What’s the meaning of “numbers” and “arithmetic operations”? We consult Georg Cantor’s turtles and look at Giuseppe Peano’s code.
Today, mathematics is regarded as a purely abstract science. On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs. Indeed, persistently trying to relate the foundations of math to reality has become the calling card of online cranks.
I find this ironic: for millennia, mathematics was essentially a natural science. We had no philosophical explanation why 2 + 2 should be equal to 4; we just looked at what was happening in the real world and tried to capture the rules. The abstractions were important, of course, but they needed to be rooted in objectivity. The early development of algebra and geometry followed suit. It was never enough for the axioms to be internally consistent; the angles of your hypothetical triangle needed to match the physical world.
That said, even in antiquity, the reliance on intuition sometimes looked untenable. A particular cause for concern were the outcomes of thought experiments that involved repeating a task without end. The most famous example is Zeno’s paradox of motion. If you slept through that class, imagine the scenario of Achilles racing a tortoise: