DOCSLIB.ORG

The Certainty of Mathematics a Philosophical Investigation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Certainty of Mathematics a Philosophical Investigation Kim-Erik Berts The Certainty of Mathematics A Philosophical Investigation Mathematics is often said to give us knowledge that it more reliable than that of other sciences. Kim-Erik Berts What do we mean by ‘mathematical certainty’? Do we have a clear understanding of this concept? Kim-Erik CertaintyThe Berts of Mathematics | 2016 | This investigation is devoted to the certainty of The Certainty of Mathematics mathematics. The starting point is that we must attend to our practice of mathematics. Through A Philosophical Investigation this approach, mathematical knowledge is seen to involve a skill in working with the concepts and symbols of mathematics, and its results are seen to be similar to rules. This normativity indicates the sense in which mathematics is certain. 9 789517 658423 Åbo Akademi University Press | ISBN 978-951-765-842-3 Kim-Erik Berts (b.1980) MA, Philosophy (Åbo Akademi University, 2006) BA, Philosophy (ÅAU, 2004) Åbo Akademi University Press Tavastgatan 13, FI-20500 Åbo, Finland Tel. +358 (0)2 215 3478 E-mail: [email protected] Sales and distribution: Åbo Akademi University Library Domkyrkogatan 2–4, FI-20500 Åbo, Finland Tel. +358 (0)2 -215 4190 E-mail: [email protected] THE CERTAINTY OF MATHEMATICS The Certainty of Mathematics A Philosophical Investigation Kim-Erik Berts Åbo Akademis förlag | Åbo Akademi University Press Åbo, Finland, 2016 CIP Cataloguing in Publication Berts, Kim-Erik. The certainty of mathematics : a philosophical investigation / Kim-Erik Berts. - Åbo : Åbo Akademi University Press, 2016. Diss.: Åbo Akademi University. ISBN 978-951-765-842-3 ISBN 978-951-765-842-3 ISBN 978-951-765-843-0 (digital) Painosalama Oy Åbo 2016 For Agnes and Gunnar Acknowledgements I have always been fascinated by mathematics. As a child, the fact that I could handle huge numbers was thrilling. Later on, this fascination was evoked by learning new things in mathematics, but it was also sometimes mixed with the frustration that can accompany the struggle with mathematical concepts. Dur- ing my studies at Åbo Akademi University, it became clear to me that it was the philosophical questions concerning mathematics that captured my interest. This philosophical interest in mathematics was also what gave me the impulse to attend university courses in mathematics to begin with. I am very grateful that I have been given the opportunity to devote so much time and attention to philosophical thinking about mathematics. This thesis is the outcome of a ten-year process. The first years were spent working full-time on the thesis. To be able to devote oneself full-time is vital to the success of this kind of project. The last four years, however, writing was done during time off from my work as a teacher of philosophy and mathematics at Vasa gymnasium. Although this has slowed down the process, teaching philosophy and mathem- atics has given me many important insights into how we learn and understand mathematics – not to mention the need to express oneself as clearly as possible. I would like to thank my supervisors Professor Lars Hertzberg and Professor Martin Gustafsson for their generous help and support during this process. Lars has created a lively research environment at the Department of Philosophy at Åbo Akademi University, and I am very lucky that I have had the chance to be a part of it. Lars’s way of doing philosophy is deeply inspiring and it has greatly influenced the work put forth in this thesis. Martin became my supervisor when he was appointed to the Philosophy chair in 2010. I was very happy to see his sharp, meticulous, and helpful comments improve the text in new ways. I have been very fortunate to have Lars and Martin as my supervisors. The first year that I was working on this thesis I spent at the Department of Philosophy at Uppsala University. During this year, I had the opportunity to discuss the philosophy of mathematics with Professor Sören Stenlund. I would like to express my gratitude to Sören for his many constructive comments on drafts of the chapters but, more importantly, for the insights into the subject that he has given me in numerous discussions. The influence of his way of doing philosophy of mathematics can be seen throughout this thesis. During the first course in philosophy that I attended at Åbo Akademi Univer- sity, Kim Solin made an interesting comment about the relation between math- ematics and the world. This comment was enough to make me head for the De- partment of Mathematics and begin my studies there. As he has also become my vii viii Acknowledgements good friend and colleague, I have had the opportunity to learn much from him – in our work together and in our many stimulating discussions. I am deeply grateful for this. I