Quotes About Mathematics

Anything that's continuous can be sliced exactly (not just approximately) into infinitely many infinitesimal pieces.

Feynman asked Wouk if he knew calculus. No, Wouk admitted, he didn't. "You had better learn it," said Feynman. "It's the language God talks.

All of calculus, and hence all of theoretical physics, hinges on this assumption of continuous space and time. That assumption of continuity has been resoundingly successful so far.

Calculus, like other forms of mathematics, is much more than a language; it's also an incredibly powerful system of reasoning.

I want to stress that—only sixty digits. That's the most we would ever need to express one distance in terms of another.

In a nutshell, calculus wants to make hard problems simpler.

If real numbers are not real, why do mathematicians love them so much?

If we're lucky and skillful enough — if we transform the equations in just the right way — we can get them to reveal their hidden implications. To a mathematician, the process feels almost palpable. It's as if we're manipulating the equations, massaging them, trying to relax them enough so that they'll spill their secrets. We want them to open up and talk to us.

Yet somehow, if the translation from reality into symbols is done artfully enough, the logic of calculus can use one real-world truth to generate another. Truth in, truth out.

In a nutshell, calculus wants to make hard problems simpler. It is utterly obsessed with simplicity.

Pi is fundamentally a child of calculus. It is defined as the unattainable limit of a never-ending process.

Thus the hexagon argument demonstrates ? > 3.

Calculus is the mathematics of change. It describes everything from the spread of epidemics to the zigs and zags of a well-thrown curveball. The subject is gargantuan—and so are its textbooks. Many exceed a thousand pages and work nicely as doorstops.

What I'm trying to say is that his calculation of ? was heroic, both logically and arithmetically. By using a 96-gon inside the circle and a 96-gon outside the circle, he ultimately proved that ? is greater than 3 + 10/71 and less than 3 + 10/70.

Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. All good problem-solvers know that hard problems become easier when they're split into chunks. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme — all the way out to infinity.

Infinity lies at the heart of so many of our dreams and fears and unanswerable questions: How big is the universe? How long is forever? How powerful is God? In every branch of human thought, from religion and philosophy to science and mathematics, infinity has befuddled the world's finest minds for thousands of years. It has been banished, outlawed, and shunned. It's always been a dangerous idea.

we've come to realize that most systems of differential equations are unsolvable, in that same sense; it's impossible to find a formula for the answer. There is, however, one spectacular exception. Linear differential equations are solvable.

That latter result, 3 + 10/70, reduces to 22/7, the famous approximation to ? that all students still learn today and that some unfortunately mistake for ? itself.

If whole numbers and their ratios couldn't even measure something as basic as the diagonal of a perfect square, then all was not number. This deflating letdown may explain why later Greek mathematicians always elevated geometry over arithmetic. Numbers couldn't be trusted anymore. They were inadequate as a foundation for mathematics.

A computer gives the average person, a high school freshman, the power to do things in a week that all the mathematicians who ever lived until thirty years ago couldn't do.

Engineering is merely the slow younger brother of physics.

Recognize, though, that graphs and equations provide an economical and effective way of expressing things that torture the tongue.

There is a spooky quality about the ability of mathematicians to get there ahead of physicists. It's as if when Neil Armstrong first landed on the moon he found in the lunar dust the footsteps of Jules Verne.

The Babylonians had achieved great competence in arithmetic, using a number system based on 60 rather than 10. They had also developed some simple techniques of algebra, such as rules (though these were not expressed in symbols) for solving various quadratic equations.