I’ve written here before about chaos theory and how it prevents us from calculating certain physical models effectively. It’s not that these models don’t accurately reflect the physics involved; it’s that any attempt to use actual numbers introduces tiny errors into the process. These cause the result to drift more and more as the calculation extends into the future.
This is why tomorrow’s weather prediction is fairly accurate but a prediction for a year from now is entirely guesswork. (We could make a rough guess based on past seasons.) Yet the Earth itself is a computer — an analog computer — that tells us exactly what the weather is a year from now.
The thing is: it runs in real-time and takes a year to give us an answer!
The problem starts when we round off our measurements to create readings we can enter into a digital computer. No matter how many digits of precision we use (“the temperature today is 54.977301284638 degrees”), these are still finite numbers, and chaos theory tells us that’s a losing hand.
Our problem repeats itself with each calculation the computer makes. Digital computers (essentially all the computers we use daily) necessarily use finite numbers — that’s what “digital” means. Because they work with finite numbers, not infinite (analog) ones, they just can’t do certain things, no matter how good they are.
But the Earth (plus some input from the moon), along with the sun (which provides power), together comprise an analog computer that calculates weather with total precision. It’s just not useful for future prediction due to running in real-time!
Speaking of the Earth, moon, and sun, calculating orbits involving multiple bodies is another extremely complex situation mathematically. There are various approaches using successive approximations, and these allow sufficiently close solutions near term. (In this case “near term” can mean millions of orbits.)
But the orbiting bodies, they figure it out as they go. The physical analog “computer” they comprise performs the calculations in real-time as part of its dynamics.
So reality itself is a kind of computer — a large analog computer that runs in real-time. Want to know where Pluto will be in ten-million years? Just ask the solar system. It’ll get back to you (in just 10,000 millennia).
This isn’t limited to large systems. The way oxygen molecules in a room behave follows specific mathematical laws. So do the elementary particles that comprise those atoms.
When we examine the physical world, we find physical computers on a variety of levels. Sub-atomic particles are computers; atoms are computers; the Earth is a computer; the solar system is a computer; the galaxy is a computer; the entire universe is a computer.
All these systems obey physical laws — laws that can be expressed mathematically — as they grind out real-time “answers” to the question: what’s next?
This leads to two ancient and related — yet still unresolved — philosophical questions regarding reality and mathematics. The first is: what came first, the chicken or the egg? The second is: just how “real” is mathematics?
Is mathematics an abstract idealization of the messy physical world? Or does reality reflect an underlying mathematical structure? Is math something we make up, or is it something we discover?
What makes the questions so compelling is the eery effectiveness of mathematics in describing our world. Our answers needs to account for this effectiveness.
Those who believe math is a fiction we use to explain reality account for this by pointing out that, of course it’s effective — it was designed to be effective. Being surprised at this is like being surprised that a hammer is good at pounding nails. And like any fiction, of course we made it up.
Those who believe math is a discovery find no surprise in the first place. For them, physical laws are just expressions of underlying mathematical frameworks. A hammer is a physical realization of the mathematics of heavy objects and nails that is tailored for human hands. A general belief here is that math has some kind of reality that awaits discovery.
An extreme view holds that reality is nothing more than mathematics. What seems to be physical reality is just an unfolding calculation that models physical reality. Not only is math real, it’s the only thing that actually is!
There is a strange undeniability to the idea that “1+1=2” once you create — or discover — the ability to count and add. On the one hand, “1+1=2” can be seen as a game involving manipulation of symbols according to a small set of made-up rules. On the other hand, if you have an apple in your left hand and an apple in your right hand, how can you not have two apples?
Likewise, the axioms of geometry seem somehow self-evident given the idea of physical space. A sphere is defined simply as ‘all points within a given distance (the radius) of the center.’ The idea is so simple it seems that it must somehow already exist waiting to be recognized. One might discover the idea even without having ever seen a physical example.
The underlying reality of mathematics, whether we invent it or discover it, and its eery effectiveness as a tool, are all topics for further discussion. Today I want mainly to highlight how the physical world acts as a computer.
There is an interesting Yin-Yang element to different levels of the physical world. At our macro level, reality seems smooth, seems analog and we can treat it as though it were. At the same time, objects in the world are discrete, individual. The foundation of mathematics lies in counting distinct objects.
As we drill down, reality becomes quantum, and matter ceases to look smooth at all. Quantum physics says all matter comes in tiny lumps. It asserts (but we haven’t reached the ability to test this) that even time and space are quantized, even they come in tiny chunks.
Yet the math that describes quantum laws uses real numbers, which are part of an utterly smooth continuum. The way a quantum system evolves over time has analog character. Only when we measure such a system do we find a discrete answer.
Information theory, which sees reality as information, reduces reality to bits — the notorious 1s and 0s of digital computers. And yet the math concerning information theory, just as does quantum theory, uses real (analog) numbers (logarithms, for example).
Regardless of whether these things reflect an underlying mathematical reality, or whether they have properties we can use invented mathematical tools to explore, these physical systems calculate answers as part of their dynamic behavior.
They are all special-purpose computers who process a single problem. A mechanical watch is a small computer for calculating time. Earth’s weather system is a giant computer for calculating weather. Our solar system is a vast computer that calculates the orbits of nine planets (yes: nine), dozens of assorted moons, and a great many smaller objects (including such things as the ISS and GPS satellites).
We live inside, and are part of, these larger computers, and to some extent we affect their calculations. Our actions on Earth do seem to affect the weather. The spacecraft we fling outwards, such as the New Horizons, which is approaching its “kiss pass” with Pluto, have a very tiny effect on the solar system.
That effect is immeasurable, but chaos theory tells us that, given billions of orbits, we may have altered their paths. Reality is an interconnected network of systems that each step of the way calculate the next step with total precision. We can be assured that the year-long calculation of the weather a year from now will be delivered to us then with 100% accuracy.