would also like to thank the participants in the Research seminar at the De- partment of Philosophy at Åbo Akademi University for many inspiring discus- sions. In particular, I would like to mention Hugo Strandberg and Göran Tor- rkulla for their valuable comments on drafts of the chapters. The atmosphere at the department is encouraging and inspiring and I am happy to be a part of it. Thanks to teachers, colleagues, and friends at the department: Jonas Ahl- skog, Benjamin Alm, Joel Backström, Stina Bäckström, Antony Fredriksson, Ylva Gustafsson, Camilla Kronqvist, Olli Lagerspetz, Mari Lindman, Yrsa Neu- man, Hannes Nykänen, Marcus Prest, Hans Rosing, Patrick Sibelius, Åsa Slotte. Thank you also to the participants in the Seminar of the Philosophy of Lan- guage at the Department of Philosophy at Uppsala University and the Seminar for the History and Pedagogy of Mathematics at the Department of Mathematics in Uppsala, in particular Anders Öberg for his thought provoking comments. I am also grateful for the support and encouragement that my parents, Mar- gareta and Lars-Erik, constantly give me. My embarking on a several-year-long project in the philosophy of mathematics, associated with a constant insecur- ity with regard to income, has not been met with suspicion, but with interested questions and encouraging words. For my wife Johanna and our children Agnes and Gunnar, who have witnessed the writing process first hand, I reserve a spe- cial ‘Thank you!’. The discussions that we have had about writing and learning mathematics have, each in their own way, given me many insights that relate directly to this thesis. During several summer holidays and many weekends, I have been working on the text instead of taking time off with my family and I am grateful that they have given me this time. Now, I can finally answer Agnes’s and Gunnar’s repeated question ‘Pappa, är din avhandling färdig nu?’ with ‘Ja!’ *** My work on this thesis has been enabled by the generous support of Stiftelsen för Åbo Akademi, Waldemar von Frenckells stiftelse, Vera och Greta Oldbergs stiftelse, The Finnish Society of Sciences and Letters, Stiftelsens för Åbo Akademi forskningsinstitut, Finnish Academy of Science and Letters, and Seniorernas råd vid Åbo Akademis studentkår. Vasa, 28 October 2016 Kim-Erik Berts Contents 1 Introduction 1 2 Certainty 7 2.1 Hilbert on Certainty 10 2.2 The Infallibility of Mathematical Methods 12 2.3 Consistency 16 2.4 Eternal Truths 19 2.5 Deductive and Empirical Sciences 20 2.6 Uncertainty 21 2.7 Certainty and Being Certain 23 3 Knowledge 37 3.1 From the Science of Quantity to a Body of Truths 39 3.2 Benacerraf and the Contemporary Discussion 45 3.3 Two Perspectives on Mathematics 49 3.4 Studying a Mathematical Object 54 3.5 Truth and Referential Semantics 57 3.6 Structuralism 61 3.7 Concluding Remarks 69 4 Formality 73 4.1 The Formal and the Intuitive 74 4.2 Historical Background to Formal Systems 78 4.3 Meaningless Signs 82 4.4 Advantages of Symbolism 89 4.5 Where is Genuine Mathematics? 95 4.6 Formal and Informal Proofs 98 5 Proof 103 5.1 The Role of Conviction 104 5.2 The Role of Understanding 111 5.3 Proof and Concept-formation 114 5.4 Proof and Experiment 121 ix x Contents 5.5 Proof and Surveyability 126 5.6 Simple Deductions 136 5.7 Concluding Remarks 142 6 Conclusion 143 Svensk sammanfattning – Swedish summary 147 Bibliography 149 1. Introduction A traditional question in philosophy concerns what, if anything, we can know with certainty. Mathematics is the only field of knowledge concerning which there has been some rough consensus that it actually gives us certain knowledge. Even though some philosophers have preferred not to limit the range of certainty to mathematics alone, this discipline is often put forward as an ideal: ‘this par- agon of reliability and truth’, as David Hilbert proclaimed.1 Certainty is an enticing concept: we want to be certain, we want to have cer- tain knowledge. We want to be sure that we have not made any mistakes and to have a safeguard against error. This is a desire that is motivated by practical reasons: life is easier if one can avoid (at least sometimes) the consequences of error or of being wrong. It is motivated by social reasons too: being wrong can be embarrassing, and being in the know is deeply satisfying. In many situations, the search for certainty makes us turn to mathematics. Mathematical techniques are often what brings certainty. They allow us to over- view situations in such a way that we see clearly what must be the case. When we have settled something using mathematical tools, we can rely on that knowledge. From a philosophical perspective, the question arises: Why is it that mathem- atics has this status? This question is raised already by the role that mathematics plays in our ordinary lives, and the urgency of it is only strengthened by the role that mathematics plays in the sciences.
