Nature of Mathematical Reasoning 80-110 - Spring 2014 Instructor: Spencer Breiner Office: Doherty Hall 4301-C Office Hours: TBD E-mail: sbreiner[at]andrew.cmu.edu Classroom: Doherty Hall 1209 Course Expectations

 Homework Set Due Date Final Exam Friday, May 9th @ 1:00 PM, Baker Hall 135 Homework 5 Wednesday, April 30th Homework 4 Wednesday, April 9th Homework 3 Wednesday, March 5 Natural Deduction Cheat Sheet Homework 2 Friday, Feb. 14 Homework 1 Monday, Jan. 27

References
 Topic Links Numbers & Sets Divide by 9 trick Infinity of primes One-to-one and onto functions Cantor's theorem Equivalence relations Sets of size |ℕ| Reduction of ℤ to ℕ Geometry and polynomials Symmetry in the plane Isometries in ℝ² Congruence Similarity The projective plane Point/Line at infinity Homogeneous coordinates Conic sections Propositional Logic Propositional logic (overview) Semantics of propositional logic Truth functions Natural Deduction First-order Logic First-order logic (overview, sections 1-5) First-order structures Partially ordered sets (w/ links to GLB/LUB and lex. order) Boolean algebras

Grading & Course Expectations
General Remarks In this course we will conduct a high-level overview of the basic elements of modern mathematics. The course will be divided into three sections. In order to understand the philosophy of mathemtical thought one must first do some mathematics, so the first section will review some facts (which you probably already know) from algebra and geometry. In the second portion we will introduce the formal vocabulary of mathematical logic, a language developed in the early 20th century to make our mathematical notions precise. In the last portion of the course we will take a step back to study the activity of mathematics from a mathematical perspective.

In one sense this class should not be very difficult; most of the graded assignments will be straightforward, assuming that you come to class and do the reading. On the other hand, I will expect you to come to class every day (almost, see below) and I will expect you to do some hard thinking while you are there.

(Rough) Schedule
 Weeks Topic Reading 1 & 2 Numbers & Sets How to Solve It, Polya 3 & 4 Polynomials & Spaces 5 & 6 Semantics What is the Name of this Book?, Smullyan 7 & 8 Syntax 9 & 10 Rings & Boolean Algebras 11 & 12 Interpretations & Metamathematics Proofs & Refutations, Lakatos 13 & 14 Intuitionistic & Constructive Logic
Reading There will be a reading assignment associated with each portion of the course. I will announce reading assignments in class and you will be expected follow along regardless of whether we discuss the material in class. Several times throughout the semester I will give pop quizzes on the assigned reading. These will have easy questions and are simply designed to ensure that you do the reading; the number of quizzes will depend on how well I trust you to complete the reading assignments on time.

In the CMU bookstore I have listed all of the books as optional. This is because, although they are required for the course, I believe that you can easily find them less expensively online. In particular the last book, \emph{Proofs \& Refutations}, is available available electronically through the CMU library and JSTOR. You do not need any particular printing or edition of any of the texts.
Books How to Solve It, George Polya

What is the Name of this Book?, Raymond Smullyan

Proofs & Refutations, Imre Lakatos

 Homework: 5 x 10% Quizzes: 15% Midterm: 15% Final: 20%

Homework sets will consist of 4-8 questions. Some of these will be mathematical questions while some will ask for explanations of mathematical phenomena. For full credit you should write your answers in full, grammatical sentences (unless otherwise specified). Each problem set will specify the relative value of different questions.

Quizzes will be 2-3 easy questions about recent reading assignments and are primarily designed to ensure that everyone does the reading before coming to class.

The midterm will be in class, probably in early March, and will resemble the homework sets.

The style of the final is yet to be determined. It could be an in-class test, a take-home test or a final paper. In a few weeks I will ask the class what you prefer and give you full details soon afterwards.

Late Policies & Attendance Late homework submissions will be accepted for reduced credit, as listed below:

 Days Late Partial Credit 1 80% 2 60% 3 50% ≥ 4 No credit

E-mail submission (pdf only, NO MS WORD FILES) is acceptable as a date stamp, but you must also submit hard copies for grading.

Each student will get one free 48 hour late paper if I (Spencer) am notified by noon on the due date.

I will not require attendence; if you need to miss a day or two that will be fine. However, I do expect you to come to class and if I notice that any individuals are regularly missing class I will set attendence policies for those students.

Please refer to the syllabus for the course's academic honesty policies.