To infinity and beyond: trying to figure out number systems

That’s Maths: Construction of set of numbers inspired by conception of physical world

Detail of The School of Athens by Raphael (Raffaello Sanzio), 1509, showing Pythagoras. Photograph: WIki/Creative Commons

Detail of The School of Athens by Raphael (Raffaello Sanzio), 1509, showing Pythagoras. Photograph: WIki/Creative Commons

 

Is space continuous or discrete? Is it smooth, without gaps or discontinuities, or granular with a limit on how small a distance can be? What about time?

Can time be repeatedly divided into smaller periods without any limit, or is there a shortest interval of time?

We don’t know the answers. There is much we do not know about physical reality: is the universe finite or infinite? Are space and time arbitrarily divisible? Does our number system represent physical space and time?

The construction of the set of numbers has been inspired by our conception of the physical world. But our ideas have evolved and changed over the centuries. At an early stage, we developed the counting numbers, 1, 2, 3 and so on. Then, subdivisions of food, goods and land led to the introduction of fractions or rational numbers, so-called because each fraction is a ratio of two whole numbers.

But not every number is rational. “As every schoolchild knows”, the length of the diagonal of a unit square is the square root of 2. This follows from the theorem of Pythagoras.

Hippasus, a follower of Pythagoras, discovered that the root of 2 cannot be expressed as a ratio of whole numbers.

Irrational numbers caused the Greeks great angst: they had developed a concept of a universe based on numbers, and they could not accept irrational numbers as legitimate.

Renaissance mathematicians had great difficulty with negative numbers, and were slow to allow them as valid quantities. They also invented or discovered “imaginary” numbers, which were needed to solve simple equations. These numbers, no more imaginary than any others, are crucial today in quantum mechanics: Schrödinger’s equation contains the root of minus 1.

German mathematician Richard Dedekind devised a technique for constructing the real numbers: each irrational number separates the rational numbers into two sets, those greater than it and those smaller. This separation, now called a Dedekind cut, serves to define the irrational numbers.

Taking the rational and irrational numbers together, we get the set of real numbers. The real numbers form a continuum, in which there are no discontinuities or gaps. Each point on the line corresponds to a real number. Or does it?

Archimedes

Georg Cantor, the founder of set theory, introduced a multitude of infinite numbers. Indeed, he showed that there is an infinitude of infinities. All of these are beyond the limits of the real line; he did not inject any new numbers into the real line.

But others, going back to Archimedes, thought about indefinitely small quantities. In stark contrast to infinities, a number that is smaller than any real number and yet larger than zero is called an infinitesimal. We can think of an infinitesimal as the inverse of an infinity.

Curiously, despite his remarkable vision of the infinite, Cantor was strongly opposed to the idea of infinitesimals and denied them as a logical possibility. This was despite their value in the field of calculus, which had been studied for 200 years before Cantor. But the innovative British mathematician John Conway, a recent victim of Covid-19, had no such inhibitions. He devised the system of surreal numbers, vastly extending the number system to what he showed was the largest possible “complete ordered field”.

And yet, there are gaps even in the surreal number system. All the systems we have discussed are collections of points, and there seems to be a fundamental difficulty defining a continuum in terms of point sets.

Peter Lynch is emeritus professor at the School of Mathematics and Statistics, UCD. He blogs at thatsmaths.com

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