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Carnegie Mellon University, Department of Philosophy,
Spring 2001

The Nature of Mathematical Reasoning

Dirk Schlimm



Class: Tuesday and Thursday, 9:00-10:20am, CFA 211

Website: http://www.contrib.andrew.cmu.edu/~dschlimm/80-110spring01

Instructor: Dirk Schlimm
Office: Baker Hall A60B
Phone: 268-5737
Email: dschlimm@andrew.cmu.edu
Office hours: By appointment. The best way to contact me is via email.

Course description: Although we spend the great bulk of our mathematical education learning how to calculate in a variety of ways, mathematicians rarely calculate anything. Instead they devote their time to clearly stating definitions, finding simple axioms, making conjectures about claims that might follow from these axioms, and then proving these claims of finding counterexamples to them. Although thinkers since Aristotle have devoted enormous time and energy to developing a theory of mathematical reasoning, it is only in the last century or so that a unified theory has emerged.

In this course, we not only consider the modern theory of mathematical reasoning, but we also consider several case studies in which a problem is simple to solve with mathematical reasoning but almost impossible to solve without it. For example, we consider how to compare the sizes of infinite sets, and how to solve the Monty Hall-Let's make a deal Problem. This allows you to get a feeling for the power of abstract reasoning, in particular when it's consequences are not so obvious at first sight, but can be validated by experience.

By learning a few facts about the evolution of mathematics from prehistory to modern times, and also by setting the relevant mathematical concepts into their historical context, you should be able to develop a basic understanding of the history of mathematics and of fundamental problems in the philosophy of mathematics.


Quizzes: A short quiz consisting of 1-5 questions concerning recent material presented in class will be handed out once a week (usually on Thursdays) at the beginning of class. It that has to be answered in class (5 minutes in total). This provides you some information about what you are expected to know, and allows you to test your knowledge of the subject. Furthermore, it provides feedback to the instructor about the difficulties of the course material.

Homework: Usually short assignments are assigned during on Thursdays and are due the following Tuesday. After each of the main topics (see the course overview) I will want you to write a short summary, but then you will have more time to complete it. Corrected assignments should be kept in the portfolio (see below).

In order to obtain a better grade assignments can be redone and handed in on the next day of classes after they were handed back. This allows you to go over your work again and the opportunity to learn from previous errors.

The assignments will be posted on the class web-site. It is your responsibility to obtain the assignment if you miss class. Attempting to give an excuse anywhere in the vicinity of: ``I didn't know there was an assignment,'' or ``I missed class and my friend gave me the wrong assignment'' will cause excessive irritation on the part of the instructor. You are free to collaborate on homework, but not to copy answers from friends. Assignments are due at the beginning of class on the date mentioned in the assignment, and have to be turned in on paper (except where explicitly mentioned). You may type them up or turn them in in legible handwriting. If you use a word-processor, make sure to use the spell-checker.

Portfolio: Students are expected to keep a portfolio about the contents of the course. This gives you the opportunity to organize the material presented in class in a neat and clear way. It will help you to keep track of where we are in the course. It will also make it easier for you to review the material and thereby help you to find out what you have really understood and what is not yet clear to you.

The portfolio must be kept in a three-ring binder made for standard size notebook paper (8 1/2 by 11 inches, preferably not less than 1 inch wide at the back). Your name should be clearly written on the outside of the binder. You may use any type of paper you like, lined or unlined, and of any color it seems good to you to use. However, please use paper that has punched holes correctly placed for insertion in the binder, and please use full-size paper.

The portfolio should include:

  1. Table of contents.
    The pages of your notes and homework should be numbered (maybe separate numberings, like H-1, H-2, for the homeworks).

  2. For each lecture:
  3. Homeworks assigned in class.

  4. Index of technical terms introduced in the course (for quick reference).
    The index should only contain a sorted list of terms, or names, and have a reference to the page where the term is defined or explained. Sometimes you may have more than one reference. In case you don't have a definition already, you may look for one.