Load more

Recommended publications
  • Testimony Using the Term “Reasonable Scientific Certainty”
    NATIONAL COMMISSION ON FORENSIC SCIENCE Testimony Using the Term “Reasonable Scientific Certainty” Subcommittee Reporting and Testimony Type of Work Product Views Document Statement of the Issue It is the view of the National Commission on Forensic Science (NCFS) that legal professionals should not require that forensic discipline testimony be admitted conditioned upon the expert witness testifying that a conclusion is held to a “reasonable scientific certainty,” a “reasonable degree of scientific certainty,” or a “reasonable degree of [discipline] certainty.” The legal community should recognize that medical professionals and other scientists do not routinely use “to a reasonable scientific certainty” when expressing conclusions outside of the courts. Such terms have no scientific meaning and may mislead factfinders [jurors or judges] when deciding whether guilt has been proved beyond a reasonable doubt. Forensic science service providers should not endorse or promote the use of this terminology. The Commission recognizes the right of each jurisdiction to determine admissibility standards but expresses this view as part of its mandate to “develop proposed guidance concerning the intersection of forensic science and the courtroom.” Forensics experts are often required to testify that the opinions or facts stated are offered “to a reasonable scientific certainty” or to a “reasonable degree of [discipline] certainty.” Outside of the courts, this phrasing is not routinely used in scientific disciplines, a point acknowledged in the Daubert decision (“it would be unreasonable to conclude that the subject of scientific testimony must be ‘known’ to a certainty; arguably, there are no certainties in science.”). Daubert v. Merrell Dow Pharms., 509 U.S. 579, 590 (1993).
    [Show full text]
  • Al-Ghazali and Descartes from Doubt to Certainty: a Phenomenological Approach
    AL-GHAZALI AND DESCARTES FROM DOUBT TO CERTAINTY: A PHENOMENOLOGICAL APPROACH Mohammad Alwahaib Abstract: This paper clarifies the philosophical connection between Al-Ghazali and Descartes, with the goal to articulate similarities and differences in their famous journeys from doubt to certainty. As such, its primary focus is on the chain of their reasoning, starting from their conceptions of truth and doubt arguments, until their arrival at truth. Both philosophers agreed on the ambiguous character of ordinary everyday knowledge and decided to set forth in undermining its foundations. As such, most scholars tend to agree that the doubt arguments used by Descartes and Al-Ghazali are similar, but identify their departures from doubt as radically different: while Descartes found his way out of doubt through the cogito and so reason, Al-Ghazali ended his philosophical journey as a Sufi in a sheer state of passivity, waiting for the truth to be revealed to him by God. This paper proves this is not the case. Under close textual scrutiny and through the use of basic Husserlian-phenomenological concepts, I show that Al-Ghazali's position was misunderstood, thus disclosing his true philosophic nature. I. Introduction This paper clarifies the philosophical relation between Al-Ghazali, a Muslim philosopher (1058--1111), and the French philosopher Rene Descartes (1596--1650), with the objective of articulating the similarities and differences in their famous journeys from doubt to certainty. Historical evidence on whether Descartes did in fact read or had knowledge of Al-Ghazali’s work will not be discussed in this paper. Instead, this paper focuses primarily on the chain of their reasoning, starting from their conceptions of truth and the arguments used by each to destroy or deconstruct the pillars of our knowledge and thus reach truth.
