Enrico Martino

Temporal and atemporal truth in intuitionistic mathematics

In Sect. 11.2, we argue that the adoption of a tenseless notion of truth entails a realistic view of propositions and provability. This view, in turn, opens the way to the intelligibility of the classical meaning of the logical constants and consequently is incompatible with the antirealism of orthodox Intuitionism. In Sect. 11.3, we show how what we call the “potential” intuitionistic meaning of the logical constants can be defined, on the one hand, by means of the notion of atemporal provability and, on the other hand, by means of the operator K of epistemic logic. Intuitionistic logic, as reconstructed within this perspective, turns out to be a part of epistemic logic, so that it loses its traditional foundational role, antithetic to that of classical logic . In Sect. 11.4, we uphold the view that certain consequences of the adoption of a temporal notion of truth, despite their apparent oddity, are quite acceptable from an antirealist point of view.

Arbitrary Reference in Mathematical Reasoning

It is claimed that the ideal possibility of picking up any object of the universe of discourse is essential not only in intuitionistic but also in classical logic and mathematics.

To Be is to Be the Object of a Possible Act of Choice

Aim of the paper is to revise Boolos’ reinterpretation of second-order monadic logic in terms of plural quantification ([4], [5]) and expand it to full second order logic. Introducing the idealization of plural acts of choice, performed by a suitable team of agents, we will develop a notion of plural reference. Plural quantification will be then explained in terms of plural reference. As an application, we will sketch a structuralist reconstruction of second-order arithmetic based on the axiom of infinite à la Dedekind, as the unique non-logical axiom. We will also sketch a virtual interpretation of the classical continuum involving no other infinite than a countable plurality of individuals.

On the ontological commitment of mereology

In Parts of Classes [1991] David Lewis argues that, like logic, but unlike set theory, mereology is “ontologically innocent”. Prima facie, Lewis’ innocence thesis seems to be ambiguous. On one side, he seems to argue that, given certain objects Xs, referring to their sum is ontologically innocent because there is not a new entity as referent of the expression “the sum of the Xs”. So, talking of the sum of the Xs would simply be a different way of talking of the Xs, looking at them as a whole. However, on the other side, Lewis’ innocence is not understood as a mere linguistic use, where sums are not reified. He himself claims that the innocence of mereology is different from that of plural reference, where the reference to some objects does not require the existence of a single entity picking up them in a whole. In the case of plural quantification “we have many things, in no way do we mention one thing that is the many taken together”. Instead, in the mereological case: “we have many things, we do mention one thing that is the many taken together, but this one thing is nothing different from the many” ([1], 87). But, due to the fact that Lewis explicitly uses sums as outright objects, we think that Lewis’ innocence thesis cannot be understood but in the sense that, even if the sum of the Xs is a well determined object, distinct from the Xs, the existence of such an object is to be necessarily accepted from whom which has already accepted the existence of the Xs. In other words, committing oneself to the existence of the Xs would be an implicit commitment to some other entities and – among them – the sum of the Xs. On the other hand, the existence of the set of the Xs would not be implicitly guaranteed by the existence of the Xs. The aim of the paper is to argue that – for a certain use of mereology, weaker than Lewis’ one – an innocence thesis similar to that of plural reference is defendable. In order to give a definite account of plural reference, we use the idea of a plural choice. Then, we propose a virtual theory of mereology, where the role of individuals is played by plural choices of atoms. A choice is not an authentic object, its existence is merely potential and it consists in the act of performing it. Accordingly, in order to interpret a formal first order mereological language, as Goodman calculus of individuals (CG), we introduce a potential semantic of plural choices. We argue that our development of virtual mereology, grounded on the notion of plural choice, is ontologically innocent in a way completely analogous to that of plural reference: our claim is that mereological sums – unlike atoms – are not real objects. Referring to a sum of atoms is nothing but a way of referring to certain atoms. Our approach is adequate to interpret a first order mereological language. It is inadequate for Lewis’ mereology, because his plural quantification on all objects is incompatible with our notion of plural choice, where just atoms are capable of being chosen.