    Here's a list of terms I expect to find in the index at the end of the semester (by then you'll be familiar with all of them!): |-, |=, 1-1 function, 7-adic system, Archimedes, Aristotle, atomic sentence, Axioms of probability, Babylonians, base case, base clause, Bayes' Theorem, binary numbers, Cantor's diagonal argument, cardinality, commensurable, completeness, conclusion, Conditional probability, constructive mathematics, countable, De Morgan rule, deductive, deductively valid, definition of even, denumerable, Egyptians, entailment, Euclid, Eudoxos, expressively complete, fallacious , final clause, form of an argument, formal, formal language, formula, function, Georg Cantor, George Boole, Gerhard Gentzen, Gottlob Frege, Hypatia, indirect, induction hypothesis, induction step, inductive, inductive clause, inductive/recursive definition, inference rule, injective, interpretation, Intuitionistic logic, irrelevant, Knights and Knaves, Kurt Gödel, Law of total probability, mathematical induction, molecular sentences, N, names of syllogism, natural deduction, neolithicum, non-classical logics, notation, numbers, objectivity, Plato, premise, proof, proof by contradiction, proof of irrationality of the square root of 2, proof that there are infinitely many prime, proposition, propositional formula, Pythagoras, Q, quantifier, R, reductio ad absurdum, semantic method to check validity of arguments, semantics, sentential/propositional logic, sound, soundness, statement, syllogism, syntax, tautology, terms in arithmetic, Thales, Thales' theorem, truth table, Z.

  5. Reading material

All materials for this course should be kept in the portfolio binder at all times. Use dividers to mark off each section. Please arrange them in the order mentioned above.

Remember: your portfolio is the embodiment of your work for this course. A complete and well presented portfolio virtually guarantees you a good grade. The opposite is also true.

The portfolio has to be handed in once during the semester and at the end of the course. It will be returned with comments and a grade. After the final exam the portfolio has to be picked up from the instructor's office. Only the end of semester grade of the portfolio will count towards the final grade. The grade for the portfolio will be based on completeness of content and clarity of exposition. What I will look for in particular is the following: Are all pages legible? Is the table of contents complete? Is there at least a page for each lecture? Are the main topics of each lecture summarized briefly? Are there personal remarks about interesting or puzzling points? Are all homeworks included (all versions of the ones which were redone)? (The content of the homeworks does not contribute to the portfolio grade, but to the grades for the homeworks.) Is the index complete? Are all terms in the index referenced correctly?

Essay: A 3-10 page essay on a topic related to the course must be handed in by April 26. The topic can be chosen by the student, but must be approved by the instructor. Every student must have picked a topic by April 12, and an outline must be presented to the instructor by April 19.

The essay provides you the possibility to study in more detail a particular subject of the course that interests you most. It should also contain a small technical part, like a brief proof.

Note that the dates mentioned above are the deadlines, no extensions will be granted. However, you can hand in the essay earlier if you like, even before the midterm exam. Starting early with the essay gives you the advantage of having more time to work on it, discuss it with the instructor, and allows you to avoid being cluttered with work at the end of the semester.

Exams: Two exams will be held during the course: on Tuesday, March 6, and Thursday, May 3. They will cover the material up to the date of the exam. The final exam will be comprehensive.

Class participation: Class participation is expected. This includes showing up regularly (missing 1 or 2 classes is reasonable, missing 7 or 8 is not -- if you have to miss a class, please tell the instructor why), showing up prepared, making an effort to answer questions posed, contribute to class discussions, and present small problems in class. It is a well-known fact that active learning (e.g., participating in discussions) is much more effective than passive learning (e.g., reading), thus you get more out of the course if you are actively involved.

The reading material only supplements the lectures, but cannot compensate for them. Most material covered in class will not be available otherwise. And although class participation will not be explicitly graded, the quizzes, which cover material presented in class and which are administered in class, do count towards your final grade.

Grading: The grade in this course depends on your continuous effort during the semester. The final grade will be based on six components according to the following weights:

16% Homeworks
16% In-class quizzes
16% Midterm
16% Final exam
18% Essay
18% Portfolio

Grades for homeworks, quizzes, exams, and portfolio will be on a scale between 0 and 10. The corresponding letter grades are: 10-9 A, 8-7 B, 6-5 C, 4-3 D, below 3 F.

Required text: The only required text is The Language of First-Order Logic by Jon Barwise and John Etchemendy which comes with the program Tarski's World for either PC or Mac. Please obtain this book within the first two weeks of classes. If you have problems ordering it, I will be happy to assist you.

Additional texts will be handed out in class.

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© Dirk Schlimm, Last modified: Tue May 1 10:32:47 EDT 2001