Mathematics as a Tool. Zentrum für interdisziplinäre Forschung Center for Interdisciplinary Research Universität Bielefeld


 Nora Koenig
 vor 1 Jahren
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1 2 Zentrum für interdisziplinäre Forschung Center for Interdisciplinary Research Universität Bielefeld ZiFGremien Authorities 3 Editorial 4 ZiFForschungsgruppe Robust Finance: Strategic Power, Knightian Uncertainty, and the Foundation of Economic Policy Advice 10 Johannes Lenhard, Martin Carrier Mathematics as a Tool 20 ZiFInterview mit Oliver Razum 25 Rückblick Review Kunst am ZiF ZiF Art 40 Diana Sprenger: Perspektivenwechsel Künstler aus Japan, Südkorea, Brasilien und Deutschland: As Time Goes By 42 Das junge ZiF The Young ZiF 44 Notizen Notes 46 Neue Veröffentlichungen aus Projekten des ZiF ZiF New Publications 47 Aktuelle ZiFProjekte Current ZiF Projects 48 ZiFKalendarium Mai bis August 2015 Upcoming Events May to August 2015 MITTEILUNGEN
2 2 ZiFGREMIEN AUTHORITIES Der Wissenschaftliche Beirat Advisory Council Prof. Dr. Lorraine Daston (Wissenschaftsgeschichte, MPI für Wissenschafts geschichte, Berlin, GER) Prof. Dr. Herbert Dawid (Wirtschaftswissenschaft, U Bielefeld, GER) Prof. Dr. Walter Erhart (Literaturwissenschaft, U Bielefeld, GER) Prof. Dr. Elena Esposito (Soziologie, U Modena und Reggio Emilia, ITA) Prof. Dr. Gerd Gigerenzer (Psychologie, MPI für Bildungsforschung, Berlin, GER) Prof. Dr. Christopher Habel (Informatik, U Hamburg, GER) Prof. Dr. Jürgen Jost (Mathematik, MPI für Mathematik in den Natur wissenschaften, Leipzig, GER) Prof. Dr. Reinhold Kliegl (Psychologie, U Potsdam, GER) Prof. Dr. Sandrine Kott (Geschichtswissenschaft, U Genf, SUI) Prof. Dr. Thomas Noll (Biotechnologie, U Bielefeld, GER) Prof. Dr. Klaus Reinhold (Biologie, U Bielefeld, GER) Prof. Dr. Helge Ritter (Informatik, U Bielefeld, GER) Prof. Dr. Birgitt RöttgerRössler (Ethnologie, FU Berlin, GER) Prof. Dr. Dr. h. c. mult. Reinhard Selten (Volkswirtschaft, U Bonn, GER) Prof. Dr. Wolfgang Spohn (Philosophie, U Konstanz, GER) Prof. Dr. Peter Weingart (Soziologie, U Bielefeld, GER) Geschäftsführende Direktorin Managing Director Prof. Dr. Ulrike Davy Das Wissenschaftliche Direktorium Board of Directors Prof. Dr. Ulrike Davy, Fakultät für Rechtswissenschaft (geschäftsführende Direktorin) Prof. Dr. Martin Egelhaaf, Fakultät für Biologie (Prorektor der U Bielefeld) Prof. Dr. Marc Ernst, Fakultät für Biologie Prof. Dr. Joanna PfaffCzarnecka, Fakultät für Soziologie Prof. Dr. Michael Röckner, Fakultät für Mathematik (stellv. geschäftsführender Direktor) Dr. Britta Padberg (Vertreterin der wissenschaftlichen Mitarbeiterinnen und Mitarbeiter) Dipl.Soz. Mary Kastner M. A. (Vertreterin der weiteren Mitarbeiterinnen und Mitarbeiter) Das ZiF fördert als Institute for Advanced Study der Universität Bielefeld heraus ragende interdisziplinäre und innovative Forschungsprojekte. Das ZiF ist eine un abhängige, thematisch ungebundene Forschungseinrichtung und steht Wissenschaftlerinnen und Wissenschaftlern aller Länder und aller Disziplinen offen. Nähere Informationen unter: The ZiF is Bielefeld University s Institute for Advanced Study and fosters outstanding and innovative interdisci plinary research projects. The ZiF is an independent thematically open research institution and is open to scholars from all disciplines and all countries. Detailed informa tion can be found at: Geschäftsführerin Executive Secretary Dr. Britta Padberg Ulrike Davy Martin Egelhaaf Marc Ernst Joanna PfaffCzarnecka Michael Röckner
3 EDITORIAL EDITORIAL 3 Als am 11. März 2011 der Atomreaktor in Fukushima in Folge des verheerenden Tsunamis havarierte, tagte am ZiF die Forschungsgruppe Katastrophenkommunikation unter der Leitung von Jörg Bergmann, Heike Egner und Volker Wulf. Was folgte, bot reichhaltiges Material für Soziologen, die daran interessiert sind, wie Katastrophen kommunikativ bewältigt werden und inwieweit Gesellschaften bereit sind, aus Katastrophen zu lernen. Mit dem Abstand von vier Jahren ist heute noch deut licher als damals sichtbar, wie stark der kollektive Umgang mit Katastrophen von kulturellen Narrativen und Ängsten bestimmt wird. Die am ZiF noch bis zum 22. Mai laufende Ausstellung As Time Goes By zeigt künstlerische Aufarbeitungen dieser Ängste und Bedrohungsszenarien. Die Angst einiger Ausstellungsbesucher, dass Exponate aus dem Katastrophengebiet von Fukushima verstrahlt sein könnten, konnte von Bielefelder Physikern mit einem Geigerzähler inzwischen zweifelsfrei ausgeräumt werden. Wie konkret über das Allgemeine gedacht werden kann, wird das Thema des Kolloquiums sein, das am 3. Juli zu Ehren der LuhmannPreisträgerin Lorraine Daston stattfindet. Und wie das Allgemeine auf das Konkrete angewandt werden kann, ist das Thema der laufenden Kooperationsgruppe Mathematics as a Tool, zu der Sie einen Beitrag von Johannes Lenhard und Martin Carrier in diesem Heft finden. Der Frühsommer wird im ZiF aber vor allem von den Aktivitäten der Forschungsgruppe Robust Finance geprägt sein, die im März ihre Arbeit aufgenommen hat. Bis Ende Juli arbeiten zwanzig Wissenschaftlerinnen und Wissenschaftler aus sieben Nationen daran, die Aufspaltung von Finanzwirtschaft, Finanzmathematik und Wirtschaftswissenschaft in getrennte Fächer und die daraus resultierende Konstellation von Financial Engineering ohne Urteilskraft und Wirtschaftspolitik ohne mathematisches Wissen zu überwinden. On March 11, 2011 when the nuclear reactor at Fukushima was destroyed as a result of the devastating tsunami, the Research Group Communicating Disaster, convened by Jörg Bergmann, Heike Egner and Volker Wulf, was meeting at the ZiF. The dire consequences of the disaster provided an extensive range of material for sociologists interested in finding out how to cope with catastrophes in terms of communication and to figure out to what extent societies are prepared to learn from disasters. More than four years later, it has now become even more clear how far collective dealing with disasters is determined by cultural narratives and fears. The exhibition As Time Goes By presented at the ZiF up to May 22 shows the artists interpretations of the theme as well as a threat scenario. Therefore, it is not surprising that even some visitors were