Axiomathes

A Note on Carnap’s Result and the Connectives Abstract Carnap’s result about classical proof-theories not ruling out non-normal valuations of propositional logic formulae has seen renewed philosophical interest in recent years. In this note I contribute some considerations which may be helpful in its philosophical assessment. I suggest a vantage point from which to see the way in which classical proof-theories do, at least to a considerable extent, encode the meanings of the connectives (not by determining a range of admissible valuations, but in their own way), and I demonstrate a kind of converse to Carnap’s result.

Fungibility in Quantum Sets Abstract It can be intuitively understood that sets and their elements in mathematics reflect the atomistic way of thinking in physics: Sets correspond to physical properties, and their elements correspond to particles that have these properties. At the same time, quantum statistics and quantum field theory strongly support the view that quantum particles are not individuals. Some of the problems faced in modern physics may be caused by such discrepancy between set theory and physical theory. The question then arises: Is it possible to reconstruct the concept of set as a collection of objects that model quantum particles rather than as a mere collection of individuals? David Deutsch has argued that identical entities can be diverse in their attributes, and that this nature, what he calls fungibility, must lie at the heart of quantum physics. In line with this idea, a set theory with fungible elements is established, and the collection of such sets is shown to be endowed with an ortholattice structure, which is better known as quantum logic.

Two Formal Interpretations of Bolzano’s Theory of Substances and Adherences Abstract Our research concerns a formal representation of Bolzano’s original concepts of Substanz and Adhärenz. The formalized intensional theory enables to articulate a question about the consistency of a part of Bolzano’s metaphysics and to suggest an answer to it in terms of contemporary model theory. The formalism is built as an extension of Zalta’s theory of abstract objects, describing two types of predication, viz. attribution and representation. Bolzano was aware about this distinction. We focus on the consistency of this formalism and the description of its semantics. Firstly, we explore the possibility to reconstruct a Russellian antinomy based on the concept of the Bolzano’s Inbegriff of all adherences. (Bolzano’s theory of ideas is often suspected of antinomial consequences.) Our aim is to show limitations of his theory that prevent a contradiction when the Inbegriff consists of non-self-referential adherences. Next, we discuss two competing semantics for the proposed theory: Scott’s and Aczel’s semantics. The first one yields a problematic result, that there are no models for the considered theory, containing a non-empty collection of all adherences. This is due to the fact that Scott’s structures verify the formula on reloading abstracts in extensional contexts. We show that Aczel’s semantics does not contain this difficulty. There are described Aczel’s models with a non-empty set of all adherences. The self-referentiality of such a collection becomes irrelevant here. Finally, we show that there are Aczel’s structures verifying the formula on reloading abstracts and we exclude them from the class of models intended for our theory.

On Why Mathematics Can Not be Ontology Abstract The formalism of mathematics has always inspired ontological theorization based on it. As is evident from his magnum opus Being and Event, Alain Badiou remains one of the most important contemporary contributors to this enterprise. His famous maxim—“mathematics is ontology” has its basis in the ingenuity that he has shown in capitalizing on Gödel’s and Cohen’s work in the field of set theory. Their work jointly establish the independence of the continuum hypothesis from the standard axioms of Zermelo–Fraenkel set theory, with Gödel’s result showing their consistency to the affirmative, while Cohen’s showing it to the negative. These results serve as the cornerstone of Badiou’s mathematical ontology. In it, drawing heavily on Cohen’s technically formidable method of forcing, Badiou makes the latter result the key to his defense of the possibility of a faithful tracing of the consequences in the ‘state’ of an ‘event’ by a ‘subject’. Whereas, Gödel’s result based on the assumption of constructability becomes the pivot for criticism of the general philosophical orientation that Badiou calls ‘constructivism’. Viewed from a position internal to mathematical formalism itself, and taking into account the twentieth century developments in the relevant field, Badiou’s stance seems to be neither appreciative of the actual course of such developments, nor just to the philosophical view point that was actually maintained by Gödel. In the present paper, this concern is intended to be substantiated through an exposition of certain facts pertaining to the said developments as well as to Gödel’s philosophical inclinations.

Mind/Brain and Economic Behaviour: For a Naturalised Economics Abstract Neuroeconomics is a science pledged to tracing the neurobiological correlates involved in decision-making, especially in the case of economic decisions. Despite representing a recent research field that is still identifying its research objects, tools and methods, its epistemological scope and scientific relevance have already been openly questioned by several authors. Among these critics, the most influential names in the debate have been those of Faruk Gul and Wolfgang Pesendorfer, who claim that the data on neural activity cannot find place in economic models, which should on the contrary be solely based on the data produced by choices. This paper aims at countering the gloomy and unsubstantiated claims of these two authors and those who believe that neuroscience cannot provide new and useful insights to the established knowledge of standard economics. The main point stressed here is that this perception is the product of a general misunderstanding of the advances made by neuroscience, which are incidentally of crucial importance.

