I've thought about this problem quite a bit and, while my initial position was the same as above (math is not "real" per se) I had to concede that integers are real, because quantity is self-evidently real.
If you have four oranges, the quantity "four" is right there. If you take away one of those oranges you know that the result cannot be split evenly without a remaining orange because of the properties of odd numbers.
If you cut the remaining orange in half then you get a rational number, but is that self-evidently real? The halves of the orange are only "halves" because we consider them in relation to their origin, which we consider to be "one" orange. So rational numbers necessarily involve the human action of relating some quantity to a reference quantity, therefore they are a higher-level abstraction built on top of the fundamental physical property of quantity.
In the end I decided that math is based on a foundation of quantity (and maybe "space" as well?) and everything else was a derived abstraction. I am very curious if anyone else has a good argument for other parts of math being fundamental.
Electromagnetic field changes are described by complex numbers. So not only you need fractions, you need irrational numbers and imaginary numbers to describe the universe. Why is counting oranges "self-evidently real" and describing electrons "kinda real"?
I'd argue the opposite - oranges never appear in the laws of physics. They are just our description of a collection of atoms sharing some pretty loosely-defined characteristic. Oranges aren't perfectly equivalent to each another, so whether you count 1 small and 1 big orange as 2 or 1.5 oranges depends on your arbitrary decision. How about 1 orange and 1 hybrid species between orange and grapefruit? How close you need to be to be considered orange? Classes of equivalence are determined by us not by the universe, and numbers are derived from that.
Electrons on the other hand are as undeniably real as anything in this universe can be.
> Electromagnetic field changes are described by complex numbers.
You can do this, but there's no need to. You can describe electromagnetism using only real numbers.
A better argument for imaginary numbers being necessary to describe the universe is quantum mechanics, since quantum interference (in particular destructive interference) means that two possible events that each have a positive probability taken in isolation can cancel each other out, implying that probabilities can combine with a minus sign. And that means that probability amplitudes, which are square roots of probabilities, can have nonzero imaginary parts.
Quantity has real concrete measurable effects that exist irrespective of the philosophical problem of classification. If I have two acorns I know I can potentially grow two very real trees. They are countable and that directly relates to the effect they can have on the world. I like to think that maybe every tree is one tree, or that all trees are part of a unity of "plants", but practically speaking seeds and trees are countable entities no matter how I classify them.
If there are two planets, we can discuss philosophically that one might be a "moon" and not a "planet", or in some sense that the planet is "continuous" with the space dust or whatever. But the existence of two distinct bodies in space will still create very specific gravitational fields from their interactions. Tides are different if you have one vs two moon, Lagrange points etc.
As for electromagnetic fields, I am not smart enough to make a judgement on that. They are described by complex numbers, but does that mean they reflect a physical embodiment of complex numbers? Or is it just that we require complex numbers in order to resolve their behavior into something measurable? I love to learn about electricity but sadly the math is beyond my ability.
> If I have two acorns I know I can potentially grow two very real trees.
There are 2-seeded acorns. And you can get more than 1 tree from 1 seed in some species by asexual reproduction. I guess it depends on how you count trees. All the possibilities I see (number of trunks, distinctive DNA, unconnected cliques of cells) are fuzzy and have unintuitive counterexamples.
4 oranges are real because we have the neural architecture to classify the oranges as belonging to the same group according to whatever our classification criteria are.
What if you can’t classify but only be conscious of input? Kinda like being in a super dreamy state (or psychedelic one). From that state of consciousness, numbers aren’t real but reality can be (in the psychedelic case).
integers are real, because quantity is self-evidently real.
But the there are more integers than there are quantifiable 'things'[1]. Are integers that are a lot larger than, say the size of the power set of all fundamental particles in the universe still "self-evidently real".
[1] Assuming a finite universe (or a finite number of finite universes) and a few other things.
If you have four oranges, the quantity "four" is right there. If you take away one of those oranges you know that the result cannot be split evenly without a remaining orange because of the properties of odd numbers.
If you cut the remaining orange in half then you get a rational number, but is that self-evidently real? The halves of the orange are only "halves" because we consider them in relation to their origin, which we consider to be "one" orange. So rational numbers necessarily involve the human action of relating some quantity to a reference quantity, therefore they are a higher-level abstraction built on top of the fundamental physical property of quantity.
In the end I decided that math is based on a foundation of quantity (and maybe "space" as well?) and everything else was a derived abstraction. I am very curious if anyone else has a good argument for other parts of math being fundamental.