There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$, by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$. The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}$. A Gentzen-style reconstruction of the Church–Fitch paradox is presented (...) following a labelled approach to sequent calculi. First, a cut-free system for classical bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between ${\mathcal {K}}$ and ${\diamondsuit}$ are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism. (shrink)

The temporality of intentions and actions in situations of social interaction can sometimes be paradoxical. I argue that in these situations it may sometimes be possible to conceive of individual acts that can, in a strong sense, be intended retroactively. This could happen when the relational patterns in social interaction literally alter the virtual structure of a participant's past corporeal intentions resulting in an odd experience of having intended something all along without knowing it. I propose that this possibility should (...) be interpreted as more than just a narrowly epistemic phenomenon. Examining this claim involves clarifying the enactive perspective on intentionality, which I do here. The enactive approach rejects the model of a causal relation between intention and action for one of an intrinsic qualitative relation between the two as facets of sense-making. I develop this idea and compare it with Merleau-Ponty's Fundierung model of the mutual relation between corporeal and reflexive intentionality to show that co-regulated moves/affections during social interaction may modulate both arcs of this relation, creating the possibility of a re-signification that alters not the actuality but the virtual tendencies that preceded the social act. (shrink)

A growing body of literature documents the important role played by moral outrage or moral anger in stakeholders’ reactions to cases of corporate social irresponsibility. Existing research focuses more on the consequences of moral outrage than a systematic analysis of how appraisals of irresponsible corporate behavior can lead to this emotional experience. In this paper, we develop and test, in two field studies, an extended model of moral outrage that identifies the cognitions that lead to, and are associated with, this (...) emotional experience. This research contributes to the existing literature on reactions to corporate social irresponsibility by explaining how observers’ evaluation of irresponsible corporate behavior leads to reactions of moral anger. The paper also helps clarify the difference between moral outrage and other types of anger and offers useful insights for managers who have to confront public outrage following cases of irresponsible corporate behavior. Finally, the analysis of the causes of stakeholders’ anger at irresponsible corporations opens important avenues for future research that are presented in the paper. (shrink)

A recent article of Chernikov, Hrushovski, Kruckman, Krupinski, Moconja, Pillay, and Ramsey finds the first examples of simple structures with formulas which do not fork over the empty set but are universally measure zero. In this article we give the first known simple $\omega $ -categorical counterexamples. These happen to be various $\omega $ -categorical Hrushovski constructions. Using a probabilistic independence theorem from Jahel and Tsankov, we show how simple $\omega $ -categorical structures where a formula forks over $\emptyset $ (...) if and only if it is universally measure zero must satisfy a stronger version of the independence theorem. (shrink)

There is a place in the Hegelian system where the notion of Objectivity is explicitly articulated. One entire section in the Science of Logic is entitled ‘Objectivity’, almost suggesting that that would be the appropriate place to unveil the question of the notion of objective thought. However, precisely that section, its content and its presence have always represented a problematic knot in the Hegelian logical system. The problem is two-fold: on the one hand, the categories of this section (Mechanism, Chemism (...) and Teleology) seem to refer to the realm of the Realphilosophie, and they seem not to respect the purity of logical thought. This makes the connection between such section and the remaining Hegelian system uncertain and questionable. On the other hand, the Objectivity section does not seem to present any character which determines its logical advancement over Subjectivity. This contribution focuses on the interpretation of the origin of this section and on the logical meaning that “objectivity” has in the Logic. This paper explores the roots of the problem attempting to undo a few of its theoretical knots via a particular interpretation of the Syllogism. The underlying thesis in this analysis is in fact that the section on Objectivity originates precisely in the internal dynamics of the Syllogism. The forms of self-reflection in the Syllogism justify the presence of the section Objectivity. In this context, Objectivity does not present itself with the characters of immediacy and externality that are normally attributed to it, but it expresses the result of the syllogistic process. The second fundamental aspect of this short analysis is that Mechanism, Chemism and Teleology as categories are not directly exported from the Realphilosophie. They are rather organizational forms in the relationships among concepts. Thus, they don’t figure as abstractions from the proper structures of nature and spirit. Rather, they express the relational structures reached by thought once the syllogism has been able to manifest a yet occult rational capacity of thought. (shrink)

We associate with any abstract logic L a family F(L) consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of L. For every countably generated [ω, ω]-compact logic L, our main applications are: (i) Elementary classes of L can be characterized in terms of $\equiv_L$ only. (ii) If U and B are countable models of a countable superstable theory without the finite cover property, then $\mathfrak{U} \equiv_L \mathfrak{B}$ . (iii) There exists the "largest" logic M such (...) that complete extensions in the sense of M and L are the same; moreover M is still [ω,ω]-compact and satisfies an interpolation property stronger than unrelativized ▵-closure. (iv) If L = L ωω(Q α ) , then $\operatorname{cf}(\omega_\alpha) > \omega$ and $\lambda^\omega for all $\lambda . We also prove that no proper extension of L ωω generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning L κλ -compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics. (shrink)

In this article, I argue that the interpretation of Kant's a priori in Marburg neo-Kantianism involved a historiographical problem concerning the Platonic interpretation of the history of exact sci...

1. Filosofia in età antica -- 2. Filosofia in età medievale e rinascimentale -- 3. Filosofia in età moderna -- 4. Filosofia in età contemporanea.