The Philosophy of Mathematical Practice by Paolo Mancosu (PDF)

Description

Contemporary philosophy of mathematics offers us an embarrassment of riches. Among the major areas of work one could list developments of the classical foundational programs, analytic approaches to epistemology and ontology of mathematics, and developments at the intersection of history and philosophy of mathematics. But anyone familiar with contemporary philosophy of mathematics will be aware of the need for new approaches that pay closer attention to mathematicalpractice. This book is the first attempt to give a coherent and unified presentation of this new wave of work in philosophy of mathematics. The new approach is innovative at least in two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just asworthy of philosophical attention as the distinction between constructive and non-constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics which escape purely formal logical treatment – such as visualization, explanation, and understanding – can nonetheless be subjected to philosophical analysis.The Philosophy of Mathematical Practice comprises an introduction by the editor and eight chapters written by some of the leading scholars in the field. Each chapter consists of short introduction to the general topic of the chapter followed by a longer research article in the area. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: diagrammatic reasoning and representation systems;visualization; mathematical explanation; purity of methods; mathematical concepts; the philosophical relevance of category theory; philosophical aspects of computer science in mathematics; the philosophical impact of recent developments in mathematical physics.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is pretty useless. A chapter conclusion summarises well the entire book: the philosophy of mathematical practice “offers a vast and virgin territory to exploration” and “Most of the work remains to be done” (p. 148). Of course no potential reader would disagree with these trivialities. It is a pity, however, that the entire book is little but a constant reiteration of this call to arms, while the work that everyone keeps pointing to remains virtually untouched.I offer a partial diagnosis as to the reasons for book’s failure: the formalistic point of view has not been rejected with sufficient zeal and conviction. The authors in this volume agree that formalism is not everything but their alternatives are meant to be modest and polite complements rather than serious alternatives. Look at the history of philosophy of mathematics: Plato, Pascal, Kepler, Descartes, Leibniz, Kant, Poincaré, Brouwer, Weyl, etc. All of these people would have said that logic and symbols and whatever formal systems you can come up with just fundamentally miss the point: what drives mathematics is the human cognitive capacities, innate intuitions, etc. This point of view predicted the shortcomings of the formalistic fad to the letter. Yet the authors in this volume refuse to draw the obvious conclusions. The cognitive perspective is condemned for no good reason for example by Avigad in his chapter on understanding:”We have all had such ‘Aha!’ moments and the deep sense of satisfaction that comes with them. But surely the philosophy of mathematics is not supposed to explain this sense of satisfaction, any more than economics is supposed to explain the feeling of elation that comes when we find a $20 bill lying on the sidewalk.” (p. 322).All the above thinkers would have agreed that “Aha” experiences are to be explained in terms of the nature of the human cognitive endowment. To say that “surely” this would be stupid, on the basis of nothing more than a silly parallel to economics, is to dismiss the major historical alternative to formalism all too easily.One further example: Tappenden argues in chapter 9 “that mathematical defining is a more intricate activity, with deeper connections to explanation, fruitfulness of research, etc. than is sometimes realised” (p. 272). Like so many other “theses” presented in this book, this is a complete triviality from the cognitive point of view. In the mind there are ideas. When these are projected onto a formal presentation it happens that some ideas map to definitions, other to theorems, others to proofs, etc. Only someone who thought that this projection onto the formal representation was not a projection but a mere equation would think that properties of the images of ideas was an accurate description of the ideas themselves, and so commit the error that Tappenden says occurs “sometimes.”

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