This truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that (...) proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities. (shrink)

This paper discerns two types of mathematization, a foundational and an explorative one. The foundational perspective is well-established, but we argue that the explorative type is essential when approaching the problem of applicability and how it influences our conception of mathematics. The first part of the paper argues that a philosophical transformation made explorative mathematization possible. This transformation took place in early modernity when sense acquired partial independence from reference. The second part of the paper discusses a series of examples (...) from the history of mathematics that highlight the complementary nature of the foundational and exploratory types of mathematization. (shrink)

Mathematics plays an inordinate role in the work of many of famous Western philosophers, from the time of Plato, through Husserl and Wittgenstein, and even to the present. Why? This paper points to the experience of learning or making mathematics, with an emphasis on proof. It distinguishes two sources of the perennial impact of mathematics on philosophy. They are classified as Ancient and Enlightenment. Plato is emblematic of the former, and Kant of the latter. The Ancient fascination arises from the (...) sense that mathematics explores something ‘out there’. This is illustrated by recent discussions by distinguished contemporary mathematicians. The Enlightenment strand often uses Kant's argot: ‘absolute necessity’, ‘apodictic certainty’ and ‘a priori’ judgement or knowledge. The experience of being compelled by proof, the sense that something must be true, that a result is certain, generates the philosophy. It also creates the illusion that mathematics is certain. Kant's leading question, ‘How is pure mathematics possible?’, is easily misunderstood because the modern distinction between pure and applied is an artefact of the 19th century. As Russell put it, the issue is to explain ‘the apparent power of anticipating facts about things of which we have no experience’. More generally the question is, how is it that pure mathematics is so rich in applications? Some six types of application are distinguished, each of which engenders its own philosophical problems which are descendants of the Enlightenment, and which differ from those descended from the Ancient strand. (shrink)

Thought experiments are a means of imaginative reasoning that lie at the heart of philosophy, from the pre-Socratics to the modern era, and they also play central roles in a range of fields, from physics to politics. The Routledge Companion to Thought Experiments is an invaluable guide and reference source to this multifaceted subject. Comprising over 30 chapters by a team of international contributors, the Companion covers the following important areas: -/- · the history of thought experiments, from antiquity to (...) the trolley problem and quantum non-locality; -/- · thought experiments in the humanities, arts, and sciences, including ethics, physics, theology, biology, mathematics, economics, and politics; -/- · theories about the nature of thought experiments; -/- · new discussions concerning the impact of experimental philosophy, cross-cultural comparison studies, metaphilosophy, computer simulations, idealization, dialectics, cognitive science, the artistic nature of thought experiments, and metaphysical issues. -/- This broad ranging Companion goes backwards through history and sideways across disciplines. It also engages with philosophical perspectives from empiricism, rationalism, naturalism, skepticism, pluralism, contextualism, and neo-Kantianism to phenomenology. This volume will be valuable for anyone studying the methods of philosophy or any discipline that employs thought experiments, as well as anyone interested in the power and limits of the mind. (shrink)

Like the latter, it unites into one and the same function the possibility of giving things a sign, of representing one thing by another, and the possibility of causing a sign to shift in relation to what it designates. The four functions that define the ...

Platonism is the most pervasive philosophy of mathematics. Indeed, it can be argued that an inarticulate, half-conscious Platonism is nearly universal among mathematicians. The basic idea is that mathematical entities exist outside space and time, outside thought and matter, in an abstract realm. In the more eloquent words of Edward Everett, a distinguished nineteenth-century American scholar, "in pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist (...) there when the last of their radiant host shall have fallen from heaven." In What is Mathematics, Really?, renowned mathematician Rueben Hersh takes these eloquent words and this pervasive philosophy to task, in a subversive attack on traditional philosophies of mathematics, most notably, Platonism and formalism. Virtually all philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Mathematical objects are created by humans, not arbitrarily, but from activity with existing mathematical objects, and from the needs of science and daily life. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of the book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Plato, Descartes, Spinoza, and Kant, to Bertrand Russell, David Hilbert, Rudolph Carnap, and Willard V.O. Quine--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, Peirce, Dewey, and Lakatos. In his epilogue, Hersh reveals that this is no mere armchair debate, of little consequence to the outside world. He contends that Platonism and elitism fit well together, that Platonism in fact is used to justify the claim that "some people just can't learn math." The humanist philosophy, on the other hand, links mathematics with geople, with society, and with history. It fits with liberal anti-elitism and its historical striving for universal literacy, universal higher education, and universal access to knowledge and culture. Thus Hersh's argument has educational and political ramifications. Written by the co-author of The Mathematical Experience, which won the American Book Award in 1983, this volume reflects an insider's view of mathematical life, based on twenty years of doing research on advanced mathematical problems, thirty-five years of teaching graduates and undergraduates, and many long hours of listening, talking to, and reading philosophers. A clearly written and highly iconoclastic book, it is sure to be hotly debated by anyone with a passionate interest in mathematics or the philosophy of science. (shrink)

This 1983 book is a lively and clearly written introduction to the philosophy of natural science, organized around the central theme of scientific realism. It has two parts. 'Representing' deals with the different philosophical accounts of scientific objectivity and the reality of scientific entities. The views of Kuhn, Feyerabend, Lakatos, Putnam, van Fraassen, and others, are all considered. 'Intervening' presents the first sustained treatment of experimental science for many years and uses it to give a new direction to debates about (...) realism. Hacking illustrates how experimentation often has a life independent of theory. He argues that although the philosophical problems of scientific realism can not be resolved when put in terms of theory alone, a sound philosophy of experiment provides compelling grounds for a realistic attitude. A great many scientific examples are described in both parts of the book, which also includes lucid expositions of recent high energy physics and a remarkable chapter on the microscope in cell biology. (shrink)