No one believes abstracta have a physical location -- they lack physical properties. The claim "2 + 2 = 4" is true -- and clearly not true invirute of anything anyone thinks... if we kill that person (/people), it is no less true.

Indeed, if numbers don't exist (for example), do we suppose that we can't communicate issues of quantity with other species (, & possible alien life, etc. etc.) ? (If we can, what shared things are we talking about when we quantify?)

It seems deeply implausible to say that our use of number is circumstantially psychological -- any description of reality is going to be indispensably quantitative --- quantity is what we are talkng about. We are not talking about ourselves.

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.

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.

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.

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.

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.

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).

Just brainstorming

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.

Groups. You can stay in your kitchen (the neutral element) or go into the bedroom, then come back (inverses). In my mind, this is as real as quantity.

There are plenty of very smart people, not just mathematicians but also physicists & scientists who are mathematical platonists.

Mathematics is indeed not a form of science. But the existence and shape of mathematics is an observable phenomenon, and so metamathematics - the study of what it is and where it comes from - can be studied scientifically. How do you know mathematics exists? Well, there's a textbook right there. Who wrote the textbook and why? A human, expressing metaphors inside their heads. How did those metaphors get inside that human's head? Ah, well, that's the interesting bit - the answer of course transpires to be "a combination of innate ideas imprinted by genetic evolution by natural selection, and sociology". And you don't have to stop there, you can explore in glorious detail exactly where each idea comes from, what innate monkey-ish tendency is being deployed, how exactly ideas like "infinity" fit in a mind designed for finding fruit and chasing things.

We can similarly bring all manner of religious beliefs under the anthropological knife. It's not a pretty process though, to the people who believe in them.

You're begging the question presupposing the non-Platonic viewpoint here. How do we know that metaphors are "inside the head"?

> There are plenty of smart people - scientists even - who believe in all kinds of deities.

Okay? This is supposed to make me feel - how exactly? I'm not inherently disdainful towards theism or theists, but if I were, I guess your remark would make me like science less, or something?

> We can similarly bring all manner of religious beliefs under the anthropological knife

I'm not really sure we can, actually. At least not in some kind of non-contentious, "objective" sense. I don't really trust individual humans to give an accurate account of why they believe their beliefs, but I trust "anthropology" and "sociology" even less. My distrust for this on an individual scale comes from the fact that many beliefs & memes exist for purposes of social signalling, group identification, etc, and it might not actually be in your interest to know exactly why you believe what you do.

But these auxiliary functions of beliefs, such as signalling etc, seem to me to scale up as you introduce groups and larger-scale activities such as "anthropology" and "sociology". Without some feedback loop keeping them honest, why would I expect anthropologists or sociologists to tell me a true story about why someone believes what they do, any more than that person or anyone else? In aerospace engineering, the feedback loop is that if your design is bad, your jet engine won't work. As a result, I generally trust aerospace engineers about jet engines. But what is there to stop sociologists, anthropologists, etc from just settling on some bullshit that agrees with their preconceived beliefs or flatters their group status and promoting it forever?

But back to math. The history of mathematical ideas is complicated and interesting, but it isn't really that relevant to the question of whether the things those ideas are about are "real", which is equivalent to asking whether mathematical platonism is true or not. The question of platonism comes down to the definition of words like "real" and "exist". It is very easy to equivocate using these words, which is why most discussions about mathematical platonism are so low quality. I think the overall question isn't that meaningful so I'm not really a platonist or an anti-platonist. In most parts of human life, when I say "x exists", I mean that I can reach out and touch x, that it has a mass, temperature, surface texture, etc. In math, when I say "x exists" I just mean that I can talk about x without creating any logical contradictions. The square root of -1 may not exist in the same sense as my laptop here, but it exists in the sense that I can do things with it, such as add, multiply, raise to powers, etc, without reaching a contradiction in my formal system. So the whole "out there" thing doesn't really matter. There doesn't need to be an "out there" in order for me to meaningfully say that the square root of -1 exists.

