The Joy of x: A Guided Tour of Math, from One to Infinity

maths always involves both invention and discovery: we invent the concepts but discover their consequences

The eminent linguistic philosopher J. L. Austin of Oxford once gave a lecture in which he asserted that there are many languages in which a double negative makes a positive but none in which a double positive makes a negative — to which the Columbia philosopher Sidney Morgenbesser, sitting in the audience, sarcastically replied, “Yeah, yeah.”

there are so many different ways to describe a part of a whole.

As a quantum physicist he would have been acutely aware that at the deepest level, nature disobeys the commutative law

beneath his rarefied taste

She’s once, twice, three times a lady

By sorting the meaningful from the generic, the arithmetic of negative numbers can help us see where the real puzzles lie.

of a purely mathematical flavor

in mathematics our freedom lies in the questions we ask — and in how we pursue them — but not in the answers awaiting us

In that sense, maths always involves both invention and discovery: we invent the concepts but discover their consequences