It is consistent with constructive set theory (without Countable Choice, clearly) that the Cauchy reals (equivalence classes of Cauchy sequences of rationals) are not Cauchy complete. Related results are also shown, such as that a Cauchy sequence of rationals may not have a modulus of convergence, and that a Cauchy sequence of Cauchy sequences may not converge to a Cauchy sequence, among others.

The inverse of the distance between two structures $\mathscr{A} \not\equiv \mathscr{B}$ of finite type τ is naturally measured by the smallest integer q such that a sentence of quantifier rank q - 1 is satisfied by A but not by B. In this way the space $\operatorname{Str}^\tau$ of structures of type τ is equipped with a pseudometric. The induced topology coincides with the elementary topology of $\operatorname{Str}^\tau$ . Using the rudiments of the theory of uniform spaces, in this elementary note (...) we prove the convergence of every Cauchy net of structures, for any type τ. (shrink)

The inverse of the distance between two structures $\mathscr{A} \not\equiv \mathscr{B}$ of finite type $\tau$ is naturally measured by the smallest integer $q$ such that a sentence of quantifier rank $q - 1$ is satisfied by $\mathscr{A}$ but not by $\mathscr{B}$. In this way the space $\operatorname{Str}^\tau$ of structures of type $\tau$ is equipped with a pseudometric. The induced topology coincides with the elementary topology of $\operatorname{Str}^\tau$. Using the rudiments of the theory of uniform spaces, in this elementary note we (...) prove the convergence of every Cauchy net of structures, for any type $\tau$. (shrink)

This is a study of the set of the Malitz-completions of a given infinite first-order structure, put in relation with properties of directed sets.

In lieu of an abstract, here is a brief excerpt of the content:Otherwise than Laïcité? Toward an Agonistic Secularism in LevinasMark Cauchi (bio)Levinas and SecularismAlong with the so-called “return of the religious” in contemporary Western philosophy and politics, there has been a renewed effort in recent years to rethink secularism, the political doctrine of the separation of religion and politics.1 It would not be difficult to show that Emmanuel Levinas has been a substantial force in the resurgence of interest in (...) religion among philosophers, but his impact on thinking about secularism is less evident. Part of the reason for this uncertainty, and part of the reason it is not taken up by scholars, is doubtless the fact that Levinas’s conception of secularism is difficult to discern.2 Although he often discusses religion, politics, and even their relationship, he rarely addresses the question of secularism directly. He certainly uses the term laïcité, and, especially in Difficult Freedom, often discusses both the place of Judaism in secular society and what he calls the “Christian atmosphere” of modernity, but he never articulates, either here or elsewhere, a complete account of the concept.In addition to the paucity of Levinas’s writings on secularism, a further difficulty is that what Levinas does say about secularism can often seem contradictory and unclear. Consider, for example, his view [End Page 187] of the most basic issue of secularism: the separation of church and state. Is such a separation wholly compatible with Levinas’s thinking? On the one hand, Levinas is unambiguous about his endorsement of the separation. He writes in “How Is Judaism Possible?” (1959) that the separation “is the citadel that protects us [French Jews] against injustice,” and therefore that “we must beware of tampering with it.” And yet, as he will immediately add, the “luminous and incontestable formula” separating church and state must be given “its best content,” as if in its common understanding it does not possess its best meaning (DF 245). This specific double move, where he simultaneously endorses and criticizes the separation of church and state, is indicative of Levinas’s more general attitude from the 1950s onward toward the whole of the standard liberal conception of secularism.What I mean by the latter conception will be fleshed out in the following sections, but it can be defined now, in a preliminary fashion, as the entire liberal enterprise of creating a culturally neutral state and public sphere for the sake of protecting the rights and liberties of the individual citizen. Because the individual citizen is defined as possessing the right to, or liberty of, thought, expression, and association—all of which together, in effect, produce freedom of religion, whether explicitly proclaimed or not—the state and public sphere must be constituted in particular ways to guarantee these liberties; in particular, they must be culturally neutralized so that any citizen, from whichever cultural background, can participate in public institutions without bias and thus without the violation of their liberties. Such neutralization is achieved by the reciprocal move wherein participants in these institutions—that is to say, individual citizens—are not allowed to bring their cultural frameworks with them into the constitutions and operations of these institutions, leading to the so-called privatization of religion.Because Levinas criticizes this model of secularism while seeming to endorse some notion of it, what I would like to do in this essay is to determine what is precisely Levinas’s criticism of secularism and, in addition, what a “Levinasian secularism” might be. My central contention will be that the latter (Levinasian secularism) is not equivalent [End Page 188] to the former (his stated views about secularism). The reason for this discrepancy is that Levinas critiques standard liberal secularism on the basis of his more general philosophy of ethical alterity, but then reinstates liberal secularism (or an unspecified version of it) as the best way to protect institutionally ethical alterity. This double movement raises the question, how can ethical alterity at once be consistent and inconsistent with liberal secularism? The question is naturally related to the much-discussed one about the relation of ethics and politics in Levinas’s thinking. Levinas’s approach to the latter relation is consistently... (shrink)

