Another striking deviation with regard to philosophical tradition consists in the fact that contemporary schools in the philosophy of mathematics, with the exception again of Brouwer's intuitionism, hardly ever refer to mathematical thought.

Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the (...) concept of intuition and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. (shrink)

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.

In contributing a foreword to this book I am complying with a wish my husband expressed a few days before his death. He had completed the manuscript of this work, which may be considered a companion volume to his book Formal Methods. The task of seeing it through the press was undertaken by Mr. J. J. A. Mooij, acting director of the Institute for Research in Foundations and the Philosophy of Science (Instituut voor Grondslagenonderzoek en Filoso:fie der Exacte Wetenschappen) of (...) the University of Amsterdam, with the help of Mrs. E. M. Barth, lecturer at the Institute. I wish to thank Mr. Mooij and Mrs. Barth most cordially for the care with which they have acquitted themselves of this delicate task and for the speed with which they have brought it to completion. I also wish to express my gratitude to Miss L. E. Minning, M. A., for the helpful advice she has so kindly given to Mr. Mooij and Mrs. Barth during the proof reading. C. P. C. BETH-PASTOOR VII PREFACE A few years ago Mr. Horace S. (shrink)

The first history of postwar mathematics, offering a new interpretation of the rise of abstraction and axiomatics in the twentieth century. Why did abstraction dominate American art, social science, and natural science in the mid-twentieth century? Why, despite opposition, did abstraction and theoretical knowledge flourish across a diverse set of intellectual pursuits during the Cold War? In recovering the centrality of abstraction across a range of modernist projects in the United States, Alma Steingart brings mathematics back into the conversation about (...) midcentury American intellectual thought. The expansion of mathematics in the aftermath of World War II, she demonstrates, was characterized by two opposing tendencies: research in pure mathematics became increasingly abstract and rarified, while research in applied mathematics and mathematical applications grew in prominence as new fields like operations research and game theory brought mathematical knowledge to bear on more domains of knowledge. Both were predicated on the same abstractionist conception of mathematics and were rooted in the same approach: modern axiomatics. For American mathematicians, the humanities and the sciences did not compete with one another, but instead were two complementary sides of the same epistemological commitment. Steingart further reveals how this mathematical epistemology influenced the sciences and humanities, particularly the postwar social sciences. As mathematics changed, so did the meaning of mathematization. Axiomatics focuses on American mathematicians during a transformative time, following a series of controversies among mathematicians about the nature of mathematics as a field of study and as a body of knowledge. The ensuing debates offer a window onto the postwar development of mathematics band Cold War epistemology writ large. As Steingart's history ably demonstrates, mathematics is the social activity in which styles of truth-here, abstraction-become synonymous with ways of knowing. (shrink)

Needless to say, Charles Parsons’s long awaited book1 is a must-read for anyone with an interest in the philosophy of mathematics. But as Parsons himself says, this has been a very long time in the writing. Its chapters extensively “draw on”, “incorporate material from”, “overlap considerably with”, or “are expanded versions of” papers published over the last twenty-five or so years. What we are reading is thus a multi-layered text with different passages added at different times. And this makes for (...) a rather bumpy read. There is another route Parsons could have taken: he could have reprinted the relevant papers with postscripts, and then top-and-tailed the collection with a preface and added concluding reflections. It must sound very ungrateful, but I rather suspect that that might have worked better. Much of the book is about arithmetic. But Parsons has woven into the discussion claims about mathematics more generally and about set theory in particular. We might well have a basic worry about this structure: for do defensible claims about the ontology and epistemology of arithmetic have to be generalizable to apply to more infinitary mathematics? For example, suppose you are attracted to a Hellman-like modal structuralist account of arithmetic: then should you think it a problem if you suspect that such an account can’t readily be extended to cope e.g. with set theory (since you boggle at the idea of possible worlds free of abstracta but with enough structure to somehow model ZFC)? Parsons himself seems to waver over such questions. So for present purposes, I’ll focus just on arithmetic, and in what follows I’ll revisit two of the most familiar Parsonian themes, his views on structuralism as an account of the ontology of arithmetic, and his exploration of the role of intuition in grounding arithmetical knowledge. (shrink)

