> My understanding was that mathematics was abstract from the very beginning.
It wasn't; but that's a common misunderstanding from hundreds of centuries of common practice.
So, how has maths gotten so abstract? Easy, it has been taken over by abstraction astronauts(1), which have existed throghout all eras (and not just for software engineering).
Mathematics was created by unofficial engineers as a way to better accomplish useful activities (guessing the best time of year to start migrating, and later harvesting; counting what portion of harvest should be collected to fill the granaries for the whole winter; building temples for the Pharaoh that wouldn't collapse...)
But then, it was adopted by thinkers that enjoyed the activity by itself and started exploring it by sheer joy; math stopped representing "something that needed doing in an efficient way", and was considered "something to think about to the last consecuences".
Then it was merged into philosophy, with considerations about perfect regular solids, or things like the (misunderstood) metaphor of shadows in Plato's cave (which people interpreted as being about duality of the essences, when it was merely an allegory on clarity of thinking and explanation). Going from an intuitive physical reality such as natural numbers ("we have two cows", or "two fingers") to the current understanding of numbers as an abstract entity ("the universe has the essence of number 'two' floating beyond the orbit of Uranus"(2)) was a consequence of that historical process, when layers upon layers of abstraction took thinkers further and further away from the practical origins of math.
> That is, numbers were specifically used to abstract over how other things behave using simple and strict rules. No?
Agree that math is built on language. But math is not any specific set of abstractions; time and again mathematicians have found out that if you change the definitions and axioms, you achieve a quite different set of abstractions (different numbers, geometries, infinity sets...). Does it mean that the previous math ceases to exist when you find a contradiction on it? No, it's just that you start talking about new objects, because you have gained new knowledge.
The math is not in the specific objects you find, it's in the process to find them. Rationalism consider on thinking one step at a time with rigor. Math is the language by which you explain rational thought in a very precise, unambiguous way. You can express many different thoughts, even inconsistent ones, with the same precise language of mathematics.
Agreed that we grew math to be that way. But there is an easy to trace history on the names of the numbers. Reals, Rationals, Imaginary, etc. They were largely named based on their relation to the language on how to relate them to physical things.
It wasn't; but that's a common misunderstanding from hundreds of centuries of common practice.
So, how has maths gotten so abstract? Easy, it has been taken over by abstraction astronauts(1), which have existed throghout all eras (and not just for software engineering).
Mathematics was created by unofficial engineers as a way to better accomplish useful activities (guessing the best time of year to start migrating, and later harvesting; counting what portion of harvest should be collected to fill the granaries for the whole winter; building temples for the Pharaoh that wouldn't collapse...)
But then, it was adopted by thinkers that enjoyed the activity by itself and started exploring it by sheer joy; math stopped representing "something that needed doing in an efficient way", and was considered "something to think about to the last consecuences".
Then it was merged into philosophy, with considerations about perfect regular solids, or things like the (misunderstood) metaphor of shadows in Plato's cave (which people interpreted as being about duality of the essences, when it was merely an allegory on clarity of thinking and explanation). Going from an intuitive physical reality such as natural numbers ("we have two cows", or "two fingers") to the current understanding of numbers as an abstract entity ("the universe has the essence of number 'two' floating beyond the orbit of Uranus"(2)) was a consequence of that historical process, when layers upon layers of abstraction took thinkers further and further away from the practical origins of math.
[1] https://www.joelonsoftware.com/2001/04/21/dont-let-architect...
[2] https://en.wikipedia.org/wiki/Hyperuranion