The greatest physicists did this to solve problems
Physics was built not on discoveries but on the tools that enabled them.
Have you ever banged your head on trying to solve a difficult problem? Maybe it was a math problem or a relationship issue. Maybe it was in a game or real life. Perhaps it had a lot of variables and moving parts, unknowns, and branching directions. Maybe it seemed like there was only one direction, but it was the wrong one.
The point was that it wasn’t easy, but you kept at it. You thought you could solve it. You seemed to make progress for a while and then you hit a road block. Something was wrong. You couldn’t make it that way and you had to backtrack. You were back to square one.
After a while of doing that, you began to despair. Could you solve this problem at all? Were you useless? Other people surely could solve problems just like this all day long. If only you were smarter, more clever, or more experienced. Maybe you even blamed other people. Your parents could have helped you out more. Maybe your teachers. If only life weren’t so full of distraction and you could just be somewhere quiet and think!
As a person who spends their life solving hard problems, I know the feeling. It happens to me over and over again. I think I can solve something. It seems obvious and straightforward. I have a great idea. And then the work starts and suddenly I’m drowning. My head is spinning. I spend hours going in circles.
Einstein said that if he had an hour to solve a problem, he would spend 55 minutes thinking about the problem and five minutes solving it. I don’t know about you, but I don’t find that quote all that helpful. After all, isn’t thinking about the problem part of solving it?
Thinking is useful but only if you know what to think about in the first place. How do we think in order to solve problems? What even is thinking?
Thinking is essentially a physical activity, but one that uses information as its basic tool. It is the transformation of information from one state to another. Like physically crafting something, thinking involves utilizing one’s skill to craft an answer to a problem through many transformations and subtle reformulations. Knowing how to think involves not banging your head, but applying skill in just the right way and using the right tools to achieve your end.
The mind is like the body in many respects. Like the body, it is weak by itself. It did not evolve to crush problems like a computer any more than the body evolved to dig like an excavator. Yet both enabled those tools to exist.
Tools are essential to thinking, but in the mind tools may not appear to be tools. Instead, we might call them heuristics, rules of thumb, or my preferred term: frameworks.
Why do we need frameworks to solve problems?
Consider a difficult problem like being stranded on a desert island. Suppose you are and you need to build shelter? What should you do? Just try to build a lean-to or should you try to maybe build something that will help you build a shelter?
After all, we humans are tool makers first and it is by our invention of tools that have no purpose in themselves but lead to something else we want to solve that we have been able to subdue the world.
With every problem, there is a right and a wrong tool and your bare hands or bare mind is usually the wrong tool. Your mind is a great instrument but without mental tools to help you, it is extremely limited in what it can do.
Your first step should always be to find out what other people have done to try to solve your problem. No sense in reinventing the wheel of a wheel does the job.
When it comes to solving problems that are unique you can’t only rely on tools that others have created. You have to build your own.
In theoretical studies such as physics and mathematics, tools are often called frameworks. For example, Isaac Newton invented his version of calculus as a tool or framework to investigate the study of motion. He needed to understand smooth, continuous change and calculus is all about change.
Albert Einstein used a readily available framework for his theory of general relativity, the curved geometry of Riemann. If, however, that mathematics had not existed at the time, he would have had to invent it, or some analog of it.
General relativity has benefited from more modern frameworks as well that allow it to be talked about without needing a coordinate system (good for mathematicians and theoreticians). The differential forms method has become the standard in the field. Einstein would find it unrecognizable in any mathematician’s textbook on the subject.
Paul Dirac, one of the fathers of quantum mechanics and quantum field theory, invented many frameworks including the bra-ket notation for understand the infinite dimensional vectors and operators of quantum mechanics and field theory. He also invented the “ladder” method for quantum harmonic oscillators, a clever way of showing raising and lowering energy levels that avoided solving the equations, something that would lead to the creation and annihilation operators of QFT.
Richard Feynman’s greatest contributions to physics are his frameworks including the path integral method of understanding quantum physics in terms of classical physics and his Feynman diagrams which are a clever way of doing perturbation calculations. These are basically calculations around a small parameter that allow for unsolvable equations to be approximated as variations around a solvable one. While perturbation theory long predated Feynman, no one had developed a simple diagrammatic way of understanding it.
Modern quantum field theory could not have been developed without the framework of group theory. And it and the renormalization group flow that make it possible could not have been well understood without the contributions of Kenneth Wilson whose lattice gauge theory and approach to renormalization as block averaging may have come after the fact of QFT but made it sensible.
In the quest to try to develop theories of everything and come up with great and lasting discoveries, I often think that physicists have forgotten the point of developing frameworks. A framework lets you solve a problem. It isn’t an hypothesis in itself although it may include some subtle ones. String theory incorporates many frameworks of course from multi-dimensional curved spaces to supersymmetry.
Not all frameworks are applicable to physics but most if not all great leaps in our understanding of the universe have come with the advent of new tools both physical and mathematical from geometry to calculus to group theory. From telescopes to computers to under ground detectors and particle accelerators, the tools are as important as the discoveries themselves.
Indeed, many Nobel Prizes have been given out to both the toolmakers as well as the discoverers. Would Roger Penrose have his prize now if he hadn’t invented his conformal diagrams? Nobody could say for sure, but every black hole physicist in the last few decades has been grateful to them.
Having the right tool at hand can make the difference between solving a problem and banging your head in frustration. I also have no doubt that if there is a theory of everything, it will depend on the development of some tools both mathematical and physical that, while they may exist now in some form, have not been combined or set up correctly or have not the sensitivity to achieve that next great leap. Will you contribute to that achievement?