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 (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.
There is a difference between using mathematics and doing mathematics.
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 day-to-day life.