What laws the dead have died to disobey
What good are dissidents? A political philosopher brought up in the Western tradition is likely to answer that dissidents are to their societies what canaries are to their miners: an alive canary signifies that the air in the mine is fit for breathing, and an alive dissident signifies that his society is fit for living. Ostensibly, this is because tolerance is a hallmark of an open society, so the presence of dissidents, the argument goes, gives the rest of us the comfort to think that if one day it be us, we too shall be spared. All that may well be true, but my intention here is less political. I want to use mathematics to illustrate (to prove, as a mathematician would say), that apart from symbolic significance, disobedience actually has an inherent pragmatic value: it is the means to advancing knowledge.
The simplest numbers known to man for thousands of years are the natural numbers: 1, 2, 3, and so on. The are the easiest for a child to grasp simply by counting fingers on her hand. The first “mathematical” thing we can do with these numbers (or any other kinds of numbers) is to come up with an operation on them, like addition; it arises each time a child wants to work out how old she will be next year. The nice thing about addition is that the sum of any two natural numbers is also a natural number. This property is important enough in mathematics to have a name: closure. Natural numbers are said to be closed under addition.
If mathematicians had been docile conformists, we would have never known numbers other than natural numbers. But somewhere along the line, a dissident came along and broke the harmony by concocting the inverse operation — the subtraction. Which right away was a problem, because the difference between two natural numbers is not always natural: 1 ➖ 2 = ? To salvage the harmony, new kind of numbers had to be invented — the negative numbers, so that 1 ➖ 2 = ➖1. Grouped with the positive numbers, they form the set of whole numbers, which have the property of closure restored under both addition and subtraction.
Another advance in mathematics came when people got tired of adding a number to itself too many times. If a farrier can shoe 3 horses a day and charges two coins per horseshoe, how much will he make in a month? Multiplication was invented to simplify just this sort of math: 3✖️30✖️4✖️2= 720 coins is the answer. So long as we stick to multiplication, whole numbers held up: the product of two whole numbers is itself a whole number. But the harmony is again broken, when the inverse operation was needed: how much must a farrier charge for each horseshoe to make 700 coins per month? The answer is 700➗(3✖️30✖️4) = 35➗18 = ? The quotient of two whole numbers is not always a whole number! Thus, the adoption of the inverse operation again necessitated the invention of a new kind of numbers: the fractions, or the rational numbers. The number of coins the farrier needs to charge per horseshoe is 35/18. And this does the trick: rational numbers are closed under addition, subtraction, multiplication and division.
Rational numbers also possess a new remarkable property: they are dense in the sense that between any two rational numbers there always exists another rational number, and, by induction, an infinite number of rational numbers. To illustrate this graphically, we have to depart from discrete points standing for whole numbers, extending infinitely to the left and to the right, and adopt the continuous line model, where each infinitely small point represents a rational number.
Or does it?
The next operation after multiplication you learned is power: any rational number r multiplied by itself a natural number of times n is r to the power of n, or rⁿ. Just like multiplication was first invented as a simple notational shortcut, but eventually led to the discovery of a new kind of numbers, — so does power. The inverse of power n is n-th root: if rⁿ = q, then r is the n-th root of q, i.e. the number which, when raised to the n-th power, yields q. Multiplying a rational number by itself a bunch of times is still a rational number, so rational numbers are closed under power. But not under root: a square root of 2 is not a rational number because there is no rational number r such that r² = 2. Mathematicians simply denote this mystery number as √2 and call this new kind of numbers (surprise) irrational. Together with the rational numbers they are called real numbers and real numbers represent every infinitely small point on a line.
The point, illustrated by these observations, is that the thinking against the received wisdom is the only thinking that leads to new wisdom. The new knowledge is yours for the asking, but the only way to discover it is to think differently from the thinking that brought you here.