If Ideas Had Shapes

A quoteblog ranging from philosophers in bathrobes to galaxy-rises

Category: Math & Stats

W. M. Priestley – Calculus: A Liberal Art (1998)

The purpose of education is to learn to tell the truth, and mathematics—if taught for its own sake—promotes this end by helping students to sharpen their intuition, to learn to reason better, to recognize valid reasoning, and to write and say more precisely what they intend. Such an education is essential to freedom, for without knowing how to tell the truth one is easily boxed in by sophistries. … Nothing is more abhorrent to Plato than the Sophists who use their art of persuasion to empower themselves through deliberately deceptive arguments with no concern for truth or other ultimate ends such as goodness and beauty.

W. M. Priestley – Calculus: A Liberal Art (1998)

A surprising amount of mathematics consists in simply saying the same thing in many different ways, until it is finally said in a way that makes it simple.

Douglas Hofstadter – Gödel, Escher, Bach (1979)

In short, Gödel showed that provability is a weaker notion than truth, no matter what axiomatic system is involved.

Douglas Hofstadter – Metamagical Themas (1985)

Imagine breaking up one second into as many tiny fragments as there are seconds in 30 years. That is how tiny a nanosecond—a billionth of a second—is. To a computer, a second is a lifetime! Of course, the computer is dawdling compared with the events inside the atoms that compose it. Take one atom. A typical electron circling a typical nucleus makes about 1015 orbits per second, which is to say, a million orbits per nanosecond.

Some stars—neutron stars—are so tightly packed that if you could remove from any of them a cube a millimeter on an edge, its mass would be about half a million tons, equal to the mass of the heaviest oil tanker every built, fully loaded!

There are about 25,000 megatons of nuclear weapons in the world [in mid-1982]. … Now if you just say to yourself “one megaton equals Paris’s doom” (or some suitable equivalent), then I think that the phrase “25,000 megatons” will become as vivid as the long string of zeros—in fact, probably more vivid.

Douglas Hofstadter – Metamagical Themas (1985)

In any case, a pretty good rule of thumb is this: Your estimate should be within ten percent of the correct answer—but this need apply only at the level of your perceptual reality. Therefore you are excused if you guessed that Rubik’s cube has 1018 positions, since 18 is pretty close to 19.5, which is about what the number of digits is.

Douglas Hofstadter – Metamagical Themas (1985)

Kurt Gödel’s famous Incompleteness Theorem in metamathematics can be thought of as arising from his attempt to replicate as closely as possible the liar paradox in purely mathematical terms.

(The liar paradox is “This sentence is false.”)

Edward Tufte – The Visual Display of Quantitative Information (1983)

If the statistics are boring, then you’ve got the wrong numbers. Finding the right numbers requires as much specialized skill—statistical skill—and hard work as creating a beautiful design or covering a complex news story.