It is argued in this study that (i) progress in the philosophy of mathematical practice requires a general positive account of informal proof; (ii) the best candidate is to think of informal proofs as arguments that depend on their matter as well as their logical form; (iii) articulating the dependency of informal inferences on their content requires a redefinition of logic as the general study of inferential actions; (iv) it is a decisive advantage of this conception of logic that it (...) accommodates the many mathematical proofs that include actions on objects other than propositions; (v) this conception of logic permits the articulation of project-sized tasks for the philosophy of mathematical practice, thereby supplying a partial characterisation of normal research in the field. (shrink)

Using Carnap’s concept explication, we propose a theory of concept formation in mathematics. This theory is then applied to the problem of how to understand the relation between the concepts formal proof and informal, mathematical proof.

The specific characteristics of mathematical argumentation all depend on the centrality that writing has in the practice of mathematics, but blindness to this fact is near universal. What follows concerns just one of those characteristics, justification by proof. There is a prevalent view that long proofs pose a problem for the thesis that mathematical knowledge is justified by proof. I argue that there is no such problem: in fact, virtually all the justifications of mathematical knowledge are ‘long proofs’, but because (...) these real justifications are distributed in the written archive of mathematics, proofs remain surveyable, hence good. (shrink)

In this paper I explore possibilities of bringing post-positivist philosophies of empirical science to bear on the dynamics of mathematical development. This is done by way of a convergent accommodation of a mathematical version of Lakatos's methodology of research programmes, and a version of Kuhn's account of scientific change that is made applicable to mathematics by cleansing it of all references to the psychology of perception. The resulting view is argued in the light of two case histories of radical conceptual (...) innovations. (shrink)

According to Kant, the axioms of intuition, i.e. space and time, must provide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjectivity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue that nevertheless (...) intuition comes into play in a fundamentally different way to that which Kant had foreseen: in the form of a formal or “categorical” yet not sensible intuition. We show further that the statement that our space is mathematically three-dimensional and locally Euclidean by no means follows from a supposed a priori nature of the sensible or subjective space as Kant claimed. In fact, the three-dimensional space can bear many different geometrical and topological structures, as particularly the mathematical results of Milnor, Smale, Thurston and Donaldson demonstrated. On the other hand, it has been stressed that even the phenomenological or perceptual space, and especially the visual system, carries a very rich geometrical organization whose structure is essentially non-Euclidean. Finally, we argue that in order to grasp the meaning of abstract geometric objects, as n-dimensional spaces, connections on a manifold, fiber spaces, module spaces, knotted spaces and so forth, where sensible intuition is essentially lacking and where therefore another type of mathematical idealization intervenes, we need to develop a new form of intuition. (shrink)

A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.

ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects or they maintain that mathematical objects are structures or positions in them. Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view.

Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that (...) this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge. (shrink)

A famous essay by Wigner is reexamined in view of more recent developments around its topic, together with some remarks on the metaphysical character of its main question about mathematics and natural sciences.

We describe recent developments in research on mathematical practice and cognition and outline the nine contributions in this special issue of topiCS. We divide these contributions into those that address (a) mathematical reasoning: patterns, levels, and evaluation; (b) mathematical concepts: evolution and meaning; and (c) the number concept: representation and processing.

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

In this paper I investigate two notions of concepts that have played a dominant role in 20th century philosophy of mathematics. According to the first, concepts are definite and fixed; in contrast, according to the second notion concepts are open and subject to modifications. The motivations behind these two incompatible notions and how they can be used to account for conceptual change are presented and discussed. On the basis of historical developments in mathematics I argue that both notions of concepts (...) capture important aspects of mathematical and scientific reasoning, and, consequently, that a pluralistic approach that allows for representing both of these aspects is most useful for an adequate account of investigative practices. (shrink)

Is God a Mathematician? By Mario Livio. New York: Simon & Schuster, 2009. 308 pp. $26.00 (hardcover). ISBN 074329405X. - CHAITIN La Lévitation [Levitation] by Joachim Boufl et. Paris: Jardin de Livres, 2006. 202 pp. ISBN 2-914569-27-0. - ALVARADO My Stroke of Insight: A Brain Scientist’s Personal Journey by Jill Bolte Taylor. New York: Viking, 2009. 183 pp. ISBN 978-0-670-02074-4. - GROSSO Intermediate States: The Anomalist 13 edited by Patrick Huyghe and Dennis Stacy. San Antonio and New York: Anomalist Books, (...) 2007. 188 pp. $12.95 (paperback). ISBN 978-1-93366-526-9. - CLARK Forbidden Science: Volume Two: Journals 1970–1979 The Belmont Years by Jacques Vallee. Documatica Research, LLC, November 2008. 547 pp. $44.42 (hardcover). ISBN 978-0-615-24974-2 - ALEXANDER The Body Electric: Electromagnetism and the Foundation of Life by Robert O. Becker and Gary Seldon. William Morrow & Co., 1985. 364 pp. (paper). ISBN 978-0-68800-123-0. - DYKSTRA Flying Saucers and Science: A Scientist Investigates the Mysteries of UFOs: Interstellar Travel, Crashes and Government Cover-ups by Stanton T. Friedman. Franklin Lakes, NJ: The Career Press, 2008. 317 pp. $16.99 (paper). ISBN 978-1-60163-011-7. - WOOD Body Snatchers in the Desert: The Horrible Truth at the Heart of the Roswell Story by Nick Redfern. Paraview-Pocket Books, a division of Simon & Schuster, 2005. 256 pp. $14.00 (paper). ISBN 0-7434-9753-8. - CARPENTER Body Snatchers in the Desert: The Horrible Truth at the Heart of the Roswell Story by Nick Redfern. Paraview-Pocket Books, a division of Simon & Schuster, 2005. 256 pp. $14.00 (paper). ISBN 0-7434-9753-8. - HEISER The Tujunga Canyon Contacts by Ann Druffel and D. Scott Rogo. San Antonio and New York: Anomalist Books, 2008. 302 pp. $16.95 (paper). ISBN 978-1-93-366533-7. - SANDOW The Invasion from Mars: A Study in the Psychology of Panic by Hadley Cantril (with a new introduction by Albert H. Cantril). New Brunswick & London: Transaction Publishers, 2005. xxxii + 224 pp. $24.95 (paper). ISBN 0-915554-45-3 - HÖVELMANN Article of Interest - ALVARADO. (shrink)