Mathematics on mathematics

by Orr Shalit


This post is the outline of a talk (or perhaps the talk is an outline of this post) that I will give on February 28 on our “open day” to prospective students in our department. This is supposed to be a story, it is intended to give a flavor, and neither the history nor the math are 100% precise, because it is a 15 minute talk! The big challenge is to take some rough ideas from this post, throw away the rest, and make that into an interesting quarter of an hour lecture. Comments are very welcome.

1. Introduction

What is mathematics? I am not going to answer that. You have all met mathematics in your life, in school, but also in other places as well, because math is everywhere. So you have some kind of idea what mathematics is about. However, I suspect that the most profound aspects of math have been hidden from you. I am here today to try to give you a taste of this mathematics which you have not yet seen. It is only fair to let you know that — for better and for worse — the mathematician’s mathematics, the mathematics that you will study if you do an undergraduate degree in math, is of a dramatically different nature from the math you learn in high-school or the math-is-everywhere kind of math which you meet in various popular accounts.

You have met various different kinds of mathematics: combinatorics, geometry, algebra, integral and differential calculus (aka HEDVA — which literally means “joy” in Hebrew). It seems as if mathematics splits into various branches, where in each branch there are different tasks that one should do. The objects of study of geometry are triangles, circles, trapezoids, etc; one has to prove that a certain triangle has this or that property, or one has to compute some angle or length or area. The objects of study in algebra are certain symbolic expressions or equations; one has to find the root of an equation, or to simplify an expression. In HEDVA the objects of study are functions; one has to compute the minimum of a function, or its anti-derivative, and so on.

The theme of this talk is that the objects of study in mathematics do not have to be only triangles or functions or equations, but they can also be geometry or analysis or algebra. Mathematics can also be used to study mathematics itself. This is profound. But perhaps more surprisingly, this has practical consequences.

Of course, I have no time to tell you precisely how this works. For this, I recommend that you come here and study mathematics.

2. Three problems in three different fields of mathematics

Geometry: It is commonly accepted that the birth place of mathematics was in ancient Greece. The Greeks did not have calculators, of course, neither did they have the decimal system. (Did you ever try to carry out multiplication or division of large numbers without using the decimal system? For us it is even hard not to think of numbers in terms of their decimal expansions). However, the ancient Greeks were experts in geometry. In fact, they could use geometry to make computations.

Suppose that we have an infinite plane (a huge white-board or piece of paper) with two points marked on it: one which is the origin, and one which is a unit distance from the origin. Suppose that we also have an infinitely long ruler (straightedge) and a compass. We also have a pen which is infinitely precise. We can use the ruler and the compass to mark points, lines and circles in the plane using the points, lines and circles which are already marked on the plane. One can use the ruler to mark a line that passes through any two points which are marked on the plane; one can use the compass to mark the circle that has a center at one point and passes through another point; and one can mark the point of intersection of two lines, two circles or a line and a circle.

It can be shown that using a sequence of the above moves, one can also preform the following moves. One can also mark on the ruler the distance d between two points, and then given a point P on a line L, mark another point Q at distance d from P on the line. Also, given a line L (containing some point P) and a point Q which is not on L, one may construct the line passing through Q which is parallel to L.

Points, lines and circles which can be constructed according to the above rules are said to be constructible. We say that a number is constructible if it is equal to the distance between two constructible points, or if it is the negative of such a distance.

To add two constructible numbers, say d and d’, simply mark one point P at distance d from the origin on some line, and then mark a point Q at distance d’ from P on the same line (and in the same direction). To multiply two constructible numbers we use this old theorem of Thales.

If x and y are constructible, then x+y, x\cdot y and (if y \neq 0) x/y is constructible.

There are several classical problems which interested the Greeks. One was this:

Given a square of area S, construct a square that has area 2 S

The side of the square has length a = \sqrt{S}, so the diagonal has length \sqrt{a^2 + a^2} = \sqrt{2} a. If we now use the diagonal as the side of the new square, its area will be (\sqrt{2} a )^2 = 2 S.

Here is another:

Given a circle with area S, construct a square that has area S.

The Greeks tried and tried, but were not able to solve this problem. This problem (which is know as the problem of “squaring the circle“) remained a challenge for more than two thousand years, until its surprising solution in the 19th century. We will come back to it later.

Algebra: While the Greeks developed geometric methods, the Babylonians developed a mathematics that was based on arithmetic. Circa 800 BC, the Babylonians already had “formulas” for solving some quadratic equations. “Formulas” for solving quadratic equations were also developed by Egyptian, Arabic and Indian mathematicians, although these formulas were not given by the neat algebraic symbolism that we use today. For example, in 628 Brahmagupta, gave the following recipe for solving the equation ax^2 + bx = c:

To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.

which is equivalent to

x = \frac{\sqrt{4ac + b^2}-b}{2a} .

(The above 8 lines are taken from this Wikipedia article, which references the quote to Brahmasphutasiddhanta (Colebrook translation, 1817, page 346)). Only in the 16th century did this seemingly simple formula reach the neat algebraic formulation that we know today, that is, that the two solutions of   ax^2 + bx + c = 0   are given by

x_{1,2} = \frac{-b\pm \sqrt{b^2-4ac}}{2a} .

Formulas for the roots of cubic equations (equations of third degree) and a quartic equations (equations of a fourth degree) were also developed, but they are terribly complicated (see this and this). There were also algorithmic methods formulated, which are equivalent to the closed form solutions but are perhaps more natural.

A natural question came up: is there a general method for finding the roots of a quintic equation (an equation of fifth degree?) What about a general algebraic equation of order n, that is, an equation of the form

a_n x^n + \ldots a_1 x + a_0 = 0 .

Is it possible to find a formula giving the roots? “Formula” could mean a lot of things, here we mean an expression involving the coefficients (a_0, \ldots, a_n) combined in some way using addition, subtraction, multiplication, division, and taking roots of some order. Though the efforts of great minds went into solving this problem, there was only partial success (in a few special cases). This problem remained unsolved until its surprising solution in the 19th century. We will come back to it later.

Calculus (Hedva): A quite different branch of mathematics is the differential and integral calculus, developed in the 17th century by Newton and Leibniz. Recall that if f is a function defined on the real line, then its derivative f' is defined as the function

f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} ,

(if that limit exists). The indefinite integral, sometimes called the anti-derivative, of a function f, denoted F(x) = \int f(x) dx, is the function F such that F'(x) = f(x). There are some functions for which we know the derivative and the anti-derivative, for example

(x^n)' = nx^{n-1} \,\, , \,\, \int x^n dx = \frac{x^{n+1}}{n+1}

(\cos x)' = \sin x \,\, , \,\, \int \cos x = \sin x

(e^{-2x})' = -2 e^{-2x} \,\, , \,\, \int e^{-2x} dx = -\frac{e^{-2x}}{2} .

An especially nice class of functions are the elementary functions: powers, roots, trigonometric functions, exponentials, logarithms, and anything that can be constructed from these using composition of functions and the four basic operations of arithmetic. If f is an elementary function, we can always compute directly f' using the rules of differentiation. Moreover, f' is again an elementary function (actually, this is part of what we mean when we say that we can compute f'). The problem of finding the indefinite integral of an elementary function turns out to be much harder. Why is this so? It turned out that it wasn’t only technically harder: there were certain elementary functions for which nobody was able to find an elementary anti-derivative. In particular, for the following elementary functions

e^{-x^2}\,\, , \, \, \frac{\sin x}{x} \,\, , \,\, x^x

there was no known formula (in terms of elementary functions) for their anti-derivative. This problem was also settled in the 19th century. We will come back to this later.

3. Field theory

And now we move to something completely different. Consider the set of all rational numbers \mathbb{Q}. This is the set of all numbers of the form q = \frac{m}{n}, where m and n are integers and n is not zero. Note that 0 and 1 are in \mathbb{Q}. The set of rational numbers has the following properties:

  1. For all x,y \in \mathbb{Q},  we have x\cdot y and x + y are in \mathbb{Q} (closure under addition and multiplication)
  2. For all x \in \mathbb{Q} ,  1 \cdot x = x and 0+ x = x (existence of neutral elements)
  3. For all x \in \mathbb{Q}  there is some y \in \mathbb{Q} such that x+y=0. If x \neq 0  then there is some z \in \mathbb{Q} such that xz = 1 (existence of inverses)
  4. For all x,y,z \in \mathbb{Q} ,  we have that

x \cdot y = y \cdot x and x+y = y+x (commutativity)

x\cdot (y \cdot z) = (x \cdot y) \cdot z and x + (y+z) = (x+y) + z (associativity)

x\cdot(y+z) = (x\cdot y) + (x\cdot z) (distibutivity)

Any set of elements which contains two distinguished elements denoted 0 and 1, and on which there are defined two operations + and \cdot for which the above four properties hold is said to be a field. Thus, the set of rationals \mathbb{Q} is a field. The set of integers \mathbb{Z} is not a field, because not every element has an inverse (there is no m \in \mathbb{Z} such that 2m = 1).

There are other fields: for example the set of all real numbers, denoted \mathbb{R}, is a field, as is the set of complex numbers \mathbb{C}. But there are many more: there are infinitely many different fields between the rational numbers and the complex numbers. How do they arise?

The field of rational numbers is closed under the operations of addition, subtraction, multiplication and division. However, it is not closed under all operations; for example, it is not closed under taking square roots. Indeed, \sqrt{2} is not a rational number. More generally, given an algebraic equation with rational coefficients

a_n x^n + \ldots a_1 x + a_0 = 0

(“with rational coefficients” means that a_0, \ldots, a_n are all rational) the solutions of this equation are not necessarily rational. The number \sqrt{2}, can be thought of as the solution to the algebraic equation

(*) x^2-2 = 0.

However, consider the set of real numbers

\mathbb{Q}[\sqrt{2}] := \{p+q\sqrt{2} : p,q \in \mathbb{Q}\}.

This turns out to be a field too — it is the smallest field containing \mathbb{Q} and \sqrt{2} — and in this field equation (*) has a solution. Not all algebraic equations with rational coefficients have a solution in \mathbb{Q}[\sqrt{2}], for example, the equation

x^2+1 = 0

has no solution in \mathbb{Q}[\sqrt{2}]. However, if we “join” the imaginary unit i = \sqrt{-1} to \mathbb{Q}[\sqrt{2}] to form the smallest field containing \mathbb{Q}[\sqrt{2}] and i, which turns out to be

\mathbb{Q}[\sqrt{2},i] = \{p+q\sqrt{2}+ri + s\sqrt{2}i : p,q,r,s \in \mathbb{Q} \}.

This discussion leads to the following definitions. If E and F are fields and E is contained in F, then we say that F is an extension of E. If F can be formed by adding finitely many solutions to algebraic equations with coefficients in E, then F is said to be a finite algebraic extension of E. If t is a n element of F which is the solution of an algebraic equation with coefficients in E, then t is said to be algebraic over E.

In classical mathematics one would study a certain field, usually \mathbb{Q} or \mathbb{R} or \mathbb{C}, and try to understand the arithmetic operations or the sovability of algebraic equations within the given field of study. In Galois Theory (named after Evariste Galois, one of the most interesting characters in the history of mathematics) one studies instead all field extensions of a certain field (Galois theory is the subject of a course which mathematics students (and only mathematics students) learn in their second or third year). We started by discussing a few rather concrete problems, but now we have brought up the issue of studying all field extension of a certain field. This seem to be going only deeper and deeper into abstraction and generalization, and indeed, we cannot go into the deep and beautiful theory that Galois and the mathematicians who followed created. But surprisingly, after Galois took this step into higher realms of generalization, it was possible to come back to the concrete problems that we discussed before, and to understand them. With every finite algebraic extension Galois associated  a group – which is a kind of finite combinatorial object. The structure of this group can often be understood in rather concrete terms, and this gives a tractable way of analyzing field extensions and solving problems related to algebraic equations. In fact, it solves other problems as well.

4. Applications to various problems

Geometry: Let us return to the problem of squaring the circle. It is easy to see, that if we can square the circle, then we must be able to construct the side of the square, which has length \sqrt{\pi}. Thus \sqrt{\pi} would be constructible, and consequently \pi would be too. On the other hand, in 1837, Wantzel (following the work of Galois) proved that if a number is constructible, then it is algebraic over \mathbb{Q}. Later, in 1882, Lindemann proved that \pi is not algebraic (I am going to prove this fact to my students next semester, in the course Infi 2 (Calculus 2)). Thus, it is impossible to square the circle. After more than 2000 years, this open problem came to its close.

Algebra: The immediate motivation for developing Galois theory was to construct a general theory of algebraic equations. Hence the connection to the problem of finding formulas for algebraic equations of various orders is not too surprising. But it is still impressive to see how Galois theory, on the one hand, explains how one may reach the known formulas for equations of third and fourth degree, and on the other hand it can explain why no such formula can exist for equations of the fifth degree or higher. Roughly, the group that is associated with the field extension that comes from adjoining the roots of an equation of the fifth degree can be very complex, and this complexity encodes in a precise way the impossibility have having a formula for solving such equations.

Calculus: Here is a more surprising application. It turns out that differentiable functions also form a kind of “field”. Just as the fields and field extensions encode number fields and various algebraic operations that can be made on these numbers, there exists a notion of a “differentiable field” which encodes algebraic operations as well as differentiation and anti-differentiation. There is a differentiable version of Galois theory that clarifies the problem of finding closed form expressions for indefinite integrals, and explains why certain indefinite integrals can not be written as an elementary function. In particular, this theory explains the fact that the indefinite integrals

e^{-x^2}\,\, , \, \, \frac{\sin x}{x} \,\, , \,\, x^x

are all inexpressible in terms of elementary functions.

5. Conclusion

We have hinted how several different classical and concrete problems of practical significance can all be approached by some unifying ideas; how deep and abstract constructions help shed light on problems which are ages old, reveal the connections between them, and point the way to their solution. It is impossible to convey in such a short time how this really works. One really has to study a lot of mathematics to fully understand this higher aspect of the subject.