afraid to be contaminated by objects imported from Fukushima. However, this fear could be distracted without any doubt by Bielefeld physicists who tested hazards generated by radiation with a Geiger counter. On July 3, we will host a colloquium in honour of Lorraine Daston, Luhmann awardee, on How to Think Concretely About the General. And how to apply the general to the concrete, that is the theme of the current ZiF Cooperation Group Mathematics as a Tool, a subject also dealt with in a contribution to this issue written by Johannes Lenhard and Martin Carrier. Early summer at the ZiF is mainly dominated by activities of the Research Group Robust Finance that started its work in March. Until the end of July, 20 scholars from seven countries will approach the subject of overcoming the separation of financial economy, financial mathematics and economic sciences into different disciplines and the resulting constellation of Financial Engineering Without Judgment and Economic Policy Without Mathematical Sophistication. Wir freuen uns auf interessante Debatten und spannende Begegnungen und wünschen eine inspirierende Lektüre! Britta Padberg We are looking forward to interesting debates and fascinating encounters and hope you enjoy reading this issue, Britta Padberg
4 4 FORSCHUNGSGRUPPE RESEARCH GROUP Robust Finance: Strategic Power, Knightian Uncertainty, and the Foundation of Economic Policy Advice Convenors: Frank Riedel (Bielefeld, GER), Patrick Cheridito (Princeton, USA), Chris Shannon (Berkeley, USA) March July 2015 Our topic is Robust Finance, a topic that is related to today s headlines in the news, as the recent events around Greece show, e.g., and a topic that will preoccupy our societies for the times to come. The research group includes fellows from Berlin, Paris, Chicago, Wuhan in China, Princeton, York, and Minneapolis, and we are proud to host them in Bielefeld, which, as a city alone, would not immediately be part of the list of city names I just mentioned. The ZiF, and this might be an important side remark for the locals, is an opportunity to put Bielefeld on the international research map. More than hundred researchers from all over the world will visit the ZiF for the various workshops around our themes of finance, economics, mathematics, and philosophy. Of course, one might ask (and this was asked in the evaluation of this proposal) if, e.g., finance and mathematics, or economics and mathematics, really imply interdisciplinarity as most of today s economic analysis is cast in mathematical terms, especially in Bielefeld where we have institutionalized this tradition with the Center for Mathematical Economics. On the other hand, in the last years, partly triggered by the financial crisis, this close collaboration has come under attack: is Economics using too much of Mathematics? Shouldn t we do economic theory in less arcane ways, more human beings, less homines oeconomici? I am not so sure: maybe even the converse is correct. It might be that economics needs to do more, but the right kind of mathematics, if we want to understand, not just talk about our problems. But this is already a point we shall discuss today, and during our research group. When I moved from Bonn to Bielefeld University in 2007, Financial Mathematics was probably at its height. This theory had transformed financial markets worldwide, it created unique opportunities for banks and investors, did contribute to economic prosperity, and, yes, made quite a number of mathematicians rich, not the worst thing of course, but also an unexpected thing for a student of mathematics who started in the late 80s of the last century. Financial Mathematics is, maybe the biggest, but definitely one of the big successes of applications of mathematics in the sciences; it is definitely a scientific revolution, and, as revolutions tend to do, it went too far, maybe... We might want to ask ourselves, in our role as mathematicians, what our contribution was and still is. Has Financial Mathematics been misused or misinterpreted deceitfully? How did it happen that we created (too many) financial engineers without economic judgment? How do we need to change the use of financial mathematics in the banks to make the system more stable? The Economics profession, at least its publicly visible part, was taken by surprise when the crisis struck in 2008: they had simply not seen it coming. Many economists had long dismissed the revolutionary developments in the financial sector as irrelevant for the real economy until
5 FORSCHUNGSGRUPPE RESEARCH GROUP 5 it turned out that financial markets are more real than they thought. This fact leads to another goal of our group: we hope that the group can contribute to overcome the unfortunate separation of (Mathematical) Finance and Economics into two fields without much communication, financial engineering without judgment and economic policy advice without proper understanding of markets. The crisis is eight years old, and if you talk to lay people, at least in Germany, where the effects are not felt that harshly, they tell you that they cannot hear the word crisis anymore. Nevertheless: it is still here let me point out three important things: links Jour fixe der FG Robust Finance im Tagungsraum Long Table rechts oben Frank Riedel, RoseAnne Dana rechts unten Patrick Beißner, Ulrich Horst, Jinniao Qiu (v. l. n. r.) 1. Greece is at the brink of collapse, and might leave the Euro zone by exit or by accident, destabilizing this currency union as it destabilized the Latin Currency Union in 1927; and to my personal disappointment, we also come to see that not every game theorist is good in playing games The European Central Bank is going to spend 60 billion Euros, and has to repeat it: 60 billion Euros per month, a total of one trillion Euros, unheard sums of in times of peace, and an experiment without a good underlying theory We have negative nominal interest rates an economic absurdity, even if you find now easily people who come up with explanations like security issues etc. Nominal interest rates are a sign of a deep economic imbalance, and they may have the power to kill the institutions that behaved sound and safe in the crisis, savings banks and insurance companies.
