Axioms and Formal Mathematics

There are two sorts of things that are called axioms in mathematics. The first sort is a set of obviously true statements about an idealized object, which statements are taken as a given starting point for logical deductions about other objects. The axioms of Euclidean geometry, the geometry we studied in high school, were self-evident truths about our idealizations of planes, lines, points, circles. The second sort depends on the fact that in mathematics a true statement about (almost) anything depends entirely on the definition of that thing. To show something is true about a mathematical object, one must show it follows logically from the object’s definition. 

The way an object is defined is to say what general category it falls in, and then add what makes this thing special – within that category. For example, a square is defined as a rectangle that has all sides of equal length. Of course one has to know beforehand what a rectangle is: a quadrilateral with interior angles all right angles.

And, of course, this process of using previously defined categories can’t go on forever. There have to be some undefined things to start with. Axioms (of the second sort) are just the statements that are assumed to be true about these undefined objects.

Mathematics that is developed from the second sort of axioms is frequently called abstract or formal mathematics. Axioms in formal mathematics are not “self-evident” truths. Logically, they would have to be seen as pure conventions. But efforts at starting with truly unmotivated and arbitrary axioms and objects have never produced anything of any interest. In practice, mathematicians have “examples” that motivate the things they study formally – so  that the facts (called theorems) derived from the formal axioms are true about their examples. And, if the theorems are going to be useful, true about other examples.

Logical deduction is a very powerful tool that is useful for studying all sorts of things. Non-Euclidean geometries – the geometries that actually model our real universe – were discovered by tweaking one of the axioms of Euclidean geometry, and studying the results formally. And, remarkably, what is self evident about our idealization of a plane and space is not self evident about our universe. And, remarkaby again, what is self evident about our idealization of the natural numbers, 1, 2, 3, ... and so on, is self evident about our universe.

Axioms of the first sort have been part of education for a long time. The second paragraph of our Declaration of Independence starts out with, “We hold these truths to be self evident … .” The entire context for axioms of the second sort (formal mathematics) is, however, typically absent from all students' education, unless they may become mathematics majors. More generally, students see proofs in their calculus books, but are never asked to do these proofs themselves. These proofs are part of formal mathematics. The computational problems students are asked to do can be understood within their intuitive mathematical education, and the deductive mathematics of high-school geometry also lies within intuitive mathematics.

So it is not surprising that many students, even some exceptionally good ones, are tripped up when courses, such as an introduction to abstract mathematics or modern algebra, taught from the perspective of formal mathematics, are first encountered. Formal mathematics is no harder or more "advanced" than the mathematics of students' previous experience. It's just different. Introduction to Proof in Abstract Mathematics and the two texts on this web site were written to introduce students of various sorts and levels to formal mathematics.

Ask any educated person where scientific knowledge comes from, and you're likely to get the answer, "From the scientific method." Ask any educated person where mathematical knowledge comes from, and you're likely not to get an answer.