80-110 Nature of Mathematical Reasoning

Spring 2002


January 14 - May 3, 2002
Mo, We, Fr  11:30-12:20pm
SH 206
Webpage: http://www.phil.cmu.edu/courses/80-110/spr02
Instructor:      Dirk Schlimm
Email: dschlimm@andrew.cmu.edu
Office: Baker Hall A 60 B
TA: Henrik Forssell
Email: jforssel@andrew.cmu.edu
Office: Baker Hall 161 A

About the course. 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.

Since the backgrounds of the students who are enrolled in this course may vary considerably, we will sometime split the Friday lectures into an elementary and an advanced track. Students may choose which one to attend.


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.


Homework: Usually a set of problems will be assigned once a week. The assignments will be posted on the class web-site. It is your responsibility to obtain the assignment if you miss class.

In order to obtain a better grade written 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.

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 when explicitly stated otherwise). 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.

Journal: A journal should be kept by the student in which notes and assignments are collected. 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.

For each lecture the main concepts and ideas should be summarized.These will be useful when reviewing the material, and for the essay you are expected to write. 

In addition, a few remarks about what you find interesting or puzzling about the lectures should be written down. Briefly answer the following questions:

Please, don't write just one-word answers, but take a few minutes after each lecture to review the lecture. By answering these questions you learn how to reflect about the material presented in class.

The journal has to be handed in once during the semester and at the end of the course. Only the version at the end of the course counts towards the final grade. Handing the journal in during the semester allows you to receive constructive feedback about your journal. The final grade for the journal will be based on completeness of content and clarity of exposition. What I will look for in particular is the following:

Although it might at first sound complicated and time consuming, keeping the journal is really just a matter of a few minutes every day. The students's responses in previous courses that I've taught has been very positive about it.

Exams: Two exams will be held during the course: a midterm and a final. They will cover the material up to the date of the exam. The final exam will be comprehensive.

Essay: A 5-10 page essay on a topic related to the course must be handed in a week before classes finish. The topic can be chosen by the student, but must be approved by the instructor. An outline must be presented to the instructor two weeks before classes finish.

The essay provides you the possibility to study in more detail a particular subject of the course that interests you most. In the introduction to the essay you should explain the relation of the chosen topic to the course. 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.

Quizzes: Short quizzes consisting of a few questions concerning recent material presented in class will be held in class on a regular basis. They do not count towards your grade, but are intended to provide to you some information about what you are expected to know, and allow you to test your knowledge of the subject. Furthermore, they provide feedback to the instructor about the difficulties of the course material.

Class participation: Class participation is expected. This includes showing up regularly (if you have to miss a class, please tell the instructor), showing up prepared, making an effort to answer questions posed, contribute to class discussions, and present short summaries and 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 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: 

Homework:  30 %
Journal: 20 %
Midterm exam:    15 %
Final exam:  15 %
Essay: 20 %
Grades for homeworks, journal, essay, and exams 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.
© Dirk Schlimm, Last modified: 1/12/02