6 FORSCHUNGSGRUPPE RESEARCH GROUP 6 We are certainly not the people who will solve all the practical and imminent problems that come with these issues. We are thinkers, and this is good so. Nevertheless, we believe that our fundamental research can be useful in overcoming successfully and lastingly the crisis. To illustrate this, let me remind you of an anecdote of the past. At the end of World War II, President Franklin D. Roosevelt asked the Director of the Office of Scientific Research and Development, Vannevar Bush, how science could best contribute to society s prosperity in the coming times of peace. The surprising answer was: by fundamental research, and leaving freedom to the scientists, having an eye on potential applications whenever new methods and ideas emerge. In this sense, we hope and believe that our research contributes to more stable financial markets in the long run. For thinkers like us, the ZiF is the ideal place for such an endeavor. Today, we are glad to open our time at the ZiF with five outstanding scholars who all combine in an excellent way their academic research with a sense of social responsibility for the greater good of society. Science needs to be communicated, and I am sure we will see outstanding examples of this difficult task today. Welcome to the ZiF! Frank Riedel, Introductory speech to the opening symposium of the ZiF Research Group 'Robust Finance' Fellows Peter Bank (TU Berlin, GER) Patrick Beißner (Universität Bielefeld, GER) RoseAnne Dana (Université Paris Dauphine, FRA) Frederik Herzberg (Universität Bielefeld, GER) Ulrich Horst (HumboldtUniversität zu Berlin, GER) Peter Klibanoff (Northwestern University Evanston, USA) Johannes Lenhard (Universität Bielefeld, GER) Qian Lin (Wuhan University, CHN) Sujoy Mukerji (University of Oxford, GBR) Daniele Pennesi (Université de CergyPontoise, FRA) Luca Rigotti (University of Pittsburg, USA) Birgit Rudloff (Princeton University, USA) JanHenrik Steg (Universität Bielefeld, GER) Jean Marc Tallon (Université Paris 1 Panthéon Sorbonne, FRA) Jacco Thijssen (University of York, GBR) Jan Werner (University of Minnesota, Minneapolis, USA) Associate Members Samuel Drapeau (TU Berlin, GER) Paolo Ghirardato (Collegio Carlo Alberto Moncalieri, ITA) Jinniao Qiu (HumboldtUniversität zu Berlin, GER) Emanuela RosazzaGianin (Università degli Studi di MilanoBicocca, ITA) Alexander Zimper (University of Pretoria, RSA) oben Nikoleta van Welbergen, Patrick Beißner, Lan Sun (v. l. n. r.) unten JeanMarc Tallon, Peter Klibanoff, Frank Riedel, RoseAnne Dana, Tolulope Fadina, Ulrich Horst, Patrick Beißner, JanHenrik Steg (v. l. n. r.)
7 FORSCHUNGSGRUPPE RESEARCH GROUP Opening Symposium Lessons of the Financial Crisis for Science and Politics Convenors: Frank Riedel (Bielefeld, GER), Patrick Cheridito (Princeton, USA), Chris Shannon (Berkeley, USA) 20 March 2015 Years after the financial crisis of 2007/08, a sufficient amount of time has passed to ask whether we have learned the lesson, and, if this is answered positively, if the right consequences have been drawn, both on the political as on the scientific side. To discuss these questions, five eminent scholars came to Bielefeld s Center for Interdisciplinary Research to share their insights with the research group on Robust Finance and 70 invited guests. The longtime director of the Ifo Institute, HansWerner Sinn, opened the conference with a talk on The Real Side of the Crisis. Sinn analyzed the current economic state of the European Union from which he derived the need for several European countries to change their relative prices and wages. Sinn first discussed dismal options. The transfer union, i.e. a unified Europe working according to similar principles as the different Länder do in Germany at the moment, will easily lead to the Dutch Disease an import of the weaknesses of one country via the subsidy mechanisms to other countries. Deflating the periphery of the European Union by austerity politics or inflating the core of Europe were also excluded by Sinn as reasonable options. The irreversible exit of countries like Greece from the Euro Zone would also have many negative externalities like bank runs, capital flight etc, as we can currently observe. What are better options? HansWerner Sinn makes three proposals: in a first step, a debt conference should cut the debts of the crisis countries; afterwards, a breathing monetary union where countries are allowed to leave the Euro zone temporarily with an option of reentry should be implemented; in the third step, harder local budget constraints after the example of the United States of America should be introduced. Peter Bank (Berlin, GER) Dietmar Bauer (Bielefeld, GER) Dirk Becherer (Berlin, GER) Geghard Bedrosian (Bielefeld, GER) Patrick Beißner (Bielefeld, GER) Stefan Berens (Bielefeld, GER) Philip Bergmann (Bielefeld, GER) Peter Bernholz (Basel, SUI) Philippe Blanchard (Bielefeld, GER) Gregor Böhl (Bielefeld, GER) Volker Böhm (Bielefeld, GER) Sonja Brangewitz (Paderborn, GER) Berno Büchel (Hamburg, GER) Rainer Buschmeier (Bielefeld, GER) Lasha Chochua (Bielefeld, GER) Oliver Claas (Bielefeld, GER) Fernando Cordero (Bielefeld, GER) RoseAnne Dana (Paris, FRA) 7
8 FORSCHUNGSGRUPPE RESEARCH GROUP 8 The Director of the MaxPlanckInstitute for Research on Collective Goods, Martin Hellwig, asked Has The Financial System Become Safer? and started right away with the remark: Safer is not safe. Indeed, in his opinion, the current changes in regulation that we have seen so far, can be compared to lowering the speed limit for trucks from 150 km/h to 140 km/h after a heavy truck accident in a tunnel has been observed. In other words: the changes of regulatory rules so far would not prevent the crisis of 2007/08! Basel III, in Hellwig s words, is rather Basel Hellwig also pointed to a failure of academia with an unwillingness to question established results. A good example how to deal with a banking crisis was set by Sweden in 1992 by a governmental takeover of the banking sector and a later reprivatization. The resolution of uncertainty and transparency in the sector could thus be restored. The former director of the Cowles Foundation for Research in Economics, John Geanakoplos, discussed the role of collateral on investment with a theoretical paper. His contribution indicated a way of thinking for economic theory in the aftermath of the crisis. His main result shows that the set of assets that may be used as collateral has effects on prices and investment in the economy. The introduction of collateralized debt obligations, e.g., can then lead to underinvestment. The economic theorist Itzhak Gilboa from Tel Aviv discussed the role of economic models as a basis to understand social sciences. In particular, the basic philosophical concept of refutability