TY - JOUR
AU - Svoboda,, David
AB - On June 24–26, 2016, a conference entitled The Emergence of Structuralism and Formalism took place in Prague under the auspices of the Institute of Philosophy of the Czech Academy of Sciences and the Catholic Faculty of Theology of the Charles University, where the proceedings took place. The conference was devoted to problems of philosophy of mathematics, regarded from the perspective of structuralism and formalism. We were very pleased to secure the presence of truly outstanding experts in the field. The organizers, Sousedik and Svoboda, highly value the fact that their invitation was accepted by the most important adherents of structuralism, Stewart Shapiro and Michael Resnik. They can be regarded as the founders of this trend in contemporary philosophy of mathematics. The other invited speakers included Michael Detlefsen, Leon Horsten, and Øystein Linnebo. Broadly speaking, the papers presented at the conference can be divided into historical and systematic groups. The historical ones discussed the key representatives of both structuralism and formalism, focusing on their emergence from prior developments in philosophy and mathematics. The more systematic papers were devoted to the proper characterization and pursuit of structuralism and formalism. They addressed current problems associated with these concepts, comparing their advantages and disadvantages, and proposing various solutions to various problems. The possibility of a Proceedings was raised by organizers and many of the participants. However, the material presented at the conference was too extensive and did not constitute a coherent whole that could be presented to a broad spectrum of readers. We were therefore forced to make a selection and offer only papers related in content for publication. Over time it became progressively clear to us that structuralism provided an appropriate theme for the project, and so we decided to exclude papers not dealing with structuralism, or having a merely tangential bearing on it. We were led to this conclusion by the conviction that structuralism is one of the best answers to the revolution that took place in mathematics in the nineteenth century and adequately reflects its contemporary form. It is not a great exaggeration to claim that the mathematical discipline had undergone such a radical transformation that one can speak of a re-foundation. Precisely this transformation gave rise to deeper reflections on the character and foundations of mathematics. Let us now indicate briefly the character of the transformation that we have in mind, and how the progress in mathematics gave rise to structuralism. To speak roughly, or as a first approximation, it is fair to state that until the end of the eighteenth century or so, mathematics required a picture or an intuitive notion, either of continuity or discrete quantities. So it was natural to hold that intuition is an essential part of the philosophical foundations of mathematics. That was the view of the Peripatetic school, and it is also found in opposing Neoplatonic thinkers. The adherents of these two opposing traditions all associated mathematical objects with imagination, albeit in different ways. The thought of Immanuel Kant, who held that geometry and arithmetic are grounded in the pure intuition of space and time, is a culmination of these older approaches. During Kant’s own lifetime, however, mathematics started to develop in directions that contradict these very traditional views. The focus slowly turned to procedures and concepts in which an image or a picture plays no part and which were for this reason poignantly called ‘blind’ by G.W. Leibniz. This was not an accidental or rare phenomenon. The transition is witnessed by the emergence of non-Euclidian geometries, the geometries of different dimensions, ideal points at infinity, and the like. Many of the efforts to rigorize the infinitesimal calculus in the nineteenth century can also be understood as developments away from intuition. There certainly existed, and still exist, attempts to keep the traditional approach alive, but for the most part they have been unsuccessful. We hold, therefore, that the philosophy of mathematics must be approached in a way different from before and that a different role must be assigned to intuition. Of course, intuition, in the present sense, cannot be entirely rejected, since even contemporary geometrical instruction can hardly be presented without it. But the intuitive pictures play a merely heuristic part and do not constitute the very foundations of mathematics. How are we to face the question of the foundation of mathematics in these circumstances? Of course, there are different alternatives. But we think that this question is quite fundamentally answered by structuralism, according to which the character of mathematical objects is determined exclusively