Course Philosophy:
Although you have been taking math courses all your academic career, you
probably would have a difficult time defining what Mathematics really is,
or saying what a mathematician actually does. Most people would probably
know that it has something to do with numbers and formulas, and problem
solving, but this is really only a small part of the story. If you ask
mathematicians what they do, most of them will tell you that their work
involves several key ideas:
 Analyzing implications of statements
 Determining essential principles
 Abstracting the general from the specific
 Expressing or quantitizing relationships
Numbers and formulas are certainly part of these processes, but they are
not the main goal nor the main tool of mathematics.
There is a difference between using
mathematics and doing mathematics.


There is a difference
between using mathematics (that is, using the relationships,
abstractions, principles and implications that have already been
determined) and doing mathematics (which is determining those
relationships, abstractions, principles and implications). For example,
you are using mathematics when you apply the quadratic formula to a
specific equation, or when you use the quotient rule to differentiate a
rational function; but you are doing mathematics when you prove that
the quadratic formula gives the answer for all quadratic equations, or when
you prove that the quotient rule is a valid identity. These abstract the
process of finding a specific answer for a specific problem to the much
broader framework of finding an answer for all problems of a common
form all at once. They express the relationship between a category of
problems and their solutions; this is what real mathematics is about.
In this course, you will get your first concentrated taste of doing
mathematics. You will begin to learn how to interpret and justify the
kinds of statements used in mathematics (e.g., "If ... then ...", "For all
x, ...", "There exists an x where ...") and their negations,
how to construct correct proofs of these statements, how to recognize flaws
in incorrect proofs, and how to use definitions and previous results in a
precise way.
These skills may seem abstract and unrelated to the real world, but they
are exactly the same skills that you need to understand and interpret the
statements that you see everywhere around you: Are the conclusions that a
manufacturer claims in his commercial justified? Are the statements a
politician makes actually related to the problem she is trying to solve, and
if so, are her conclusions valid? Are the trends reported in a news
article a reasonable interpretation of the data? Do you have enough
information to determine whether an investment will be profitable? The
approaches that we use in this course to justify mathematical statements
can also be brought to bear on any problem that involves determining
consequences or justifying conclusions, and in that way, they represent
significant and powerful tools for your daytoday life.

Math 99 (Fall 2000) web pages
Created: 26 Aug 2000
Last modified: 27 Aug 2000 14:35:12
Comments to: dpvc@union.edu



 