    [Show full text]
  • Mathematical Scepticism: the Cartesian Approach Luciano Floridi
    Mathematical Scepticism: the Cartesian Approach1 Luciano Floridi Wolfson College, Oxford, OX2 6UD, UK [email protected] - www.wolfson.ox.ac.uk/~floridi Introduction Paris, 15 February 1665: Molière’s Don Juan is first performed in the Palais-Royal Hall. Third Act, First Scene: the most daring of Don Juan’s intellectual adventures takes place. In a dialogue with his servant Sganarelle, Don Juan makes explicit his atheist philosophy: SGANARELLE. I want to get to the bottom of what you really think. Is it possible that you don’t believe in Heaven at all? D. JUAN. Let that question alone. SGANARELLE. That means you don’t. And Hell? D. JUAN. Enough. SGANARELLE. Ditto. What about the devil then? D. JUAN. Oh, of course. SGANARELLE. As little. Do you believe in an after life? D. JUAN. Ha! ha! ha! SGANARELLE. Here is a man I shall have a job to convert. […] SGANARELLE. But everybody must believe in something. What do you believe in? D. JUAN. What do I believe? SGANARELLE. Yes. D. JUAN. I believe that two and two make four, Sganarelle, and four and four make eight. SGANARELLE. That’s a fine thing to believe! What fine article of faith! Your religion is then nothing but arithmetic. Some people do have queer ideas in their heads, and those that have been educated are often the silliest. I never studied, thank God, and no one can boast he taught me anything. But, to my poor way of thinking, my eyes are better than books. I know very well that this world we see around us is not a mushroom grown up in a single night.
    [Show full text]
  • On Certainty (Uber Gewissheit) Ed
    Ludwig Wittgenstein On Certainty (Uber Gewissheit) ed. G.E.M.Anscombe and G.H.von Wright Translated by Denis Paul and G.E.M.Anscombe Basil Blackwell, Oxford 1969-1975 Preface What we publish here belongs to the last year and a half of Wittgenstein's life. In the middle of 1949 he visited the United States at the invitation of Norman Malcolm, staying at Malcolm's house in Ithaca. Malcolm acted as a goad to his interest in Moore's 'defence of common sense', that is to say his claim to know a number of propositions for sure, such as "Here is one hand, and here is another", and "The earth existed for a long time before my birth", and "I have never been far from the earth's surface". The first of these comes in Moore's 'Proof of the External World'. The two others are in his 'Defence of Common Sense'; Wittgenstein had long been interested in these and had said to Moore that this was his best article. Moore had agreed. This book contains the whole of what Wittgenstein wrote on this topic from that time until his death. It is all first-draft material, which he did not live to excerpt and polish. The material falls into four parts; we have shown the divisions at #65, #192, #299. What we believe to be the first part was written on twenty loose sheets of lined foolscap, undated. These Wittgenstein left in his room in G.E.M.Anscombe's house in Oxford, where he lived (apart from a visit to Norway in the autumn) from April 1950 to February 1951.
    [Show full text]
  • Value in Experience, Thoughts on Radical Empiricism; Reflections on Frankenberry and Stone1
    Value In Experience, Thoughts on Radical Empiricism; Reflections on Frankenberry and Stone1 David W. Tarbell Modern philosophy, going back to Descartes and Locke, has given preferred status to the kind of judgments that we find within the mathematical sciences and has made it difficult to support judgments of value and meaning. In Chapter Four of his book, The Minimalist Vision of Transcendence, Jerome Stone describes what he calls a "Generous Empiricism." He is building on the tradition of radical empiricism, and presents a transactional understanding of experience that allows him to support empirically grounded value judgments. I argue that a critique from the perspective of Heidegger and others of the continental tradition may present problems to Stone's approach. I then propose a modification of Stone's notion that I think would answer such criticisms. This leads me to the notion that our value judgments have an aesthetic component, and that we can find pragmatic support for the belief that there is an aesthetic tendency in the world of which we are a part. There is something divine and possibly transcendent in that thought. I. The Value Problem: One of the factors affecting the moral, ethical, and political climate of our present culture is diversity of judgment on matters of value. Some have said that the presence of so much diversity argues against our ability to resolve questions of truth in these areas. Their positions vary among skepticism, denying that we can know moral truth, nihilism, questioning that there is such a thing as moral truth, or relativism, claiming that moral truth is conditional on social and/or cultural circumstances.
