Is the space continuous or discrete? Is it smooth, without gaps or discontinuities, or grainy with a limit to the smallest possible distance? And the time ?
Can the time be repeatedly split into smaller periods without any limit, or is there a shorter time interval?
We don’t know the answers. There is a lot that we don’t know about physical reality: is the universe finite or infinite? Are space and time arbitrarily divisible? Does our digital system represent physical space and time?
The construction of the set of numbers was 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 the 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 integers.
But not all numbers are rational. “As any schoolboy knows,” the length of the diagonal of a unit square is the square root of 2. This follows from the Pythagorean theorem.
Hippasius, a disciple of Pythagoras, discovered that the root of 2 cannot be expressed as a ratio of whole numbers.
Irrational numbers caused great anguish to the Greeks: they had developed a concept of the 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 accept them as valid quantities. They also invented or discovered “imaginary” numbers, needed to solve simple equations. These numbers, no more imaginary than the others, are today crucial in quantum mechanics: the Schrödinger equation contains the root of negative 1.
German mathematician Richard Dedekind developed a technique for constructing real numbers: each irrational number separates rational numbers into two sets, larger and smaller. This separation, now called the Dedekind cut, is used to define irrational numbers.
By taking the rational and irrational numbers together, we get the set of real numbers. 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 he do it?
Georg Cantor, the founder of set theory, introduced a multitude of infinite numbers. Indeed, he showed that there is an infinity of infinities. It all goes beyond the limits of the real line; it did not inject new numbers into the real line.
But others, going back to Archimedes, thought of indefinitely small quantities. Unlike infinites, a number that is smaller than any real number and yet greater than zero is called an infinitesimal. We can consider an infinitesimal as the inverse of an infinity.
Oddly enough, despite his remarkable view of infinity, Cantor was strongly opposed to the idea of infinitesimals and denied them as a logical possibility. This was despite their value in the arena 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 designed the Surrealist Number System, greatly expanding the number system to what he showed to be the largest “complete ordered field” possible.
And yet there are loopholes even in the surreal digital system. All the systems we have discussed are sets of points, and there seems to be a fundamental difficulty in defining a continuum in terms of sets of points.
Peter Lynch is Professor Emeritus at the School of Mathematics and Statistics, UCD. He blogs at thatsmaths.com