was discussed in detail. The idea of refuting a theory by an experiment as derived from Physics by Popper might not be a useful approach for the social sciences. As Amos Tversky pointed out, our theories are embarrassed, not refuted. The leading mathematician Walter Schachermayer from Vienna discussed the historical background of Mathematics in Finance ; starting with an overlooked thesis by Louis Bachelier in 1903, the theory of Brownian motion made a spectacular career in Finance with the help of Paul Samuelson and Robert Merton. This allowed to spawn the spectacular success of mathematics in finance, but led also to misdevelopments as the theory was applied to fields where its assumptions are simply not satisfied. Schachermayer asked for more modesty in applying our mathematical models. The five talks reflect the broad spectrum of the research group and led to a vivid discussion which will inspire the fellows for their time at the ZiF. Frank Riedel Ghislain Herman Demeze (Bielefeld, GER) Frederik Diermann (Bielefeld, GER) Bernhard Eckwert (Bielefeld, GER) Jürgen Eichberger (Heidelberg, GER) Tolulope Fadina (Bielefeld, GER) Giorgio Ferrari (Bielefeld, GER) Bettina Fincke (Bielefeld, GER) Florian Gauer (Bielefeld, GER) John Geanakoplos (New Haven, ZSA) Itzhak Gilboa (Tel Aviv, ISR) Simon Grant (Acton, AUS) Michael Günther (Bielefeld, GER) Philipp Harting (Bielefeld, GER) Tim Hellmann (Bielefeld, GER) Tobias Hellmann (Bielefeld, GER) Martin Hellwig (Bonn, GER) Niklas Herzig (Bielefeld, GER) JanOtmar Hesse (Bielefeld, GER) Ulrich Horst (Berlin, GER) Hermann Jahnke (Bielefeld, GER) Stefan Jonitz (Kirchlengern, GER) Klebert Kentia Tonleu (Berlin, GER) Hartmut Kliemt (Frankfurt am Main, GER) Philipp Külpmann (Bielefeld, GER) Jabob Landwehr (Bielefeld, GER) Andreas Lange (Hamburg, GER) Yuanyuan Li (Bielefeld, GER) Nadja Maraun (Paderborn, GER) Sujoy Mukerji (Oxford, GBR) Igor Muraviev (Bielefeld, GER) Elena Orlova (Bielefeld, GER) Daniele Pennesi (CergyPontoise, FRA) Lavinia PerezOstafe (Zürich, SUI) Luca Rigotti (Pittsburgh, USA) Walter Schachermayer (Wien, AUT) Willi Semmler (Bielefeld, GER) Gerlinde Sinn (München, GER) Hans Werner Sinn (München, GER) Mathias Staudigl (Bielefeld, GER) JanHenrik Steg (Bielefeld, GER) Nina Stephan (Paderborn, GER) Yuliia Stupnytska (Bielefeld, GER) Lan Sun (Bielefeld, GER) Andreas Szczutkowski (Bielefeld, GER) JeanMarc Tallon (Paris, FRA) Jacco Thijssen (York, GBR) Johannes Tiwisina (Bielefeld, GER) Hale Utar (Bielefeld, GER) Sander van der Hoog (Bielefeld, GER) Nikoleta van Welbergen (Bielefeld, GER) Carl Christian von Weizsäcker (Bonn, GER) Stefan Weber (Hannover, GER) Nicole Ziemnicki (Bielefeld, GER) Benteng Zou (Luxemburg, LUX) Volker Böhm, Peter Bernholz, Carl Christian von Weizsäcker (v. l. n. r.) Peter Bank (l.), Ulrich Horst
9 FORSCHUNGSGRUPPE RESEARCH GROUP Tagungsbeiträge Contributions Frank Riedel The ZiF Research Group Robust Finance: Strategic Power, Knightian Uncertainty and the Foundations of Economic Policy Advice HansWerner Sinn The real side of the crisis Martin Hellwig Has regulatory reform made the financial system safer? John Geanakoplos Collateral and Investment Itzhak Gilboa A model of modeling Walter Schachermayer Mathematics in finance Anfragen contact zur ZiFForschungsgruppe Robuste Finanzmärkte beantwortet der Forschungsgruppenassistent Oliver Claas Tel. +49(0) Informationen Further Information zur Forschungsgruppe Robuste Finanzmärkte p links Chris Shannon, Frank Riedel, Oliver Claas unten John Geanakoplos oben Martin Egelhaaf begrüßt als Prorektor der Universität Bielefeld die Gäste der Eröffnungstagung der neuen ZiFForschungsgruppe links HansWerner Sinn
10 10 Johannes Lenhard, Martin Carrier (Department of Philosophy, Bielefeld University) Mathematics as a Tool Johannes Lenhard does research in philosophy of science with a particular focus on the history and philosophy of mathematics and statistics. During the last years his research has concentrated on simulation modeling. He argues that it can be philosophically characterized as a new type of mathematical modeling. Currently, he works in a ZiF cooperation group on Mathematics As a Tool. Martin Carrier has worked on various fields in the philosophy of science History of Early Modern Physical Theory, Theory Change: Problems of Methodological Comparison and Confirmation Theory, SpaceTime Philosophy. His present chief research field is the methodology of applied research and the methodological changes imposed on science by the pressure of practice. Philosophy of science often focuses on the characteristics of fundamental research without taking into account that a large part of scientific research today is commissioned research, industrial research, or applied research, performed so as to accomplish shortterm practical goals. In such instances, the aims of research do not grow out of the smooth development of a discipline but are shaped by nonscientific problems, and the relevant timeframes are narrow. Kinds of bias may emerge under such conditions that are lacking in basic research but merit closer philosophical scrutiny. Mathematik als Werkzeug Unsere Kooperationsgruppe untersucht die Mathematik in ihrem Gebrauch als Werkzeug, weil wir an der Nützlichkeit der Mathematik interessiert sind, und zwar in solchen Bereichen, in denen die fundamentalen Gegenstände und ihre Beziehungen gerade nicht selbst schon von mathematischem Charakter sind und nicht mathematisch formulierten Naturgesetzen folgen. Eine Vielzahl wissenschaftlicher Disziplinen von der Klima wissenschaft über die Epidemiologie bis zur Linguistik arbeitet verstärkt mit mathematischen (insbesondere Computer) Modellen, ohne freilich dem klassischen Bild der mathematischen Physik zu folgen. Wie lässt sich die Funktion der Mathematik dort charakterisieren? Die Mathematik bietet einen Zugang zu allgemeinen Strukturen unabhängig von deren physikalischem (oder ontologischem) Charakter. Man kann die Mathematik als Instrument gebrauchen, so unsere These, um neue und sehr praktische Einsichten zu gewinnen in Gebieten, obwohl deren grundlegende Prinzipien sich einer Mathematisierung widersetzen. Anders als in der klassischen Konzeption der modernen Wissenschaft, gemäß der es bei der Mathematisierung darum geht, das»buch der Natur«(Galilei) zu entziffern, geht es bei der Mathematisierung in unserem Sinne eher um praktische, instrumentelle Vorzüge. Der folgende Aufsatz beginnt mit einem philosophischhistorischen Teil, der unseren Beitrag in die Diskussion um die Rolle der Mathematisierung in den Wissenschaften einordnet. Im zweiten, systematisch orientierten Teil werden fünf Merkmale der Mathematik als Werkzeug hervorgehoben, unter ihnen die Vermittlung zwischen Theorie und Beobachtungen, datengetriebene Forschung und die Rolle von Idealisierungen.
