Supporters of the modified Platonic view of mathematics like to point out that, over the centuries, mathematicians have produced {or discovered} numerous objects of pure mathematics with absolutely no application in mind. Decades later, these mathematical constructs and models were found to provide solutions to problems in physics. Penrose tilings and non-Euclidean geometries are beautiful testimonies to this process of mathematics unexpectedly feeding into physics, but there are many more.
Advocates of the adjusted Platonic perspective on mathematics highlight that throughout history, many mathematicians have invented or uncovered pure mathematical entities without any specific practical purpose. Often, years later, these abstract mathematical concepts have been revealed to hold significant relevance in solving real-world issues in physics. This phenomenon illustrates the unpredictable bridge between pure mathematics and practical applications.
Examples such as Penrose tilings and non-Euclidean geometries exemplify this intriguing occurrence, showcasing how seemingly esoteric mathematical principles can eventually contribute to our understanding of physical phenomena. There is a vast array of similar instances in the annals of mathematics that reinforce the idea of mathematics as a foundation that can unexpectedly inform and enhance scientific exploration.
