Syllabus

Title
0508 Philosophy of Mathematics
Instructors
ao.Univ.Prof. Dr. Gabriele Mras
Contact details
gabriele.mras@wu.ac.at
Type
FS
Weekly hours
2
Language of instruction
Englisch
Registration
09/06/18 to 09/28/18
Registration via LPIS
Notes to the course
Subject(s) Doctoral/PhD Programs
Research Seminar in Main Subject I - Philosophy
Research Seminar in Main Subject II - Philosophy
Research Seminar in Main Subject III - Philosophy
Research Seminar in Main Subject IV - Philosophy
Dissertation-relevant theories - Philosophy
Research Seminar - Philosophy
Research Seminar - Philosophy
Dates
Day Date Time Room
Tuesday 10/02/18 12:00 PM - 03:00 PM D4.0.034
Tuesday 11/13/18 12:00 PM - 03:00 PM D4.0.034
Tuesday 11/20/18 12:00 PM - 03:00 PM D4.0.034
Tuesday 11/27/18 12:00 PM - 03:00 PM D4.0.034
Tuesday 12/04/18 12:00 PM - 03:00 PM D4.0.034
Tuesday 12/11/18 12:00 PM - 03:00 PM D4.0.034
Tuesday 01/08/19 12:00 PM - 02:00 PM D4.0.034
Tuesday 01/22/19 12:00 PM - 02:00 PM D4.0.034
Tuesday 01/29/19 12:00 PM - 02:00 PM D4.0.034

On this page:

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Contents

The course „Philosophy of Mathematics“ is concerned with versions of logicism, i.e. the view that mathematical truths are truths of logic. We will focus on Frege‘s and Boole‘s answers to the questions „What are numbers?", „ What is equality“ and "How can arithmetic be useful, if it is about objects that are not material or physical?“ 

The first part will present the infamous axiom V from Frege‘s  „Grundgesetze der Arithmetik“ and Russell’s objection to it. We will then look back to the reasoning that led to the possibility to construct an antinomy. We will discuss: „If numbers have their properties necessarily what account of ‚equality‘ could do justice to this fact?“ „What is the major difference between Frege‘s and Boole‘s view in respect to expressions of mathematical equations?“ „What is Frege‘s and Boole‘s view of the depiction of logical relations?“ 

In the third part we will focus on Cantor‘s , Schroeder‘s and Frege‘s understanding of ‚set‘ and discuss Frege‘s famous definition of number in the „Grundlagen der Arithmetik“ , his understanding of „Umfang“ and his reasoning for the introduction of „ Wertverläufe“ .

In last part we will have a look at the discusion of the significance of Russell‘s antinomy for understanding what ‚set‘ is and what could not count as definition of number.

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Learning outcomes

One of the learning outcomes is to make students familiar with the philosophical enquiries concerning the foundations of arithmetic, resp. the logicism that was defended by Frege and Boole. ; the overall aim and structure of their philosophical projects, and the key arguments of their major works. the more general learning outcomes are: to develop general philosophical skills: analyzing philosophical texts and defending positions in discussion and writing argumentative essays.

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Attendance requirements

Attendance Requirement: 75%

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Teaching/learning method(s)

The course is designed as a seminar, not as a lecture. The major part of the classes will be devoted to discussion.

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Assessment

Throughout the term we will have discussions on class readings. Part of the grade for this course depends on active participation in these discussions. In addition it is required to write a paper and to give a 30 minutes presentation.

  • paper 30 %
  • presentation 40 %
  • final exam 30 % 
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Prerequisites for participation and waiting lists

knowledge of basic concepts in logic

some familiarity with the history pf philosophy and 19th century philosophy of mathematics

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Availability of lecturer(s)

ao.Univ.Prof.Dr. Gabriele M. Mras
Office hours: Thursday 14:00 - 15:00,
Building D4, 3rd floor, room number D4.3.020
Tel.: 01-31336-4257
Email: gabriele.mras@wu.ac.at

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Unit details
Unit Date Contents
1

