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”.
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.
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″.
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″.