A Brief Guide to the Great Equations by Robert Crease Read Free Book Online

Pythagorean theorem can teach. For those who do not merely learn how to prove it, but manage to come up with a new proof, the experience teaches the thrill of creativity itself. The person who does this is not merely watching the proof come into being as a spectator watches the unfolding of a little play – that person has become a playwright, doing what mathematicians do, practicing mathematics as a creative art, experiencingthe joy of creation, discovering that the true essence of mathematics is doing more mathematics. Such a person has discovered the power of discovery.
    For Plato, Hobbes, Descartes, Hegel, Schopenhauer, Loomis, Einstein, Fraser, and countless others, the Pythagorean theorem was far more than a means to compute the length of hypotenuses. To someone who follows the reasoning, something more than the bare result becomes evident. In the experience of one thing – the content, the mathematics – there is a moment of manifestation in which something else, a structure of reasoning, also comes to appearance. It is a rugged, hardy, stubborn piece of knowledge that no religious conviction can dispel, no political ideology can disguise, no academic artifice can conceal.
    In a similar way that 1 + 1 = 2 imparts the idea of addition, so the Pythagorean theorem imparts the idea of proof making. It makes possible what philosophers call categorical intuition: one can see in it more than bare content, but a structure of the understanding. It involves a journey short enough that its stages can be taken in at a glance to illustrate the journey of knowledge itself. It is a proof that demonstrates Proof.

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‘The Soul of Classical Mechanics’:
NEWTON’S SECOND LAW OF MOTION
F
=
ma
    [Newton’s own] description: A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.
    DISCOVERER : Isaac Newton
    DATE : 1684–87
    Newton’s second law of motion,
F
=
ma
, is the soul of classical mechanics.
– Frank Wilczek,
Physics Today
    The equation
F
=
ma
is shorthand for Newton’s second law of motion. It is the 1 + 1 = 2 of classical mechanics. It seems obvious and straightforward. The equation appears simply to translate an ordinary experience into measurable terms: push something and it either starts to move or moves differently.
    Yet, like 1 + 1 = 2,
F
=
ma
erupts into mystery when looked at closely. It does not, in fact, refer to ordinary experience but to an abstract world of zero resistance: in the real world we have to continue pushing things like desks and carts to keep them moving at the same speed. The equation does not incorporate Einstein’s famous discoveryof the interchangeability of mass and energy. It gives centre stage to force – a concept absent from most formulations of contemporary theories like relativity and quantum mechanics. Finally, the equation seems, in a contradictory way, to be both a name and a description. It seems both to define force, mass, and motion, and to state an empirically discovered and testable relationship among them.
    How can such an elementary equation about something as ordinary as motion conceal so many complexities? The answer can be gleaned in the remarkable historical journey that led from ancient times to the equation’s formulation in the seventeenth century. To arrive at this equation, human beings had to train themselves to look at motion in new ways – to learn to look at different aspects of it, and to change how they thought about what they saw. In the course of this vast journey, new sights slowly and progressively came into view, occupied centre stage, and then vanished off the horizon, with each familiar landscape slowly yielding to another, until the travelers found themselves in an entirely new world.
Greek Notions of Motion and Change
    The journey began in primitive times, when human beings saw the world as ruled by deities. This was natural and inevitable, perhaps the simplest and