Diversity Reading List

Abstract:
According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. I then propose a fallibilist account of mathematical justification. I show that the main function of mathematical justification is to guarantee that the mathematical community can correct the errors that inevitably arise from our fallible practices.

Abstract:
In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice. In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education.

Abstract:
Some personal thoughts and opinions on what “good quality mathematics” is and whether one should try to define this term rigorously. As a case study, the story of Szemer´edi’s theorem is presented.

Abstract:

A source of tension between Philosophers of Mathematics and Mathematicians is the fact that each group feels ignored by the other; daily mathematical practice seems barely affected by the questions the Philosophers are considering. In this talk I will describe an issue that does have an impact on mathematical practice, and a philosophical stance on mathematics that is detectable in the work of practising mathematicians. No doubt controversially, I will call this issue ‘morality’, but the term is not of my coining: there are mathematicians across the world who use the word ‘morally’ to great effect in private, and I propose that there should be a public theory of what they mean by this. The issue arises because proofs, despite being revered as the backbone of mathematical truth, often contribute very little to a mathematician’s understanding. ‘Moral’ considerations, however, contribute a great deal. I will first describe what these ‘moral’ considerations might be, and why mathematicians have appropriated the word ‘morality’ for this notion. However, not all mathematicians are concerned with such notions, and I will give a characterisation of ‘moralist’ mathematics and ‘moralist’ mathematicians, and discuss the development of ‘morality’ in individuals and in mathematics as a whole. Finally, I will propose a theory for standardising or universalising a system of mathematical morality, and discuss how this might help in the development of good mathematics.

Publisher’s Note:
Plato's Sun-Like Good is a revolutionary discussion of the Republic's philosopher-rulers, their dialectic, and their relation to the form of the good. With detailed arguments Sarah Broadie explains how, if we think of the form of the good as 'interrogative', we can re-conceive those central reference-points of Platonism in down-to-earth terms without loss to our sense of Plato's philosophical greatness. The book's main aims are: first, to show how for Plato the form of the good is of practical value in a way that we can understand; secondly, to make sense of the connection he draws between dialectic and the form of the good; and thirdly, to make sense of the relationship between the form of the good and other forms while respecting the contours of the sun-good analogy and remaining faithful to the text of the Republic itself.

Publisher’s Note:

Hao Wang (1921-1995) was one of the few confidants of the great mathematician and logician Kurt Gödel. A Logical Journey is a continuation of Wang's Reflections on Gödel and also elaborates on discussions contained in From Mathematics to Philosophy. A decade in preparation, it contains important and unfamiliar insights into Gödel's views on a wide range of issues, from Platonism and the nature of logic, to minds and machines, the existence of God, and positivism and phenomenology. The impact of Gödel's theorem on twentieth-century thought is on par with that of Einstein's theory of relativity, Heisenberg's uncertainty principle, or Keynesian economics. These previously unpublished intimate and informal conversations, however, bring to light and amplify Gödel's other major contributions to logic and philosophy. They reveal that there is much more in Gödel's philosophy of mathematics than is commonly believed, and more in his philosophy than his philosophy of mathematics. Wang writes that "it is even possible that his quite informal and loosely structured conversations with me, which I am freely using in this book, will turn out to be the fullest existing expression of the diverse components of his inadequately articulated general philosophy." The first two chapters are devoted to Gödel's life and mental development. In the chapters that follow, Wang illustrates the quest for overarching solutions and grand unifications of knowledge and action in Gödel's written speculations on God and an afterlife. He gives the background and a chronological summary of the conversations, considers Gödel's comments on philosophies and philosophers (his support of Husserl's phenomenology and his digressions on Kant and Wittgenstein), and his attempt to demonstrate the superiority of the mind's power over brains and machines. Three chapters are tied together by what Wang perceives to be Gödel's governing ideal of philosophy: an exact theory in which mathematics and Newtonian physics serve as a model for philosophy or metaphysics. Finally, in an epilog Wang sketches his own approach to philosophy in contrast to his interpretation of Gödel's outlook.

Abstract: Revealed preference approaches to modelling agents’ choices face two seemingly devastating explanatory objections. The no self-explanation objection imputes a problematic explanatory circularity to revealed preference approaches, while the causal explanation objection argues that, all things equal, a scientific theory should provide causal explanations, but revealed preference approaches decidedly do not. Both objections assume a view of explanation, the constraint-based view, that the revealed preference theorist ought to reject. Instead, the revealed preference theorist should adopt a unificationist account of explanation, allowing her to escape the two explanatory problems discussed in this paper.

Publisher's Note: The role of science in policymaking has gained unprecedented stature in the United States, raising questions about the place of science and scientific expertise in the democratic process. Some scientists have been given considerable epistemic authority in shaping policy on issues of great moral and cultural significance, and the politicizing of these issues has become highly contentious.

Since World War II, most philosophers of science have purported the concept that science should be “value-free.” In Science, Policy and the Value-Free Ideal, Heather E. Douglas argues that such an ideal is neither adequate nor desirable for science. She contends that the moral responsibilities of scientists require the consideration of values even at the heart of science. She lobbies for a new ideal in which values serve an essential function throughout scientific inquiry, but where the role values play is constrained at key points, thus protecting the integrity and objectivity of science. In this vein, Douglas outlines a system for the application of values to guide scientists through points of uncertainty fraught with moral valence.

Following a philosophical analysis of the historical background of science advising and the value-free ideal, Douglas defines how values should-and should not-function in science. She discusses the distinctive direct and indirect roles for values in reasoning, and outlines seven senses of objectivity, showing how each can be employed to determine the reliability of scientific claims. Douglas then uses these philosophical insights to clarify the distinction between junk science and sound science to be used in policymaking. In conclusion, she calls for greater openness on the values utilized in policymaking, and more public participation in the policymaking process, by suggesting various models for effective use of both the public and experts in key risk assessments.

Abstract: In this paper I argue that Poincare's acceptance of the atom does not indicate a shift from instrumentalism to scientific realism. I examine the implications of Poincare's acceptance of the existence of the atom for our current understanding of his philosophy of science. Specifically, how can we understand Poincare's acceptance of the atom in structural realist terms? I examine his 1912 paper carefully and suggest that it does not entail scientific realism in the sense of acceptance of the fundamental existence of atoms but rather, argues against fundamental entities. I argue that Poincare's paper motivates a non-fundamentalist view about the world, and that this is compatible with his structuralism.

Abstract: Poincare is well known for his conventionalism and structuralism. However, the relationship between these two theses and their place in Poincare's epistemology of science remain puzzling. In this paper I show the scope of Poincare's conventionalism and its position in Poincare's hierarchical approach to scientific theories. I argue that for Poincare scientific knowledge is relational and made possible by synthetic a priori, empirical and conventional elements, which, however, are not chosen arbitrarily. By examining his geometric conventionalism, his hierarchical account of science and defence of continuity in theory change, I argue that Poincare defends a complex structuralist position based on synthetic a priori and conventional elements, the mind-dependence of which precludes epistemic access to mind-independent structures.