Computational Construction of the Reality: Abstraction and Exploration-Driven Strategies in Constructing Protein–Protein Interfaces Abstract Computational modeling is one of the primary approaches to constructing protein–protein interfaces in the laboratory. The algorithm-driven computational protein design has been successfully applied to the construction of functional proteins with improved binding affinity and increased thermostability. It is intriguing how a computational protein modeling approach can construct and shape the reality of new functional proteins from scratch. I articulate an account of abstraction and exploration-driven strategies in this computational endeavor. I aim to show that how a computational modelling approach, which is laden with mathematics and algorithms, can have a constructive force on the target protein.

Einstein’s 1905 ‘Annus Mirabilis’: Reconciliation of the Basic Research Traditions of Classical Physics Abstract To make out in what way Einstein’s manifold 1905 ‘annus mirabilis’ writings hang together one has to take into consideration Einstein’s strive for unity evinced in his persistent attempts to reconcile the basic research traditions of classical physics. Light quanta hypothesis and special theory of relativity turn out to be the contours of a more profound design, mere milestones of implementation of maxwellian electrodynamics, statistical mechanics and thermodynamics reconciliation programme. The conception of luminiferous ether was an insurmountable obstacle for Einstein’s statistical thermodynamics in which the leading role was played by the light quanta paper. In his critical stand against the entrenched research traditions of classical physics Einstein was apparently influenced by David Hume and Ernst Mach. However, when related to creative momenta, Einstein’s 1905 unificationist modus operandi was drawn upon Mach’s principle of economy of thought taken in the context of his ‘instinctive knowledge’ doctrine and with promising inclinations of Kantian epistemology presuming the coincidence of both constructing theory and integrating intuition of Principle.

Embedded Pragmatic Effects and Conversational Implicatures Abstract In a recent paper Mandy Simons has argued that in a Gricean framework there is room for embedded pragmatic effects. One of her goals has been to demonstrate that an argument put forward by François Recanati to the effect that it is not possible to apply Gricean reasoning to generation of local pragmatic effects is mistaken. In his commentary Recanati maintains that the view suggested by Simons can be called Gricean only in a very broad sense and insists that the process responsible for local effects is essentially different from the one which generates conversational implicatures. In my view their exchange highlights important issues concerning the way in which local pragmatic effects are generated and is worthy of a careful analysis. In what follows I critically examine Simons’s and Recanati’s views and then suggest the view that in the light of this analysis seems to me the most adequate. In particular, I argue—against Recanati—that during the interpretation process interpreters construct literal propositional nuclei, which usually fell short of being truth-evaluable but which play an important role in the interpretation. I claim that the view that assumes that such literal propositions are constructed in the process of interpretation is more universal and may be used to analyse a wider range of examples than the view that does not postulate such constructions. Nevertheless, I maintain—against Simons—that the global and local pragmatic inferences are importantly different.

How to Overcome Antirealists’ Objections to Scientific Realism Abstract Van Fraassen contends that there is no argument that rationally compels us to disbelieve a successful theory, T. I object that this contention places upon him the burden of showing that scientific antirealists’ favorite arguments, such as the pessimistic induction, do not rationally compel us to disbelieve T. Van Fraassen uses the English view of rationality to rationally disbelieve T. I argue that realists can use it to rationally believe T, despite scientific antirealists’ favorite arguments against T.

Epistemology and the History of Science: The Problem of Historical Epistemology in the Italian Debate of the Twentieth Century Abstract The essay, based also on unpublished writings, analytically reconstructs the Italian debate concerning the problem of historical epistemology and of the different relationships that can be established between epistemological reflection and the history of science. We start from awareness, à la Lakatos, that a “history of science without philosophy of science is blind, while a philosophy of science without the history of science is empty”. However, during the twentieth century Italian different theoretical positions emerged. Giulio Preti began by underlining how the history of science should be understood as the history of scientific thought. This position was close to that expressed by Giovanni Gentile for whom the history of science had to be reduced to the history of philosophy. Against this neo-realist claim, an epistemologist like Ludovico Geymonat reacted by underlining how science has its own history as a science. The critical debate between Preti and Geymonat has finally led the first to underline how the history of science must then be articulated in different conceptual traditions, while the latter ended up sharing the need to study the history of science as a history of scientific thought.