Wilholt T. Review of José Ferreiros and Jeremy J Gray (eds.): The Architecture of Modern Mathematics: Essays in History and Philosophy. Bulletin of Symbolic Logic. 2007;13(2):368-370

Wilholt, Torsten

Publications at Bielefeld University

368 REVIEWSThe architecture of modern mathematics: Essays in history and philosophy, edited by Jose´Ferreiro´s and Jeremy J. Gray, Oxford University Press, Oxford 2006, xii + 442 pp.In this volume, Ferreiro´s and Gray have brought together a selection of 14 new essays(only one of them published before) that share a commitment to integrating the philo-sophical reflection of mathematics with the close historical study of the development ofmathematical practices and ideas. In the resulting volume, philosophy and history do notplay exactly symmetrical roles: All contributions (with the possible exception of the intro-duction) present themselves as historical essays that highlight some philosophical aspectsrather than the other way round. The papers are concerned with mathematical developmentsbetween 1800 and 1970, and the majority of them focus on the contributions of one or twogreat mathematicians within a specific area of mathematics.The book opens with a long and spirited introduction by the editors, in which they adopta critical stance towards contemporary mainstream philosophy of mathematics. They seeit dominated by attempts to apply “systematic philosophy” to mathematics. The resultingparadigm is criticized for various shortcomings: It tends to ignore the conceptual andmethodological dynamics of mathematics; it over-emphasizes one element of mathematicalpractice (viz., proving theorems within the framework of a given axiomatic system) at theexpense of other, philosophically no less interesting ones (like conjecturing, explaining, orvisualizing); it is obsessed with its pet theories, logic and set theory, and therefore oftenfails to acknowledge the richness of mathematical concepts and methods; it has sufferedfrom “excessive attention to all-inclusive claims” (p. 18). Such deficiencies, the editorssuggest, can be overcome by means of a closer integration of the history and philosophyof mathematics, geared to an ideal of “philosophical reflection” rather than “systematicphilosophy”. To be sure, similar criticisms have been raised before (for example in Kitcherand Aspray’s introduction to their Minnesota Studies volume of 1988), and the editorsexplicitly acknowledge this. Yet one can hardly contest their diagnosis that, by and large,the received picture of the philosophy of mathematics has not changed much during the last50 years (p. 4 f.). This is despite the many valuable investigations along the lines suggestedby Ferreiro´s and Gray that have of course already been undertaken (among others, by manyof the contributors to their volume). Given this, the editors’ initiative to present a goodselection of recent samples of this work is all the more welcome.Many of the contributions focus on the philosophical ideas developed by great mathe-matical practitioners in the context of their work. For example, Jamie Tappenden alertshis readers to the fact that during the time of Frege’s mathematical upbringing, which oc-curred in a context where analysis was widely perceived as a core discipline of mathematics,there was no consensus about what exactly analysis was. In particular, a Weierstrassiantradition with an emphasis on rigor and avoidance of appeal to geometric intuition clashedwith a Riemannian one that permitted the application of geometric tools to analysis andpursued conceptual clarity and simplicity as a primary goal. Frege’s siding with Riemann’sunderstanding of complex analysis is then shown to be continuous with some of his logico-philosophical insights—e.g., that a function is not an expression and that definitions shouldbe representation-independent.Riemann himself and his philosophical ideas are subjected to a close investigation inanother contribution. Jose´ Ferreiro´s argues that the key to Riemann’s philosophy of mathe-matics is his involvement with physics, which for a while was so serious that he regarded hisquest for a unified theory of physical forces as his main occupation. According to Ferreiro´s,this led Riemann to view mathematics as a part of the natural sciences, which implied theabandonment of its a priori status.Michael Beaney’s paper invites reflections on the historicity of philosophical ideas inmathematics itself, as he reviews the role of historical references in Frege’s work and howthey relate to his pursuit of the elucidation (“Erla¨uterung”) of expressions. The use ofREVIEWS 369historical reference-points like Kant is shown often to be an important part of Fregeanelucidation, which is itself an indispensable task as not every expression can be defined interms of more basic ones.The volume also contains papers on more unusual and idiosyncratic philosophical ideasof mathematicians. For example, Erhard Scholz portrays Hermann Weyl as an advocateof a “symbolic realism”, according to which mathematics acquires meaning from contextsof scientific, technical and social practices, while questions concerning its fundamental on-tology should be regarded in an agnostic way. Moritz Epple juxtaposes Felix Hausdorffwith his literary alter ego Paul Mongre´, under which name he published several pieces offiction and philosophical reflections in a broadly Nietzschean vein. Epple tries to unearth thecommon epistemological concerns of the mathematician and the literary writer. Alongsidesuch investigations of individual mathematicians and their philosophical agenda, the volumealso presents some interesting and more encompassing efforts to correlate and compare thephilosophical reactions of several different mathematicians to certain mathematical and sci-entific developments. An example is Jeremy Gray’s closing essay on the profound changesthat transformed mathematics after 1880. He concentrates on geometry and identifies aLeibnizian and a Kantian strategy of coping with the deep philosophical questions aboutmathematical truth raised by, among other things, the advent of non-Euclidean geometry.How can such historical studies of mathematicians’ philosophical ideas (there are severalmore among the papers within this volume) support the philosophical reflection on math-ematics demanded by the editors? I think they can play very helpful roles in this respect,especially if they are concerned with influential, established, or at least coherently defensiblephilosophical positions. They can serve to demonstrate the contingency of the ideas thatdominate the philosophical terrain, in the sense that they show how positions have emergedfrom changes and tensions within mathematics at