Recent headlines, such as those in the journal Nature, declared “Paradox at the heart of mathematics makes physics problem unanswerable,” and “Gödel’s incompleteness theorems are connected to unsolvable calculations in quantum physics.” Indeed, the degree to which mathematics describes, constrains, or makes predictions about reality is sure to be a fertile and important discussion topic for years or even centuries to come.
In 1931, mathematician Kurt Gödel determined that some statements are “undecidable,” suggesting that it is impossible to prove them either true or false. From another perspective, in his first incompleteness theorem, Gödel recognized that there will always be statements about the natural numbers that are true, but that are unprovable within the system. We now leap forward more than eighty years and learn that Gödel’s same principle appears to make it impossible to calculate an important property of a material, namely the gaps between the lowest energy levels of its electrons. Although this finding seems to concern an idealized model of the atoms in a material, some quantum-information theorists such as Toby Cubitt suggest that this finding limits the extent to which we can predict the behavior of certain real materials and particles.
Prior to this finding, mathematicians also discovered unlikely connections between prime numbers and quantum physics. For example, in 1972, physicist Freeman Dyson and number theorist Hugh Montgomery discovered that if we examine a strip of zeros from Riemann’s critical line in the zeta function, certain experimentally recorded energy levels in the nucleus of a large atom have a mysterious correspondence to the distribution of zeros, which, in turn, has a relationship to the distribution of prime numbers.
Of course there is a great debate as to whether mathematics is a reliable path to the truth about the universe and reality. Some suggest that mathematics is essentially a product of the human imagination, and we simply shape it to describe reality.
Nevertheless, mathematical theories have sometimes been used to predict phenomena that were not confirmed until years later. Maxwell’s Equations, for example, predicted radio waves. Einstein’s field equations suggested that gravity would bend light and that the universe is expanding. Physicist Paul Dirac once noted that the abstract mathematics we study now gives us a glimpse of physics in the future. In fact, his equations predicted the existence of antimatter, which was subsequently discovered. Similarly, mathematician Nikolai Lobachevsky said that “there is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.”
Mathematics is often in the news, particularly as physicists and cosmologists make spectacular advances and even contemplate the universe as a wave-function and speculate on the existence of multiple universes. Because the questions that mathematics touches upon can be quite deep, we will continue to discuss the implications of the relationship between mathematics and reality perhaps for as long as humankind exists.