Why reality doesn't matter in mathematics...
It's All About the Assumptions
Maybe you’ve heard it said:
“Mathematics describes a universe. Physics describes this universe.”
But what does that really mean?
Mathematics begins with axioms—assumptions accepted without proof. From these, entire logical systems unfold through internal consistency. Crucially, math is formal and syntactic: it manipulates symbols according to rules, independent of meaning or measurement. It tells you what must be true if the assumptions are.
Whether those assumptions correspond to reality? That’s a different matter.
This is why mathematics is a tool for consequence, not a claim about the real world.
Physics, by contrast, seeks to describe this world. It uses mathematics as its modeling language—but then subjects it to empirical scrutiny. The goal is to identify which equations, derived from which assumptions, actually predict how Nature behaves.
But here’s the catch:
Even the most elegant mathematics will fail if the starting assumptions are wrong. A single flawed axiom can ripple outward—generating contradictions, failed predictions, or entire theories that obscure more than they reveal.
That’s why physics isn’t just about solving equations. It’s about choosing the right ones.
Einstein understood this tension.
“We cannot solve our problems with the same thinking we used when we created them.”
“Do not worry about your difficulties in mathematics; I can assure you that mine are still greater.”
And so we arrive at what may be the most problematic axiom ever imported into physics:
A = A.