Why Beauty is Truth by Ian Stewart Read Free Book Online

axioms for a piece of mathematics are the rules of the game.
    Anyone who objects to the axioms can change them if they wish: however, the result will be a different game. Mathematics does not assert that some statement is true: it asserts that if we make various assumptions, then the statement concerned must be a logical consequence. This does not imply that the axioms are unchallengeable. Mathematicians may debate whether a given axiomatic system is better than another for some purpose, or whether the system has any intrinsic merit or interest. But these discussions are not about the internal logic of any particular axiomatic game. They are about which games are worthwhile, interesting, or fun.

    The consequences of Euclid’s axioms—his long, carefully selected chains of logical deductions—are extraordinarily far-reaching. For example, he proves—with logic that in his day was considered impeccable—that once you agree to his axioms you necessarily must conclude that:
    â€¢Â  The square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.
    â€¢Â  There exist infinitely many prime numbers.
    â€¢Â  There exist irrational numbers—not expressible as an exact fraction. An example is the square root of two.
    â€¢Â  There are precisely five regular solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
    â€¢Â  Any angle can be divided exactly into two equal parts using only straightedge and compass.
    â€¢Â  Regular polygons with 3, 4, 5, 6, 8, 10, and 12 sides can be constructed exactly using only straightedge and compass.
    I have expressed these “theorems,” as we call any mathematical statement that has a proof, in modern terms. Euclid’s point of view was rather different: he did not work directly with numbers. Everything that we would interpret as properties of numbers is stated in terms of lengths, areas, and volumes.

    The content of the Elements divides into two main categories. There are theorems, which tell you that something is true. And there are constructions, which tell you how to do something.
    A typical and justly famous theorem is Proposition 47 of Book I of the Elements , usually known as the Pythagorean theorem. This tells us that the longest side of a right triangle bears a particular relationship to the other two sides. But it does not, without further effort or interpretation, provide a method for achieving any goal.

    The Pythagorean theorem.
    A construction that will be important in our story is Proposition 9 of Book I, where Euclid solves the “bisection problem” for angles. Euclid’smethod of bisecting an angle is simple but clever, given the limited techniques available at this early stage of the development.

    How to bisect an angle with straightedge and compass.
    Given (1) an angle between two line segments, (2) place your compass tip where the segments meet, and draw a circle, which crosses the segments at two points, one on each (dark blobs). Now (3) draw two circles of equal radius, one centered at each of the new points. They meet in two points (only one is marked), and (4) the required bisector (dotted) runs through both of these.
    By repeating this construction, you can divide an angle into four equal pieces, or eight, or sixteen—the number doubles at each step, so we obtain the powers of 2, which are 2, 4, 8, 16, 32, 64, and so on.

    As I mentioned, the main aspect of The Elements that affects our story is not what it contains but what it doesn’t. Euclid did not provide any method for:
    â€¢Â  Dividing an angle into three exactly equal parts (“trisecting the angle”).
    â€¢Â  Constructing a regular 7-sided polygon.
    â€¢Â  Constructing a line whose length is equal to the circumference of a given circle (“rectifying the circle”).
    â€¢Â  Constructing a square whose area is equal to that of a given circle