11 JOHANNES LENHARD, MARTIN CARRIER MATHEMATICS AS A TOOL What is meant when mathematics is called a tool? More than once, we received the spontaneous reply that the phrase would be inadequate because mathematics is much more than a tool. Rather than claiming that mathematics is merely a tool, the title of our ZiF cooperation group shall indicate that it is fruitful to look at mathematics insofar as it is used as a tool in the sciences. An investigation that is guided by this perspective will have to take into account two interrelated questions. How is this tool actually used and what characterizes mathematics as a tool? Both a philosophical conception and the consideration of actual practices are relevant. Then, however, we have to deal with a tension between empirical and normative viewpoints. On the one hand, the status and impact of mathematics should be explored by reconstructing scientific practice rather than by relying on general philosophical conceptions. The latter dominated philosophy of science a couple of decades ago. On the other hand, one should not uncritically accept every hype, but rather insist on critical reflection. Both aspects taken together circumscribe the rationale of our group. That mathematics plays an important role in many sciences is a commonly accepted fact. For a long time, leading disciplines of science featured or aimed at establishing fundamental mathematization. In a nutshell, the use of mathematics is understood as an important facet of what makes the hard sciences hard. Does the use of mathematics indicate fundamental significance? Is it a necessary condition for deciphering the workings of nature? And, if so, does mathematics play a fundamental role in constituting scientific knowledge? The investigation of important questions like these has some impact on how we conceive of science and scientific rationality. The role mathematics plays in the sciences has received different and conflicting assessments. A main distinction is the one between a strong and a weak view. Put very roughly, the former holds that mathematically formulated laws of nature refer to or reveal the rational structure of the world. The weak view denies that these fundamental laws are of an essentially mathematical character, and rather suggests that mathematics is merely a tool for systematizing observational knowledge about these laws and make additional use of these laws. Our point is that both views are not correct, precisely because both do not take mathematics as a tool into account. Let us work out this perspective in more detail, partly in (illuminating) contrast to popular accounts of mathematics and mathematization. We understand our endeavor as a fundamentally interdisciplinary one that needs to be informed from at least three angles: the recent practice of using mathematics, history, and philosophy of science. As a kind of preliminary result, we want to put forward a position that combines features of both, strong and weak, viewpoints. This position is supposed to characterize the use of mathematics in certain specific areas where mathematical reasoning is employed. It is not meant as a general position in the philosophy of mathematics, but rather intended to bring out characteristic features of making practical use of mathematical instruments. The position can be seen as a strong view about how mathematics functions as a tool. It is strong insofar as it assigns an active and even shaping role to mathematics. But at the same time it refrains from any claims about the mathematical structure of the universe that is (allegedly) mirrored by mathematical theories in the physical sciences. Employing mathematics as a tool is independent of the possible mathematical structure of the objects under consideration. Hence the tool perspective is contextual rather than ontological. When mathematics is used as a tool, it cannot be guided exclusively by internal mathematical reasoning. Instead, adequate tooluse is also a matter of the problem at hand and its context. Consequently, tooluse has to respect conditions like suitability, efficacy, optimality, and others. The notion of tool also stresses that there often is a spectrum of means to tackling a particular task that 11
12 JOHANNES LENHARD, MARTIN CARRIER MATHEMATICS AS A TOOL 12 will usually differ in how good they serve particular purposes associated with this task. The traditional philosophical viewpoint recognizes the permanent validity of mathematical theorems as a pivotal feature. The tool perspective, in contrast, underlines the inevitably provisional adequacy and validity of mathematics any tool can be changed, made better, or lose its adequacy. Mathematics as the Language of Nature the Promises and Limitations of Mathematization Let us circumscribe the traditional account of mathematics, elucidating the toolaccount by way of differences. The use of mathematical laws for describing and explaining natural phenomena is among the chief epistemic achievements of the Scientific Revolution of the seventeenth century. Medieval scholarship had joined Aristotle in emphasizing the difference between ideal mathematical postulates and real physical phenomena and had considered it impossible, for this reason, to accomplish a mathematical science outside of the ideal realm of celestial bodies. By contrast, the pioneers of the Scientific Revolution, such as Galileo Galilei, René Descartes, and Johannes Kepler, suggested seeking for mathematical laws of nature and conceived physics as a mathematized science. Galileo s law of freely falling bodies, Descartes s (or Snel s) law of refraction and his law of inertia, or Kepler s laws of planetary motion implemented this new idea of mathematical laws of nature. Underlying this new approach was the assumption that nature exhibits a mathematical structure. As Galileo put it, the book of nature is written in mathematical language; or in Kepler s words, God used geometrical figures in creating the world. In an influential historical account, Alexandre Koyré featured a Platonic vision of a mathematically structured nature as a methodological key element of the Scientific Revolution (Koyré 1968, 1978). Newton s Principia mathematica philosophia naturalis is often regarded as an early climax of the endeavor to capture the blueprint of the universe in mathematical terms. Michael Mahoney aptly pointed out that according ontological force to mathematical structure was a significant move during the socalled Scientific Revolution that brought nature into the realm of mathematics (Mahoney 1998). A second vision of the Scientific Revolution consisted in connecting understanding and intervention. Francis Bacon and Descartes developed the idea of an applied science in which knowledge about natural processes provides a basis of technology. The best way to take nature into the service of humans is to elucidate her inner workings. This idea has proven highly effective for mathematical science. The precision of mathematical laws of nature makes accurate predictions possible, which in turn make such laws suitable for supporting technical intervention. When the outcome of technical procedures needs to be anticipated with high precision, mathematical laws are promising to meet such requirements. However, the success of the mathematical sciences is not unlimited. It is true, these sciences have managed to enhance enormously their grip on complicated phenomena in the past 150 years. But: even if mathematical treatment was often able to strip off constraints of idealizations and controlled laboratory conditions and to cope with ever more complex systems, the full complexity and intricacy of realworld situations, until the present day, still poses various difficulties and obstacles to their mathematical treatment. An early example of a complexity problem is the socalled threebody problem that Henri Poincaré demonstrated to be unsolvable around The difficulty concerns the calculation of the motion of bodies under the influence of their mutual gravitational attraction. This dynamical model proved to be so complex that no analytical solution could be derived. Consequently, in spite of the fact that the underlying mathematical laws are known comprehensively, the longterm motions that result from their rule cannot be foreseen.
