Michael Spivak once said that the definitions in math should be hard and the theorems should be easy. This is because the definitions are benchmarks for different levels of abstraction. Theorems are essentially chaining definitions together to get new definitions.

Gian-Carlo Rota also put it eloquently with the following quote:

The results of mathematics are seldom directly applied; it is the

definitionsthat are really useful. Once you learn the concept of a differential equation, you see differential equations all over, no matter what you do. This you cannot see unless you take a course in abstract differential equations. What applies is the cultural background you get from a course in differential equations, not the specific theorems. If you want to learn French, you have to live the life of France, not just memorize thousands of words. If you want to apply mathematics, you have to live the life of differential equations. When you live this life, you can then go back to molecular biology with a new set of eyes that will see things you could not otherwise see.

In one of my favorite books, *Sapiens* by Yuval Noah Harari, the idea of
the steam engine is discussed:

In 1825, a British engineer connected a steam engine to a train of mine wagons full of coal. The engine drew the wagons along an iron rail some twenty kilometres long from the mine to the nearest harbour. This was the first steam-powered locomotive in history. Clearly, if steam could be used to transport coal, why not other goods? And why not even people? On 15 September 1830, the first commercial railway line was opened, connecting Liverpool with Manchester. The trains moved under the same steam power that had previously pumped water and moved textile looms. A mere twenty years later, Britain had tens of thousands of kilometres of railway tracks. Henceforth, people became obsessed with the idea that machines and engines could be used to convert one type of energy into another. Any type of energy, anywhere in the world, might be harnessed to whatever need we had, if we could just invent the right machine. For example, when physicists realised that an immense amount of energy is stored within atoms, they immediately started thinking about how this energy could be released and used to make electricity, power submarines and annihilate cities.

Part of the problem with early energy conversion was that there was no concept of “energy” as such. Isolating that concept from other concepts by naming it and defining it as “ability to do work” gives rise to a new way of thinking about how to move things and, with some imagination, the idea that all movement is actually the same in a certain sense (i.e. E = mc\^2).

From there, it’s a much smaller step to try to convert more exotic sorts of energy.