Today, mathematics is regarded as an 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 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:
Catch me if you can.
We can reason that after a while, Achilles will catch up to the turtle’s original position (red dot); however, by the time he gets there, the animal will have moved some distance forward (yellow dot):
And they would have gotten away with it too…
Next, consider the time needed for Achilles to reach the yellow dot; once again, by the time he gets there, the turtle will have moved forward a tiny bit. This process can be continued indefinitely; the gap keeps getting smaller but never goes to zero, so we must conclude that Achilles can’t possibly win the race.
Amusingly, the problems caused by infinity lingered on the periphery of mathematics for centuries, fully surfacing only after we attempted to fix them with calculus. Calculus gave us a rigorous solution to the ancient puzzle: an infinite sum of time slices can be a finite number, so Achilles does catch up to the tortoise. Yet, to arrive at that result, the new field relied on the purported existence of infinitely small numbers (infinitesimals). The founders struggled to explain how to construct such entities, where to find them on the real number line (you can’t), and whether they’re safe to mix with real number algebra in the first place.
Over time, this prompted a number of mathematicians to try and build a more general model of mathematics, starting from the ground up — that is, from the principles of formal logic. In particular, one prominent faction of the movement sought to define numbers and arithmetic operations in a way that was fully independent of the physical realm.
By golly, Mr. Peano, it all adds up
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