*[This is a transcript of the video embedded below.]*

There’s a lot of mathematics in physics, as you have undoubtedly noticed. But what’s the difference between the math that we use to describe nature and nature itself? Is there any difference? Or could it be that they’re just the same thing, that everything *is math? That’s what we’ll talk about today.

I noticed in the comments to my earlier video about complex numbers that many people said oh, numbers are not real. But of course numbers are real.

Here’s why. You probably think I am “real”. Why? Because the hypothesis that I am a human being standing in front of a green screen trying to remember that the “h” in “human” isn’t silent explains your observations. And it explains your observations better than any other hypothesis, for example, that I’m computer generated, in which case I’d probably be better looking, or that I’m a hallucination, in which case your sub consciousness speaks German und das macht igendwie keinen Sinn oder?

We use the same notion of “reality” in physics, that something is real because it’s a good explanation for our observations. I am not trying to tell you that this is The Right Way to define reality, it’s just for all I can tell how we use the word. We can’t actually see elementary particles, like the Higgs-boson, with our own eyes. We say they are real because certain mathematical structures that we have come up with describe our observations. Same thing with gravitational waves, or black holes, or the particle spin.

And numbers are just like that. Of course we don’t see numbers as objects walking around, but as attributes of objects, like the spin that is a property of certain particles, not a thing in and by itself. If you see three apples, three describes what you see, therefore it’s real. Again, if that is not a notion of reality you want to use, that’s totally okay, but then I challenge you to come up with a different notion that is consistent and agrees with how most people actually use the word.

Interestingly enough, not all numbers are real. The example I just gave was for integers. But if you look at all numbers with infinitely many digits after the decimal point we don’t actually need all those digits to describe observations, because we cannot measure anything with infinite accuracy. In reality we only ever need a finite number of digits. Now, all these numbers with infinitely many digits are called the real numbers. Which means, odd as it may sound, we don’t know whether the real numbers are, erm, real.

But of course physics is more difficult than just number. For all we currently know, everything in the universe is made of 25 particles, held together by four fundamental forces: gravity, the electromagnetic force, and the strong and weak nuclear force. Those particles and their forces can be mathematically described by Einstein’s Theory of General Relativity and Quantum Field Theory, theories which have been remarkably successful in explaining what we observe.

For what the science is concerned, I’d say that’s it. But people often ask me things like “what is space-time?” “what is a particle?” And I don’t know what to do with questions like this.

Space-time is a mathematical structure that we use in our theories. This mathematical structure is defined by its properties. Space-time is a differentiable manifold with Lorentzian signature, it has a distance measure, it has curvature, and so on. It’s a math thing. We call it “real” because it correctly describes our observations.

It’s a similar story for the particles. A particle is a vector in a Hilbert space that transforms under certain irreducible representations of the Poincare group. That’s the best answer we have to the question what a particle is. Again we call those particles “real” because they correctly describe what we observe.

So when physicists say that space-time 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. Now that may partly be because they’re looking for a simple answer and there just isn’t one. But I think there’s another reason, it’s that they intuitively think there must be something more to space-time and matter, something that distinguishes the math from the physics. Something that makes the math real or, as Stephen Hawking put it “Breathes fire into the equations”.

But those mathematical structures in our theories already describe all our observations. This means just going by the evidence, you don’t need anything more. It’s therefore possible that reality actually is math, that there is no distinction between them. This idea is not in conflict with any observation. The origin of this idea goes all the way back to Plato, which is why it’s often called Platonism, though Plato thought that the ideal mathematical forms are somehow beyond human recognition. The idea has more recently been given a modern formulation by Max Tegmark who called it the Mathematical Universe Hypothesis.

Tegmark’s hypothesis is actually more, shall we say, grandiose. He doesn’t just claim that actually reality is math but that all math is real. Not just the math that we use in the theories that describe our observations, but all of it. The exponential function, Mandelbrot sets, the number 18, they’re all real as you and I. If you believe Tegmark.

But should you believe Tegmark? Well, as we have seen earlier, the justification we have for calling some mathematical structures real is that they describe what we observe. This means we have no rationale for talking about the reality of mathematics that does not describe what we observe, therefore the mathematical universe hypothesis isn’t scientific. This is generally the case for all types of the multiverse. The physicists who believe in this 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’s real, but this is unnecessary to describe what we observe and therefore unscientific.

Let me be clear that this doesn’t mean it’s wrong. It isn’t wrong to say the exponential function exists, or there are infinitely many other universes that we can’t see. It’s just that this is a belief-based statement, not supported by evidence. What’s wrong is to claim that science says so.

Then what about the question whether we are made of math? Well, you can’t falsify this hypothesis. Suppose you had an observation that you can’t describe by math, it could always be that you just haven’t found the right math. So the idea that we’re made of math is also not wrong but unscientific. You can believe it if you want. There’s no evidence for or against it.

I want to finish by saying I am not doing these videos to convince you to share my opinion. I just want to introduce you to some topics that I think are thought-stimulating, and give you a starting point, in the hope it will give you something interesting to think about.