I was recently walking amongst the Flatirons at dusk, and I suddenly came to an interesting intellectual discovery. It seemed fairly profound to me, though ultimately the nature of the realization is fairly simple, once you consider it for a second. Before I describe the precise nature of the discovery, perhaps I should give some level of indication why it seemed so profound. On a personal level.
In an increasing number of my pursuits, I’ve recognized that there seems to be a conversation of sorts going on between different aspects of reality. In particular, I tend not to be one of the participants in the conversation, but either my impulsive actions or intellectual activity seems to be the medium by which this conversation occurs. I suppose my latest discovery is one particular instantiation of this phenomenon. If nothing else, simply observing this conversation is somewhat remarkable. The trick, I suppose, is realizing that the conversation is even happening in the first place. Only then can you actually witness it as it occurs. It took me about 21 years to actually figure out it was happening. My simply describing it will make it patently obvious, and perhaps some of you that have been involved in research-based endeavor have already come to a similar realization.
In order to explain my realization, I first have to explain the nature of physics. Particularly, what theoretical physics is all about. Now, I recognize that some might have a viscerally negative reaction to my even bringing up physics, likely because of some poor experience in high school. First of all, let me say that I’m not actually going to be describing physics from a high level, rather than actually doing it. So don’t stop reading even if you feel some degree of repulsion towards physics. Second of all, before I even describe physics, let me say that in my experience, physics is either profoundly frustrating and difficult, or shockingly easy. There’s very little in-between. More specifically, if you take the time to develop an intuition for the most basic principles governing a particular theory, working with the theory becomes remarkably easy. And if you don’t develop that intuition, you’re literally screwed. Physics becomes exponentially more difficult. In my experience, this is why so many people believe they are “bad” at physics. The American school system (especially high school, but certainly college as well) teaches students to memorize facts. It does not teach students to develop intuition for the first principles governing a particular topic. Thus, students are typically totally unprepared to engage with something like physics, wherein the material becomes exponentially more difficult if you opt to approach it using a “memorize everything” learning framework. Hmm, I’m going down a rabbit hole. All that is to say that physics might not be as hard as you think, and it’s possible you didn’t approach it correctly.
Anyway, if you actually know me, I’ve probably already given you this particular speech, and it’s certainly not the main body of that which I wish to discuss at the particular moment. So, let’s move on to what’s actually interesting.
In order to describe how theoretical physics works, I first need to describe how math works. In essence, one first defines particular entities of different types. Then one describes concrete rules for how the entities interact with one another. Finally, one typically manipulates the entities in question according to the rules provided which leads the creation of a new state that logically follows according to the definitions used to describe the entities. So, for example, a number is an example of an abstract entity defined within the context of mathematics. One then can define operations such as addition and multiplication, both of which take two numbers and produce a new number. Not only that, but rules such as the “distributive property” (which you probably remember from basic algebra) define basic rules that govern relationships between addition and multiplication.
Ok, the actual particulars of math aren’t important. Basically, all you need to understand is that math provides a framework for creating abstract (typically symbolic) entities, and defining rigorous, deterministic rules for how these entities interact.
Ok, so now I’m in a position to describe what theoretical physics actually does. It’s actually pretty simple. First, you observe something in the physical world. This could be anything from gravity, to temperature, to windspeed. Really anything. Then, you observe how different mathematical structures behave. More precisely, you study what happens to different mathematical entities when you apply previously defined rules to these entities. Thirdly, you attempt to find a mathematical structure that behaves similarly to the physical phenomenon you observed (in some capacity). Finally, you manipulate the mathematical entity and then map it back onto the physical system, which effectively allows you to predict how the physical system will behave.
I think probably I’ll give a related example to explain this more clearly. What is a map? A map is a simple two-dimensional image that is a representation of geographical features in a particular area. We’re all familiar with maps. Now let me ask you this. Is the map the same thing as the actual land it represents? Obviously not. Is the map perfect? No. So why is it still useful? It’s useful because it’s a good enough representation to help you perform some action. Maybe let’s be more specific. Let’s say you’re in Denver, and you’re trying to drive to Salt Lake City. If you don’t have a map, it becomes substantially more difficult to acquire the information necessary to make it to Salt Lake City. Which exit do you take? What’s the right highway? However, if you possess a map, and the map is a sufficiently faithful representation of the different features you’ll encounter on your way to Utah, then you can make accurate predictions about how you should best approach your journey. In other words, if you want to get somewhere, you need information about what lies between you and your destination. The map allows you to easily acquire this information because the map is incredibly accessible. Far more accessible than acquiring this information by interacting physical terrain.
