There’s a lot of math in physics, as you’ve probably noticed. But what is the difference between the mathematics we use to describe nature and nature itself? Is there a difference? Or could it be that they are just the same, that everything * is math? That’s what we’ll talk about today.
I noticed in the comments on my previous video on complex numbers that a lot of people said, oh, numbers aren’t real. But of course numbers are real.
Here’s why. You probably think that I am “real”. Why? Because the hypothesis that I am a person who stands in front of a green screen and tries to remember that the “h” in “Mensch” is not mute explains your observations. And it explains your observations better than any other hypothesis, for example that I am computer generated, in which case I would probably look better, or that I am a hallucination, in which case your subconscious speaks German and that doesn’t make any sense, does it?
We use the same term “reality” in physics that something is real because it is a good explanation for our observations. I’m not trying to tell you this is the right way to define reality, it’s just for all I can tell how we use the word. We cannot see elementary particles like the Higgs boson with our own eyes. We say they are real because certain mathematical structures that we have made up describe our observations. The same is true for gravitational waves or black holes or the particle spin.
And numbers are just like that. Of course, we don’t see numbers as objects moving around, but as attributes of objects, such as spin, which is a property of certain particles, not a thing in and of itself. If you see three apples, three describe what you see, so it’s real. Again, if this is not a term from reality that you want to use, that’s perfectly fine, but then I urge you to come up with another term that is consistent and in line with how most people actually use the word .
Interestingly, not all numbers are real. The example I just gave was for whole numbers. But if you look at all the numbers with an infinite number of decimal places, we don’t actually need all of these digits to describe observations because we cannot measure anything with infinite accuracy. In reality we only ever need a finite number of digits. All these numbers with an infinite number of digits are now called the real numbers. That said, as strange as it sounds, we don’t know if the real numbers are, um, real.
But of course, physics is more difficult than just numbers. From what we currently know, everything in the universe consists of 25 particles that are held together by four fundamental forces: gravity, electromagnetic force, and strong and weak nuclear forces. These particles and their forces can be mathematically described by Einstein’s general theory of relativity and quantum field theory, theories which have been remarkably successful in explaining our observations.
As for science, I’d say it is. But people often ask me things like “what is spacetime?” “What is a Particle?” And I don’t know what to do with such questions.
Spacetime is a mathematical structure that we use in our theories. This mathematical structure is defined by its properties. Spacetime is a differentiable manifold with a Lorentzian signature, it has a measure of distance, it has a curvature and so on. It’s a math thing. We call it “real” because it correctly describes our observations.
It is similar with the particles. A particle is a vector in a Hilbert space that transforms itself under certain irreducible representations of the Poincare group. That is the best answer to the question of what a particle is. We call these particles “real” again because they correctly describe what we are observing.
So when physicists say that spacetime is real or the Higgs boson is real, they mean that a certain mathematical structure correctly describes observations. But many people seem to find this unsatisfactory. Part of that may be because they are looking for a simple answer and there simply isn’t any. But I think there is another reason too, which is that they intuitively think that space-time and matter must be more, something that sets mathematics apart from physics. Something that makes math real or, as Stephen Hawking put it, “breathes fire into the equations”.
But these mathematical structures in our theories already describe all of our observations. This means that all you need to do is look at the evidence, you don’t need anything. It is therefore possible that the reality is actually math, that there is no difference. This idea does not contradict any observation. The origin of this idea goes all the way back to Plato, which is why it is often referred to as Platonism, even though Plato thought that the ideal mathematical forms were somehow beyond human knowledge. The idea has recently been formulated in a modern way by Max Tegmark, who called it the mathematical universe hypothesis.
Tegmark’s hypothesis is actually, shall we say, more grandiose. Not only does he claim that reality is actually math, but that all math is real. Not just the math we use in the theories that describe our observations, but everything. The exponential function, Mandelbrot sets, the number 18, they are all real like you and me. If you believe Tegmark.
But should you believe Tegmark? Well, as we have seen before, we have the justification for calling some mathematical structures real, that they describe what we are observing. This means that we have no rationale for talking about the reality of mathematics that does not describe what we are observing, so the mathematical universe hypothesis is not scientific. This is generally the case for all species in the multiverse. The physicists who believe in it argue that unobservable universes are real because they are in their math. But just because you have math for something doesn’t mean it’s real. You can just assume it is real, but this is unnecessary to describe what we are observing and therefore unscientific.
Let me make it clear that this doesn’t mean it is wrong. It is not wrong to say that the exponential function exists or that there are an infinite number of other universes that we cannot see. It’s just that this is a belief-based statement that isn’t backed by evidence. What is wrong is to say that science says so.
Then what of the question of whether we are made of math? Well, you can’t falsify this hypothesis. Assuming you have an observation that math cannot describe, it is always possible that you just couldn’t find the right math. So the idea that we are made up of mathematics is not wrong either, it is unscientific. Believe it if you want. There is no evidence for it or against it.
In conclusion, I want to say that I am not making these videos to convince you to share my opinion. I just want to introduce you to a few topics that I think are food for thought and give you a starting point in hopes that it will give you something interesting to think about.