Decidability

How do we know if a problem is solvable or a theorem provable? This mode of reasoning takes us into the heart of mathematics. Obviously, if something cannot be solved or proved, we should not be wasting our time trying to find a solution or proof; that is, if it can be shown to be undecidable, that is the end of the matter. This is a central principle on which computer science is built. If something can be programmed to produce a certain output, then the algorithm used shows that it is decidable in the first place; if it does not produce an output, then it is undecidable. This chapter deals with the question of decidability and its basis in logic, culminating with Kurt Gödel’s remarkable proof that within any formal system of logic there are propositions (statements) that can be neither proved nor disproved. Since then, mathematicians have embraced a much more flexible way to do mathematics, without discarding the previous ways.

Infinity

The paradoxes of Zeno in antiquity might seem like sophisms, but, as it has turned out, they shed light on mathematical infinity and have led to many derivative ideas in mathematics and science. Zeno’s paradoxes portray movement in terms of discrete points on a number line, where a move from A to B is done in separate (discrete) steps. But the gap in between is continuous. So, to resolve the paradoxes a basic distinction between discrete and continuous is required—a gap that was filled with the calculus much later. The infinity concept became in the nineteenth century the basis for the discovery of new numbers—by the German mathematician Georg Cantor—which seemed at the time to be counterintuitive. This chapter looks at the paradox of infinity and at Cantor’s ingenious discoveries.

i

What kind of number is √−1? In a way that parallels the unexpected discovery of √2 by the Pythagoreans, when this number surfaced as a solution to a quadratic equation, mathematicians asked themselves what it could possibly mean. Not knowing what to call it, René Descartes named it an imaginary number. Like the irrationals, the discovery of i led to new ideas and discoveries. One of these was complex numbers—numbers having the form (a + bi), where a and b are real numbers and i is √−1. Incredibly, complex numbers turn out to have many applications. They are used to describe electric circuits and electromagnetic radiation and they are fundamental to quantum theory in physics. This chapter deals with imaginary numbers, which constitute another of the great ideas of mathematics that have not only changed the course of mathematics but also of human history.

e

The number e, which is equal to 2.71828…, might seem like something trivial—a play on numbers by mathematicians. Nothing could be further from the truth. It is defined as the limit of the expression (1 + 1/n)n as n becomes large without bound. What possible connection does this number have with other areas of mathematics? As it turns out, it forms the base of natural logarithms; it appears in equations describing growth and change; it surfaces in formulas for curves; it crops up frequently in probability theory; and it appears in formulas for calculating compound interest. It is another example of how the ideas in mathematics are not isolated ones, but highly interrelated. The purpose of this chapter is, in fact, to link e to other great ideas, showing how mathematical discovery forms a chain—a chain constructed with a handful of fundamental ideas that appear across time and space, finding form and explanation in the writings and musings of individual mathematicians.

The algorithm

Mathematics has formed a partnership with computer science, since at least the 1940s. Devising algorithms and turning them into computer programs is akin to deconstructing a problem, a proof, or a theory into its key components. In other words, a computer can be used to attack the problem of decidability, discussed in the previous chapter, on the basis of the validity (or not) of some algorithm. This chapter deals with this key notion, completing the chain of ideas described in this book that started with Pythagoras. Although not named in this way, the concept of algorithms goes back to Euclid and his Fundamental Theorem of Arithmetic. Computers are adept at solving all kinds of problems that involve pattern and structure. They do so by computing all possibilities for a problem, rather than providing a theoretical explanation for the solution. The latter is left up to the programmer—in this case, the mathematician.

Exponents

Mathematicians have devised notations and symbols to compress information and to explore mathematics via the symbols themselves. In the 1500s the invention of exponential notation, which was devised as a type of shorthand to facilitate the cumbersomeness of just reading repeated multiplications of the same digit, was a watershed event. Exponential notation not only saves space and lessens the mental energy required to process the relevant information, it is critical for writing larger and larger numbers. Moreover, mathematicians started to play with exponential notation in an abstract way, discovering new facts about numbers, leading to the notion of logarithms and all the discoveries that this, in turn, has brought about. This chapter will deal with exponents and logarithms, and their significance to the history of mathematics—a history often characterized by problems of notation that have led serendipitously to new ideas and branches.

Zero

Numerals are symbols that represent numbers. The most commonly used numerals, which are easy to read after one has learned to use them, are the decimal ones. The principle used to construct them is an efficient one—the position of each digit in the numeral indicates its value as a power of ten. But for such numerals to work this system requires a symbol as a place-holder for a position that has “nothing” in it. That symbol is 0, which makes it possible to differentiate between numbers such as “eleven” (= 11) “one hundred and one” (= 101), and “one thousand and one” (= 1001) without the need for additional numerals. The 0 tells us, simply, that the position is “empty.” This chapter looks at the origin of this extraordinary symbol, which over time became a number like any other, but with peculiar properties. It led to concepts such as negative numbers and the number line, which became crucial to the evolution of mathematics itself.

π‎ (Pi)

Pi is defined formally as the ratio of the circumference (C) of a circle to twice the radius (r). This was known in antiquity even before mathematicians tackled this ratio. As such, it is an amazing discovery, which tells us that no matter the size of a circle, the ratio of the circumference to the diameter (twice the radius) never changes. This might strike us as obvious, but the obvious often holds many hidden treasures. Like the Pythagorean theorem, π‎ established something that people knew practically and mathematicians gave this knowledge an abstract form, so that it could be used and studied further. And it has produced incredible results. This chapter deals with π‎, and its appearances in other part of mathematics and in Nature, reminding us that it is there, and defying us to understand why. The more technologically advanced we become and as our picture of π‎ grows ever more sophisticated, the more its mysteries grow.

Prime numbers

The Pythagoreans developed many of the ideas related to numbers that have become so familiar to us, including even and odd numbers, square numbers, triangular numbers, and so on. They also discovered that some integers cannot be decomposed into factors. These are called prime numbers and they constitute the building blocks of all the other integers, called composite. This chapter deals with the prime numbers in a general non-technical way, since much of the writing about them is quite specialized. The prime numbers have remarkable properties, many of which are still resistant to being proved. Prime numbers matter deeply to mathematics, not to mention to the progress of human knowledge generally. Pythagoras believed that prime numbers were part of a secret code which, if deciphered, would allow us to unlock the mysteries of the cosmos itself.

The Pythagorean theorem

The Pythagorean theorem constitutes one of the first great ideas of mathematics. Its importance is evident in the fact that it has been taught in schools throughout human history; it has had many applications in science and engineering; it has cropped up in numerous other mathematical ideas; it has led to discoveries, such as Fermat’s last theorem; and the list could go on and on. In a phrase, the theorem is one of those ideas that have mattered to mathematics and to human history. This chapter looks at the theorem and at why it is so profoundly important. It also deals with some key concepts that the theorem has introduced, including irrational numbers.