Quotes About Mathematics

Mathematics catalogues everything that is not self-contradictory; within that vast inventory, physics is an island of structures rich enough to contain their own beholders.

When I was ten years old, all I gave my sweetheart was a pair of projections that turned the group of rotations in four dimensions into principal bundles over the three-sphere. Ancient constructions, though I did rediscover them for myself.' 'How were they received?' 'She liked them so much, she extended them to larger spaces and gave me back the result.

Remember this rule: Like most reality shows, integers have no point whatsoever.)

many reasons to learn how to program: To understand our world. To study and understand processes. To be able to ask questions about the influences on their lives. To use an important new form of literacy. To have a new way to learn art, music, science, and mathematics. As a job skill. To use computers better. As a medium in which to learn problem-solving.

Todo cuanto tuviera que ver con Claire merecía ser repetido a cámara lenta, sin que las matemáticas fueran una excepción

Poetry is as precise a thing as geometry.

This is still the legacy of Euclidean geometry, where space has three dimensions, a plane has two, a line has one, and a point has zero.

Mandelbrot moved beyond dimensions 0,1,2,3…to a seeming impossibility: fractional dimensions. The notion is a conceptual high-wire act. For nonmathematicians it requires a willing suspension of disbelief. Yet it proves extraordinarily powerful.

The fractal dimension of a metal's surface, for example, often provides information that corresponds to the metal's strength.

As the physicist Murray Gell-Mann once remarked: "Faculty members are familiar with a certain kind of person who looks to the mathematicians like a good physicist and looks to the physicists like a good mathematician. Very properly, they do not want that kind of person around.

Most seductive of all was an image that the authors called a strange attractor.

Mathematical Ideas in Biology

Where Newton was reductionist, Goethe was holistic. Newton broke light apart and found the most basic physical explanation for color. Goethe walked through flower gardens and studied paintings, looking for a grand, all-encompassing explanation. Newton made his theory of color fit a mathematical scheme for all of physics. Goethe, fortunately or unfortunately, abhorred mathematics.

Sometimes he and his father would work out puzzles together. Once they came upon a particularly difficult problem that turned out to be insoluble. That was acceptable, his father told him: you can always try to solve a problem by proving that no solution exists. Lorenz liked that, as he always liked the purity of mathematics

The mathematics applied to fluid systems and to electrical systems. But almost no one in the classical era suspected the chaos that could lurk in dynamical systems if nonlinearity was given its due.

In PM, as Gödel said, "one can prove any theorem using nothing but a few mechanical rules.

There must be truths, that is, that cannot be proved—and Gödel could prove it.

Incompleteness was real. It meant that mathematics could never be proved free of self-contradiction.

Shannon used a phrase he had never used before: "information theory.

By contrast, a twentieth-century fluid dynamicist could hardly expect to advance knowledge in his field without first adopting a body of terminology and mathematical technique. In return, unconsciously, he would give up much freedom to question the foundations of his science.

believed, as deeply as he believed anything, that mathematics should be something all by itself. With self-containment came clarity. And clarity, too, went hand in hand with the rigor of the axiomatic method. Every serious mathematician understands that rigor is the defining strength of the discipline, the steel skeleton without which all would collapse. Rigor is what allows mathematicians to pick up a line of thought that extends over centuries and continue it, with a firm guarantee.

But Yorke had offered more than a mathematical result. He had sent a message to physicists: Chaos is ubiquitous; it is stable; it is structured. He also gave reason to believe that complicated systems, traditionally modeled by hard continuous differential equations, could be understood in terms of easy discrete maps.

To find the new x, the rule was to take the old y, add 1 and subtract 1.4 times the old x squared. To find the new y, multiply 0.3 by the old x. That is: xnew = y +1 – 1.4x2 and ynew = 0.3x. Hénon picked a starting point more or less at random, took his calculator and started plotting new points, one after another, until he had plotted thousands.

THE ATTRACTOR OF HÉNON. A simple combination of folding and stretching produced an attractor that easy to compute yet still poorly understood by mathematicians. As thousands, the millions of points appear, more and more detail emerges. What appear to be single lines prove, on magnification, to be pairs, then pairs of pairs. Yet whether any two successive points appear nearby or far apart is unpredictable.