Dans l'œuvre immense de Benedetto Croce (1866-1952) l'esthétique occupe une place centrale. Philosophe peu orthodoxe inspiré par les pensées de Kant et de Vico il a profondément marqué la culture italienne. Associé souvent à Gentile avec lequel il a rompu en 1925, en s'opposant au fascisme, il a réfléchi toute sa vie à une philosophie immanentiste de l'Esprit où l'esthétique trouverait sa place à côté d'une logique, d'une philosophie de la pratique et d'une économie ou philosophie de l'utile. L'influence de (...) Croce s'exerça non seulement sur les philosophes mais aussi sur des historiens de l'art comme Lionello Venturi, Carlo Ludovico Ragghianti, Julius Schlosser, Hans Sedlmayr ou encore Herbert Read et Pierre Francastel qui en discutèrent les thèses. Le Bréviaire d'esthétique (1912) rassemble quatre conférences destinées à un public nord-américain à l'invitation du " Rice Institute " qui ouvrait alors ses portes au Texas. Ces conférences jamais prononcées furent néanmoins publiées en anglais dès 1913, et marquent une étape importante dans la réflexion esthétique du philosophe napolitain qui qualifie alors l'art d'" intuition lyrique ". À ces conférences furent ajoutés plus tard deux autres textes ici traduits. Lire Croce aujourd'hui permet de revenir à la source d'un ensemble de questions auxquelles l'esthétique contemporaine est confrontée tout en découvrant une pensée qui, bien que souvent hostile à certaines manifestations de l'art de son temps (D'Annunzio ou les futuristes par exemple), restait paradoxalement ouverte à ses transformations qu'elle recommandait de saisir dans leur singularité et leur mouvement propre. D'où une conception particulière de la critique et de ses rapports avec l'histoire de l'art que l'on trouvera ici exposée. (shrink)

We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$, as $\kappa (...) $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$ ). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert’s axiom of completeness. (shrink)

In this detailed study of the republican tradition in the development of the Enlightenment, the central problem of utopia and reform is crystallized in a discussion of the right to punish. Describing the political situation in Europe in the seventeenth and eighteenth centuries, the author shows how the old republics in Italy, Poland and Holland stagnated and were unable to survive in the age of absolutism. The Philosophes discussed the ideal of republicanism against this background. They were particularly influenced by (...) the political and religious radicalism of John Toland, which had survived the English Restoration and was then reaching Europe. Professor Venturi traces the debate on the penal laws and the attempt to relate utopian ideas of society to the practical problem of dealing with man in society, which culminated in the assertion by many Philosophes that an unjust social system necessitated harsh penal laws, thereby rejecting the possibility of reform. (shrink)

Likert scales are useful for collecting data on attitudes and perceptions from large samples of people. In particular, they have become a well-established tool in soundscape studies for conducting in situ surveys to determine how people experience urban public spaces. However, it is still unclear whether the metrics of the scales are consistently interpreted during a typical assessment task. The current work aims at identifying some general trends in the interpretation of Likert scale metrics and introducing a procedure for the (...) derivation of metric corrections by analyzing a case study dataset of 984 soundscape assessments across 11 urban locations in London. According to ISO/TS 12913-2:2018, soundscapes can be assessed through the scaling of 8 dimensions: pleasant, annoying, vibrant, monotonous, eventful, uneventful, calm, and chaotic. The hypothesis underlying this study is that a link exists between correlations across the percentage of assessments falling in each Likert scale category and a dilation/compression factor affecting the interpretation of the scales metric. The outcome of this metric correction value derivation is introduced for soundscape, and a new projection of the London soundscapes according to the corrected circumplex space is compared with the initial ISO circumplex space. The overall results show a general non-equidistant interpretation of the scales, particularly on the vibrant-monotonous direction. The implications of this correction have been demonstrated through a Linear Ridge Classifier task for predicting the London soundscape responses using objective acoustic parameters, which shows significant improvement when applied to the corrected data. The results suggest that the corrected values account for the non-equidistant interpretation of the Likert metrics, thereby allowing mathematical operations to be viable when applied to the data. (shrink)

