The Philosophy of Mathematical Practice
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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 mathematical practice. 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 as worthy 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 a 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 representational 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.
visualizable curve, e.g. Weierstrass’s continuous but nowhere differentiable function,⁷ or f deﬁned on [0, 1] thus: f (x) = x. sin(1/x) if 0 < x 1; f (0) = 0 So the visual thinking does not provide us with a form of reasoning that is applicable to all − δ continuous functions. The matter is discussed further in Giaquinto (1994). 1.4 Other uses of visual thinking Even if visual thinking in analysis cannot be a means of proof or discovery (in the sense in which I am using that term), it is
achieved by ‘reading off’ the descriptions of the category speciﬁcation, since we have no direct access to those descriptions. Rather, as a result of having the category speciﬁcation, we have a number of dispositions which, taken together, give some indication of the kind of structured set it represents. These are dispositions to answer certain questions one way rather than another. For example: Given any two marks, must one precede the other? Yes. Do the intermark spaces vary in length? No. Is
structure of [0, 1] under the ‘less than’ ordering. So we must make do with geometrical points. But thinking of a line as composed of geometrical points leads to numerous paradoxes. For instance, the parts of a line either side of a given point would have to be both separated (as the given point lies between them) and touching (as there is no distance between the two parts, the given point being extensionless). A related puzzle is that a line segment must have positive extension; but as all its
for example, three- or four-sided, may depend on the initial data of the construction in a manner sufﬁciently inscrutable to thwart control. One purpose of Euclid’s treating numerous construction problems in Book I may therefore be to underwrite the availablity of diagrams which propositions might require us to consider. It is probably prudent to regard the availability of diagrams satisfying complicated co-exact conditions as potentially a sore point in traditional practice; and potentially
reasoning: although ‘exact’ conclusions are not read off from the diagram, diagram use also remains in need of probing, due to the limitations of diagram control. Because of this, the long-range stability of traditional geometry—across centuries, participants, and the extended body of accumulated geometrical argument—cannot be understood without taking into account the constructive critical attitude towards geometrical argument expressed through probing. In our philosophical account of