certain points in time (as in the case ofFrege). They can even highlight alternative paths (as, arguably, in the case of Riemann).However, if the proposed convergence of history and philosophy of mathematics is toshow its whole potential, it must go beyond a mere history of philosophy of mathematics. Itmust also prove its potential to use the history of mathematics in order to lead to convincing,contemporary philosophical insights. The actual mathematical practices of the historicalprotagonists often offer more important material than their own philosophical reflections.As an illustration of this, Jeremy Avigad has contributed a paper on Dedekind’s theory ofideals that exemplifies the great promise of an integrated history and philosophy of mathe-matics. Among the more fascinating aspects of mathematical practice that has received toolittle attention from traditional philosophy is its considerable freedom: Not only are theremyriads of new concepts and structures that could in principle be introduced and investi-gated, but each of them can also be defined and approached in many very different ways.A case in point is the theory of ideals and its different early treatments, most importantlyby Kronecker and Dedekind. Avigad traces the development of Dedekind’s several versionsof the theory in order to identify the mathematical values that informed his methodologicalchoices. In unexpressed analogy to the so-called “epistemic values” that are familiar from thephilosophy of the natural sciences, Avigad gives a precise description of principles of gener-ality, uniformity, and familiarity as they appear in Dedekind’s work. The insights enabled bythis investigation promise to give rise to many interesting and potentially fruitful historicaland philosophical questions—e.g., how did certain mathematical values become establishedin the mathematical community rather than others, how do they relate to the values of othersciences and even to other (“extra-scientific”) values?Most of the many interesting discussions in this volume offer impulses for philosophicalreflection rather than their own philosophical conclusions. An exception is the introduction,where Ferreiro´s and Gray put forward a tentative “hypothetical conception of mathematicalpractice”, which is designed to take the historical richness of the subject into account. It is370 REVIEWSmeant to reflect an insight that most mathematical propositions cannot be taken to be “true inthe world we live in”, but must be seen as parts of systems based on hypotheses. How exactlythis is to be spelled out—and, in particular, how it relates to views that have been discussedat length under the name “deductivism” or “if-thenism”—is not clear. Prima facie, the hardproblem for a hypothetical conception is to explain how a whole mathematical theory “iserected on” its hypothetical basis (to use the authors’ expression), without loading all theexplosive philosophical freight onto an uncritical use of the notion of logical consequence(as deductivism has been criticized of doing). Clearly, systematic philosophizing aboutmathematics will have to be part and parcel of an integrated enterprise of the history andphilosophy of mathematics. Ferreiro´s and Gray’s volume shows that the real convergencehas yet to occur, but it also offers enough samples of interesting and insightful work to makethe endeavor look promising.Torsten WilholtDepartment of Philosophy, Bielefeld University, P.O. Box 100 131, D-33501 Bielefeld,Germany. twilholt@uni-bielefeld.de.Harry J. Gensler. Historical dictionary of logic. Historical Dictionaries of Reli-gions, Philosophies, and Movements, vol. 65. The Scarecrow Press, Lanham (Maryland),Toronto, and Oxford, 2006, xliv + 307 pp.The book under review is a sort of an encyclopedia of logic. The author declares in thePreface that he understands logic as a discipline analyzing arguments and helping us to seewhich ones are valid. The aim of the book is to introduce the central concepts of logic ina series of brief, nontechnical articles. The book emphasizes deductive logic but there arealso entries on areas like inductive logic, fallacies, definitions as well as key concepts fromepistemology, mathematics and set theory.Logic is nowadays developed by philosophers, mathematicians and computer scientists. Inthe book under review the philosophical branch of logic is emphasized. Therefore the areasin which philosophers are interested and which they teach are first of all taken into account.The mathematical and computer-science items as more technical and more specialized havenot been treated so extensively.The main part of the book is a dictionary section consisting of 352 entries. They are ar-ranged alphabetically and are devoted to fundamental concepts and problems of logic. Themost significant theories and issues arising out of Western philosophy and other traditions,the more common fallacies and argument forms are presented and explained. There areentries on historical periods and figures, including ancient logic, medieval logic, Buddhistlogic, Aristotle, Ockham, Boole, Frege, Russell, Go¨del, Peano, Quine, etc. One finds thereinformation on propositional logic, modal logic, deontic logic, temporal logic, set theory,many-valued logics, mereology, paraconsistent logic. There are also entries concerning appli-cations of logic to various fields such as philosophy, mathematics or computer science. Thepresupposed character of the book excluded very technical expositions but the author suc-ceeded to explain technical problems—as, for example, Go¨del’s incompleteness theorems—inan adequate way not using complicated technical apparatus. The entries vary in length froma sentence or two to several pages.One should stress that the book under review provides the potential reader not only withknowledge about well established canon of logical notions and concepts but it indicates andexplains also many modern, still being developed and studied theories and achievementsin logic, in particular those connected with computer science or fuzzy logic as well as withcognitive science.The front of the book has three important parts: (1) a short notation section, (2) achronology starting from the ancient Greece and going to 2001, (3) an introduction where anoverall view of logic is given (it should provide a broader context for the dictionary entries).

Review of José Ferreiros and Jeremy J Gray (eds.): The Architecture of Modern Mathematics: Essays in History and Philosophy

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Review of José Ferreiros and Jeremy J Gray (eds.): The Architecture of Modern Mathematics: Essays in History and Philosophy

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