An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...) are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

This book offers a historical explanation of important philosophical problems in logic and mathematics, which have been neglected by the official history of modern logic. It offers extensive information on Gottlob Frege’s logic, discussing which aspects of his logic can be considered truly innovative in its revolution against the Aristotelian logic. It presents the work of Hilbert and his associates and followers with the aim of understanding the revolutionary change in the axiomatic method. Moreover, it offers useful tools to understand (...) Tarski’s and Gödel’s work, explaining why the problems they discussed are still unsolved. Finally, the book reports on some of the most influential positions in contemporary philosophy of mathematics, i.e., Maddy’s mathematical naturalism and Shapiro’s mathematical structuralism. Last but not least, the book introduces Biancani’s Aristotelian philosophy of mathematics as this is considered important to understand current philosophical issue in the applications of mathematics. One of the main purposes of the book is to stimulate readers to reconsider the Aristotelian position, which disappeared almost completely from the scene in logic and mathematics in the early twentieth century. (shrink)

This book is an exploration and defense of the coherence of classical theism’s doctrine of divine aseity in the face of the challenge posed by Platonism with respect to abstract objects. A synoptic work in analytic philosophy of religion, the book engages discussions in philosophy of mathematics, philosophy of language, metaphysics, and metaontology. It addresses absolute creationism, non-Platonic realism, fictionalism, neutralism, and alternative logics and semantics, among other topics. The book offers a helpful taxonomy of the wide range of options (...) available to the classical theist for dealing with the challenge of Platonism. It probes in detail the diverse views on the reality of abstract objects and their compatibility with classical theism. It contains a most thorough discussion, rooted in careful exegesis, of the biblical and patristic basis of the doctrine of divine aseity. Finally, it challenges the influential Quinean metaontological theses concerning the way in which we make ontological commitments. (shrink)

A philosophical position may be characterized in different ways. Here we try to say how the naturalist answers certain . The questions come from very different areas; the spectrum of subjects is therefore quite mixed. There are, however, aspects of order: We start with (questions about) abstract subjects like logic, mathematics, metaphysics, then turn to problems of realism. And since in general naturalists are realists, the following questions on truth, laws of nature, origin of the universe, cosmology, evolution, body-mind-problem, freedom (...) of the will, religion, moral, and pseudoscience, are answered against a realistic background. German Eine philosophische Haltung kann man auf verschiedene Weisen charakterisieren. Hier versuchen wir zu sagen, wie der Naturalist auf bestimmte Kernfragen antwortet. Die Fragen stammen aus sehr verschiedenen Gebieten; das Themenspektrum ist deshalb recht bunt. Es gibt aber durchaus Ordnungsgesichtspunkte: Wir beginnen mit (Fragen zu) eher abstrakten Gegenständen wie Logik, Mathematik, Metaphysik, kommen dann zu Problemen des Realismus. Und da Naturalisten im Allgemeinen Realisten sind, werden die anschließenden Fragen über Wahrheit, Naturgesetze, Weltentstehung, Kosmologie, Evolution, Leib-Seele-Problem, Willensfreiheit, Religion, Moral und Pseudowissenschaften vor einem realistischen Hintergrund beantwortet. (shrink)

The goal of this paper is to explain the nature of philosophy as a distinct science with its own subject-matter. This is achieved through a comparative analysis of mathematical and philosophical knowledge that reveals a profound similarity between mathematics and philosophy as mutually complementary sciences exploring the field of abstract entities that can be comprehended only by purely a priori theoretical inquiry. By considering this complementarity, a general definition of philosophy can be obtained by dualizing the traditional Aristotelian definition of (...) mathematics as the “science of quantity”. Philosophy should thus be interpreted as an a priori science of the pure qualitative attributes of being. (shrink)

This paper argues that in mathematical practice, conjectures are sometimes confirmed by “Inference to the Best Explanation” as applied to some mathematical evidence. IBE operates in mathematics in the same way as IBE in science. When applied to empirical evidence, IBE sometimes helps to justify the expansion of scientists’ ontological commitments. Analogously, when applied to mathematical evidence, IBE sometimes helps to justify mathematicians' in expanding the range of their ontological commitments. IBE supplements other forms of non-deductive reasoning in mathematics, avoiding (...) obstacles sometimes faced by enumerative induction or hypothetico-deductive reasoning. Both platonist and non-platonist interpretations of mathematics ought to accommodate explanation in mathematics and ought to recognize IBE in mathematics, though these interpretations disagree on the ontological commitments that mathematicians ought to have. This paper offers an inductive account of why mathematical IBE tends to lead to mathematical truths. (shrink)

In the Grundlagen , Frege offers eight main arguments, together with a series of more minor supporting arguments, against Mill’s view that numbers are “properties of external things”. This paper reviews all eight of these arguments, arguing that none are conclusive.

The starting point of the following considerations is a case study concerning the discovery of Halley's comet and the theoretical description of its path. It is shown that the set of objects involved in that system and the time interval during which their paths are observed are determined in a theory dependent way – thereby making use of the very theory later used for that system's theoretical description. Metatheoretical consequences this fact has with respect to the structuralist view of empirical (...) theories are discussed in detail. German Den Ausgangspunkt der folgenden Überlegungen bildet eine Fallstudie, die der Entdeckung und theoretischen Beschreibung des Halleyschen Kometen gewidmet ist. Es wird gezeigt, dass bei diesem System die Menge der involvierten Objekte und die Zeitintervalle, über die sich die Beobachtung der Bahnen dieser Objekte erstreckt, in theorieabhängiger Weise bestimmt werden – und zwar in Abhängigkeit von genau derjenigen Theorie, die anschließend zur theoretischen Beschreibung des Systems eingesetzt wird. Metatheoretische Konsequenzen, die sich aus diesem Umstand insbesondere für das strukturalistische Theorienkonzept ergeben, werden im Detail analysiert. (shrink)

The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.

Critical notice of C. Pincock's Mathematics and Scientific Representation (2012).

We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the (...) brain can transform and organize its perceptual intake. It is not necessary for a geometrical form to be perfectly instantiated in order for perception of such a form to be the basis of a geometrical concept. (shrink)

This paper aims to provide a basic explanation of existence, fundamental aspects of reality, and consciousness. Existence in its most general sense is identified with the principle of logical consistency: to exist means to be logically consistent. The essence of the principle of logical consistency is that every thing is what it is and is not what it is not. From this principle follows the existence of intrinsic, indescribable identities of things and relations between them. There are three fundamental, logically (...) necessary relations: similarity, composition and instantiation. Set theory, mathematics, logic and science are presented as relational descriptions of reality. Qualities of consciousness (qualia) are identified with intrinsic identities of things or at least a certain subset of them, especially in the context of a dynamic form of organized complexity. (shrink)