In the Tractatus, it is stated that questions about logical formatting cannot be meaningfully formulated, since it is precisely the application of logical rules which enables the formulation of a question whatsoever; analogously, Wittgenstein’s celebrated infinite regress argument on rule-following seems to undermine any explanation of deduction, as relying on a logical argument. On the other hand, some recent mathematical developments of the Curry-Howard bridge between proof theory and type theory address the issue of describing the “subjective” side of logic, (...) that is, the concrete manipulation of rules and proofs in space and time. It is advocated that such developments can shed some light on the question of logical formatting and its apparently unintelligible paradoxes, thus reconsidering Wittgenstein’s verdict. (shrink)

This paper contributes to the generalization of lattice-valued models of set theory to non-classical contexts. First, we show that there are infinitely many complete bounded distributive lattices, which are neither Boolean nor Heyting algebra, but are able to validate the negation-free fragment of \. Then, we build lattice-valued models of full \, whose internal logic is weaker than intuitionistic logic. We conclude by using these models to give an independence proof of the Foundation axiom from \.

How does our language relate to reality? This is a question that is especially pertinent in set theory, where we seem to talk of large infinite entities. Based on an analogy with the use of models in the natural sciences, we argue for a threefold correspondence between our language, models, and reality. We argue that so conceived, the existence of models can be underwritten by a weak notion of existence, where weak existence is to be understood as existing in virtue (...) of language. (shrink)

We analyze axioms and postulates as speech acts. After a brief historical appraisal of the concept of axiom in Euclid, Frege, and Hilbert, we evaluate contemporary axiomatics from a linguistic perspective. Our reading is inspired by Hilbert and is meant to account for the assertive, directive, and declarative components of modern axiomatics. We will do this by describing the constitutive and regulative roles that axioms possess with respect to the linguistic practice of mathematics.

In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how `intuitively plausible' an axiom is, whereas extrinsic justification supports an axiom by identifying certain `desirable' consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated nor (...) sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation. (shrink)

In this paper, we argue for an instrumental form of existence, inspired by Hilbert’s method of ideal elements. As a case study, we consider the existence of contradictory objects in models of non-classical set theories. Based on this discussion, we argue for a very liberal notion of existence in mathematics.

In this paper, we examine the logics of essence and accident and attempt to ascertain the extent to which those logics are genuinely formalizing the concepts in which we are interested. We suggest that they are not completely successful as they stand. We diagnose some of the problems and make a suggestion for improvement. We also discuss some issues concerning definability in the formal language.

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second order number theory is forcible (...) from some $$T\supseteq \mathsf {ZFC}+$$ large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of $$H_{\omega _1}$$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $$\Delta _0$$ -properties and for all universally Baire sets of reals. We will extend these results also to the theory of $$H_{\aleph _2}$$ in a follow up of this paper. (shrink)

In this article we analyze the method of forcing from a more philosophical perspective. After a brief presentation of this technique we outline some of its philosophical imports in connection with realism. We shall discuss some philosophical reactions to the invention of forcing, concentrating on Mostowski’s proposal of sharpening the notion of generic set. Then we will provide an overview of the notions of multiverse and the related philosophical debate on the foundations of set theory. In conclusion, we connect this (...) modern debate and Mostowski’s proposal, suggesting a way to analyze the notion of genericity within the framework of a multiverse structure. (shrink)

In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing. This technique will be used to suggest a unified perspective on generic absoluteness principles.

This article outlines a semantic approach to the logics of unknown truths, and the logic of false beliefs, using neighborhood structures, giving results on soundness, completeness, and expressivity. Relational semantics for the logics of unknown truths are also addressed, specically the conditions under which sound axiomatizations of these logics might be obtained from their normal counterparts, and the relationship between refexive insensitive logics and logics containing the provability operator as the primary modal operator.

We present a case study for the debate between the American and the Australian plans, analyzing a crucial aspect of negation: expressivity within a theory. We discuss the case of non-classical set theories, presenting three different negations and testing their expressivity within algebra-valued structures for ZF-like set theories. We end by proposing a minimal definitional account of negation, inspired by the algebraic framework discussed.

