Maths and Physics #2: Back to Greece

Max Planck is known as the founder of quantum theory. He came up with the idea of the quantum as “an act of desperation”, to explain weird experimental observations that could not be explained by theory. He found he could explain the observations “only by butchering the mathematics of the underlying theory”, by assuming the existence of “quanta”. But to him, quanta were just mathematical constructs, not real-world constituents.

 

Albert Einstein, in trying to explain the photoelectric effect, concluded that the energy of light (and all electromagnetic waves) was quantized. Quantization was real, argued Einstein, not just a mathematical convenience.

 

Many had noted that Maxwell’s laws were “symmetrical” in certain mathematical ways. Einstein went further than others. Not just Maxwell’s laws, he said, (mathematical) symmetry applies to all universal laws of nature. Conversely, he said, if a universal law isn’t symmetrical, it’s wrong. So far, all experiments show the universe is governed by mathematically symmetrical laws.

 

Until this point, Einstein believed that physics was a “concrete, intuitive science” and maths was a tool to understand things, nothing more. All kinds of advanced maths, he believed, was irrelevant to physics. Then Einstein tried to explain gravity within the context of his special theory of relativity. And realized the maths he knew wasn’t enough! A geometrical understanding was as much necessary as the algebraic understanding. From here on, he changed his mind – advanced maths was essential for a physicist to know.

 

If the path to the general theory of relativity lay via maths, Hilbert, one of the great mathematicians of the era, felt he had a chance at it (Einstein had explained the work-in-progress theory in a talk). A race began – Einstein v Hilbert. Each was strong in one area (physics or maths) but weak at the other. Einstein won.

 

Even though Hilbert had lost, other mathematicians sensed they could do physics… via maths. One such mathematician, Hermann Weyl, thought he’d found a way to unify the maths of Einstein’s relativity with Maxwell’s electromagnetism. Sadly, the unified maths made predictions which contradicted reality. But Weyl worked on, fixed those mismatches, and ended up framing the gauge theory, “one of the foundations of our understanding of nature”.

 

Another mathematician, Emmy Noether, found that the mathematical symmetry of various physics theories implied that certain quantities had to be conserved always. Thus, she said, the famous law of conservation of energy was one such example of a more general pattern of conservation.

 

As Farmello says, the wheel had come full circle. The Greeks believed one could understand the world by pure thought, without observations and experiments. Galileo and Newton had reversed that view. And now mathematicians (with support of some physicists, like Einstein) were again taking us back to the Greek view!