In constructive mathematics it is of interest to consider a more general, but classically equivalent, notion of linear order, a so-called pseudo-order. The prime example is the order of the constructive real numbers. We examine two kinds of constructive completions of pseudo-orders: order completions of pseudo-orders and Cauchy completions of ordered groups and fields. It is shown how these can be predicatively defined in type theory, also when the underlying set is non-discrete. Provable choice principles, in particular a generalisation of (...) dependent choice, are used for showing set-representability of cuts. (shrink)

We give a new account of Björling’s contribution to uniform convergence in connection with Cauchy’s theorem on the continuity of an infinite series. Moreover, we give a complete translation from Swedish into English of Björling’s 1846 proof of the theorem. Our intention is also to discuss Björling’s convergence conditions in view of Grattan-Guinness’ distinction between history and heritage. In connection to Björling’s convergence theory we discuss the interpretation of Cauchy’s infinitesimals.

We seek to elucidate the philosophical context in which the so-called revolution of rigor in inifinitesimal calculus and mathematical analysis took place. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at that time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg Neo-Kantian philosophy formulated a sophisticated (...) position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the "great triumvirate" of Cantor, Dedekind, and Weierstrass. Expressed in terms of modern mathematics, the Marburg philosophers saw the introduction of both infinitesimals and limits as completions whose prototype was Dedekind's of the rational number system resulting in the real numbers. At least partially,, this idea of "completions" can be captured in terms of a category-theoretical description of the conceptual development of modern mathematics. The feasibility of such a modern reformuation may be taken as evidence that the philosophical resources of Marburg neo-Kantianism may be of interest even for contemporary philosophy of mathematics. (shrink)

We interpret the concept of determinism for a classical system as the requirement that the solution to the Cauchy problem for the equations of motion governing this system be unique. This requirement is generally believed to hold for all autonomous classical systems. Our analysis of classical electrodynamics in a world with one temporal and one spatial dimension provides counterexamples of this belief. Given the initial conditions of a particular type, the Cauchy problem may have an infinite set of solutions. Therefore, (...) random behavior of closed classical systems is indeed possible. With this finding, we give a qualitative explanation of how classical strings can split. We propose a modified path integral formulation of classical mechanics to include indeterministic systems. (shrink)

A constructive definition of the continuum based on formal topology is given and its basic properties studied. A natural notion of Cauchy sequence is introduced and Cauchy completeness is proved. Other results include elementary proofs of the Baire and Cantor theorems. From a classical standpoint, formal reals are seen to be equivalent to the usual reals. Lastly, the relation of real numbers as a formal space to other approaches to constructive real numbers is determined.

By splitting idempotent morphisms in the total and base categories of fibrations we provide an explicit elementary description of the Cauchy completion of objects in the categories Fib) of fibrations with a fixed base category \ and Fib of fibrations with any base category. Two universal constructions are at issue, corresponding to two fibered reflections involving the fibration of fibrations \.

We continue the investigation of generalized Bosbach states that we began in Part I, restricting our research to the commutative case and treating further aspects related to these states. Part II is concerned with similarity convergences, continuity of states and the construction of the s-completion of a commutative residuated lattice, where s is a generalized Bosbach state.

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum (...) with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others. (shrink)

Every second-countable regular topological space X is metrizable. For a given “computable” topological space satisfying an axiom of computable regularity M. Schröder [10] has constructed a computable metric. In this article we study whether this metric space can be considered computationally as a subspace of some computable metric space [15]. While Schröder's construction is “pointless”, i. e., only sets of a countable base but no concrete points are known, for a computable metric space a concrete dense set of computable points (...) is needed. But there may be no computable points in X. By converging sequences of basis sets instead of Cauchy sequences of points we construct a metric completion of a space together with a canonical representation equation image, equation image. We show that there is a computable embedding of in with computable inverse. Finally, we construct a notation of a dense set of points in with computable mutual distances and prove that the Cauchy representation of the resulting computable metric space is equivalent to equation image. Therefore, every computably regular space has a computable homeomorphic embedding in a computable metric space, which topologically is its completion. By the way we prove a computable Urysohn lemma. (shrink)

