Imagine Earth were to shrink to the size of a marble. We might be in trouble, but the planet would continue its smooth course around the sun while the moon would maintain its orbit, circling Earth once a month.
Isaac Newton proved Earth’s gravitational pull would be the same even if all the mass were concentrated in a single point. But the density at that point would be infinite, a condition physicists and mathematicians call a singularity.
Such singularities are found in black holes, stars that have collapsed under their own weight. According to general relativity, mass concentrations curve space-time, inducing the force of gravity. With enough matter in a small enough volume, gravity becomes infinitely strong.
In 1916, just months after Albert Einstein’s general relativity appeared, Karl Schwarzschild discovered a solution of the equations with a singularity. Decades later, this idea led to the theory of black holes, crushed stars with spherical boundaries that trap anything falling inside, including light rays.
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There is now abundant evidence that black holes exist, but do they really represent space-time singularities? Most physicists believe the singularities are mathematical artefacts, and would vanish in a more fundamental theory incorporating quantum effects.
Physical equations enable us to predict the future, but singularities imply a lack of predictability; theory just breaks down. It was hoped that quantum effects would eliminate infinities, but current versions of quantum gravity are plagued with singularities. It seems that infinite quantities are inherent and unavoidable.
German physicist Hermann Weyl opened his essay, Levels of Infinity, with the statement “mathematics is the science of the infinite”.
Infinity is at the core of mathematics. We can gain a first impression of it by placing all the counting numbers, 1, 2, 3 ... in a row stretching towards the right without end. Including the negative integers extends the row to the left. But there are gaps in the row, crying out to be filled. We can insert an infinity of fractions between any two whole numbers but, while the gaps become ever-smaller, their number grows without limit: they never go away.
Towards the end of the 19th century, two mathematicians, Richard Dedekind and Georg Cantor, found ways to define quantities known as real numbers, filling all the gaps and producing a mathematical continuum. But this may or may not correspond to the points on a physical line; we have no way of knowing whether we have too few or too many numbers for these points.
Cantor proved many startling results. There is not just one infinity, but an entire hierarchy of transfinite quantities, increasing without limit. Around 1970, John Conway discovered an entirely new way of defining numbers, which includes all the familiar numbers, all Cantor’s transfinite numbers and a breathlessly vast universe of new numbers, both infinitely large and infinitesimally small. These are the surreal numbers.
So far, the surreal numbers have not been used in physical theories. But this is typical; new mathematical developments often find applications only years or decades after their discovery.
Given that fundamental physical theories involve singularities, and infinite quantities are natural elements of the surreal numbers, these exotic numbers may prove valuable in future theories of quantum gravity. Perhaps physicists should embrace infinity rather than trying to banish it from their theories.
Peter Lynch is emeritus professor at the School of Mathematics & Statistics, University College Dublin. He blogs at thatsmaths.com
