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, remarkably 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.
Mathematics has two fundamental aspects: (1) discovery/logical deduction and (2) description/computation.
Discovery/deductive mathematics asks the questions:
1. What is true about this thing being studied?
2. How do we know it is true?
On the other hand, descriptive/computational mathematics asks questions of the type:
3. What is the particular number, function, and so on, that satisfies ... ?
4. How can we find the number, function, and so on?
In descriptive/computational mathematics, typically some pictorial, physical, or business situation is described mathematically, and then computational techniques are applied to the mathematical description, in order to find values of interest. The foregoing is frequently called “problem solving”. Examples of the third question such as “How many feet of fence will be needed by a farmer to enclose ...” are familiar. The first two questions, however, are unfamiliar to most. The teaching of computational techniques continues to be the overwhelming focus of mathematics education. For most people, the techniques, and their application to real world or business problems, are mathematics. Mathematics is understood only in its descriptive role in providing a language for scientific, technical, and business areas.
Mathematics, however, is really a deductive science. Mathematical knowledge comes from people looking at examples, and getting an idea of what may be true in general. Their idea is put down formally as a statement—a conjecture. The statement is then shown to be a logical consequence of what we already know. The way this is done is by logical deduction. The mathematician Jean Dieudonne has called logical deduction “the one and only true powerhouse of mathematical thinking” (Note 1 below), and
learning how to construct a logical argument is at least as worthy a component of a general education as is learning how to compute.
The deductive and descriptive aspects of mathematics are complementary—not antagonistic—they motivate and enrich each other. The relation between the two aspects has been a source of wonder to thoughtful people (Note 2 below).
This text presents a system designed to enable students to find and construct their own logical arguments. The system is first applied to elementary ideas about sets and subsets and the set operations of union, intersection, and difference—which are now generally introduced prior to high school. These set operations and relations so closely follow the logic used in elementary mathematical arguments, that students using the system are naturally prepared to prove any (true) conjectures they might discover about them. It is an easy entry into the world of discovery/deductive mathematics. It enables students to verify the validity of their own conjectures—as the conjectures are being made.
The system is based on a bottom-up approach. Certain things are best learned from the bottom up: programming in a specific programming language, for example, or learning how to play chess. In the bottom-up learning, there ought to be no doubt of what constitutes a valid chess move on a valid chess board. Other things, such as speaking in one's own native language, are learned from the top down. As we learn to speak, grammar (which would be analogous to the rules of the game for chess) is not even part of our consciousness. Grammatical rules are followed only because they are used implicitly by those that we imitate. If the people around us use poor grammar, we nevertheless learn to feel it is “right”—and we speak the same way.
The system in this text is based on a number of formal inference rules that model what a mathematician would do naturally to prove certain sorts of statements. The rules make explicit the logic used implicitly by mathematicians (Note 3 below). After experience is gained, the explicit use of the formal rules is replaced by implicit reference. Thus, in our bottom-up approach, the explicit precedes the implicit. The initial, formal step-by-step format (which allows for the explicit reference to the rules) is replaced by a narrative format—where only critical things need to be mentioned. Thus the student is lead up to the sort of narrative proofs traditionally found in text books. At every stage in the process, the student is always aware of what is and what is not a proof—and has specific guidance in the form of a "step discovery procedure" that leads to a proof outline.
Notes
1 J. Dieudonne, Linear Algebra and Geometry, Hermann, Paris, 1969, page 14.
2 John Polkinghorne in his The Way the World Is (Wm B. Eerdmans, Grand Rapids, MI, 1984, page 9) states, “Again and again in physical science we find that it is the abstract structures of pure mathematics which provide the clue to understanding the world. It is a recognized technique in fundamental physics to seek theories which have an elegant and economical (you can say beautiful) mathematical form, in the expectation that they will prove the ones realized in nature. General relativity, the modern theory of gravitation, was invented by Einstein in just such a way. Now mathematics is the free creation of the human mind, and it is surely a surprising and significant thing that a discipline apparently so unearthed should provide the key with which to turn the lock of the world.”
3 Although the rules resemble those of formal logic, they were developed solely to help students struggling with proof—without any input from formal logic.