I think that a lot of philosophy is like this too, when you mentally zoom in really closely on a problem, it often reduces to some kind of equivocation or inconsistent language usage.

Btw I don't really consider anthropology or sociology to be real intellectual disciplines, and I'm pretty on the fence about psychology and economics. I realize that is an unpopular opinion but I've thought about it a lot and I'm pretty certain that it's correct. Aerospace engineering is real because it attaches to some fundamental reality, namely that of the spinning fan blades, the combusting fuel, etc. If you get your engineering wrong, the fan blades won't spin. Likewise, math is attached to systems of axioms. When your do your math wrong, you get a contradiction. Sociology and anthropology don't attach to anything, they're like a closed loop, like theology. If you get your anthropology wrong, nothing really happens.

In the game Hearts, if you take most of the spades you lose. However, if you manage to take all the spades you win, and they call it "shooting the moon".

In a similar fashion, when you reject everything as an unreal system of metaphors, Platonism "shoots the moon" by having us reexamine what we thought we meant by "real" in the first place.

Not to say I agree with GP, but I don’t think it will be so easy to prove GP wrong either

I do not like this conclusion. Mathematics has always been something of a religion for me. But I can find no flaw with the argument. From a scientific perspective, mathematics bottoms out at "what goes on in human noggins".

Firstly, note the word "direct" here. If it has any relevance, then the authors have assumed the burden of explaining either that there are only direct experiences, or why indirect experiences don't count.

Secondly, what are the premises here? If this is supposed to be axiomatic, then there is literally no reason to either accept or reject it, and claims that the issue has been settled are just statements of belief; otherwise, the argument needs to have premises that are not begging the question in some way. As it stands, this claim is not an argument; it is more of an intuition pump.

Metaphysical discussions tend to (always?) end up as being about the meaning of words like 'real' and 'true'. Whether such discussions can really tell us anything about what must be true is arguably the most meta question in metaphysics.

Aside, but this is also Aristotle's exact argument against Platonism in general, though when he makes it in the Nichomachean Ethics he is specifically talking about ethical Good (if the definition/actual taking place of the Good lies in some other plane, we can't participate in it so no one is or can be good), but the idea is the same even when he's talking about what a soul is in De Anima. Aristotle doesn't believe in 'souls' in the way we think of them as religio-spiritual entities that exceed the capacity of the body; a 'soul' for Aristotle is the body but in a way that radically challenges the idea of a body as mere shell or vessel - soul is what any form of life repeats doing, as a body, in order to continue being itself. It should be noted that a lot of time at Aristotle's Academy was spent in Zoology, studying animals and their anatomy.

I'd say the patterns you mentioned in an earlier comment are a way for math (or parts of it e.g. some integers) to be "out there". If humans embody mathematics, then analogously so do those patterns.

Now following Hume and Locke "induction" is often treated as something "invalid", a problem to be solved. If induction is however is not a problem (see for example, Groarke, 2009, An Aristotelian Account of Induction). Aristotelian approaches are reasonable. Hence, numbers and other mathematical concepts can be very real.

2) While the vast majority of the discourse on the interpretation of mathematics oscillates (fruitlessly) between Platonism and Nominalism, I tend towards a more Aristotelian view. See for example

- Franklin: An Aristotelian Realist Philosophy of Mathematics

- Keith Hossack: Knowledge and the Philosophy of Number.

Note that these two authors do not converge on exactly the same interpretation.

Apropos: North (2021): Physics, Structure, and Reality explores the relationship between "mathematical structure" reality and theoretical physics.

Really? What is the empirical evidence for it?

Or you can say they exist but in a different way to physical reality.

I mean pi probably still was 3.14159... before humans evolved so it's not our fault really.

Personally I think maths not only exists but physical reality is a subset. I mean why else is there something rather than nothing? Scientifically it's the only hypothesis that works for that really.

have you looked there already? :)

Even if you stop “believing in” math, your proofs are still either correct or incorrect.

It can only have any meaning if you adhere to some scientific model.

At this point you've deviated so much from the OP discussion that you could as well talk about angels dancing in pinheads. Any quantification of them is as real as your proof.