This is a translation of Jacob Klein's study "Die Griechische Logistik und die Entstehung der Algebra" which appeared in 1934-1936. His principal thesis is that the Renaissance mathematicians of the sixteenth century did not simply continue the work of the Greek and Arab mathematicians but in the process of developing ancient mathematics introduced a radically new conception of number which has since guided modern mathematical thought. The central figure in this revolution is Vieta. Klein traces the influence of (...) Vieta's ideas upon Stevin, Descartes, Wallis, and other figures of the scientific revolution, after discussing the conception of number and arithmetic in Plato, Aristotle, and other Greek sources. Persons reading this book with a primary interest in the philosophical ideas involved will be frustrated by the mass of historical detail which often obscures rather than illuminates the philosophical issues. But the book deserves its reputation as an important historical study.--R. H. K. (shrink)

This long-awaited volume is a must-read for anyone with a serious interest in philosophy of mathematics. The book falls into two parts, with the primary focus of the first on ontology and structuralism, and the second on intuition and epistemology, though with many links between them. The style throughout involves unhurried examination from several points of view of each issue addressed, before reaching a guarded conclusion. A wealth of material is set before the reader along the way, but a reviewer (...) wishing to summarize the author's views crisply will be frustrated. The chapter-by-chapter survey below conveys at best a very incomplete and imperfect impression of the work's virtues, and even of its contents, falling short even of supplying a full menu for the banquet of food for thought that Parsons serves up to his readers. (shrink)

With reference to an already existing and relatively widespread use of the expression in question, mathematical “thought experiments” (“TEs”) involve mathematical reasoning in which visualisation plays a relatively more important role. But to ensure an unambiguous and consistent use of the term, certain conditions have to be met: (1) Contrary to what has happened so far in the literature, the distinction between logical-formal thinking and experimental-operational thinking must not be ignored; (2) The separation between the context of (...) discovery and the context of justification is to be rejected, at least in one of the main senses in which it was defended by the logical empiricists and Popper (this excludes any position which, ascribing to mathematical TEs only a heuristic role, regards them an intermediate step to attain more traditional forms of rigour); (3) The distinction between mathematical TEs and formal proofs must be regarded as one of degree, and not as a qualitative one, although this distinction may be used in a de facto way for particular or local purposes. (shrink)

A uniquely integrative work, this volume provides a much needed compilation of primary source material to researchers from basic neuroscience, psychology, developmental science, neuroimaging, neuropsychology and theoretical biology. * The ...

Since the beginning of the twentieth century, prominent authors including Jean Piaget have drawn attention to Edmond Goblot’s account of mathematical thought experiments. But his contribution to today’s debate has been neglected so far. The main goal of this article is to reconstruct and discuss Goblot’s account of logical operations (the term he used for thought experiments in mathematics) and its interpretation by Piaget against the theoretical background of two open questions in today’s debate: (1) the relationship (...) between empirical and mathematical thought experiments and (2) the question of whether mathematical thought experiments can play a justificatory function in proofs. The main corollary of this analysis is that Piaget’s interpretation is seriously flawed and insufficiently appreciative of important theses of Goblot’s account. First, Goblot can be easily defended against Piaget’s main criticism, and second, Goblot developed ideas about mathematical thought experiments that still deserve attention. (shrink)

This long-awaited volume is a must-read for anyone with a serious interest in\nphilosophy of mathematics. The book falls into two parts, with the primary focus of\nthe first on ontology and structuralism, and the second on intuition and\nepistemology, though with many links between them. The style throughout involves\nunhurried examination from several points of view of each issue addressed, before\nreaching a guarded conclusion. A wealth of material is set before the reader along\nthe way, but a reviewer wishing to summarize the author’s views (...) crisply will be\nfrustrated. The chapter-by-chapter survey below conveys at best a very incomplete\nand imperfect impression of the work’s virtues, and even of its contents, falling\nshort even of supplying a full menu for the banquet of food for thought that Parsons\nserves up to his readers. (shrink)

