When I was a kid I had a teacher ask a class to write down the definition of math. He went on to define math as the formal study of patterns. I’ve rarely seen that kind of definition anywhere else, and it’s certainly not easy to formalize, but I think it has a kind of meta-rational integrity that has kept it at the forefront of my understanding of math ever since.
If a thing has patterns associated with it in any way, then there is math associated with it through those patterns. If a thing is random, then there is math about what randomness looks like and does. Because of that, to say a thing could exist “without math” feels like a confusion to me. But that doesn’t mean the math is some attribute of a thing, it’s more like implicit structure that’s part of every aspect of the thing, or maybe implicit in the category boundary you drew to define the thing.
We could have the same conversation about all the “defining features” you list, too. I could show how the color and texture of, say, steel, arise from imperfect crystalline arrangements of iron and carbon atoms, which exist because of how subatomic particles behave, and so on (I have physics and materials science degrees, so to a significant extent I actually can do this). If I knew enough about biology and how minds work, I could show how smell arises from interactions between me and small molecules evaporating from the surface of a thing. These kinds of attributes are not ontologically fundamental, they arise (inexorably) from processes that can be understood, aka from patterns, and when we abstract those patterns away from their instantiation in a specific context we call those patterns math.
When I was a kid I had a teacher ask a class to write down the definition of math. He went on to define math as the formal study of patterns. I’ve rarely seen that kind of definition anywhere else, and it’s certainly not easy to formalize, but I think it has a kind of meta-rational integrity that has kept it at the forefront of my understanding of math ever since.
If a thing has patterns associated with it in any way, then there is math associated with it through those patterns. If a thing is random, then there is math about what randomness looks like and does. Because of that, to say a thing could exist “without math” feels like a confusion to me. But that doesn’t mean the math is some attribute of a thing, it’s more like implicit structure that’s part of every aspect of the thing, or maybe implicit in the category boundary you drew to define the thing.
We could have the same conversation about all the “defining features” you list, too. I could show how the color and texture of, say, steel, arise from imperfect crystalline arrangements of iron and carbon atoms, which exist because of how subatomic particles behave, and so on (I have physics and materials science degrees, so to a significant extent I actually can do this). If I knew enough about biology and how minds work, I could show how smell arises from interactions between me and small molecules evaporating from the surface of a thing. These kinds of attributes are not ontologically fundamental, they arise (inexorably) from processes that can be understood, aka from patterns, and when we abstract those patterns away from their instantiation in a specific context we call those patterns math.
There are quite a lot of articles on this site that carry bits and pieces of additional information relevant to what I’m trying to point to. A few that immediately come to mind are Trust in Math, How to Convince Me That 2+2=3, Mysterious Answers to Mysterious Questions, and Reductionism.