1. Intro to class, administrative details, course overview.     
Russell`s antinomy: Russell`s letter to Frege 1902 and Frege`s reaction
Readings:
Bell, D.: Russell's correspondence with Frege; in: McGuinness, B.: Abridged from the German edition, Oxford: Brasil Blackwell; Chicago: Univerity of Chicago Press 1980, p. 159-170.
Frege, G.: Frege -  Löwenheim. Frege - Russell; in: Frege, G.: Wissenschaftlicher Briefwechsel; Hamburg: Felix Meiner Verlag 1976; p. 159-161, 201-201, 211-217.
Frege, G.: Frege - Hilbert; in: Gabriel, G., Kambartel, F. and Thiel, C.: Gottlob Freges Briefwechsel mit D. Hilbert, E. Husserl, B. Russell, sowie ausgewählte Einzelbriefe Freges. Hamburg: Felix Meiner Verlag 1980; p. 1-24.
Klement, K.C.: The Russell-Dummett Correspondence on Frege and His Nachlaß; in: The Bertrand Russell Society Bulletin, Fall 2014, Number 150, p. 25-29.
Russell, B.: The Logical and Arithmetical Doctrines of Frege; in: The Principles of Mathematics, New York London: W.W. Norton & Company, p. 501-522.
Sittler, Tom M.: Bertrand Russells Brief an Gottlob Frege, in dem er die 'Russellsche Antinomie' beschreibt (1902) – Tom M. Sittler – A blog about: philosophy, economics, altruism, anything not on this list https://thomas-sittler.github.io/frege/
download
Reaktor: 16. Juni 1902: Gottlob Frege erhält einen Brief http://redaktor.de/16-juni-1902-gottlob-frege-erhaelt-einen-brief-audio/
download
2

2.    The concept of a number in die Grundlagen der Arithmetic:
Object, Property, Concept, Property of a Concept
Readings:
Blanchette, P. A.: Gottlob Frege; in: Förster, M. and Gjesdal K.: Oxford Handbook of 19th – Century German Philosophy, p. 1-23.
Hylton, P.: Logic in Russell´s Logicism; in: Propositions, Functions, and Analysis Selected Essays on Russell`s Philosophy, Oxford: Clarendon Press 2005, p. 49-82.
Lotter, D.: Der Mathematiker als Philosoph; in: Logik und Vernunft Freges Rationalismus im Kontext seiner Zeit, München: Verlag Karl Alber Freiburg 2004, p. 33-78.
Sluga, H.: Frege Against the Booleans; in: Notre Dame Journal Logic, Volume 28, Number 1, January 1987, P. 80-98.
Essler, W. K.; Brendel, E.: Algebra der Klasse und Theorie der natürlichen Zahlen; in: Grundzüge der Logik II Klassen – Relationen – Zahlen, Frankfurt am Main: Vittorio Klostermann 1993, p. 51-72, 227-275.
Porst S. S.: Seminar zu Freges Grundlagen der Arithmetik bei Prof. Dr. Felix Mühlhölzer „Protokoll“ von Seven-S. Porst, Sommersemester 1999, p. 1-21.
Kutschera, F.: Das Leben Freges; in: Gottlob Frege: Eine Einführung in sein Werk, Berlin/New York: Walter de Gruyter 1989,  p. 1-21.
Zalta, E. N.: Frege`s Theorem and Foundations for Arithmetic; in: Stanford Encyclopedia of Philosophy 2008, p. 1-43.

 
3

3.    Cantor and Dedekind on “What is a number”
Readings:
Cantor, M.: Babylonier, die Ägypter; Arithmetisches und Geometrisches;  in: Vorlesungen über Geschichte der Mathematik von Moritz Cantor, Erster Band, von den älteren Zeiten bis zum Jahre 1200 n. Chr. 3. Auflage, Leipzig: Druck und Verlag von B. G. Teubner 1907, p. 1-90.
Cantor, M.: Zahlentheorie, Combinatorik, Wahrscheinlichkeitsrechnung, Reihen bis 1736, Reihen seit 1737, Eulers Introductio, Band I ;  in: Vorlesungen über Geschichte der Mathematik von Moritz Cantor, Dritter Band, von 1668-1758. Mit 147 in den Text gedruckten Figuren. 2. Auflage, Leipzig: Druck und Verlag von B. G. Teubner 1901, p. 610-711.
Dauben, J. W.: Georg Cantor und die Mächtigkeit der Mengen; in: Faltings, G.: Moderne Mathematik, Spektrum Akademischer Verlag, p. 124-134.
Reck, E. H.: Frege, Dedekind, and the Origins of Logicism; in: History and Philosophy of Logic, Taylor Francis 2013, p. 242-265.
Reck, E. H: Dedekind`s Contributions to the Foundations of Mathematics; in: Stanford Encyclopedia of Philosophy, 2008, p. 1-14.
Wilson, M.: Frege and Russell: Does Science Talk Sense? EUJAP, Vol. 3, No. 2, 2007, p. 179-190.
Tait, W.W.: Frege versus Cantor and Dedekind: On the Concept of Number, p. 1-46.

