Why physics forces science to admit that abstract numbers are real

The Problem of Abstract Reality

In everyday experience, physical reality seems clearly divided between tangible things and concepts. Stars, rocks, and electrons possess physical mass, occupy coordinates in spacetime, and participate in causal interactions. In contrast, numbers, sets, and functions seem abstract. They cannot be bumped into, weighed, or observed with telescopes. This distinction long fueled the philosophical position of nominalism, which asserts that abstract objects do not exist independently in the world, but are merely useful labels, mental constructs, or linguistic tools invented by human minds.

Conversely, mathematical realism, often called Platonism, maintains that mathematical entities enjoy an objective existence independent of human thought. Historically, this realism faced a significant epistemological puzzle: if numbers are causally inert and exist outside spacetime, how could human beings ever interact with or acquire reliable knowledge about them? For decades, this question left realists vulnerable to the charge that their philosophy required a quasi-mystical form of perception to access the mathematical realm.

During the mid-twentieth century, American philosophers Willard Van Orman Quine and Hilary Putnam completely reframed this impasse. Instead of appealing to abstract intuition, they argued that the justification for mathematical realism rests entirely upon the ordinary methods of empirical science. If modern physics cannot formulate its most successful descriptions of reality without mathematical objects, science itself forces an acceptance of mathematical reality.