Acceptance of Augustin Fresnel's wave theory of light posed numerous questions for early nineteenth-century physicists. Among the most pressing was the problem of the properties of the luminiferous ether. Fresnel had shown that light waves were transverse. Therefore, since, among ordinary materials, only solids support transverse vibrations, there existed striking likenesses between highly tangible solids and the highly intangible ether. Accordingly, such men as Augustin-Louis Cauchy, James MacCullagh, Franz Neumann, and George Green constructed various theories of an elastic-solid ether.1 At (...) the same time, however, the disconcerting implausibilities of an all-pervasive solid provoked considerable apprehension in regard to the elastic-solid tradition. Thomas Young found the concept ‘perfectly appalling’ and argued that ‘the hypothesis [that fluids can support transverse vibrations] remains completely open for discussion, notwithstanding the apparent difficulties attending it.’2 John Herschel, probably the most important English advocate of the wave theory, regarded the concept of a solid ether as only a temporary device, useful ‘till the real truth shall be discovered.’3 Consequently, despite the accomplishments which helped to make the elastic-solid theory ‘the most celebrated special form of the wave theory’,4 there were important voices of reservation. (shrink)

Motivated by the recent wave of investigations on plane domain wall spacetimes with nontrivial topologies, the present paper deals with (probably) the most simple source field configuration which can generate a spatially planary symmetric static spacetime, namely a minimally coupled massless scalar field that depends only upon a spacelike coordinate. z. It is shown that the corresponding exact solutions (ℳ. g±) are algebraically special, type D-[S-3T] (in11), and represent globally pathologic spacetimes with a G4-group of motion acting on R2 × (...) R orbits. In spite of the model simplicity, these Φ-generated worlds possess naked timelike singularities (reached within a finite universal time by normal nonspacelike geodesics). are completely free of Cauchy surfaces, and contain, in the t-leveled sections, points that cannot be joined by C1-trajectory images of oblique non-spacelike geodesies. Finally, we comment on the possibility of deriving from (ℳ, g±) two other physically interesting Φ-generated” spacetimes by appropriate joining conditions in the (z=0)-plane. (shrink)

This manuscript investigates fixed point of single-valued Hardy-Roger’s type F -contraction globally as well as locally in a convex b -metric space. The paper, using generalized Mann’s iteration, iterates fixed point of the abovementioned contraction; however, the third axiom of the F -contraction is removed, and thus the mapping F is relaxed. An important approach used in the article is, though a subset closed ball of a complete convex b -metric space is not necessarily complete, the convergence of the Cauchy (...) sequence is confirmed in the subset closed ball. The results further lead us to some important corollaries, and examples are produced in support of our main theorems. The paper most importantly presents application of our results in finding solution to the integral equations. (shrink)

In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure - Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if computable is replaced by primitive recursive (p. r., for short), these definitions lead to a number of different (...) concepts, which we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b -adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (shrink)

In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure – Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if “computable” is replaced by “primitive recursive” , these definitions lead to a number of different concepts, which we (...) compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b -adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (shrink)

It is common in accounts of the secularization of Western thought to make reference to the name of Nietzsche. Nietzsche is undeniably a critic of religion, but he is equally a critic of the secular. It is for this reason that I propose thinking about Nietzsche’s philosophy as postsecular. This term is one that has evolved over the last couple decades in response to the so-called “return of the religious” in society, social theory, and philosophy and suggests that secularity and (...) religiosity are not antithetical and therefore that we must move beyond the forms of thinking that would regard them as such. First, I focus on his concepts of will to power, interpretation, perspectivism, and history, and sketch how they are all interrelated and how these ground his interpretation of religion. Based on this view, I show, second, that while religion is, for Nietzsche, a historical perspective, it is also, for him, part of the historical genealogy of secularity. This point leads to my final one on the relationship between secularity and religion in Nietzsche’s thought, where I argue that, on his own terms, Nietzsche’s anti-Christianity is a hyper-Christianity, even while rejecting facets of Christianity. (shrink)

In recent continental philosophy of religion there has been significant attention paid to the Abrahamic doctrines of creation ex nihilo and divine omnipotence, especially by deconstructive thinkers such as Derrida, Caputo, and Keller. For these thinkers, the doctrine represents a form of agency that does violence to various forms of alterity. While broadly supportive of their fundamental philosophical and ethico-political views, especially about the primordiality of alterity, I differ from them in that I argue that creation ex nihilo articulates the (...) very structure of the alterity they are concerned with. The essay proceeds through a reading of Derrida’s representation of the doctrine and a “deconstruction” of his view by means of a reading of Augustine and Anselm. (shrink)

La fondation d'une revue comme Dialogue s'accompagne de crainte et d'espoir. Crainte de ne pouvoir la faire vivre et progresser, crainte « inavouée » que la production philosophique ne soit pas à la hauteur de l'opinion que nous avons de nous-mêmes ou de ce que le monde extérieur attend de nous… Espoir cependant de consolider les structures d'une Association canadienne de philosophie à peine naissante, chancelante, mais vivant déjà d'un idéal exigeant et dont la revue porte le nom.