This is precisely the same reason why a physicist tries to find a mathematical structure that behaves in the same ways as some physical phenomenon. Instead of trying to drive from Denver to Utah, the physicist is interested in figuring out how some physical system is going to behave. More specifically, the physicist is interested in figuring out how some system will behave in different environments and situations. There are two ways of acquiring this information. First, you can simply empirically observe how the entity behaves in a variety of different situations. More specifically, if you’re trying to figure out what something does in a particular situation, you can simply attempt to observe that thing under those conditions. So let’s say you’re interested in how electrons behave at very low temperatures. One way to acquire this information is to simply observe electrons at very low temperatures. Ok, great.
But what’s the problem with this approach? Any time you want to figure out how something will behave in a new situation, you need to find a way to observe the system under those particular conditions. Typically, this is either difficult or effectively impossible. What if you’re trying to figure out how electrons behave at the center of the sun? That’s a pretty tough one to directly observe.
Luckily, there’s another option. Instead of just empirically writing down how electrons behave under a wide variety of conditions, let’s say you instead try to find a mathematical structure that behaves in the same way as those electrons under different conditions. Let’s say you find such a mathematical structure. That’s pretty great! Why? Because if you wish to acquire information about how the electron behaves in a new environment, instead of observing it in that new environment, you can simply simulate the behavior of the electron using the mathematical structure you’ve found. In particular, instead of observing the electron in the new environment, observe the equivalent mathematical structure in an equivalent mathematical environment. And why would you do this? Because working with the math is much easier than working with the actual system.
Just like the example with the geographical map, it is much easier to acquire information from a mathematical structure than a physical structure.
Ok, so just to reiterate, this is literally all that happens in theoretical physics. First you observe something. Then you find a mathematical structure that behaves in the same way. This mathematical structure gives you an easier way to acquire information about the physical system in question. Boom. Done.
That ended up taking longer to describe than I previously thought it might. Well, whatever. Now let’s get to my realization.
I’ve presented physics in a fairly simple light. Observe something. Find math that behaves similarly. Use the math to predict something about the thing you observed.
However, I’ve left out one crucial step in the process. Namely, I haven’t talked about how to efficiently work with the mathematical structures you find.
Now it is true that math should behave according to certain rigorously defined rules. However, typically in something like physics, you have something you’re trying to find, and you need to mess around with the math to get it in a form that gives you the information you want. And this can be pretty monstrously difficult.
I imagine you’ve likely experienced this, regardless of whether you hate math or if you’re a theoretical mathematician. Typically, in school, math question questions are frequently phrased as “Here’s some stuff we know. Now solve for X.” And it typically requires some brain power to solve for X.
In fact, given the rules of math, in any given situation, there are usually an exponentially large number of different options available for how you can possibly manipulate an equation. So the question is, how do you figure out how to manipulate the math to get the information you want?
In my experience, I’ve found that perhaps the most powerful way to do this is by developing an intuition for how to mathematical entities in question should behave. Then you can let your intuition serve as a guide for how you should perhaps manipulate certain equations.
But here’s the question: what is that intuition? What is actually going on when you follow your intuition?
Here’s the thing I realized. If I told you to imagine a ball spinning in front of you, I imagine you’d be able to conjure up this apparition in your head. In fact, it could likely be more than just a 2d image in your head. You could probably imagine holding the ball, touching it. Engaging with it with your different senses.
I realized that when I talk about using my “intuition” to solve some problem in math, what I’m really doing is conjuring up some physical representation of the mathematical quantity in question to give me insight into how I should proceed. In other words, my intuition is almost precisely the opposite of what we do in physics.
In theoretical physics, you find math that behaves in the same way as a system. However, in using intuition to work with mathematical structures, you’re basically conjuring up some physical representation of the mathematical structure to give you insight into how you should manipulate the given mathematical structure.
When you’re dealing with physics, typically the subject of your intuition is actually the physical system you’re describing in the first place.
Ok, this post is stretching on a bit long, but here’s what I’ll say. Up until my little walk in the Flatirons, I wrongly conceptualized the process of mathematical modeling as being simply simple one-time back-and-forth between the physical world and the mathematical world. Namely, you attempt the convert the physical world into the mathematical world, you work with the mathematical world, and then you convert back to the physical world.
I’ve realized that in this sort of process, there’s actually much more back and forth between mathematical knowledge and physical intuition. And at a certain point, it begins to feel as though the physical and mathematical worlds are engaged in conversation, with your brain being the medium by which this exchange takes place.
I find this quite compelling.