The real question is whether 1 + 1 = 2 is true independent of us recognizing it. If the answer is no, then math really is just a system of ideas, and you’ve slipped into psychologism, where truth depends on minds.
But take one thing and then another: you have two things. That’s true whether or not anyone notices. Some mathematics is a human system of ideas, but some of it isn’t. Arithmetic reflects real patterns in the world. Logic, too, is not merely invention, it formalizes cause and effect. Numbers, in the Pythagorean sense, aren’t just marks on paper or symbols of order; they are the order inherent in reality, the ratios and structures through which the world exists at all.
At bottom, this debate is about the logos: what makes the universe intelligible at all, and why it isn’t simply chaos. When people say “math is real,” they mean it in the Platonic sense, not that numbers are rocks, but that they belong to the intelligible structure underlying reality.
God enters the picture not as a bolt-on explanation, but as the consequence of taking mathematical order seriously. If numbers and geometry are woven into reality itself, then the question isn’t whether math is real, it’s why the universe is structured so that it can be read mathematically at all. Call that intelligible ground the logos, or call it God; either way, it’s not an extra mystery but the recognition that reason and order are built into the world.
Calling math “just useful” misses the point. Why is the universe so cooperative with our inventions in the first place? The deeper issue is the logos: that the world is intelligible rather than chaos. That’s what people mean when they say math is real, not that numbers are physical things, but that the order they reveal is woven into reality itself.
> But take one thing and then another: you have two things. That’s true whether or not anyone notices.
You cannot justify this statement without equally justifying my position.
Say you conceive of a counterfactual world without any humans in it. You know that within this world there could be a rock and another rock, you understand that this would be two rocks, and so you are reassured that one and one is two, even though no one is watching within this counterfactual world.
All of this happened in your mind. All along, you were the observer of the supposedly unobserved world you conceived of.
You are the unavoidable human observer of any counterfactual world you conceive of. You intend the world to have no human observers, but your intention fails. It is impossible. The properties of a truly unobserved world are unknowable to you.
This is why the Enlightenment left Platonism behind centuries ago. We can't say what the world would be without us, because any attempt is not only constructed within the mind, but also contemplated and observed through the mind. You can't escape projecting your systems of ideas onto everything you think about.
Once this is taken into account, Platonism has no explanatory power and is nothing more than superfluous metaphysical mystification.
> But take one thing and then another: you have two things.
This isn't true in general, because for example you can take two equal volumes of a material and put them together, you will have less than two times the volume because of gravity. The mathematical statement that 1+1=2 follows by definition, and it's useful in applications only when the conditions are met that make it accurate, or accurate enough for the given purposes.
Mathematics is useful because the physical world exhibits regularities in its structure. Talking about logos or God adds an air of mystery to that but I don't know what more it adds