The human brain finds it extremely hard to cope with a new level of abstraction. This is why it was well into the eighteenth century before mathematicians felt comfortable dealing with zero and with negative numbers, and why even today many people cannot accept the square root of minus-one as a genuine number.

There can be very little of present-day science and technology that is not dependent on complex numbers in one way or another.

I certainly do care about measuring educational results. But what is an 'educational result?' The twinkling eyes of my students, together with their heartfelt and beautifully expressed mathematical arguments are all the results I need.

I saw the first one [video with Hans Rosling ] when he did - I think it was his first one - in 2006, a TED Talk. And for the first time in my life, I thought here's someone who can take statistics that most people regard as dull and boring and bring it alive.

In fact, we tend to think things have been getting much worse. In fact, over the last 50 years, almost everything in the world on a global scale has got better. And the way that Hans [ Rosling] did this - it was very good.

I've never met Hans Rosling, but I just knew him through his many YouTube videos, and they were absolute dynamite.

The increased abstraction in mathematics that took place during the early part of this century was paralleled by a similar trend in the arts. In both cases, the increased level of abstraction demands greater effort on the part of anyone who wants to understand the work.

The whole apparatus of the calculus takes on an entirely different form when developed for the complex numbers.

In fact, when you try to use [Hans Rosling] data to predict the future, all sorts of problems arise. But what it does do is say, hey, just catch your breath a minute and see what's really been going on. We do have reason to feel good about the fact we've made progress.

Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence.

What makes it possible to learn advanced math fairly quickly is that the human brain is capable of learning to follow a given set of rules without understanding them, and apply them in an intelligent and useful fashion. Given sufficient practice, the brain eventually discovers (or creates) meaning in what began as a meaningless game.

I firmly believe that mathematics does not exist outside of humans. It is something we, as a species, invent.

In addition to its use in arithmetic and science, the Hindu-Arabic number system is the only genuinely universal language on Earth, apart perhaps for the Windows operating system, which has achieved the near universal adoption of a conceptually and technologically poor product by the sheer force of market dominance.

A PhD in Mathematics is three years of guessing it wrong, plus one week of getting it right and writing a dissertation.

Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience.