechanics using hyperfinitely many worlds. A hyperfinite model allows one to consider idealized measurements of observables with continuous-valued spectra where different outcomes are associated with possibly infinitesimal probabilities. One can also prove hyperfinite formulations of Everett’s limiting relative-frequency and randomness properties, theorems he considered central to his formulation of quantum mechanics. Finally, this model provides an intuitive framework in which to consider no-collapse formulations of quantum mechanics more generally.

We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for discrete first-order real rosy theories.

We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form $$\phi ^\mathcal {M}\le r$$, and the open diagram, which encapsulates strict inequalities of the form $$\phi ^\mathcal {M}< r$$. We show (...) that the closed and open $$\Sigma _N$$ diagrams are $$\Pi ^0_{N+1}$$ and $$\Sigma ^0_N$$ respectively, and that the closed and open $$\Pi _N$$ diagrams are $$\Pi ^0_N$$ and $$\Sigma ^0_{N + 1}$$ respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal. (shrink)

For a locally compact group G, we show that it is possible to present the class of continuous unitary representations of G as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how nondegenerate ∗-representations of a general ∗-algebra A (with some mild assumptions) can be viewed as an elementary class, in a many-sorted language, and use the correspondence between continuous unitary representations of G and nondegenerate ∗-representations of L1(G). We relate the notion (...) of ultraproduct of logical structures, under this presentation, with other notions of ultraproduct of representations appearing in the literature, and we characterize property (T) for G in terms of the definability of the sets of fixed points of L1 functions on G. (shrink)

The randomization of a complete first-order theory [Formula: see text] is the complete continuous theory [Formula: see text] with two sorts, a sort for random elements of models of [Formula: see text] and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of [Formula: see text]. We show that if [Formula: see text] has the exchange property and [Formula: see text], then [Formula: see text] has a strict independence (...) relation in the home sort, and hence is real rosy. In particular, if [Formula: see text] is o-minimal, then [Formula: see text] is real rosy. (shrink)

In the paper Randomizations of Scattered Sentences, Keisler showed that if Martin’s axiom for aleph one holds, then every scattered sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is “yes”. It follows that the absolute Vaught conjecture holds if and only if every \-sentence with few separable randomizations has countably many countable models.

We study the relationship between fields of transseries and residue fields of convex subrings of non-standard extensions of the real numbers. This was motivated by a question of Todorov and Vernaeve, answered in this paper.

This paper discusses the logical space of alternative conceptual engineering projects, with a specific focus on (1) the processes, (2) the targets and goals, and (3) the methods of such projects. We present an overview of how these three aspects interact in the contemporary literature and discuss those alternative projects that have yet to be explored based on our suggested typology. We show how choices about each element in a conceptual engineering project constrain the possibilities for the others, thereby giving (...) rise to distinct groupings of possible projects under the banner of conceptual engineering. We conclude with a critical reflection on the potential ethical issues that arise as a result of effectively putting conceptual engineering into practice. (shrink)

Putting God First: Jewish Humanism after Heidegger tackles the challenge of maintaining Jewish identity in a world dominated by Western humanism. It argues that the Holocaust reflects more broadly on contemporary humanism than the Jewish world has ever dared to acknowledge. It advances the view that the establishment of the State of Israel presents a profound historical opportunity to disentangle Jewish thought from elements of the Western humanist tradition that threaten Jewish survival and conceal from view the plausibility of core (...) Jewish ideas and values. The work proposes that a healthy and peaceful relationship between Westernism and a robust Jewish identity can be achieved only by unraveling the two, and it presents a philosophical path for achieving this. Following a precise analysis of modern and postmodern thought that includes discussions of Immanuel Kant, Jacques Derrida, Michel Foucault, Ludwig Wittgenstein, George Steiner, Bruno Latour, Hannah Arendt, Jonathan Haidt, Daniel Kahneman, and others, the centerpiece of the book offers a close reading of the masterwork Being and Time, which Martin Heidegger composed in the period when his thought was gravitating toward his endorsement of Hitler and Nazism. This deep dive into Heidegger's ontological understanding of how the individual (Dasein) interacts with others in space and time yields a method for articulating the onto-theological meanings of key Jewish concepts such as the Nefesh (psyche), the Neshamah (soul), Tzelem Elohim (human being in God's image), Am Segulah, (Jewish chosenness), Shabbat, the Land of Israel, and more in texts such as the Tanya, the Sefat Emet, the Nefesh HaChayim, the Kedushat Levi, and in the writings of Rav Kook and the Maharal of Prague. Ultimately the book proposes a disentangled spiritual/political vision for a Jewish state that is characterized by the politics of Tikun Olam" -- From the publisher. (shrink)