    [Show full text]
  • To a Moral Certainty: Theories of Knowledge and Anglo-American Juries 1600-1850 Barbara J
    Hastings Law Journal Volume 38 | Issue 1 Article 2 1-1986 To A Moral Certainty: Theories of Knowledge and Anglo-American Juries 1600-1850 Barbara J. Shapiro Follow this and additional works at: https://repository.uchastings.edu/hastings_law_journal Part of the Law Commons Recommended Citation Barbara J. Shapiro, To A Moral Certainty: Theories of Knowledge and Anglo-American Juries 1600-1850, 38 Hastings L.J. 153 (1986). Available at: https://repository.uchastings.edu/hastings_law_journal/vol38/iss1/2 This Article is brought to you for free and open access by the Law Journals at UC Hastings Scholarship Repository. It has been accepted for inclusion in Hastings Law Journal by an authorized editor of UC Hastings Scholarship Repository. "To A Moral Certainty": Theories of Knowledge and Anglo-American Juries 1600-1850 by BARBARA J. SHAPIRO* In many American jurisdictions the jury is instructed that the prose- cution must prove the defendant's guilt "beyond a reasonable doubt and to a moral certainty." At first glance, "to a moral certainty" appears to be one of those typical redundancies of common law that seek to cure ambiguity by compounding it. Because it troubles us that "beyond rea- sonable doubt" conveys no very precise meaning, we add a second phrase, "to a moral certainty," but it conveys even less meaning and makes our whole problem worse. Only a few quite well-educated older people who have read a great deal of nineteenth-century literature are likely even to have said, "I am morally certain that you left your coat in the restaurant" or "Are you morally certain that you came into the room before he did?" It is the kind of phrase that a screenwriter might put in the mouth of a country storekeeper to suggest a slightly bookish, straight-laced, religious old man still living in an earlier age.
    [Show full text]
  • Augustine's Certainty in Speaking About Hell and His Reserve In
    Chapter 2 Augustine’s Certainty in Speaking about Hell and His Reserve in Explaining Christ’s Descent into Hell Paul J. J. van Geest 1 Introduction About two years after Augustine was ordained to the priesthood – in 391, when he was 37 years old – a controversy arose surrounding Origen, one of the most admired and authoritative Christian authors in those days.1 Origen was ac- cused of denying that punishment in hell is for all eternity. We may deduce just from the titles of the works that Augustine wrote before or around this time, such as De ordine, De beata vita, Contra Academicos or a volume of De libero arbitrio, that he was not particularly interested in this matter. It was not until his ordination as a bishop, in 395, that Augustine began to formulate thoughts on hell and Christ’s descent there. At this time the subdeacon Asterius and the deacon Presidius, who acted as Jerome’s messengers, explained to him the ob- jectionable nature of Origen’s points of view.2 Then Augustine asked Jerome, in Epistula 40, to send him a list of Origen’s ‘errors’.3 By reading Contra Rufinum, Augustine subsequently learned that Jerome considered Origen’s ideas on the restoration of everything and everyone (ἀποκατάστασις) to be incompatible with the apostolic faith in hell and its eternal pain. It turned out that Jerome thought that this was the onset of heresy (haec maxime haeretica).4 1 Cf. L. Clark, The Origen Controversy: The Cultural Construction of an Early Christian Debate (Princeton: Princeton University Press, 1992), passim.
    [Show full text]
  • Ratio, Reason, Rationalism (Ideae)
    Ratio, reason, rationalism (ideae) for Saint Augustine through the Ages: an Encyclopedia, edited Allan Fitzgerald, Eerdmans, 1999, 696-702. Reason, as Augustine understood it, has been obscured to us by fundamental changes in the relations between reason and revelation, and between philosophy, theology and religion. There is now mutual incomprehension and indifference, division and opposition between them. In Augustine, there was, across religious divisions, a common reason in respect even to theological questions (c.Acad. 3,20,43; conf. 7; ciu.dei 8 & 10). Within the Christian religion, reason, faith and understanding were different modes of apprehending a single truth. The end of religion was to pass from faith to understanding, i.e. to intellectual vision. Reason was necessary for faith to have an object and to assist in the passage to vision (sol. 1,6,13-7,14; quant. 33,76; lib.arb. 2,2,5-6; doc.chr. 2,12,17; praed.sanct. 2,5; s. 118,1; ep. 120,3; trin. 8,5,8; 12,12,25; 14,1,3). For Augustine, reason (ratio) characterizes the human, and it, or mind (mens), is the best part of soul (c.Acad. 1,2,5; retr. 1,1,2). It proceeds and ends by self-knowledge and the knowledge of God, which are inescapably intertwined, and include the knowledge of all else (uera rel. 39,72; conf. 7,1-2 & X; en.Ps. 41,6- 8;145,5; trin. 14,12,15f.). Augustine can say that he wants to know only God and the soul (sol. 1,2,7).
    [Show full text]
  • Beyond the Edge of Certainty: Reflections on the Rise of Physical Conventionalism
    Beyond the Edge of Certainty: Reflections on the Rise of Physical Conventionalism Helmut Pulte Abstract. Until today, conventionalism is mainly regarded from the point of view of geometry, both in historical as in philosophical perspective. This paper, which corresponds in a certain way with earlier studies of J. Giedymin, aims at a broader interpretation. The importance of the so-called ‘physics of principles’, rooted in the tradition of analytical mechanics, for Poincaré’s conventionalism is emphasized. It is argued that important elements of phys- ical conventionalism can already be found in this tradition that underwent a fundamental change in the course of the 19th century. Résumé. Jusqu’à présent, le conventionnalisme a toujours été considéré à point de vue de la géométrie, que ce soit dans une perspective historique ou philosophique. Cet article, qui reprend dans une certaine mesure les travaux antérieurs de J. Giedymin, vise une interprétation plus large. L’importance de la ‘physique des principes’, qui s’inscrit dans une tradition de la mécanique an- alytique est mise en valeur par le conventionnalisme de Poincaré. J’y défends la thèse selon laquelle d’importants éléments du conventionnalisme physique peuvent déjà être trouvés dans cette tradition qui a subi un changement fon- damental au cours du 19ème siècle. Zusammenfassung. Der Konventionalismus wird auch heute noch in aller Regel — sowohl in philosophischer als auch historischer Perspektive — vom Standpunkt der Geometrie aus betrachtet. Dieser Beitrag, der in mancher Hin- sicht an frühere Untersuchungen J. Giedymins anknüpft, zielt auf eine breitere Interpretation ab. Die Bedeutung der sogenannten ‘Physik der Prinzipien’, die ihre Wurzeln in der Tradition der analytischen Mechanik hat, für Poincaré’s Konventionalismus wird unterstrichen.
    [Show full text]
  • Knowledge and Certainty
    phis_136 phis2007.cls (1994/07/13 v1.2u Standard LaTeX document class) 7-22-2008 :809 PHIS phis_136 Dispatch: 7-22-2008 CE: N/A Journal MSP No. No. of pages: 23 PE: Kristen 1 Philosophical Issues, 18, Interdisciplinary Core Philosophy, 2008 2 3 4 5 6 7 8 9 KNOWLEDGE AND CERTAINTY 10 11 12 13 Jason Stanley 14 Rutgers University 15 16 17 18 What is the connection between knowledge and certainty? The question 19 is vexed, in part because there are at least two distinct senses of “certainty”. 20 According to the first sense, subjective certainty, one is certain of a propo- 21 sition if and only if one has the highest degree of confidence in its truth. 22 According to the second sense of “certainty”, which we may call epistemic 23 certainty, one is certain of a proposition p if and only if one knows that 24 p (or is in a position to know that p) on the basis of evidence that gives 25 one the highest degree of justification for one’s belief that p. The thesis that 26 knowledge requires certainty in either of these two senses has been the basis 27 for skeptical arguments. For example, according to one kind of skeptical 28 argument, knowledge requires epistemic certainty, and being epistemically 29 certain of a proposition requires having independent evidence that logically 30 entails that proposition. Since we do not have such evidence for external 31 world propositions, we do not know external world propositions. According 32 to another kind of skeptical argument, due to Peter Unger (1975), knowledge 33 requires subjective certainty, and we are never subjectively certain of any 34 proposition.