by their mutual relations. That is why these objects, as Resnik poignantly noted, have no inner properties, but merely external, i.e., relational ones. Thus, mathematics is not a science of some self-standing objects, whose ontological status gives rise to unsolvable disputes (and has done so since antiquity). Its subject is structures. We think that this turn helps to create an entirely new framework in which discussions regarding the character of mathematics can take place. The papers in the present issue aim to aid this process. We hope that it will stimulate debates of two kinds. In the first place, it can give rise to external criticism or debate by philosophers inclined to a different view from the one presented by structuralism. Second, it can also stimulate internal debates among the structuralists. We hope that both will contribute to a deeper and more precise articulation of the structuralist positions. We were delighted that the editorial board of Philosophia Mathematica agreed to a special issue for our conference, focusing on structuralism. We thoroughly agree with their policy that all of the papers were to be refereed in the usual way, with the now common standards of blind review. The conference organizers and editors of the proposed special issue were not involved in the refereeing process. That would be handled internally, in the usual manner, by the journal. If enough papers were accepted, then there would be special issue (and, if not, the accepted ones would be published as separate, stand-alone entries). The result was the four papers of this special issue. Resnik’s classic papers ‘Mathematics as a science of patterns: Ontology and reference’ (Noûs 15 (1981), 529–550) and ‘Mathematics as a science of patterns: Epistemology’, (Noûs 16 (1982), 95–105) introduced structuralism into contemporary philosophy of mathematics. He and Shapiro are commonly taken to be ‘realist’ structuralists, holding that structures themselves exist as abstracta, and that mathematical objects are places in these structures. These views are opposed by an eliminative structuralism, one that denies that structures exist as stand-alone entities. Such views can be thought of as both structuralist articulations of nominalism, and as nominalist articulations of structuralism. In his paper here, Resnik describes the evolution of his own views away from the realism into a ‘non-ontological’ structuralism, one that accepts W.V.O. Quine’s theme of ontological relativity. Ladislav Kvasz addresses the realist views of (the early) Resnik and Shapiro, with respect to the much discussed problems for ontological realism from Benacerraf’s ‘Mathematical truth’ (Journal of Philosophy 70 (1973), 661–679). The problem, in sum, is that mathematical objects, as traditionally conceived, are abstract, and thus causally inert. So mathematical objects, and the structures posited by the realist structuralists, are inaccessible to human cognition. So, the argument concludes, we cannot know anything about them, and thus we cannot know any mathematics. Kvasz introduces a distinction between abstract and ideal objects, and argues that mathematical objects, as conceived by realist structuralism, are primarily ideal, and not abstract. He then shows how mathematical practice can indeed result in epistemic access to such objects. Fiona Doherty’s paper is on the historical side, but uses this history to illuminate the contemporary discussion. She presents David Hilbert’s much discussed 1900 position as a species of non-eliminative structuralism, in contrast with his later formalist views. Doherty shows how the structuralist themes allow Hilbert to respond to the objections by Gottlob Frege in the course of the famous Frege-Hilbert controversy. She then shows how Hilbert’s particular brand of non-eliminative structuralism has the resources to overcome the objection that realist structuralism cannot accommodate indiscernible objects, such as the two square roots of |$-1$| in complex analysis. Realist structuralism has been charged with incoherence over the thesis that mathematical objects have no properties, or no essential properties, beyond their relations to other objects from the same structure. Michael Dummett once called such views ‘mystical’. Leon Horsten provides a philosophical treatment of structures adapted from Kit Fine’s ‘arbitrary objects’ (e.g., Reasoning with Arbitrary Objects. Oxford: Basil Blackwell, 1985). He argues that the result is a most attractive philosophy of mathematics. August 2019 © The Authors [2019]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - Introduction to Special Issue: The Emergence of Structuralism
JO - Philosophia Mathematica
DO - 10.1093/philmat/nkz019
DA - 2019-10-01
UR - https://www.deepdyve.com/lp/oxford-university-press/introduction-to-special-issue-the-emergence-of-structuralism-ce8Lhu30U5
SP - 299
VL - 27
IS - 3
DP - DeepDyve
ER -