The works of Hannah Arendt and Albert Camus--two of the most compelling political thinkers of the "resistance generation" that lived through World War II--can still provide penetrating insights for contemporary political reflection. Jeffrey C. Isaac offers new interpretations of these writers, viewing both as engaged intellectuals who grappled with the possibilities of political radicalism in a world in which liberalism and Marxism had revealed their inadequacy by being complicit in the rise of totalitarianism. According to Isaac, self-styled postmodern (...) writers who proclaim the death of grandiose ideologies often fail to recognize that such thinkers as Camus and Arendt had already noted this. But unlike many postmodernists, these two sought to preserve what was worthy in modern humanism--the idea of a common human condition and a commitment to human rights and the dignity of individuals. Isaac shows that both writers advanced the idea of a democratic civil society made up of self-limiting groups. Although they criticized the typical institutions of mass democratic politics, they endorsed alternative forms of local and international organization that defy the principle of state sovereignty. Isaac also shows how Arendt's writings on the Middle East, and Camus's on Algeria, urged the creation of such institutions. The vision of a "rebellious politics" that Arendt and Camus shared is of great relevance to current debates in democratic theory and to the transformations taking place in Europe and the states of the former Soviet Union. (shrink)

In the last book of The Consolation of Philosophy, Boethius develops his solution to the problem of divine foreknowledge and free choice. Interpreters standardly hold that this problem and his solution to it presuppose causal indeterminism. In this paper, I argue that Boethius, following a Neoplatonist view found in Proclus, is a causal determinist and compatibilist and maintains that God’s providential knowledge ensures the occurrence of all the events he knows. This alternative interpretation offers a better fit with Boethius’s text (...) and its historical antecedents and provides resources to address the main criticisms of his solution. (shrink)

We present a novel model of individual people, online posts, and media platforms to explain the online spread of epistemically toxic content such as fake news and suggest possible responses. We argue that a combination of technical features, such as the algorithmically curated feed structure, and social features, such as the absence of stable social-epistemic norms of posting and sharing in social media, is largely responsible for the unchecked spread of epistemically toxic content online. Sharing constitutes a distinctive communicative act, (...) governed by a dedicated norm and motivated to a large extent by social identity maintenance. But confusion about this norm and its lack of inherent epistemic checks lead readers to misunderstand posts, attribute excess or insufficient credibility to posts, and allow posters to evade epistemic accountability—all contributing to the spread of epistemically toxic content online. This spread can be effectively addressed if people and platforms add significantly more context to shared posts and platforms nudge people to develop and follow recognized epistemic norms of posting and sharing. (shrink)

For the past fifty years anxiety over naturalism has driven debates in social theory. One side sees social science as another kind of natural science, while the other rejects the possibility of objective and explanatory knowledge. _Interpretation and Social Knowledge_ suggests a different route, offering a way forward for an antinaturalist sociology that overcomes the opposition between interpretation and explanation and uses theory to build concrete, historically specific causal explanations of social phenomena.

I present and explain the Bohmian account of collapse in quantum mechanics.

Presents Newton's unifying idea of gravitation and explains how he converted physics from a science of explanation into a general mathematical system.

This first critical edition of Isaac Abravanel’s correspondence opens a window into the cultural, political and commercial world of one of the first Jewish humanists of the quattrocento. Jewish leader of the expelled Sephardim after 1492, commentator of the Bible, Abravanel is a legendary figure of the Sephardic history. The edition of the letters along with the introductive essay that reconstructs their cultural background intends to connect the legendary figure of Abravanel to the major reason of his remarkable career: (...) his successful assimilation of Renaissance humanism. (shrink)

My aim in this paper is to give a philosophical analysis of the relationship between contingently available technology and the knowledge that it makes possible. My concern is with what specific subjects can know in practice, given their particular conditions, especially available technology, rather than what can be known “in principle” by a hypothetical entity like Laplace’s Demon. The argument has two parts. In the first, I’ll construct a novel account of epistemic possibility that incorporates two pragmatic conditions: responsibility and (...) practicability. For example, whether subjects can gain knowledge depends in some circumstances on whether they have the capability of gathering relevant evidence. In turn, the possibility of undertaking such investigative activities depends in part on factors like ethical constraints, economical realities, and available technology. In the second part of the paper, I’ll introduce “technological possibility” to analyze the set of actions made possible by available technology. To help motivate the problem and later test my proposal, I’ll focus on a specific historical case, one of the earliest uses of digital electronic computers in a scientific investigation. I conclude that the epistemic possibility of gaining access to scientific knowledge about certain subjects depends (in some cases) on the technological possibility for making responsible investigations. (shrink)

This major work challenges some widely held positions in epistemology - those of Peirce and Popper on the one hand and those of Quine and Kuhn on the other.

The Mathematical Principles of Natural Philosophy.

Reproduces the text of Newton's dissertation on the nature and properties of light.