index 0e16eac..095b5a8 100644
@@ -12,7 +12,7 @@ in 1960, physicist eugene wigner wrote a famous essay asking why math works so w
consider: mathematicians developed non-euclidean geometry in the 1800s as a pure intellectual exercise, asking "what if parallel lines could meet?" decades later, einstein needed exactly that math for general relativity. the universe, it turned out, actually uses non-euclidean geometry. the math was ready and waiting.
-this keeps happening. complex numbers were invented to solve polynomial equations. they turned out to be essential for quantum mechanics. group theory was developed as abstract algebra. it turned out to describe the fundamental symmetries of particle physics. mathematicians keep building tools that physicists later discover the universe was using all along.
+this keeps happening. complex numbers were invented to solve polynomial equations. they turned out to be essential for quantum mechanics. group theory was developed as abstract algebra. it turned out to describe the fundamental symmetries of particle physics — [[structural/symmetry-and-groups|symmetry and groups]] explains how noether's theorem connects every symmetry to a conservation law, one of the deepest results in all of physics. mathematicians keep building tools that physicists later discover the universe was using all along.
## newton's laws are differential equations
@@ -40,7 +40,7 @@ if you know [linear algebra](/wiki/structural/linear-algebra-as-thinking), you a
gravity isn't a force — it's the curvature of spacetime. mass tells spacetime how to curve; curvature tells matter how to move. the math behind this is riemannian geometry: metrics, tensors, curvature, geodesics.
-einstein's field equations relate the curvature of spacetime (a geometric quantity) to the distribution of matter and energy (a physical quantity). solving these equations predicted gravitational lensing, black holes, and gravitational waves — all confirmed experimentally, decades after the math predicted them.
+einstein's field equations relate the curvature of spacetime (a geometric quantity) to the distribution of matter and energy (a physical quantity). solving these equations predicted gravitational lensing, black holes, and gravitational waves — all confirmed experimentally, decades after the math predicted them. this is where [[structural/multivariable-calculus-as-thinking|multivariable calculus]] reaches its most extreme form — the field equations are systems of coupled nonlinear partial differential equations in curved spacetime.
## the deep point