High-School Geometry, Axioms, and Formal Mathematics

The Role of Sketches in High-School Geometry

The book I used as a high-school geometry student emphasized deductive proof—done in a step/reason format. In order to show the need for such an approach, the text began by showing the limitations of drawing inferences from pictures. Later, when proof had been well developed, the text suggested a step-by-step procedure that students could use to develop a proof of some proposition offered for proof. Step 1 was to draw and label a picture! It never struck me at the time, but what the book was suggesting as the first step was to create something that it had previously been at pains to show could not be trusted.

The thing that's going on here is that for some things in high-school geometry a sketch is to be trusted—and for some things it is not. This was never made specific—and I never noticed, at the time, the lack of an explanation. But years later the situation left an uneasy feeling. It should not be left as an apparent contradiction.

In short, pictures in high-school geometry may be used to draw topological conclusions, but not metric conclusions. That is the rule that is implicitly operative (but never, as far as I know, explicitly stated). Here are two examples to show what we mean by "topological".

(1) Draw a circle on a paper and put a dot (point) inside the circle. Now suppose that what you thought was paper was really a thin rubber sheet. By stretching the sheet in various ways you can deform the circle so that it is no longer a circle—but it is, and will remain (no matter how you stretch the sheet) a closed curve. Being a closed curve is a topological property. The point you put inside the circle will remain inside the closed curve. This property of the point is a topological property.

(2) Draw a line on a rubber sheet and label points A, B, C, and D in that order along the line. You cannot, by stretching the sheet change the order to A, C, B, D. The order of points on the line is a topological property. Topological properties are those that are invariant under stretching the sheet.

Metric properties are those that involve distance: the actual size of an angle or length of a line segment, for example. In high-school geometry, you are never allowed to assert two line segments to be of equal length without proof. That was the point made at the beginning of the book.

The fact that pictures in high-school geometry may be used to draw topological conclusions, but not metric conclusions has a use beyond merely fixing up an omission in a text. It provides the clue for us to see what will be wrong in those supposed proofs of silly propositions. For example, I have seen a supposed proof that all triangles are isosceles. I don't remember the proof at all, but I do know where the error is to be found. Somewhere, along the line of constructions and argument, we are conned into accepting a picture that is not topologically equivalent to the correct picture.

In fact the "right" picture may be difficult to determine: who could tell whether a point defined by the intersection of two lines in our construction fell inside or outside a circle of a certain radius with center at some other point in our picture?

In high-school geometry it is legitimate (and necessary) to draw topological inferences from sketches. It is legitimate to draw metric inferences only from axioms and propositions proven from the axioms. David Hilbert has provided axioms for the needed topological inferences. So all propositions follow from the axioms—and it is not legitimate, in Hilbert's geometry, to draw any inferences from sketches. (But when constructing a proof, you had better have a sketch—so you will know what is going on!)

Axioms

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 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.

Hilbert thought, and predicted, that all mathematics could be formalized. The mathematician Kurt Gödel proved that Hilbert was wrong about this point. There are actually two things going on at the same time in mathematics.

Non-Euclidean geometries – the geometries that actually model our real universe – were discovered by tweaking one of the axioms of Euclidean geometry. Euclidean geometry is, of course, the stuff we studied in high school. The axioms of Euclidean geometry were self-evident truths about our idealizations of planes, lines, points, circles. And, remarkably, what is self evident about our idealization of a plane and space is not 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) has, however, been absent from education up to a point seen as appropriate for the few 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 to formal mathematics.

It is, I think, very interesting to note that the entities of "higher mathematics" are all, almost universally, based on the idea of a set. And the idea of a set is part of intuitive mathematics, but the useful theorems about sets used elsewhere in formal mathematics are proved formally. And the philosophers haven't got it all figured out yet.