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Hello Jonathan Swift encapsulated the counter-intuitive character of infinity with unsusional style.

So, a naturalist observed a flea, he wrote, has smaller fleas on him that prey.

And these have smaller fleas to bite him, and so proceed add infinitum.

Alas, the developing utility which mathematicians put to the idea of infinity didn't find the English philosopher Thomas Hobbes quite so relaxed.

When confronted with a diagram depicting an infinite solid whose volume was finite, he wrote,

to understand this for sense, it is not required that a man should be a geometrician or logician, but it should be mad.

Yet philosophers and mathematicians have continued to grapple with the unending, and it's a core concept in modern maths.

So, what is mathematical infinity? Are some infinities bigger than others, and does infinity exist in nature?

With me to discuss the mathematics of the infinity, of the infinite is Ian Stewart, professor of mathematics at the University of Warwick,

Sarah Rees, reader in pure mathematics at the University of Newcastle, and Robert Kaplan, co-founder and co-director of the Math Circle at Harvard University,

and author of a new book, The Art of the Infinite, are lost language of numbers.

Ian Stewart, let's start this with Zeno, who illustrated the problem by using the infinitesimal,

and one of his paradox, the best known, is the tortoise and the hare, Achilles and the hare, the race between them,

and how Achilles or the hare can never catch the tortoise according to his mathematics.

That's right, Zeno, what he was really trying to do was to explain and show up certain logical problems with assumptions about the structure of space and time.

So, just to show that this little story has a serious philosophical point.

Achilles and the tortoise start off, they're having a race, the tortoise gets head start, but Achilles can run much faster.

And so, we all know that Achilles will catch the tortoise up very rapidly, be out in front, but Zeno says, no, he won't.

Think about what happens, the tortoise is out in front, some specific amount in front.

Achilles has to get to the point where the tortoise starts from.

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