    [Show full text]
  • The Relationship Between Knowledge, Belief, and Certainty: Preliminary Report
    The Relationship Between Knowledge, Belief, and Certainty: Preliminary Report .Joseph Y. Ha.lpern IBM Almaden Resea.rch Center San .Jose, CA 95120 [email protected] (ARPA/CSNET) [email protected] (BITNET) Abstract: We consider the relation between lt is well known that we can capture different co knowledge and certainty, where a fact is known if it notions of knowledge by varying the nditions on is true at all worlds an agent considers possible and the acceuibility relation which defines the set of rob is certain if it holds with p ability 1. We iden­ worlds that an agent consid ers poRsiblc (sec [HM85) tify certainty with probabilistic belief. We show for an overview). One particular set of require­ that if we assume one fixed probability assignment, ments on the accessibility relation, namely tnat it then the logic KJH5, which has been identified as be uri a/, Eudidr.an, and tran�itivc (we define these c perhaps the most appropriate for belief, provides terms helow) resu lts in the logi K015 which has o a complete axiomatization for reasoning about cer­ been considered the logic most appr priate for be­ tainty. Just as an agent may believe a fact although lief [Lev81a, FH88a). IP i!l false, he may be certain that a fact tp is true al­ We can also give a probabilistic interpretation to s though IP is false. However, it is ea y to see t.hat an belief. The greater the probability of'{' (according agent can have such false (probabilistic) beliefs only to an agent's subjective probability function), the rob b at a set of worlds of p a ility 0.
    [Show full text]
  • Wittgenstein on Mathematics and Certainties
    international journal for the study of skepticism 6 (2016) 120-142 brill.com/skep Wittgenstein on Mathematics and Certainties Martin Kusch University of Vienna, Austria [email protected] Abstract This paper aims to contribute to the debate over epistemic versus non-epistemic read- ings of the ‘hinges’ in Wittgenstein’s On Certainty. I follow Marie McGinn’s and Dan- iele Moyal-Sharrock’s lead in developing an analogy between mathematical sentences and certainties, and using the former as a model for the latter. However, I disagree with McGinn’s and Moyal-Sharrock’s interpretations concerning Wittgenstein’s views of both relata. I argue that mathematical sentences as well as certainties are true and are propositions; that some of them can be epistemically justified; that in some senses they are not prior to empirical knowledge; that they are not ineffable; and that their primary function is epistemic as much as it is semantic. Keywords Wittgenstein – Marie McGinn – Daniele Moyal-Sharrock – mathematics – certainties • • • I’ll teach you differences. Shakespeare, King Lear, Act 1, Scene 4 • • • To resolve these philosophical problems one has to compare things which it has never seriously occurred to any-one to compare. wittgenstein 1978: vii 15 ∵ © koninklijke brill nv, leiden, 2016 | doi 10.1163/22105700-00603004 <UN> Wittgenstein On Mathematics And Certainties 121 1 Introduction This paper aims to contribute to the debate over epistemic versus non- epistemic readings of the ‘certainties’ (also known as ‘hinges’ or ‘hinge propo- sitions’) of Wittgenstein’s last notebooks, posthumously published as On Cer- tainty (oc) in 1969. Interpreters on both sides of this debate have sometimes connected it to a further discussion: to wit, the discussion over the correct ren- dering of Wittgenstein’s views on mathematical sentences.
    [Show full text]