Science writer Steve Nadis explains:

The equations that govern black holes were true before there were black holes. That claim is hotly contested, and cuts through one of the deepest fault lines in the philosophy of mathematics. On one side are those who hold that mathematical structures, including well-established principles and basic geometric shapes like the tetrahedron, exist independently of human thought – not as a language we invented to describe reality, but rather as the substrate of reality itself. On the other side of the debate are those who argue that mathematics is the product of human labours, imposed on a world that would be wholly indifferent to it were we not here.

“Mathematics is out there,” May 20, 2026

Nadis focuses on Princeton University math prof Sergiu Klainerman, a champion of this point of view:

Early in graduate school, Klainerman came across an essay by the Nobel Prize-winning physicist Eugene Wigner – ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’ (1960) – that would stay with him for the rest of his career. Wigner’s subject was the uncanny way that mathematical concepts developed for one purpose keep turning up, unbidden, in other settings that seem to be wholly unrelated. …

What he wanted to write about, above all, was the mystery that had haunted him since he first read Wigner as a graduate student: why does mathematics work so unreasonably well and keep rearing its head, time and again, in the unlikeliest of places? Klainerman agrees with Wigner that this qualifies as a genuine mystery – deep, unresolved and woven into fabric that connects abstract thought, mathematical principles and physical reality.” ”Out there,”

An essay well worth reading, especially when we consider where we are now (the war on math rages still):

Theoretical physics has arrived at a place where some of its fundamental objects are so far removed from ordinary experience that direct observation may be impossible in principle, not merely in practice. The quarks of particle physics are permanently confined within hadrons, such as protons and neutrons, never appearing in isolation. The strings posited by string theory are estimated to be roughly a 100 billion billion times smaller than a proton – a scale at which our ordinary notions of observation lose their meaning entirely. And collapsed stars called black holes, though now believed to abound in the heavens, are defined by the fact that nothing – not even light – can carry information out from their interiors. They will thus remain forever shrouded, with their secrets secured within. ”Out there,”

The more we learn about the universe, the further it seems to stray from straightforward materialism. If mathematics is discovered, not invented, the foundation of our universe is abstract, not concrete. That should make a huge difference in our approach to it