This is Part 3 in a series on the mathematical universe hypothesis. Check out Parts 1 and 2.
Beginning in the 1980’s, another physicist, Roger Penrose, came up with what he called the Triangle of Reality, which sounds like the nerdiest cult in history (and when later I get to talk about the Pythagoreans you’ll see that I’m right). The three corners of the triangle are math, matter, and mind. We have math, which appears to be a set of objective, abstract truths (2+2=4 kind of stuff). We have our minds, our subjective experience of the world, and we have matter, which is…the world.
So which is responsible for which? Do our minds create math? Does matter create mind? How does it all work? The mathematical universe concept breaks the triangle. It’s just math. Math IS the universe, it IS matter.
And that means math is mind.
According to the mathematical university hypothesis, we – you and me and every conscious being – are just…equations. Relationships. Logical structures. There’s nothing more to us. What we think is our conscious, subjective experience of the world (growing older, enjoying a slice of cheese, having that awkward moment when you miss a high-five) is an illusion. It’s all just…math. A structure that never changes, never evolves.
Consider the Mandelbrot set. It’s a fractal. You zoom in on any one part of it and you just get more of the same repeating patterns. It’s all governed by an extremely simple equation, but it contains an enormous amount of complexity.
Imagine writing down an equation that creates a fractal so complex, so intricate that if you zoomed down into it far enough, you saw an extremely complicated pattern emerge, a structure that was so twisted up on itself that it was aware of itself and its environment.
Tegmark calls this the difference between the bird perspective and the frog perspective. From the viewpoint of the bird, we can see the whole complex manifestation of the math. We can see these self-aware substructures, just like we can see this beautiful pattern appear in the Mandelbrot set. But we don’t get that perspective. We’re IN the math (because we ARE made of math), so we’re just frogs. We can’t zoom out to see how our little corner of the mathematical pattern is just one structure amongst many. We think we have our minds and the universe around us.
Yes, this means that there is no free will. There’s not even past and future. There’s no passage of time. There’s not even space. There’s just math, just equations. A set of logical, orderly relationships that exist. The pattern of your life already exists. If you were a bird, you could rise above and see your whole life played out.
I’m going to use an analogy and I have to admit that it’s a bad one. But I’m using it to illustrate just how wild this idea is. So take my hand, because we’re going to crazytown. It’s like we’re characters in a video game. From our perspective, the world exists, we exist, stuff exists. We talk and laugh and move around. But it’s all just software, just logic and programming. There is no “us” – there’s just bits of code.
This analogy is bad because you immediately think of either a) the matrix or b) the simulation argument. And that’s because for this analogy to work you don’t just need the software, you need the hardware, the computer. In the mathematical universe, there is no hardware, no computer. There’s just code.
Here’s a slightly better analogy. It’s more like we’re the characters in a story. From our perspective the story unfolds page by page, and we never know what comes next. But the book is already written, bound, and printed. It’s done – it exists. Yes we are real, just like the characters in a book are real, but we’re really just words on a page.
2+2=4. You can design a computer program to compute this. You can shove a bunch of rocks together to demonstrate this. But it’s still true, 2+2=4 even in a world without computers and without rocks. The mathematical structure doesn’t need a substrate (it doesn’t need an engine) to exist.
That’s….heavy stuff, man. Like I said, fun to think about. But as we start to think about it, some questions come up (which is also why this kind of stuff is fun to think about).
And the first question that comes to mind is….hey, isn’t there a lot of math out there? Like, A LOT a lot? There are a vast array of POTENTIAL logical systems that could describe ALL SORTS of potential universes. Like universes with 5 spatial dimensions, or 2 time dimensions, or 47 forces, or 15 versions of the electron, or a speed of light that’s half of what we experience, and on and on and on and on.
Tegmark has an answer for you. And that answer is second verse same as the first: the multiverse! That’s right, there’s a universe for everybody. Every possible mathematical structure is realized and existing all in parallel. We just happen to occupy this one – we are self-aware little subsets in a larger mathematical framework - because it’s the kind that allowed sentient beings to arise.
Hey, isn’t that just the anthropic principle? The universe is the way it is because if it wasn’t the way it is then we wouldn’t be here to see it?
Why, yes it is.
And the second question comes to us from none other than the final boss of mathematics himself, Kurt Godel. Now Godel didn’t actually critique Tegmark’s mathematical universe because Godel has been dead for a long time, but in the early 20th century he did something remarkable: he showed that for any sufficiently complex formal system, like mathematics, there are statements that are TRUE but cannot be proven.
I am definitely not going to get into the weeds of this one, but hey I’m up for the challenge if anyone wants to ask. The point that matters for us is that there is no mathematical system that is COMPLETE (yes there are many, many technicalities that I’m glossing over, deal with it math nerds).
So look, buddy, you want the universe to be made of math. That’s cute and all. but there’s no mathematical system that is complete, so how can you have a complete universe that actually functions as, you know, a universe?
Tegmark has an answer for you here too. Maybe it’s not ALL possible mathematical structures make up the multiverse, but only the small subset of mathematical structures that are computable. In other words, the math that is our universe isn’t complex enough to run into any Kurt Godel problems. We’re actually juuuuust complex enough to have sentience but simple enough to escape any problems with completeness.
Ta-da: a universe made of math that answers some major criticisms. We cool?