13 JOHANNES LENHARD, MARTIN CARRIER MATHEMATICS AS A TOOL A second, similarly famous example, and an early one for the study of complex systems, goes back to the meteorologist Edward Lorenz. He found in the early 1960s that a given set of differential equations produced different results depending on tiny variations in the initial conditions. Even minute dissimilarities, fluctuations, and uncertainties in these initial conditions led to quite distinct subsequent states. Lorenz s discovery is called deterministic chaos today: the time evolution of relevant physical systems depends so sensitively on the precise initial conditions that no longterm prediction is feasible. Although the nature of the system and the laws governing its time development are known without remainder, its future course cannot be anticipated reliably. More general, systems get complex when individual entities interact on one level and their interaction leads to emergent phenomena on a higher level, especially when the investigation of the higherlevel phenomena requires taking into account the details of lowerlevel interactions. In other examples of complexity, the means of mathematical modeling are insufficient for deriving useful solutions. The NavierStokes equations are believed to completely capture the behavior of fluids. But these equations cannot be solved in general; only for special cases solutions can be given. Likewise, Schrödinger s equation is taken to account comprehensively for nonrelativistic quantum phenomena. Yet already the helium atom can be treated only approximately. The complexity of the circumstances drives the mathematization of nature to its limits at least regarding the predictions of the course of nature and the targeted interventions in this course. In sum, the examples show that the concept of complexity has many facets. In a different vein, some fields of science appear much less suitable to mathematical treatment quite independently of their complexity. An important area are the life sciences. Molecular biology is mostly directed at disclosing mechanisms which are governed by spatial structures. Molecules mesh with each other and occupy receptors in virtue of the geometric properties. To be sure, all these mechanisms may be based on quantum laws, and spatial features can be expressed geometrically. But, up to these days, mathematical approaches hardly contribute significantly to understanding the relevant interactions. Spatial figures, mechanisms, and feedback loops are the essential concepts. Admittedly, mathematization has led to a couple of statistical rules, like Mendel s laws, or yielded reaction rates of biomolecules. But it is an open question whether mathematics will be able to add important further insights to the life sciences. In short, nature seems not to welcome mathematization indiscriminately. Thus, the adequacy of the strong view regarding the mathematization of nature seems to be constrained to rather narrow limits. Mathematical laws of nature and the option of putting them to use are restricted to a closely encircled range of sciences and phenomena. However, a look into recent sciences reveals that mathematical procedures flourish and fulfill various tasks outside this range. Fields such as molecular biology or systems biology make ample use of mathematical methods, though without embodying the vision of the strong view, i.e. without pretending to provide access to the inner workings of the pertinent mechanisms. This observation motivated us to change the perspective: 13 Mathematics as a Tool Mathematical analysis can be helpful in practical respect even if the pertinent fundamental processes cannot be understood or caught productively in mathematical terms. Mathematics deals with structures and such structures can be found at various places, not alone in the makeup of bodies and their interactions. It is part of the power of mathematical methods to be able to disclose features and identify patterns in all sorts of data, independently of their nature and origin.
14 JOHANNES LENHARD, MARTIN CARRIER MATHEMATICS AS A TOOL 14 As a result, mathematics is essential in bringing out structures in data and making use of them, as well as in establishing links between theory and data. As to the first item, mathematics is helpful for identifying data patterns and thus makes data driven research possible. The increase of computing power in the past decades has opened up a new path toward analyzing data, a path that does not lead through constructing theories and building models. As to the second item, mathematics is suitable for forging links between theories and phenomena. Measurement procedures can be accounted for mathematically even if the constitution of the pertinent objects and interactions is not governed by mathematical laws. For instance, schematic screening is dependent on mathematical methods. Highthroughput procedures or DNAarrays require mathematical assistance in registering and processing the data. Automated pattern recognition or statistical analysis are indispensable for gaining access to relevant features of the data. We speak of mathematics as a tool in order to designate the usefulness of mathematics even in areas where the fundamental objects and their interactions seem not to be governed by mathematical laws of nature. Mathematics as a general approach to disclosing and handling general structures irrespective of their physical nature can yield productive insights even for such areas. Using mathematics as a tool provides new opportunities for taking advantage of mathematical methods. Employing mathematics as an instrument in this sense can break new ground in practical respect even in fields whose basic principles are resistant to mathematization. Thus mathematization is analyzed regarding its instrumental virtues. It is not customary to look at mathematics from this perspective. The standard account of mathematics is centered on deducibility and provability and associated with coherence and certainty. The tool account is different in that mathematics is analyzed as an instrument, and, consequently, cast for an auxiliary role. If mathematics is used as a tool, the agenda is set by other fields; mathematics helps to solve problems that emerge within these fields. Consequently, the particular internal coherence of mathematics does not by itself guarantee progress. For example, mathematical considerations like deriving new consequences from already accepted model assumptions are only a first step in considering the adequacy of a mathematical model. Further, as our group has learned (or come to know) in a number of cases, the instrumental use of mathematics often proceeds in a rough and ready way and seems to be part of endeavors that look tentative and deficient in epistemic respect. Still, it is a virtue rather than a vice of mathematics that it proves to be helpful even under conditions that leave much to be desired in epistemic respect. Using mathematics as a tool is in no way meant in a pejorative sense. It rather emphasizes the productivity of mathematics in coping with structures of arbitrary nature. Using mathematics as a tool is by no means a recent development. On the contrary, the ancient slogan saving the phenomena was popular to describe the instrumental use of mathematics in astronomy. Mathematics was used to show the compatibility of observations with cosmological and physical principles and with additional auxiliary assumptions. This compatibility was established by reproducing and anticipating astronomical observations on their basis. However, many features of this mathematical account were not taken seriously as a description of nature. The narrow empirical basis of terrestrial observers was assumed to be insufficient for disclosing the true motions of the planets. As a result, in the tradition of saving the phenomena, the success of explaining and predicting celestial phenomena was not attributed to the truthful representation of the universe. Mathematics was used as a tool. Although the tool perspective is relevant for large parts of mathematics as it occurs in the sciences, it has not yet received much scholarly attention. This might come as a surprise given the
15 JOHANNES LENHARD, MARTIN CARRIER MATHEMATICS AS A TOOL recent practical turn in philosophy of science. One reason is that this perspective requires adopting an interdisciplinary approach that looks at mathematics in connection with other disciplines. The recent work on mathematical practice, however, primarily looks at what mathematicians do rather than focusing on how mathematics functions in a wider context (Mancosu 2008, van Kerkhove 2010). Among the factors that contributed to eclipsing the instrumental use from the philosophical and (to a lesser extent) from the historical view is the traditional distinction between pure and applied mathematics. This distinction suggests a division of labor between the pure branch that creates and validates mathematical knowledge and the applied branch that draws particular conclusions from this body of knowledge so as to solve extramathematical problems. On this approach, construction and application of mathematics look like two separate activities and the applied branch is marked as being completely epistemically reliant on the pure branch. A different approach maintains the distinction between pure and applied mathematics while reversing the hierarchical relation. Mathematical structures are then claimed to be abstractions from structures in the real world on which pure mathematics would build its theories. Mathematics, accordingly, would be primarily a natural science (Kitcher 1983, Maddy 1997). We do not share either view. The counterpoint we will develop abandons this assumption of unidirectional dependence and assumes, on the one hand, that using mathematics as a tool has an impact on the corpus of mathematical knowledge. New mathematical knowledge springs from developing methods designed to cope with practical challenges. Yet we also believe, on the other hand, that the system of mathematical knowledge shapes many practical or instrumentalist solutions. Mathematics and the various sciences we consider both benefit from proceeding in reciprocal dependence. Mathematics is no readymade repository simply to be tapped. On the contrary, problems and the tools to their solution coevolve. We want to propose five characteristic aspects of mathematics as a tool that exhibit significant differences to the standard account mentioned. These five partially overlapping topics will be discussed in the following section. 