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Anatoly_Vorobey comments on The Limits of Intelligence and Me: Domain Expertise - Less Wrong
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Electric charge is precisely the sort of example that makes me think aliens could conceivably be doing OK without negative numbers. There are two kinds of charges, call them white charge and red charge. White charges create white fields, while red charges create red fields, and the white and red fields coexist in space. These fields exert forces on white and red charges according to well-defined equations. We find it very convenient to identify white with + and red with -, and speak of a single electromagnetic field, but I don't think (though I might be missing something) that this description is physically essential. That is, not only is the choice of electron as - and proton as + arbitrary, but the decision to view these two kinds of charges as positive and negative halves of a single notion of charge is arbitrary as well. It does seem very convenient mathematically, but without that convenience the equations of motion would not be significantly more difficult.
Charge conservation makes a lot more sense in the + and - context than in the red and white context.
It's more convenient, but "a lot more sense"? I don't know. I have bread and cheese in my kitchen, which I only use to make cheese sandwiches. I don't have a bread conservation law, and I don't have a cheese conservation law, but I have a "bread and cheese conservation law", which says that the amount of bread that will go missing is the same as the amount of cheese that will go missing, up to a constant factor. Do I really need to introduce a notion of "beese", viewing bread as positive beese and cheese as negative beese? I could do that, and I will then have a beese conservation law, but it's not evident to me that my "bread and cheese conservation law" is less suitable for solving practical problems than the "beese conservation law". If I didn't need negative numbers for other things and didn't already know about them, I suspect I could get by with my "bread and cheese conservation law".
Indeed, people talk about the conservation of bee minus ell without labelling it anything else. So what?
You can, but if you get a guest who's gluten-intolerant and who will eat your cheese ignoring the bread, the "beese conservation law" will be broken.
If you can show that the charge conservation law could be broken, the argument for positive/negative would become much weaker. That's a pretty large "if", however, more or less Nobel-sized :-)
That's just the poverty of my analogy, not of the underlying argument. In the white/red formulation of electromagnetism, the law of white and red charge conservation says that whenever any amount of red charge goes missing, the same amount of white charge must disappear with it. There's no inherent need to use negative magnitudes and sum up anything to 0.
In a similar way you can call numbers less than zero red numbers and numbers greater than zero white numbers.
So you've changed the labels, but did anything more important happen?
http://arxiv.org/pdf/1303.6576
I came across this in a Hacker News discussion. It's a rigorous derivation of (positive) real numbers without using 0 or negative numbers at all. In other words, pretend that you don't know what 0 and negative numbers are, come up with a slightly different axiom set for what is essentially a positive part of an ordered field, etc.
Interestingly, this isn't stated as an explicit goal in the article, you need to read it between the lines.
The paper is weak evidence of what I was talking about in this thread; weak because actual aliens probably wouldn't discover real numbers this way. But it does show it's possible to quite easily talk and reason about them w/o ever employing negative numbers, or even 0.
The point is - is it possible to get to a working theory without inventing negative numbers.
So with charges and my white-red charge conservation law, I never need to subtract 5 reds from 3 reds. Unlike e.g. loaning money, this sort of problem doesn't seem to arise with charges. When we use positive and negative charge, a large part of the algebraic machinery made available to us by negative numbers sits unused (we don't multiply charges either; that is we do in terms of Coulomb's law, but that's a notational convenience). That's why I said that electric charge is a good example of why aliens could conceivably get by w/o negative numbers. If they didn't have them for other reasons by the time they got around to investigate electricity, they might get by with the white-red formalism just fine.
If you already know negative numbers, then sure, it's easy to imagine just relabeling them and nothing much changes. But to people in the first millennium AD, they were a very real and tangible invention. When ancient Greeks said that something like "x+4=2" is an obviously absurd equation w/o a solution, they meant it. They didn't go "oh, I have these I-OWE-U numbers that I use to count my debts but don't call them "negative", anyway, the solution is I-OWE-U-2".
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".