Eduardo Castro

Some recent literature [Hicks, M. T. and van Elswyk. P., (2015) pp. 433-443, 2015; Bhogal, H. (2017), pp. 447-460] has argued that the non-Humean conceptions of laws of nature have a same weakness as the Humean conceptions of laws of nature. That is, both conceptions face an explanatory circularity problem. The argument is as follows: the Humean and the non-Humean conceptions of laws of nature agree that the law statements are universal generalisations; thus, both conceptions are vulnerable to an explanatory circularity problem between the laws of nature and their instances. In this paper, I argue that Armstrong’s necessitarian view of laws of nature is invulnerable to this explanatory circularity problem.

State of art paper on the topic laws of nature, around the problem of identification what is to be a law of nature. The most prominent theories of contemporary philosophical literature are discussed and analysed, such as: the simple regularity theory, from Hume; the Mill-Ramsey-Lewis best systems theory; the Dretske-Tooley-Armstrong theory of laws as relations among universals; Ellis’s essentialist theory; Cartwright’s theory of laws as ceteris paribus laws; the anti-reductionist theories of Lange, Maudlin and Carroll, the anti-realist theories of Mumford, van Fraassen and Giere; etc.

Opinionated state of the art paper on mathematical explanation. After a general introduction to the subject, the paper is divided into two parts. The first part is dedicated to intra-mathematical explanation and the second is dedicated to extra-mathematical explanation. Each of these parts begins to present a set of diverse problems regarding each type of explanation and, afterwards, it analyses relevant models of the literature. Regarding the intra-mathematical explanation, the models of deformable proofs, mathematical saliences and the demonstrative structure of mathematical induction are addressed. Regarding the extra-mathematical explanation, modal, abstract and deductive models are addressed.

I propose a deductive-nomological model for mathematical scientific explanation. In this regard, I modify Hempel’s deductive-nomological model and test it against some of the following recent paradigmatic examples of the mathematical explanation of empirical facts: the seven bridges of Königsberg, the North American synchronized cicadas, and Hénon-Heiles Hamiltonian systems. I argue that mathematical scientific explanations that invoke laws of nature are qualitative explanations, and ordinary scientific explanations that employ mathematics are quantitative explanations. I analyse the repercussions of this deductivenomological model on causal explanations.

The Benacerraf’s problem is a problem about how we can attain mathematical knowledge: mathematical entities are entities not located in space-time; we exist in spacetime; so, it does not seem that we could have a causal connection with mathematical entities in order to attain mathematical knowledge. In this paper, I propose a solution to the Benacerraf’s problem supported by the Quinean doctrines of naturalism, confirmational holism and postulation. I show that we have empirical knowledge of centres of mass and of entities outside of our light cone and that these entities are inefficacious causality entities, at least,with us. At the end, I defend the existential knowledge of centres of mass and of entities outside of our light cone against the Eleatic principle of Cheyne that we only could attain existential knowledge of entities by a causal connection.

This discussion paper is a reply to Stathis Psillos’ paper “Induction and Natural Necessities” :327–340, (2017), published in this journal. In that paper, he attempts to refute David Armstrong’s solution to the problem of induction. To accomplish this desideratum, he proposes that the best explanation for our observed regularities is a sort of “best before date” necessity. That is, necessary connections may break down and are not by default timeless. He develops arguments against my :67–82, (2014) defence of the necessitarian solution regarding a previous paper by Beebee :504–527, 2011.. He alleges that best before date necessity is no worse than timeless necessity; his proposal does not imply any further inductive generalisation to timeless necessity; and inductive inferences are justified. In this discussion paper, I provide arguments against these three claims.

Mathematical proofs aim to establish the truth of mathematical propositions by means of logical rules. Some recent literature in philosophy of mathematics alleges that some mathematical proofs also reveal why the proved mathematical propositions are true. These mathematical proofs are called explanatory mathematical proofs. In this paper, I present and discuss some salient problems around mathematical explanation: the existence problem, the normative problem, the explanandum problems of truth value and psychological value, the logical structure problem, the regress problem and the modelling problem. At the end, I sum up two contemporary models for mathematical explanation – the deductive-nomological model and the model of Steiner. I analyse these models against the previous problems.

Abstract: This paper defends David Armstrong’s solution to the problem of inductionb against Helen Beebee’s attack on that solution. To solve theproblem of induction, Armstrong contends that the timeless necessity explanation is the best explanation of our observed regularities, whereas Beebee attempts to demonstrate that the time-limited necessity explanation is an equally good explanation. Allegedly, this explanation blocks Armstrong’s solution. I demonstrate that even if the time-limited ecessity explanation were an equally good explanation of our observed regularities, this explanation does not block Armstrong’s solution. I argue that, in fact, the timeless necessity explanation is a better explanation of our observed regularities than is the time-limited necessity explanation.

This is a dissertation of philosophy of mathematics, in the analytical tradition, about the Quine-Putnam mathematical indispensability argument, that we ought to have ontological commitment to mathematical entities that are indispensable to our best scientific theories. It is an argument for the metaphysical mathematical realism supported by Quinean doctrines such as naturalism and holism. My overall aim is to make a discussion of the argument. The argument will be defended against generic objections or some of its detractors such as Azzouni, Maddy, Cheyne and Balaguer. Matters connected with the indispensability of mathematics, such as the epistemic problem of Benacerraf and the agnosticist view of Balaguer, will be discussed. Primarily, the discussion is ontological; secondarily, the discussion is epistemological and metaphysical.

State of the art paper on the problem of induction: how to justify the conclusion that ‘all Fs are Gs’ from the premise that ‘all observed Fs are Gs’. The most prominent theories of contemporary philosophical literature are discussed and analysed, such as: inductivism, reliabilism, perspective of laws of nature, rationalism, falsificationism, the material theory of induction and probabilistic approaches, according to Carnap, Reichenbach and Bayesianism. In the end, we discuss the new problem of induction of Goodman, raised by the grue predicate.

This paper is a reply to Benjamin Smart’s : 319–332, 2013) recent objections to David Armstrong’s solution to the problem of induction : 503–511, 1991). To solve the problem of induction, Armstrong contends that laws of nature are the best explanation of our observed regularities, where laws of nature are dyadic relations of necessitation holding between first-order universals. Smart raises three objections against Armstrong’s pattern of inference. First, regularities can explain our observed regularities; that is, universally quantified conditionals are required for explanations. Second, if Humean’s pattern of inference is irrational, then Armstrong’s pattern of inference is also irrational. Third, universal regularities are the best explanation of our observed regularities. I defend Armstrong’s solution of induction, arguing against these three claims.

State of the art paper on the topic realism/anti-realism. The first part of the paper elucidates the notions of existence and independence of the metaphysical characterization of the realism/anti-realism dispute. The second part of the paper presents a critical taxonomy of the most important positions and doctrines in the contemporary literature on the domains of science and mathematics: scientific realism, scientific anti-realism, constructive empiricism, structural realism, mathematical Platonism, mathematical indispensability, mathematical empiricism, intuitionism, mathematical fictionalism and second philosophy.

This paper defends the Quine-Putnam mathematical indispensability argument against two objections raised by Penelope Maddy. The objections concern scientific practices regarding the development of the atomic theory and the role of applied mathematics in the continuum and infinity. I present two alternative accounts by Stephen Brush and Alan Chalmers on the atomic theory. I argue that these two theories are consistent with Quine’s theory of scientific confirmation. I advance some novel versions of the indispensability argument. I argue that these new versions accommodate Maddy’s history of the atomic theory. Counter-examples are provided regarding the role of the mathematical continuum and mathematical infinity in science.