Ethan Siegel, writing in Forbes, concludes that No, the Universe is Not Purely Mathematical in Nature.
His argument, unless I’ve badly misunderstood him, is that many purely mathematical models of the universe have turned out to be wrong, and one needs observations to guide the building of better models.
I think he has this exactly backward.
If our Universe is uniquely woven from some special fabric, then it at least might (or might not) be possible to discern some of its properties by pure reason.
But if our Universe is a purely mathematical structure, then it is surely one of a great many purely mathematical structures (we know this, because we are familiar with a great many purely mathematical structures). This means that only observations can help us determine which of those mathematical structures we inhabit.
Siegel’s article is well written and fun to read. But I think his arguments constitute evidence for exactly the opposite of the conclusion he wants to draw.
Did not like the title, but generally agree with his point, which actually seems to be your point: there are infinite mathematical models to explain what we perceive, and we have to keep comparing to reality if we are hoping to pick the right one.
Almost sounds like the definition of science to me.
So while I liked the article, I found his continual anguish that a specific model did not fit to be jarring.
To be fair, though, this article gave mad props to Hilbert, who probably deserves mad props. But, again, he seemed to think Hilbert was something of a failure as a scientist, when in actuality he was a success at studying something that transcends our reality.
“Lost in Math” by Sabine Hossenfelder is (I thought) a decent argument about (as the subtitle says) “how beauty leads physics astray”. It’s written for general readers, though, so while it talks about the “beautiful math”, it doesn’t actually show you the beautiful math.
Her more recent book, “Existential Physics”, goes into detail on “ascientism”, defined as “religion masquerading as science under the guise of mathematics”. Enjoyed that, too. (I didn’t agree with everything, but I can enjoy arguments without agreeing.)
Siegel did not write the title. The essay does not say anything about whether the universe is purely mathematical in nature.
Theoretical physicists do a lot of work today developing mathematical models, in the absence of evidence. This includes string theory, quantum gravity, multiverse, black hole interiors, etc. There is a debate over whether this has ever worked.
Siegel says no. As you say, his examples are not particularly convincing.
“if our Universe is a purely mathematical structure, then it is surely one of a great many purely mathematical structures” — Big “if”. Yes, but some of those advances involved mathematical theories that no one would have listed as a possibility.
This line of argument seems to put a thumb on the scale on the side of mathematical realism, but I see it as pointing in a different direction. It superficially appears unlikely that the universe is so strange that it could not be brought into correspondence with a purely mathematical structure. That it does correspond to a mathematical structure does not allow us to conclude that mathematical constructs in general exist or are “real” in the same way that our universe is.
Perhaps the greatest indication that the universe is made of maths, and that the universe is only a subset of mathematical possibility is this. When we think of complexity in general, we think of a lower level complexity and an upper level complexity. The lowest level complexity would be something containing just a single bit of information. But mathematics has no upper limit – you can always get more complex, which indicates that mathematics is more primary than physics.
The trouble is, mathematics may not have a limit to its upper complexity, but it also distils its meaning from sentience, in that mathematics needs an up and running mind on which its properties can be conceived. No mind = no maths.
Moreover, while mathematics has no upper limit to its complexity, it also doesn’t seem to contain the properties (by itself) to resolve the ‘something not nothing’ problem – which suggests that to resolve that problem, we need something underpinning maths – i.e., sentience. Now, once we get into the realms of an upper level mathematical complexity, we find that there really is no limit to how complex complexity can get – and then we are into the realms of infinite complexity with sentience and no upper limit – and that seems to lead us to God.