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Hello, Galileo wrote this grand book, The Universe, is written in the language of mathematics. It was said before Galileo and has been said since, and in the last decade of the 20th century, it's been said again, most emphatically.

So how important is maths in relation to other sciences at the end of the 20th century? What insight can you give us into the origins of life and the functioning of our brains? And what does it mean to say that mathematics has become more visual?

Joining me is Ian Stewart who's professor of mathematics, Gresham professor of Geometry at the University of Warwick, and one of the country's most prolific popularizers of mathematics having written or co-authored over 60 books on the subject.

Also an active research mathematician with over 120 published papers, his most recent book was Life's Other Secret, The New Mathematics of the Living World, published last year.

Brian Butterworth is professor of cognitive neuroscience at University College London. His book The Mathematical Brain has just been published. It looks at the way in which the brain deals with numbers and how this is influenced by cultural factors.

Currently he's conducting research in Britain, Italy and China on arithmetical processes in normal and brain-damaged adults and children.

Ian Stewart, you quote Karl Friedrich Gauss, who you considered to be one of the greatest mathematicians who said in the 18th century that maths should be about notions, not notations. Can you enlarge on this?

Yeah, I think he Gauss was trying to counter the idea that what math is about is doing calculations with symbols, algebra, and these very formal things.

Gauss was saying, no, it's about ideas and those ideas can be very, very broad. And what we're seeing now is that this point of view is unfolding in a way that nobody would have anticipated at the time Gauss said it, which is that mathematics in certain senses is providing all sorts of areas you wouldn't expect it to turn up in.

People listening, and myself, I'm listening, think that maths is about earlying up, subtracting, long revision and so on. And people are much more sophisticated and that will still take it.

So when you say ideas, can you give us an example?

Okay, let's take for example the question of when a living creature grows, say a baby growing in the womb, it grows from an embryo, it turns into a fetus, it turns into a child.

And there are a whole lot of processes going on in changes of shape, changes of form, changes of internal chemistry, all of these things.

Mathematics has quite a lot to say about that kind of behavior, that kind of process, and I think in the future it will have a lot more to say about that.

So that's the idea, that is an idea of the way that life develops rather than a simple adding up.

You say that it's also becoming more visual. Can you tell us what you mean by that?

Mathematicians have spent the last 50 years or so trying to turn visual intuition, informal visual intuition, the feeling we have in our heads for the shapes of things, into a kind of logically precise way of thinking and things you can write down.

And so, instead of just using numbers and doing arithmetic, there's a kind of free flowing mathematics of form, but it is just as mathematical as it is just as logical.

The invention of topology, which is rubbish geometry, the idea that a coffee cup is the same as a donut is what the standard cliché for topology.

Because a coffee cup has one hole in it, it's the hole in the handle where you put your finger through, an American style donut has a hole in it.

You can make a plasticine donut and deform it slowly into a coffee cup.

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