In my journey to rebuild everything that I learnt from the ground up, I was stuck at a question which was troubling, “what are numbers really?”
A few popular schools of thought are: (i) platonism: numbers are abstract quantities– just like abstract objects as opposed to concerete [3], (ii) nominalist: they represent real objects in the world (fails at points like square root of -1), and (iii) fictionalist: math is just false (it’s only a useful fiction, just because it’s effective doesn’t make it true).
I align myself mostly to a fictionalist. I believe that the math we use is false in the absolute sense in that none of these mathematical are intrinsic to the universe. While a subset of the numbers do represent the world really (like natural numbers), there’s a set that doesn’t (like sqrt -1). So overall, I’d say it’s false, but subparts are good way to represent the world.
Another way to think about this is that we’re creating an isolation box where we have assumptions, and given the assumptions math is true. But there’s not reason to believe that the assumptions are true. Or we could say we can create numbers as we want. It’s kinda like making a story up, did the hare win the race? yes, but it’s only in the story not in reality. So we’re pretty much making up a story of numbers that we use to explain phenomena in the world.
This begs the question, why then does it seem like the universe obeys math quite respectfully? For example, gravitation and inverse square law. Weiner [4] thinks out loud about the same in his 1960 article where he says that it’s unreasonable that math is this effective. I don’t have an answer for why, but Bertand Russell and Max tegmark’s explanations seem plausible. Max argues that the universe itself is a math structure, hence our basic math reliably estimates the advanced structure. Russell argues that we still know very little and we might have only discovered structures that (by coincidence) obey math properties with came up with. One of these two could explain the unreasonable effectiveness.
Disclaimer:
I’m a beginner and what I write is to the best of my (very limited) knowledge. It’s certainly possible that I would change my mind on all these concepts soon.
References:
Watched [1] for getting an overview on various schools of thoughts on the philosophy of numbers. [2] has a much more extensive discussion that I did not go through. That’s because I believe it’s better for me to form my view after limited reading and keep changing it as I read more, instead of waiting to read every article online before making up my view.
- Do Numbers Exist? - Numberphile
- https://en.wikipedia.org/wiki/Philosophy_of_mathematics
- https://en.wikipedia.org/wiki/Abstract_and_concrete
- https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