We offer tableaux systems for logics of essence and accident and logics of non-contingency, showing their soundness and completeness for Kripke semantics. We also show an interesting parallel between these logics based on the semantic insensitivity of the two non-normal operators by which these logics are expressed.

Despite their controversial ontological status, the discussion on arbitrary objects has been reignited in recent years. According to the supporting views, they present interesting and unique qualities. Among those, two define their nature: their assuming of values, and the way in which they present properties. Leon Horsten has advanced a particular view on arbitrary objects which thoroughly describes the earlier, arguing they assume values according to a sui generis modality, which he calls afthairetic. In this paper, we offer a general (...) method for defining the minimal system of this modality for any given first-order theory, and possible extensions of it that incorporate further aspects of Horsten’s account. The minimal system presents an unconventional inference rule, which deals with two different notions of derivability. For this reason and the failure of the Necessitation rule, in its full generality, the resulting system is non-normal. Then, we provide conditional soundness and completeness results for the minimal system and its extensions. (shrink)

It is prima facie uncontroversial that the justification of an assertion amounts to a collection of other (inferentially related) assertions. In this paper, we point at a class of assertions, i.e. mathematical assertions, that appear to systematically flout this principle. To justify a mathematical assertion (e.g. a theorem) is to provide a proof—and proofs are sequences of directives. The claim is backed up by linguistic data on the use of imperatives in proofs, and by a pragmatic analysis of theorems and (...) their proofs. Proofs, we argue, are sequences of instructions whose performance inevitably gets one to truth. It follows that a felicitous theorem, i.e. a theorem that has been correctly proven, is a persuasive theorem. When it comes to mathematical assertions, there is no sharp distinction between illocutionary and perlocutionary success. (shrink)

In this paper we study a new operator of strong modality ⊞, related to the non-contingency operator ∆. We then provide soundness and completeness theorems for the minimal logic of the ⊞-operator.

In this article I propose to look at set theory not only as a founda­tion of mathematics in a traditional sense, but as a foundation for mathemat­ical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some example of this use (...) of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results and discussing some recent results that show how it is possible to “complete” the structures H(ℵ1) and H(ℵ2). Then I will argue that a set theoretical foundation of mathematics can be relevant also for the philosophy of mathematical practice, as long as some axioms of set theory can be seen as explanations of mathematical phenomena. In the end I will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundation of mathematics. (shrink)

In this article I propose to look at set theory not only as a founda­tion of mathematics in a traditional sense, but as a foundation for mathemat­ical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some example of this use (...) of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results and discussing some recent results that show how it is possible to “complete” the structures H(ℵ1) and H(ℵ2). Then I will argue that a set theoretical foundation of mathematics can be relevant also for the philosophy of mathematical practice, as long as some axioms of set theory can be seen as explanations of mathematical phenomena. In the end I will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundation of mathematics. (shrink)

: In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one (...) of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution. (shrink)

The present work is focused on a specific question: is time permanent or not in Kant’s first Critique? I tried to find out my answer on the papers written by authors from Anglo-American tradition, who don’t seem so interested in this topic. Indeed, the former problem was to find remarks about it. Then I highlighted two different points of view: ones think that time is permanent. They tried to warrant this, using especially what Kant says about it in the second (...) edition section dedicated to first Analogy. Others think that time flows, in particular when it’s considered as medi- ate external sense. Finally, I tried to explain why, on my point of view, it’s possible to find in Kant this two contrary position and how it is possible to get out from this situation. (shrink)

ABSTRACT We present and discuss a change in the introduction of Hilbert’s Grundlagen der Geometrie between the first and the subsequent editions: the disappearance of the reference to the independence of the axioms. We briefly outline the theoretical relevance of the notion of independence in Hilbert’s work and we suggest that a possible reason for this disappearance is the discovery that Hilbert’s axioms were not, in fact, independent. In the end we show how this change gives textual evidence for the (...) connection between the notions of independence and simplicity. (shrink)