 
4

4.    I) The concept as a function with a special value
Readings:
Frege, G.:  § 1-11; in: Angelelli, I.:  Begriffsschrift und andere Aufsätze Zweite Auflage mit E. Husserls und H. Scholz Anmerkungen, Hildesheim/Zürich/New York: Georg Olms Verlag 2014, p. 1-19.
Frege, G.: Funktion und Begriff; in: Frege, G.: Funktion, Begriff, Bedeutung, Göttingen: Vandenhoeck & Ruprecht, p. 18-39.

II) Syllogistic vs. Modern Logic
Readings:
Lukasiewicz, J.: Elements of the system and theses of the system; in: Aristotle´s Syllogistic from the standpoint of modern formal logic, Oxford: Clarendon Press 1951, p.1-42.
Frege, G.:  § 1-11; in: Angelelli, I.:  Begriffsschrift und andere Aufsätze Zweite Auflage mit E. Husserls und H. Scholz Anmerkungen, Hildesheim/Zürich/New York: Georg Olms Verlag 2014, p. 1-19.
Schütte K., Schwichtenberg, H.:  Mathematische Logik; in: Scharlau, W.: Dokumente zur Geschichte der Mathematik, Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn, 1990, p. 717-740.
Strobach, N.: Neuere Interpretationen der Aristotelischen Syllogistik, p. 1-16.
Wermuth, U.: Typesetting the „Begriffsschrift“ by Gottlob Frege in: plain TEX, TUGboat, Volume 36, No. 3, 2015, p. 243-256.
Gao, S.: Why is Humes Principle not good enough for Frege? 2011, p. 1-11.
Anellis, I.H.: How peircean was the “Fregean ‘revolution’ in logic? p. 1-51.
MacFarlane, J.: Frege, Kant, and the Logic in Logicism; in: The Philosophical Reviews, Vol 111, No. 1, 2002, p. 25-65.

 
5

5.    I) The concept of a number again — Die Grundgesetze der Arithmetik
II) Concepts, functions, extensions, value ranges,  identity
Readings:
Frege, G.: Funktion und Begriff; Über Sinn und Bedeutung; Über Begriff und Gegenstand. in: Frege, G.: Funktion, Begriff, Bedeutung, Göttingen: Vandenhoeck & Ruprecht, p. 18-39, 40-65, 66-80.
Frege, G.: Vorwort, §53-55, Band 2 § 147 und Nachwort; in Frege, G: Gottlob Frege Grundgesetze der Arithmetik Begriffsschrift abgeleitet Band I und II, Paderborn: Mentis 2009,  p.1-18, 446-447, 548-563.
Klement, K. C.: Frege´s Changing Conception of Number; in: Theoria, 78, 2012, p. 146-167.
Rossberg, M., Green, J.J., Ebert, P.A.: The convenience of the Typesetter; Notation and Typography in Frege`s Grundgesetze der Arithmetik; in: The Bulletin of Symbolic Logic, Volume 21, Issue 01, 2015, p. 15-30.

 
6

6.    The axiom V in Die Grundgesetze der Arithmetik
Readings:
Frege, G.: Vorwort, §53-55, Band 2 § 147 und Nachwort; in Frege, G: Gottlob Frege Grundgesetze der Arithmetik Begriffsschrift abgeleitet Band I und II, Paderborn: Mentis 2009,  p.1-18, 446-447, 548-563.
Blanchette P. A.: Axioms in Frege; in: Ebert, P.; Rossberg, M.: This is a preliminary version of an essay that appears (or will appear) in a volume on Grundgesetze, Oxford University Press, p. 1-29.
Sluga, D. H.: Frege as a Rationalist; in: Schirn, M.: Studien zu Frege I Logik und Philosophie der Mathematik, Stuttgart-Bad Cannstatt 1976, p. 1-31.

 
7

7.    Discussion of the consequences of Russell`s antimony
Readings:
Utrecht, D. v. D.: Der Grundlagenstreit zwischen Brouwer und Hilbert; in: Eichhorn, Thiele, E. J.: Vorlesungen zum Gedenken an Felix Hausdorff, Berlin: Heldermann Verlag 1994, p. 207-212.
Linnebo, O.: Philosophical logic and the philosophy of mathematics, p. 1-6.
Anscombe, G.E.M., Rhees, R., von Wright, G.H.: Teil III 1939-1940, in: Ludwig Wittgenstein Bemerkungen über die Grundlagen der Mathematik, Werkausgabe Band 6, Oxford: Suhrkamp 1984, p. 143-159.
Soames, S.: What ist the Frege/Russell Analytics of Quantification? in: Analytic Philosophy in America, APA, Chapter 7, p. 1-11.
Last edited: 2018-10-25



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