On the Brouwerian concept of negative continuity

So at first sight Theorem 0.1 seems to be superseded by Theorem 0.3; and in fact 0.1 does not occur in modern intuitionistic analysis. However we contend that Brouwer’s direct proof of 0.1 has a remarkable foundational significance. Brouwer himself observes that 0.1 is an immediate consequence of the intuitionistic point of view. This is not the case for 0.2. Brouwer says he had knowledge of 0.1 since 1918 and though this result suggested to him the conjecture of 0.2, nevertheless he did not succeed in proving 0.2 until much later. Yet, in my opinion, Brouwer’s proof of 0.1 has not been fully understood, possibly because of its rather informal and elliptical exposition. Recently several authors tried to reconstruct Brouwer’s argument: see [4, 5, 7, 91. Still, these reconstructions though in some ways interesting fail to shed light on the sense of the immediacy of 0.1 claimed by Brouwer. For instance this immediacy is completely obscured by Veldman. In fact his reconstruction rests on

Brouwer's equivalence between virtual and inextensible order

Brouwer’s theorem of 1927 on the equivalence between virtual and inextensible order is discussed. Several commentators considered the theorem at issue as problematic in various ways. Brouwer himself, at a certain time, believed to have found a very simple counterexample to his theorem. In some later publications, however, he stated the theorem in the original form again. It is argued that the source of all criticisms is Brouwer’s overly elliptical formulation of the definition of inextensible order , as well as a certain ambiguity in his terminology. Once these drawbacks are removed, his proof goes through.

Grounding Megethology on Plural Reference

In Mathematics is megethology (Lewis, Philos Math 1:3–23, 1993) Lewis reconstructs set theory combining mereology with plural quantification. He introduces megethology, a powerful framework in which one can formulate strong assumptions about the size of the universe of individuals. Within this framework, Lewis develops a structuralist class theory, in which the role of classes is played by individuals. Thus, if mereology and plural quantification are ontologically innocent, as Lewis maintains, he achieves an ontological reduction of classes to individuals. Lewis’work is very attractive. However, the alleged innocence of mereology and plural quantification is highly controversial and has been criticized by several authors. In the present paper we propose a new approach to megethology based on the theory of plural reference developed in To be is to be the object of a possible act of choice (Carrara, Stud Log 96: 289–313, 2010). Our approach shows how megethology can be grounded on plural reference without the help of mereology.

Connection between the ‘principle of inductive evidence’ and the bar theorem

I introduced the “principle of inductive evidence” PIE in my paper “Creative subject and bar theorem” (Martino 1982). Because of a misunderstanding in my correspondence with the editors, the published version of the above paper is not the final revised draft, but a first outline of the article which needs some corrections and explications. I shall refer to the published version as CS. In CS, I asserted somewhat rashly the absolute equivalence of PIE and the monotonic bar theorem \(BI_{M}\) by means of all too sketchy proof in the course of which I introduced in passing a rather problematic assumption without explaining it properly. Therefore, I shall present here a more adequate treatment of the connection between PIE and \(BI_{M}\). In fact, I shall assume acquaintance with Sects. 4.1 and Theorem 4.1 of CS and provide a revised version of Theorem 4.2.

On the Infinite in Mereology with Plural Quantification

In “Mathematics is megethology,” Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural quantification are, in some ways, particularly relevant to a certain conception of the infinite. More precisely, though the principles of mereology and plural quantification do not guarantee the existence of an infinite number of objects, nevertheless, once the existence of any infinite object is admitted, they are able to assure the existence of an uncountable infinity of objects. So, if—as Lewis maintains—MPQ were parts of logic, the implausible consequence would follow that, given a countable infinity of individuals, logic would be able to guarantee an uncountable infinity of objects. §

Classical and Intuitionistic Semantical Groundedness

Kripke’s notion of semantical groundedness for classical logic is developed in an intuitionistic framework. It is argued that semantical groundedness yields the most natural solution of the semantical paradoxes.