Since the beginning of the twentieth century, prominent authors including Jean Piaget have drawn attention to Edmond Goblot’s account of mathematical thought experiments. But his contribution to today’s debate has been neglected so far. The main goal of this article is to reconstruct and discuss Goblot’s account of logical operations (the term he used for thought experiments in mathematics) and its interpretation by Piaget against the theoretical background of two open questions in today’s debate: (1) the relationship (...) between empirical and mathematical thought experiments and (2) the question of whether mathematical thought experiments can play a justificatory function in proofs. The main corollary of this analysis is that Piaget’s interpretation is seriously flawed and insufficiently appreciative of important theses of Goblot’s account. First, Goblot can be easily defended against Piaget’s main criticism, and second, Goblot developed ideas about mathematical thought experiments that still deserve attention. (shrink)

The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no matter whether the (...) proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)

This paper examines the role of the notion of structure in Leibniz’s mathematical thought. I show (i) how Leibniz’s structuralist understanding of truth and reason conditions his methodological formalism; (ii) how Leibniz’s prioritization of arithmetic over geometry is founded on his relational and structural conception of number; (iii) how his mathematics of infinity, i. e. his calculus, relies on the structural traits of infinity; and lastly, (iv) how the concept of structure is the indispensable partner of the concept (...) of expression that Leibniz considers universal and metaphysically fundamental. I conclude that the power and innovative character of his mathematical thought stems from a mathematical structuralism that is supported by a structural conception of truth and reason, and an even more fundamental metaphysical structuralism. (shrink)

This paper provides an analysis of the use of axioms in Banach’s Ph.D. and their role in the progression of Banach’s mathematical thought. In order to give a precise account of the role of Banach’s axioms, we distinguish two levels of activity. The first one is devoted to the overall process of creating a new theory able to answer some prescribed problems in functional analysis. The second one concentrates on the epistemological role of axioms. In particular, the notion (...) of norm completeness, as it appears in Banach’s text, can be interpreted as an epistemic linchpin between several a priori inhomogeneous domains of mathematical thought. (shrink)

The ideal of mathematical exactness is strongly paradigmatic for modern science, for which Mathematics practically functions as a metaphysical foundation. This strongly influenced Philosophy. In our century, however, critical voices arise, even from the ranks of scientists. Reflection on the foundations of Mathematics has produced a deeper insight into its nature. The tendency to judge content by structure becomes less predominant. Metaphysics is no longer rejected as only producing constructions with unjustified claims to necessity. This book combines contemporary Philosophy (...) of Mathematics with classical philosophical themes. It replaces the anti-metaphysical tendencies of modern and postmodern Philosophy by an unrestricted horizon of epistemic perspectives. (shrink)

The truth of it all -- The forward-backward method -- On definitions and mathematical terminology -- Quantifiers I: the construction method -- Quantifiers II: the choose method -- Quantifiers III: specialization -- Quantifiers IV: nested quantifiers -- Nots of nots lead to knots -- The contradiction method -- The contrapositive method -- The uniqueness methods -- Induction -- The either/or methods -- The max/min methods.

As in science and philosophy, thought experiments in mathematics link a problem to new epistemic resources that are unavailable in a given practice, e.g., Euclidean geometry. Thought experiments invite us to perform an imaginary scenario involving counterfactual, deductive and sensory elements. This chapter aims to pinpoint the beneficial peculiarities of thought experiments in mathematics in comparison with inferences, diagrams and calculative procedures. Reflection about thought experiments assists us to realize both the limits and opportunities in (...) class='Hi'>mathematical thinking. Henceforth, the analysis of examples suggests a broader understanding of mathematical, thought and experiment. (shrink)

This paper exposes the relation between the different mathematical orientations, on the one hand, and the modes of documentary film on the other. When we take, with Badiou, mathematics as ontology, and mathematical orientations as orientations to Being, we find in the structural similarity of mathematics and documentary an equivalence: between modes of documentaryand mathematical-ontological decisions, regarding the inscription of 'what is'. From here we move to consider Badiou's notion of 'in-appearing' through a reading of Alain Resnais' (...) documentary Night and Fog. (shrink)