15 Five Characteristic Features We present five features that are intended to sketch characteristic modes of using mathematics as a tool. Each of them is rich enough to justify separate treatment, but there is also much overlap among them so that in practice concrete instances usually involve more than one feature. (i) Mathematics as a Mediator between Theory and Observation Mathematical theories are indispensable for mediating between theory and observation. This applies to measurement, i.e., to connecting theoretical quantities with data, as well as to technology. The use of mathematics is essential for registering phenomena and for generating and shaping phenomena. In the former case, mathematics is employed as observation theory, in the latter, it is used as an instrument of intervention in the course of nature. Mathematical theories of observation have been in use for a long time. Isaac Newton measured the gravitational attraction of the Earth by bringing mathematical laws of mechanics to bear on pendulum oscillation. Measuring electric current intensity by means of a galvanometer relies on Maxwell s equations. Similarly, the use of mathematics for ameliorating technological intervention has a long tradition. In 1824, Sadi Carnot was the first to apply highbrow theoretical principles (caloric theory) to the operation of advanced technical devices (the steam engine). His
16 JOHANNES LENHARD, MARTIN CARRIER MATHEMATICS AS A TOOL 16 analysis supported the conclusion that the only way to enhance the efficiency of the steam engine was to increase the relevant temperature differences. We will not dwell on such familiar ground. We rather aim to address observation theories of a hitherto unfamiliar sort. The pertinent mathematical procedures are not tied to specific physical circumstances or phenomena. Mechanics, electrodynamics, or thermodynamics are physical theories that are employed in registering mechanical, electrodynamical, or thermodynamical quantities. By contrast, the mathematical observation theories that interest us are independent of any particular realm of objects. Using neural networks, statistical procedures, or Bayesian formulas for surveying, exploring and interpreting the data, does not hook up with any properties of the objects under scrutiny. In other words, these mathematical procedures do not follow the causal path from the objects and their interactions to the display of certain values (as in the case of substantive observation theories). Rather, certain patterns in the data are identified irrespective of their physical nature and causal origin. The use of mathematics as an instrument of intervention is often not strictly separated from its use as an observation theory. An example is presented by testing fertilizers in agriculture where a mathematical model is used to conceptualize the situation by defining parameters that capture the dynamics and make it accessible for experimentation. Certain parameters, for instance the fraction of a certain substance, might be varied in a systematic way. A certain way of mixing and of distributing the fertilizer may prove (economically) optimal. Then the same mathematical procedures are utilized for observing and for intervening. The point can be generalized: The quest for certainty has repeatedly been discussed as a characteristic feature of modern science with mathematization playing a central role in providing certainty. The instrumental perspective transforms this viewpoint. Now, mathematics is related more to a quest for optimality, providing systematic help to plan and carry out interventions. (ii) Datadriven Research and the Use of Big Data Generalizing these considerations leads to the phenomenon of datadriven research. Datadriven research can be contrasted with modeldriven research, in which theoretical expectations or a microcausal model distinguishes certain patterns in the data as significant. Datadriven research is different in that it starts off from the data without an underlying model that decides about their significance. For instance, in pharmaceutical research, large amounts of potential agent substances are scanned automatically (highthroughput screening). The question posed merely is whether a certain substance binds with a certain intensity to a certain receptor. Masses of data are produced by such a largescale screening process, and the result always is that certain patterns are selected. Notwithstanding the great hype of Big Data that is related to the large collections of data (amazon customers, google map online users, etc.), the crucial question in methodological respect is whether datadriven research is really tantamount to shaking profoundly the heuristic and explanatory strategies in mathematical science. Our ongoing research on this point will hopefully provide a fuller account in the near future. (iii) Tuning Models Mathematically Theoretical mathematical models are rarely perfect. Even if a model is very good, it regularly provides a kind of mathematical skeleton that includes a number of parameters whose quantitative assignment has to be read off from empirical data. A case in point is the gravitational constant that does not follow from mathematical arguments though it does follow mathematically that there is a gravitational constant. Such parameter evaluations are normal and not specific for using mathematics as a tool.
17 JOHANNES LENHARD, MARTIN CARRIER MATHEMATICS AS A TOOL However, it is a different matter when important features of models are determined by the particular problem at hand, using mathematical procedures for enriching models with refinements, adjustments, and correction factors. Tuning then plays an essential role and therefore these procedures are to be distinguished from filling numerical gaps left in theorybased models. There are many options in between where neither the mathematical structure determines the behavior nor is learning or adaptation completely open. Typically, and arguably most importantly, tuning is involved whenever the discrete structure of computer models has to be fit to theoretically described or empirically measured phenomena. 17 (iv) Using Alternative Routes to Solving Equations Solving differential equations by a computer simulation proceeds by not literally solving these equations but rather by calculating values of its discretized proxy. Solutions are calculated point by point at a grid and for specific parameter values. As a result, using computational models does not simply mean to enhance the performance of mathematical models but rather changes the ways in which these models are constructed and the modes in which they operate. Digital computers require discrete versions of all relevant objects and operations. This requirement presents a kind of instrumental imperative that drives traditional mathematical modeling into new pathways. Most notably, traditional mathematics is full of continuous quantities and relationships. Yet they need to be remodeled as discrete entities so as to become tractable by digital computers. Typically, the discretization produces unwanted effects that then have to be neutralized by artificial measures. An early example is the artificial viscosity that John von Neumann came up with to reintroduce the possibility of very steep wavefronts after discretization of supersonic waves (Winsberg 2003, for a case in meteorology see Lenhard 2007). Parameterization schemes arise from the need to find discrete counterparts to the quantities in continuous models. A telling example is the dynamics of clouds whose (micro)physical basis is to a large part not yet known. However, cloud dynamics forms an important part of the general circulation models of the atmosphere. They involve the calculation of this dynamics from values defined at grid points. Clouds, however, are subgrid phenomena (a typical grid may work with horizontal cells of 100 km x 100 km). Hence the effects of clouds for the whole dynamics need to be expressed as a kind of net effect at the grid points, i.e. they have to be parameterized. What are appropriate parameterization schemes? What are effective parameters that adequately summarize cloud dynamics? How are these parameters estimated reliably? These questions are interdependent and can be solved only by extensive simulation procedures that assume parameter values, observe the model dynamics, readapt the values, etc. Hence adequate parameterization has to find a balance between accurate description and effective manipulability. Admittedly, this point is closely related to tuning. As we said, the issues are partly overlapping. Another example are socalled multiscale methods. To describe the binding behavior of, say, an enzyme, researchers employ a couple of different models that are based on different theoretical approaches. An enzyme is built from amino acids and contains typically about 8000 atoms, but only a small fraction of those atoms, maybe 50, is relevant for the binding behavior. These atoms are modeled with quantum mechanical methods that provide the most detailed view and the most precise results but are computationally too expensive to cover the entire enzyme. Even for these about 50 atoms one has to reside to relatively cheap modeling strategies, like density functional theory. The remaining majority of atoms is modeled via (classic) force fields, i.e. with socalled molecular dynamics methods. This is much coarser and ignores quantum effects, but experience has shown that this does not result in a significant loss in adequacy of model behavior.
