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Gurkenglas comments on The Limits of Intelligence and Me: Domain Expertise - Less Wrong
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Getting physics to space travel tier without doing subtractions is implausible, and the integers are the subtraction-closure of the natural numbers.
Look at any oscillation function, say one of the solutions of the simplest (most universally considered) differential equations, sin(x). It has negative values, and if you were to work only with positive numbers [edit: I meant taking its absolute value], it wouldn't be differentiable everywhere, which would be a pain.
Complex numbers come from closing the real numbers algebraically, and if you don't have complex numbers, you won't go to space today.
You seem very confident about all these assertions, but I don't understand where the confidence is coming from. Subtraction is clearly needed, but having imaginary entities that you use for the results of subtraction operations that are clearly invalid seems fanciful. It took a lot of time and effort for people to realize that it only looks fanciful, but in fact is very convenient and useful.
You don't need negative numbers to define a limit of a sequence; in fact the notion of a distance between a and b is more natural than the formula |a-b|. You can define the distance between a and b as the subtraction of the smaller between them from the larger.
Therefore you don't need negative numbers to define the derivative of a smooth function at x. You just say that at x the function is growing with derivative such-and-such, or shrinking with derivative such-and-such, or holding steady. Working with such a definition might be more cumbersome and might break down to more special cases, but I don't think it's hugely more cumbersome.
You can do quantum mechanics without complex amplitudes.
Which assertions would that be?
Fanciful? So, which criteria are we using to decide whether something like negative numbers are a reasonable concept?
By the way, I think people used negative numbers for a very long time, it's just that they called them "debt" or "shortfall".