Philosophy’s Loss of Logic to Mathematics: An Inadequately Understood Take-Over

Cham, Switzerland: Springer Verlag (2018)
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Abstract

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.

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Chapters

On the Motivations of Gödel’s Ontological Proof

In recent years there has been a surge of interest in Gödel’s ontological proof of the existence of God. In spite of all this extensive concern, it is not certain whether there is any improvement in understanding the motivations of Gödel’s ontological proof. Why was Gödel so preoccupied with complet... see more

Between Bernays and Carnap

Bernays has not drawn scholarly attention that he deserves. Only quite recently, the reevaluation of his philosophy, including the projects of editing, translating, and reissuing his writings, has just started. As a part of this renaissance of Bernays studies, this chapter tries to distinguish caref... see more

Friedman on Implicit Definition: In Search of the Hilbertian Heritage in Philosophy of Science

Michael Friedman’s project both historically and systematically testifies to the importance of the relativized a priori. The importance of implicit definitions clearly emerges from Schlick’s General Theory of Knowledge . The main aim of this paper is to show the relationship between both and the rel... see more

Introduction

This is a collection of my papers on the history and philosophy of logic and mathematics published for the last thirty years. Virtually all the chapters tackle some particular logical, methodological, epistemological, and ontological issues that are not entirely clear in official history of modern l... see more

Scotus, Frege, and Bergmann

If N. Cocchiarella’s recent discussion of Frege’s function-correlate is correct, we have reason to assimilate Frege’s ontology to the Avicennian-Scotistic tripartite ontology of individuals, universals, and common natures in themselves. Further, to the extent that Scotus’ ontology is similar to Freg... see more

Zermelo and the Axiomatic Method

This chapter intends to examine the widespread assumption, which has been uncritically accepted, that Zermelo simply adopted Hilbert’s axiomatic method in his axiomatization of set theory. What is essential in that shared axiomatic method? And, exactly when was it established? By philosophical refle... see more

Ontological Regress of Maddy’s Mathematical Naturalism

This chapter is an attempt to probe the question as to why Maddy gave up mathematical realism and moved to her own version of mathematical naturalism. According to one widespread hypothesis, Maddy’s change of mind was brought up by her criticism of Quine-Putnam indispensability argument. Though quit... see more

On Cocchiarella’s Retroactive Theory of Reference

I attempt to get further insights from Cocchiarella’s history and philosophy of logic in understanding the contrast of Aristotelian and Fregean logic. Recently Cocchiarella proposed a conceptual theory of the referential and predicable concepts used in basic speech and mental acts . This theory is i... see more

Patterson on Tarski’s Definition of Logical Consequence

We still do not know against what historical/philosophical background and motivation Tarski’s definition of logical consequence was introduced, even if it has had such a strong influence. In view of the centrality of the notion of logical consequence in logic and philosophy of logic, it is rather sh... see more

Frege’s Distinction Between “Falling Under” and “Subordination”

Frege frequently complains that others are ignorant of the distinction between “falling under” and “subordination”. This criticism is not only directed against the philosophers who are under the influence of Aristotelian logic but also against the mathematicians of his time. I shall show that this d... see more

What if Haecceity Is not a Property?

In some sense, both ontological and epistemological problems related to individuation have been the focal issues in the philosophy of mathematics ever since Frege. However, such an interest becomes manifest in the rise of structuralism as one of the most promising positions in recent philosophy of m... see more

Biancani on Scientiae Mediae

We can witness the recent surge of interest in the controversy over the scientific status of mathematics among Jesuit Aristotelians around 1600. Following the lead of Wallace, Dear, and Mancosu, I propose to look into this controversy in more detail. For this purpose, I shall focus on Biancani’s dis... see more

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Woosuk Park
Korea Advanced Institute of Science & Technology (KAIST)