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In symmetry, 1 + 1 = 1, where 1 (or c1) represents a 360 degree rotation around an axis. Also, 2 + 2 = 1, and 4 + 4 + 4 + 4 = 1 (180 degree rotation = 2, 90 degree = 4).

You are representing an abstract concept (circles) using base 360. That’s unrelated to the fundamental concept of adding 1 and 1 to get 2

fundamental is just whatever axioms you choose, not your gut intuition of what "makes sense"

mind you Platonism concerns the existence of mathematical entities, not their referents. If you take "1" to represent a natural number in one case and a rotation in another, that's not what matters. You could use hieroglyphics or emoji if you wanted. The question is, is the mathematical object and relation itself real? Formalists claim that mathematical expressions represent nothing at all, they're just syntax.

The platonic question is, is there a mathematical symmetry where the identity operation does not hold? Is there Euclidian mathematics where a 360 degree rotation does not return you to your original state, where one thing is not equal to itself, and so on.

I mentioned this in another comment, but Platonism only is concerned with how humans are able to communicate with each other using shared concepts, if the "entities" exist as you say, most people who study philosophy would agree that insofar as they exist, they exist in people's heads (since, something existing in your mind is still real in so far as it has a material reality).

That's not true. In the Philosophy of Mathematics when people refer to Platonism they make two claims. That mathematical objects are 1. abstract and 2. independent. (also this is the case in traditional Platonism).

An object is abstract if it is not spatiotemporal or causal. Asking what the number 4 weighs or does makes no sense. It is not mental, i.e. mind independent, in that it exists outside of any agent's thoughts. A Platonist would argue that even an alien civilization is going to discover, not invent, logic and natural arithmetic. Unlike say, Inglourious Basterds which is a mental product that did not exist before Tarantino thought it up.

It is debatable if that is the case in traditional Platonism, the evidence for artificial forms comes from a single epistle (the 7th) that is arguably fabricated. I've read some of this epistle in the original, and stylistically it is very far removed from Plato's regular style, and the language resembles (to me) Koine Greek and not Attic Greek.

In any case, just because someone calls themselves a platonist doesn't mean they've carefully studied Plato. In the same fashion as Kant, asserting time and space precede the apprehension of objects that are necessarily within time and space does not imply that time and space are necessarily part of the world itself, but only that one's means of judgement begins with the notions of time and space. In the same way, whether or not you agree with Kant, it is entirely plausible to assume instead that there is a shared faculty in the minds of humans that allows us to conceive of things like 4, and addition and subtraction and mathematical functions in general, and this faculty would neither be abstract nor independent but biological and empirically observable. If this latter case is true, then it would prove to be far more useful for scientific investigation than simply assuming that the reason why we can communicate about the number 4 is because it exists abstractly and independently, because that doesn't tell us anything more about why we have shared concepts of numbers in the first place.

It really doesn’t matter what Plato believed in. “Platonism” in philosophy of mathematics has a specific meaning regardless of that.

If they are unfamiliar with natural numbers, they may believe natural number is an inconsistent concept and won't believe that 1+1=2. Similar to how people believed round earth is inconsistent.

It seems misleading to call this "symmetry" because what you wrote is not a group.

The only idempotent element in a group is the identity. And in abelian groups where the binary operation is given by "+" the identity is denoted by "0". Yet you have "1" being idempotent, so conclude that this is not a group (or that this is a sleight of hand because you've made the identification that 1 = 0).

But they might start with different axioms (e.g. non-euclidean), exploring in different directions, and perhaps discover truths new to us that affect our own mathematics.

By definition, this is hard to imagine; I mean, try to imagine a mathematics that isn't even based on numbers...!

Numbers are just this strange paint that humans apply to some things. Real computers are not based on numbers, they are based on physics of the natural world, which we only model with numbers. There is no ontological confirmation that reality is numerical; we just have a small handful of experiences which indicate that it can be understood as such.

This hits to the core question of whether math is real or fantasy. If we built a computer to calculate 1+1=2 and prints true for some millennium until one day when it suddenly prints false; what type of problem is that? Is math now wrong, or is it safe?

what if one day 1+1 becomes meaningless to us? The computer prints 2 but suddenly it’s as static to us? This seems like a downgrade, but what if this is accompanied by direct knowledge or an upgraded type of model to understand reality? It’s not seen as 2, but the underlying natural mechanisms are perceived directly? Math is broken, it doesn’t even seem to exist, yet the world continues on and we perceive it in some new strange predictive certainty?

I think mathematics has problems with continuum, because it tries to describe our analog universe with digital paradigm.

The claim that the universe is fundamentally analog is on a somewhat shaky ground in postmodern (quantum) physics, especially once you bring (discretized!) information into it, which has to happen quite "soon" if you want to deal with statistically emergent phenomena like temperature.

They quantize because the underlying solution is a circular standing wave, which is stable if it consists of a whole number of periods. The wave itself is of course analog, quantization emerges through many hoops and provides stable states with certain potential well like any stable state, and can squiggle a little under perturbation or squiggle a lot under, say, Zeeman effect.

Can we really speak of the wave(let) "being" analog when it is impossible (according to our current understanding of physics) to see that experimentally ?

The theory says it's analog and experiments so far agree with the theory. There are continuous phenomena too.

> I can't imagine a universe where 1 + 1 = 2 is incorrect

I think it really depends on lots of factor, like how their biology works, how their civilization works, how their perception works, what are their goals etc. There could be a civilization where they don't use "logic" (as we know it) as a tool to develop their civilization, maybe they have a form of logic where true can sometimes be false based on mood or state of the world (but of course we won't call that "logic" when we see it)

Yeah, I sometimes forget about it, but Gödel did make a mathematical argument for the existence of God !

https://plato.stanford.edu/entries/ontological-arguments/#Go...

> I can't imagine a universe where 1 + 1 = 2 is incorrect

I can. For instance, I was playing Civilization one day, I thought my empire should have 32768 gold but apparently I was 32768 in debt! So such universe can certainly exist, especially if it's a simulation.

> If you can establish the axioms of zero, 1, adding, multiplication, etc., you can then continue into the rest of mathematics and also computation.

It just sounds like you don't know enough set theory. Do you believe in the Axiom of Choice?

You can imagine infinity but reality is not imagination

is the list of the natural numbers (0, 1, 2, and so on) real?

is the length of any list real?

The context of the conversation is about imagining universes with different properties, however.

for me, yes, 1+1 = 2 is platonic. I start going all formalist when thinking about the continuum hypothesis [0]. The more I try to understand what's going on with the number and size of infinities, the more it seems that they're no longer something platonic, just something that drops out of a rule engine operating on a weird formal system.

Disclaimer: not a mathematician so it might be obvious to others that these things exist in platonic space.

[0] https://en.wikipedia.org/wiki/Continuum_hypothesis

Um, you don't believe in infinity.... how do computers actually work in your universe?

Computers are a real object that have to deal with the halting problems and other infinities in a bound manner to do useful things for us.

You can't imagine a universe with quantum physics either, but it exists.

In a 1-bit universe 1+1=0.