Mathematica

doing it without ever noticing. We start with human language, shift to mathematical language for reasoning, and return to human language. We do this each time we formulate hypotheses and try to draw conclusions from them. This day-to-day activity

really don’t amount to much. Language is the most difficult thing of all to learn. It’s an incredibly long, frighteningly difficult process. At eighteen months old, hardly anything we babble is intelligible. And yet we keep on trying all day long. It’s enough to get you down, but we never give it up. No one ever says: “Language, that’s really not m

The only way to get there is to go beyond words. Replacing “the sum of whole numbers from 1 to 100” with “1 + 2 + 3 + . . . + 98 + 99 + 100” is a good start. You might have the impression of seeing the sum in a more tangible and concrete way. But that’s only ever an illusion. In reality, you’ll be missing most of the numbers, those hidden by the el

My intuition isn’t any less fallible than yours. It’s always getting things wrong. I have, however, learned never to be ashamed of it. I don’t disdain my mistakes, I don’t push them aside, because I don’t think that they betray my intellectual inferiority or some cognitive biases hardwired in my brain. On the contrary. Nothing’s more exciting than

Rationality has as bad a reputation as mathematics. And like the latter, it exists in two versions. The visible side of rationality presents itself in the form of established knowledge, science and technology, and well-structured and logically sound arguments. Schools spend a lot of time teaching it, with mixed results. The flip side of rationality

Fear of mistakes and fear of the truth is one and the same thing. The person who fears being wrong is powerless to discover anything new. It’s when we fear making a mistake that the error which is inside of us becomes immovable as a rock.

undeniable that mathematical creation feels magical and supernatural. But behind all that there’s necessarily a human reality that is neither supernatural nor magical.

Contrary to common belief, it’s never abstraction that makes math difficult to understand. Abstraction is our universal mode of thinking. The words that we use are all abstractions. Speaking, making sentences, is to manipulate and assemble abstractions. In that respect, four-dimensional geometry isn’t any more abstract than two-dimensional geometry

Before ending this chapter, let’s take the time to clarify what Cartesian doubt really is, and the personal benefits everyone can get from it. After all, this is the best possible introduction to another key concept that up to now we’ve only begun to address: the concept of the mathematical proof. At school we’re taught that Cartesian doubt is a me