# In what sense is the physical world effectively described by mathematics?

I'm not sure whether this belongs here or in the Physics SE - please let me know if I'm mistaken!

After reading Wigner's article on "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", I'm not sure what it means for the physical laws in our universe to be amenable or subject to mathematical effectiveness. Can we imagine a universe which isn't such?

Here's my attempt: imagine a physical world where some object replicates itself between 0 (disappears) and 9 times (produces 9 copies of itself) every millisecond according to the digits of some irrational number chosen by the creator, say $$\pi$$.

Presumably humans living in this world would never be able to predict the next number in the sequence and therefore how many copies of the object will be present in the next moment. But even this is amenable to mathematical analysis and humans could effectively conclude after sufficiently long observation that the replication process is almost surely unpredictable and behaves like a ten-sided dice, in the same way that the measurement of position or spin on the quantum scale is probabilistic.

So my attempt failed. Can anyone imagine a world which isn't amenable to mathematical inquiry, or is Wigner's point that the formulae obtained for a wide range of physical processes in our world are surprisingly simple in that they don't require 100 terms to write down?

• Your title asks one question, the description seems to ask a much different question. Perhaps a title closer to your core question? Perhaps: "Can an example be given of a universe that is not effectively described by mathematics?" Mar 8 at 22:44
• "A wide range" certainly doesn't cover everything. We invite you to take our tour and read-up in the help center about how we work. You'll find we like a single focused question and you'll need to edit yours to fit with that - it might work here - and welcome to worldbuilding. Mar 8 at 23:13
• If you are going to ask a question about the implications of (or variations upon) an article, it's generally considered polite to provide a (free) link to that article. Mar 8 at 23:26
• If the universe was different, math would be different to describe it. So it's kind of a self-fulfilling prophesy. Math does a great job describing how the universe works because math is derived from how the universe works. I think this is more of a physics or mathematics question than worldbuilding. Mar 8 at 23:49
• I'd argue that intelligent life would be impossible in such a world. But well, let's spin it: the laws of nature you learn at school are nothing more than a fit that works well in isolated and macroscopic circumstances. Even calculating the energy levels in a helium atom accurately would take infinitely long. In a way, our world is not predictable by mathematics precisely. The error bars on high accuracy calculations of simple molecular properties are insane. So you're lucky, no world can be predicted by mathematics without error bars. Depending on the desired accuracy, 100 terms is nothing Mar 8 at 23:50

As mentioned by L. Dutch, we have a lot more mathematics than is used to described physics in out world. I think my take would be that any world where physical processes follow a physical law can be described by mathematics. I can only think of two ways to stretch the answer a bit that will allow a world not describable by mathematics:

• A world where the physical processes are described by mathematics, but those mathematics are so complex that no humans (or other intelligent beings) can fathom them. Mind you, any living organisms need some degree of predictability in order to survive in their environment, so a world that cannot be predicted might not be capable of supporting life. If you take it even further to things like atomic binding models, then you might not even have suns and planets and the like.
• A world not governed by physical principles, like the realm of the Fair Folk in the Exalted role-playing setting. Basically a world governed entirely by some sort of magic. There, local reality is whatever the most being around would like it to be. Battles become a matter of changing the environment to most suit you, and you become more powerful by telling the most interesting stories.
• I guess my point is - what kind of physical process would not follow a physical law that can be described by mathematics? Your first point only invokes the possibility that humans don't understand the math, while the second doesn't quite seem satisfying: 'local reality is whatever the most being around would like it to be' really is a mathematical statement about majority vote regarding desire, which can again be formulated as a specific physical pattern in the brain or something similar. Mar 18 at 20:14
• With our current knowledge, it doesn't seem like human brain activity can be described by mathematical expressions to the degree where you can predict what a specific human being will do in a specific situation. That is what the situation in the second point would require in order for it to be predictable by mathematics. It's going from modelling the physical world to modelling a sentient brain to the degree where you can exactly predict every action someone would take. Mar 20 at 6:23
• Right, so your answer is basically that despite the brain being amenable to mathematical inquiry/understanding/description, one cannot predict much given the intrinsic complexity of the system... So in this sense, Wigner's point is about the surprising mathematical simplicity of physical equations/laws, which does not seem to apply to other areas (like neuroscience)? Mar 20 at 9:40
• It's a question of emergent complexity. Even though we understand the physics and chemistry of a brain, it in no way means that we can use mathematical models to automatically predict what a sentient being will do with any kind of certainty. Mar 21 at 13:06
• en.wikipedia.org/wiki/Emergence, youtube.com/watch?v=htmntSoCasg, I'd say that emergent complexity is the difference between knowing every detail of solid objects, classical mechanics and neuroscience, and then thinking that you can go from that to predicting an entire chess match between two grandmasters before the game begins just because you know the physical laws governing that system. Mar 24 at 16:19

As Galileo said

Philosophy [i.e. natural philosophy] is written in this grand book — I mean the Universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.

We can develop much more mathematics than physics, in the sense that we can explore mathematical paths which are not physically bound to any existing system: for example it is possible to model a planetary system where the central force scales with $$1/r$$ instead of $$1/r^2$$: this doesn't mean that that system will ever exists.

Out of all the mathematical formulations, physics will pick the one which better allows to describe the system it is modelling. The same will happen in the case you describe.