An article published in May in Quanta Magazine had the following remark as its lead:
A surprising new proof is helping to connect the mathematics of infinity to the physical world.
My first thought was that the mathematics of infinity is already connected to the physical world. But Natalie Wolchover’s opening few paragraphs were inviting:
With a surprising new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary.
The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: “finitistic” ones, which can be proved without invoking the concept of infinity, and “infinitistic” ones, which rest on the assumption — not evident in nature — that infinite objects exist.
Mapping and understanding this division is “at the heart of mathematical logic,” said Theodore Slaman, a professor of mathematics at the University of California, Berkeley. This endeavor leads directly to questions of mathematical objectivity, the meaning of infinity and the relationship between mathematics and physical reality.
It is becoming increasingly clear to me that harmonizing the finite and the infinite has been an almost ever-present human enterprise, at least as old as the earliest mythical descriptions of the worlds we expected to find beyond the boundaries of the day-to-day, worlds that were below us or above us, but not confined, not finite. I have always been provoked by the fact that mathematics found greater precision with the use of the notion of infinity, particularly in the more concept-driven mathematics of the 19th century, in real analysis and complex analysis. Understanding infinities within these conceptual systems cleared productive paths in the imagination. These systems of thought are at the root of modern physical theories. Infinite dimensional spaces extend geometry and allow topology. And finding the infinite perimeters of fractals certainly provides some reconciliation of the infinite and the finite, with the added benefit of ushering in new science.
Within mathematics, the questionable divide between the infinite and the finite seems to be most significant to mathematical logic. Wolchover’s article addresses work related to Ramsey theory, a mathematical study of order in combinatorial mathematics, a branch of mathematics concerned with countable, discrete structures. It is the relationship of a Ramsey theorem to a system of logic whose starting assumptions may or may not include infinity that sets the stage for its bridging potential. While the theorem in question is a statement about infinite objects, it has been found to be reducible to the finite, being equivalent in strength to a system of logic that does not rely on infinity.
Wolchover published another piece about disputes among mathematicians about the nature of infinity that was reproduced in Scientific American in December 2013. The dispute reported on here has to do with a choice between two systems of axioms.
According to the researchers, choosing between the candidates boils down to a question about the purpose of logical axioms and the nature of mathematics itself. Are axioms supposed to be the grains of truth that yield the most pristine mathematical universe? … Or is the point to find the most fruitful seeds of mathematical discovery…
Grains of truth or seeds of discovery, this is a fairly interesting and, I would add, unexpected choice for mathematics to have to make. The dispute in its entirety says something intriguing about us, not just about mathematics. The complexity of the questions surrounding the value and integrity of infinity, together with the history of infinite notions is well worth exploring, and I hope to do more.