Jonathan Bain Humanities and Social Sciences Polytechnic Institute of New York University

---

MA 1114 - Conceptual Mathematics

[Home]

Prof:  Jonathan Bain Fall 2006 Office:  RH 201D              T/Th 9:00-10:50 Off. hr:  W 1-2pm  phone:  260-3688 I. Description This is a course on the conceptual foundations that underlie mathematical reasoning.  These foundations are given an explicit formulation in what is called category theory.  In barest outline, category theory is about two basic notions:  objects and maps.  It turns out that much if not all of mathematics can be expressed in terms of these notions.  To fully appreciate this, we will begin with a very basic concept that plays a central role in the development of mathematics; namely, the concept of infinity.  In the first part of the course, we will trace the development of the concept of infinity in the history of mathematics, from the ancient Greeks, through the Calculus, and culminating in late 19th century attempts to rigorize it in the notion of a set.  In the second part of the course, we will be very gently introduced to category theory, which attempts to generalize the notion of a set to a more primitive construction that many mathematicians believe is at the basis of most if not all of mathematics. The course is geared towards students with no prior college-level background in science or mathematics, and requires only elementary arithmetical skills.  In this course, you will not learn how to balance a checkbook, or how to calculate the charge to mass ratio of the electron (for instance).  Rather, you will learn how to apply general principles that underlie problem-solving techniques in mathematics and many other fields of inquiry. I. Required Text 1. Moore, A. M. (2001) The Infinite, 2nd Ed., Routledge. 2. Lawvere, F. W., and S. H. Schanuel (1997) Conceptual Mathematics, Cambridge Univ. Press. III. Course Requirements 1. There will be an assignment handed out every Thursday (except for Thanksgiving) that will be due at the beginning of class on the following Thursday.  Late assignments will not be accepted.  As compensation, only your best 10 of 12 scores on weekly assignments will be used in the calculation of your end-of-term grade. 2. There will be a midterm, and a final. Makeups will only be given in very extenuating circumstances and only for legitimate reasons. (Holiday scheduling is not a legitimate reason: plan accordingly!) 3. Attendance is mandatory.  Participation is very important in this class. While you will not be graded on attendance, class participation will figure positively into end-of-term grade adjustments: To encourage such participation, your grade will be rounded up by as much as one half of a letter grade based on how much you participate. (On the other hand, class participation will not adversly affect your grade; i.e., you will not be penalized for little or no participation.) IV. Grade Distribution There will be two grading options: Plan 1 Plan 2  (Essay option) Attendance:  10% Attendance:  10% Assignments:  30%       Assignments:  20% Exam:  30% Exam:  20% Final:  30% Final:  25% Essay:  25% V.  Topics Concepts:  Infinity Greeks Calculus Set theory Foundations:  Category Theory VI. Class Schedule: The following schedule may need to be revised over the course of the semester. Week 1 Tues 9/5 Intro:  The Branches of Mathematics Thurs 9/7 Paradoxes of the Infinite:  Astound Your Friends and Neighbors! Week 2 9/12 Greeks and Infinity:  Potentialities, Actualities, Achilles, Turtles 9/14 The Differential Calculus:  Dividing Infinitesimals Week 3 9/19 The Integral Calculus:  Adding Infinitesimals 9/21 Rigorization and Proof:  What Makes Mathematics, Mathematics... Week 4 9/26 Naive Set Theory:  Intuitive Infinity? 9/28 Ordinal Numbers:  Gobs of 'Em Week 5 10/3 NO CLASS 10/5 Cardinal Numbers:  Infinities of Infinities Week 6 10/10 Axiomatic Set Theory:  Rigorous Infinity? 10/12 Conceptual Headaches I:  The Lowenheim-Skolem Theorem Week 7 10/17 Conceptual Headaches II:  Godel's Incompleteness Theorems 10/19 MIDTERM Week 8 10/24 Category Theory:  What's More Primitive than a Set? 10/26 Isomorphisms, Sections, and Retractions:  Structure-Preserving Maps and their Bits and Pieces Week 9 10/31 More Categories:  Dynamical Systems and Graphs 11/2 More Categories:  Calling All Monoids Week 10 11/7 Functors and n-Categories:  Categorizing Categories and What it's Good For 11/9 Objectifying Objects:  Making Good on All the Hyperbolae Week 11 11/14 Terminal Objects:  I'll be Baaack... 11/16 Object Arithmetic I:  Multiplying Objects Week 12 11/21 Initial Objects:  ...kcaaaB eb ll'I 11/23 NO CLASS Thanksgiving Week 13 11/28 Object Arithmetic II:  Adding Objects 11/30 Map Objects and Exponentiation Week 14 12/5 Cantor’s Diagonal Argument, Again 12/7 Categorical Logic:  Tippy the Topos Week 15 5/3 NO CLASS Reading Day Week 16 Final (Date to be announced by Registrar)