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

5/7 - The final is posted here (due Wed. May 9th @ 1:00 PM in Baker Hall 135).
4/16 - Homework 5 is posted below (due Wed. April 30th).
3/31 - Homework 4 is posted below (due Wed. April 9th). I have also posted some references for first-order logic.
3/31 - Here are links to the Lakatos readings for the next few weeks, as well as your reading for this Friday:
Lakatos: Part 1,Part 2, Part 3, Part 4
Benacerraf: What Numbers Could Not Be .
2/28 - Homework 3 is posted below (due Wed. March 3) along with the cheat sheet on natural deduction.
2/28 - Someone requested that I post the reading assignments here. Today's assignment is Smullyan, Ch. 13 ("Logic and Life"). Other recent assignments include Smullyan Ch. 3 & 4, and the section in Polya on "Induction and Mathematical Induction."
1/21 - Reading assignment 3 for Friday, 1/24. Please read the following sections: "Analogy", "Generalization", "Inventor's Paradox" and "Notation".
1/16 - Here is reading assignment 2 for Monday, 1/20. Please read the following sections: "Definition", "Heuristic", "Heuristic Reasoning", "Problems to find, problems to prove", "Reductio ad absurdem and indirect proof", "Terms, old and new" and "Why proofs?".
1/14 - Here is a copy of the syllabus and the reading assignment for Friday, 1/17.
1/13 - First day of class!

Homework SetDue 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

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
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
1 & 2Numbers & SetsHow to Solve It, Polya
3 & 4Polynomials & Spaces
5 & 6SemanticsWhat is the Name of this Book?, Smullyan
7 & 8Syntax
9 & 10Rings & Boolean Algebras
11 & 12Interpretations & MetamathematicsProofs & Refutations, Lakatos
13 & 14Intuitionistic & 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%

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 LatePartial Credit
≥ 4No 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.