18 JOHANNES LENHARD, MARTIN CARRIER MATHEMATICS AS A TOOL 18 The largest errors occur through the ways these two regimes are coupled together. The coupling cannot be determined theoretically as the two regimes are not compatible. The only solution is to introduce a tentative coupling and to adapt it so that the coupled model attains a good fit to already known cases. Such loops of adaptations, checks, and readaptations treat the coupling mechanism as a variable, without these loops multiscale methods could hardly be used at all. In short, mathematics is used as a tool for establishing a goodenough, or even optimal, compromise between accuracy and tractability. (v) NonRepresentational Idealizations Idealizations are intimately connected to the role of mathematics in the sciences. All mathematical operations inevitably deal with objects or models that are in some sense idealized. The point is what sense of idealization is the relevant one in our context. According to a straightforward view, we can think of idealization as a departure from complete, veridical representation of realworld phenomena (Weisberg 2013, 98). All mathematical models suit this description. They can be adequate without having to be completely faithful representations. The interesting point rather is in what ways they can deviate from the representational ideal. We want to introduce a type of idealization that cuts across the various suggestions to capture different sorts of idealized models. All these suggestions deal with objectrelated idealizations. Idealizations create a simpler version of the relevant objects and their relationships so that mathematical models of them become more tractable. In principle, these simplifications could be removed by deidealizing the model step by step and thereby making the models more and more complex. However, from our perspective, the most important distinction is another one, namely, between objectrelated and toolrelated idealizations. The latter result from the properties of the tool and make sure that the tool can be used in the first place. Toolrelated idealizations exhibit no clear relation to issues of representation. The question of how adequate the tool works in a certain situation needs to be tackled independently. Philosopher of science Robert Batterman (2010, see also the chapter in this volume) has pointed out that asymptotic limits or singularities regularly destroy the representation relation, because the objects covered by the model simply do not exist. Synthesis We have discussed five modes of using mathematics as a tool which partly overlap and are partly heterogeneous in kind. The instrumental use of mathematics is not governed by a single scheme. Rather, the heterogeneity of tools and of tool usages teaches a lesson about the limits of mathematization. Though our group is still making progress and this essay is hence of a preliminary nature, we would like to highlight two preliminary findings separately: Control replaces explanation, and validation is accomplished by use. Control often is a practical goal, for instance when a satellite has to be navigated into a stable orbit. The clincher of using mathematics as a tool lies in providing knowledge about how to control processes or systems. In the traditional understanding of the role of mathematics in the sciences, its usefulness in formulating comprehensive, unifying accounts of the phenomena is stressed. Mathematics provides the deductive links that connect the first principles with their empirical consequences and thus produces a pyramidshaped body of knowledge. The rule of first principles over the phenomena is established by mathematical derivations. In the present context of mathematics as a tool, however, the deductive links are often tenuous, restricted to a narrow
19 JOHANNES LENHARD, MARTIN CARRIER MATHEMATICS AS A TOOL realm of conditions, forged in an adhoc manner, or intransparent. As a result, the explanatory power conferred on the principles by applying mathematics is considerably weakened. Yet the power of prediction need not be reduced at the same time. On the contrary, accurate predictions are an essential criterion for judging the quality of mathematical tools. One of the conclusions to be drawn from such consideration is that predictive power may not be completely dependent on insights into the inner workings of nature. Instead, predictive power is often established by using mathematics as an instrument. Consequently, the validation of such tools is accomplished by examining their practical achievements. The pivot of validation is practical use. We think a general lesson can be learned from regarding mathematics as a tool. Namely, it challenges the philosophical viewpoint that modern science is characterized by a uniform method and also calls into question the role mathematics plays in arguments about science and rational order. 19 Acknowledgement We would like to mention that this essay profited greatly from the discussion in our cooperation group that includes Philippe Blanchard, Jürgen Jost, and Michael Röckner. Cited literature Batterman, Robert W. (2010) On the Explanatory Role of Mathematics in Empirical Science, British Journal for the Philosophy of Science 61, Kitcher, Philip (1983) The Nature of Mathematical Knowledge, New York: Oxford University Press. Koyré, Alexandre (1968) Newtonian Studies, Chicago: University of Chicago Press. Koyré, Alexandre (1978) Galileo Studies, Hassocks: Harvester Press. Lenhard, Johannes (2007) Computer Simulation: The Cooperation Between Experimenting and Modeling, Philosophy of Science 74, Maddy, Penelope (1997) Naturalism in Mathematics, Oxford: Clarendon Press. Mahoney, Michael S. (1998) The Mathematical Realm of Nature, in D.E. Garber et al.(eds.), Cambridge History of SeventeenthCentury Philosophy, Cambridge: Cambridge University Press, Vol. I, pp Mancosu, Paolo (ed.) (2008) The Philosophy of Mathematical Practice, Oxford: Oxford University Press. Van Kerkhove, Bart, Jonas de Vuyst, and Jean Paul van Bendegem (eds.) (2010) Philosophical Perspectives on Mathematical Practice, London: College Publications. Weisberg, Michael (2013) Simulation and Similarity: Using Models to Understand the World, Oxford: Oxford University Press. Winsberg, Eric (2003) Simulated Experiments: Methodology for a Virtual World, Philosophy of Science 70,
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