What are you, really?

Last month, I reviewed Jim Holt’s book “Why does the world exist?” This question immediately brings up another question: What exists anyway? Holt does not seem to be very sympathetic to the idea that mathematical objects exist, or at least he makes fun of the idea:

“A majority of contemporary mathematicians (a typical, though disputed, estimate is about two-thirds) believe in a kind of heaven – not a heaven of angels and saints, but one inhabited by the perfect and timeless objects they study: n-dimensional spheres, infinite numbers, the square root of -1, and the like. Moreover, they believe that they commune with this realm of timeless entities through a sort of extra-sensory perception.”

There’s no reference for the mentioned estimate, but what’s worse is that referring to mathematical objects as “timeless” implies a preconceived notion of time already. It makes perfect sense to think of time as a mathematical object itself, and to construct other mathematical objects that depend on that time. Maybe one could say that the whole of mathematics does not evolve in this time, and we have no evidence of it evolving in any other time, but just claiming that mathematics studies “timeless objects” is sloppy and misleading. Holt goes on:

“Mathematicians who buy into this fantasy are called “Platonists”… Geometers, Plato observed, talk about circles that are perfectly round and infinite lines that are perfectly straight. Yet such perfect entities are nowhere to be found in the world we perceive with our sense… Plato concluded that the objects contemplated by mathematicians must exist in another world, one that is eternal and transcendent.”

It is interesting that Holt in his book comes across as very open-minded to pretty much everything his interview partners confront him with, including parallel-worlds, retrocausation and panpsychism, but discards Platonism as a “phantasy.”

I’m not a Platonist myself, but it’s worth spending a paragraph on the misunderstanding that Holt has constructed because this isn’t the first time I’ve come across similar statements about circles and lines and so on. It is arguably true that you won’t find a perfect circle anywhere you look. Neither will you find perfectly straight lines. But the reason for this is simply that circles and perfectly straight lines are not objects that appear in the mathematical description of the world on scales that we see. Does it follow from that they don’t exist?

If you want to ask the question in a sensible way, you should ask instead about something that we presently believe is fundamental: What’s an elementary particle? Is it an element of a Hilbert space? Or is it described by an element of a Hilbert space? Or, to put the question differently: Is there anything about reality that cannot be described by mathematics? If you say no to this question, then mathematical objects are just as real as particles.

What Holt actually says is: “I’ve never seen any of the mathematical objects that I’ve heard about in school, thus they don’t exist and Platonism is a phantasy. “ Which is very different from saying “I know that our reality is not fundamentally mathematical.” With that misunderstanding, Holt goes one to explain Platonism by psychology:

“And today’s mathematical Platonists agree. Among the most distinguished of them is Alain Connes, holder of the Chair of Analysis and Geometry at the College de France, who has averred that “there exists, independently of the human mind, a raw and immutable mathematical reality.”… Platomism is understandably seductive to mathematicians. It means that the entities they study are no mere artifacts of the human mind: these entities are discovered, not invented… Many physicists also feel the allure of Plato’s vision.”

I don’t know if that’s actually true. Most of the physicists that I asked do not believe that reality is mathematics but rather that reality is described by mathematics. But it’s very possibly the case that the physicists in my sample have a tendency towards phenomenology and model building.

Most of them see mathematics as some sort of model space that is mapped to reality. I argued in this earlier post that this is actually not the case. We never map mathematics to reality. We map a simplified system to a more complicated one, using the language of mathematics. Think of a computer simulation to predict the solar cycle. It’s a map from one system (the computer) to another system (the sun). If you do a calculation on a sheet of paper and produce some numbers that you later match with measurements, you’re likewise mapping one system (your brain) to another (your measurement), not some mathematical world to a real one. Mathematics is just a language that you use, a procedure that adds rigor and has proved useful.

I don’t believe, like Max Tegmark does, that fundamentally the world is mathematics. It seems quite implausible to me that we humans should at this point in our evolution already have come up with the best way to describe nature. I used to refer to this as the “Principle of Finite Imagination”: Just because we cannot imagine it (here: something better than mathematics) doesn’t mean it doesn’t exist. I learned from Holt’s book that my Principle of Finite Imagination is more commonly known as the Philosopher’s Fallacy.

“[T]he philosopher’s fallacy: a tendency to mistake a failure of the imagination for an insight into the way reality has to be.”

Though Googling "philopher's fallacy" brings up some different variants, so maybe it's better to stick with my nomenclature.

Anyway, this has been discussed since some thousand years and I have nothing really new to add. But there’s always somebody for whom these thoughts are new, as they once were for me. And so this one is for you.

xkcd: Lucky 10000.