Mathematical explanation -- What is naturalism? -- Perception, practice, and ideal agents: Kitcher's naturalism -- Just metaphor?: Lakoff's language -- Seeing with the mind's eye: the Platonist alternative -- Semi-naturalists and reluctant realists -- A life of its own?: Maddy and mathematical autonomy.
There is sufficient evidence that intellectual property rights are corrupting medical research. One could respond to this from a moral or from an epistemic point of view. I take the latter route. Often in the sciences factual discoveries lead to new methodological norms. Medical research is an example. Surprisingly, the methodological change required will involve political change. Instead of new regulations aimed at controlling the problem, the outright socialization of research seems called for, for the sake of better science. I (...) appeal to an analogy between socialized medicine and socialized research. †To contact the author, please write to: Department of Philosophy, University of Toronto, Toronto, Ontario M5R 2M8, Canada; e‐mail: email@example.com. (shrink)
1. Introduction : the mathematical image -- 2. Platonism -- 3. Picture-proofs and Platonism -- 4. What is applied mathematics? -- 5. Hilbert and Gödel -- 6. Knots and notation -- 7. What is a definition? -- 8. Constructive approaches -- 9. Proofs, pictures and procedures in Wittgenstein -- 10. Computation, proof and conjecture -- 11. How to refute the continuum hypothesis -- 12. Calling the bluff.
I reply to a number of papers (published in Croatian Journal of Philosophy 7 , 29-92 and in this issue) that stem from a conference in Rijeka on thought experinlents. These are papers by Ana Butković, Dave Davies, Boris Grozdanoff, Dunja Jutronić, Nenad Miščević, Ksenija Puškarić, and Irina Starikova. Their criticisms of my views are diverse, but one theme, perhaps inevitably, dominates the criticisms: the unworkability of my Platonism. I try to defend this and to adequately answer other criticisms, as (...) well. (shrink)
Most disciplines make use of thought experiments, but physics and philosophy lead the pack with heavy dependence upon them. Often this is for conceptual clarification, but occasionally they provide real theoretical advances. In spite of their importance, however, thought experiments have received rather little attention as a topic in their own right until recently. The situation has improved in the past few years, but a mere generation ago the entire published literature on thought experiments could have been mastered in a (...) long weekend. Now the subject is beginning to flourish. Given the relative newness of the field, it might be useful to have several examples at one’s finger tips, so a number of great ones will be described. Attention will also be drawn outside physics and philosophy. In mathematics there is something analogous to thought experiments -- visual reasoning and picture proofs. I will look briefly at this class of thought experiments and try using them to make a case for possibly settling the continuum hypothesis. After this, I will return to thought experiments in the sciences and propose an account of how they work. Finally, I will end with a sketch of a topic I am currently working on, a kind of progress report which, I hope, will be an inducement to others. (shrink)
Examples of classic thought experiments are presented and some morals drawn. The views of my fellow symposiasts, Tamar Gendler, John Norton, and James McAllister, are evaluated. An account of thought experiments along a priori and Platonistic lines is given. I also cite the related example of proving theorems in mathematics with pictures and diagrams. To illustrate the power of these methods, a possible refutation of the continuum hypothesis using a thought experiment is sketched.
Recent years have seen a number of naturalist accounts of mathematics. Philip Kitcher’s version is one of the most important and influential. This paper includes a critical exposition of Kitcher’s views and a discussion of several issues including: mathematical epistemology, practice, history, the nature of applied mathematics. It argues that naturalism is an inadequate account and compares it with mathematical Platonism, to the advantage of the latter.
There has been a sharp rise in private funding of medical research, especially in relation to patentable products. Several serious problems with this are described. A solution involving the elimination of patents and public funding administered through extended national health care systems is proposed.
Philosophy of Mathematics is clear and engaging, and student friendly The book discusses the great philosophers and the importance of mathematics to their thought. Among topics discussed in the book are the mathematical image, platonism, picture-proofs, applied mathematics, Hilbert and Godel, knots and notation definitions, picture-proofs and Wittgenstein, computation, proof and conjecture.
According to the standard view of definition, all defined terms are mere stipulations, based on a small set of primitive terms. After a brief review of the Hilbert-Frege debate, this paper goes on to challenge the standard view in a number of ways. Examples from graph theory, for example, suggest that some key definitions stem from the way graphs are presented diagramatically and do not fit the standard view. Lakatos's account is also discussed, since he provides further examples that suggest (...) many definitions are much more than mere convenient abbreviations. (shrink)
Everyone appreciates a clever mathematical picture, but the prevailing attitude is one of scepticism: diagrams, illustrations, and pictures prove nothing; they are psychologically important and heuristically useful, but only a traditional verbal/symbolic proof provides genuine evidence for a purported theorem. Like some other recent writers (Barwise and Etchemendy ; Shin ; and Giaquinto ) I take a different view and argue, from historical considerations and some striking examples, for a positive evidential role for pictures in mathematics.
A number of issues connected with the nature of applied mathematics are discussed. Among the claims are these: mathematics "hooks onto" the world by providing models or representations, not by describing the world; classic platonism is to be preferred to structuralism; and several issues in the philosophy of science (reality of spacetime, the quantum state) are intimately connected to the nature of applied mathematics.
This book adds to the growing literature on thought experiments. There are numerous examples drawn from the sciences and philosophy. The principle claim is that thought experiments are a limiting case of real experiments. It is a moderate empiricist view, in contrast to, e.g., the Platonism of Brown or the strict empiricism of Norton. Highly recommended.
In Smoke and Mirrors , James Robert Brown fights back against figures such as Richard Rorty, Bruno Latour, Michael Ruse and Hilary Putnam who have attacked realistic accounts of science. This enlightening work also demonstrates that science mirrors the world in amazing ways. The metaphysics and epistemology of science, the role of abstraction, abstract objects, and a priori ways of getting at reality are all examined in this fascinating exploration of how science reflects reality. Both a defense of science and (...) knowledge in general and a defense of a particular way of understanding science, Smoke and Mirrors will be provocative and lively reading for all those who have an interest in how science works. (shrink)