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Comments (78)
Just as an aside, and not to criticize your frustration at your grade school math teacher, it may be worth spending some time thinking about whether negative numbers in fact exist and what exactly do you mean when you confidently assert that they do.
I expect the math teacher wasn't making any kind of philosophical argument such as "do any numbers exist, and if so in what sense?" There is a different connotation, for my idiolect anyway, between "no such thing as X" and "X does not exist".
It's possible that the only numbers that exist are the complex numbers, and that more familiar subsets such as the hilariously named "real" and "natural" numbers are invented by humans. I appreciate that this story is usually told the other way round.
Yeah, I'm sure the teacher wasn't making a philosophical argument. I can easily devil's-advocate for the teacher who may have thought, with some justification, that you first need to explain to children why "3 - 4" doesn't make sense and is "illegal", before you introduce negative numbers. A lot depends on the social context and the behavior of little Chris Hallquist, but it's not unusual that precocious little know-it-alls insist on displaying their advanced knowledge to the entire class, breaking up the teacher's explanations and confusing the rest of the kids. What Chris saw as a stupid authority figure may have been a teacher who knew what negaive numbers were and didn't want them in their classroom at that time.
Re: the existence of negative numbers - I was thinking more of the status of negative numbers compared to natural numbers. Negative numbers are an invention that isn't very old. A lot of very smart people throughout history had no notion of them and would have insisted they didn't exist if you tried to convince them. While natural numbers seem to arise from everyday experience, negative numbers are a clever invention of how to extend them without breaking intuitively important algebraic laws. Put it like this: if aliens come visit tomorrow and share their math, I'm certain it'll have natural numbers, and I think it likely it'll also have negative numbers, but with much less certainty.
As to the teacher, yeah that sounds plausible. If Chris wants to satisfy our curiosity he can expand a little on how that conversation went. In my experience, teachers can really be dicks about that kind of thing.
AFAIK, integers (including negative integers) occur in nature (e.g. electrical charge) as do complex numbers. Our everyday experience isn't an objective measure of how natural things are, because we know less than John Snow about nearly everything.
I'd bet any aliens who get here know more than us about the phenomena we currently describe using general relativity and quantum mechanics. If they do all that without negative or complex numbers I'll be hugely surprised. But then I'd be super surprised they got here at all :)
Integers, sure, but can you give some examples for complex numbers occurring in nature?
Wave functions are complex, as are impedance values. (The former might be closer to "ontologically basic" than the latter)
However, I believe there are alternatives.
You are talking about maps -- about human minds finding it convenient to describe certain natural phenomena through complex numbers. I read the original claim as saying that complex numbers are part of the territory.
Are there square roots of -1 in nature?
Why do you think wave functions are part of the map but electric charge is part of the territory? (I'm assuming you agree with scav's claim that electric charge is an example of negative integers occurring in nature.)
Hmm... I don't have a good answer. My intuition is that integers are "simple enough" and, in particular, sufficiently unambiguous, to be part of the territory, but complex numbers are not. However even a tiny bit of reflection shows that my idea of "simple enough" is arbitrary.
I guess we've fallen into the "is mathematics real?" tar pit. Probably shouldn't thrash around too much :-)
Umm... I'm gonna punt this one.
Numbers don't occur in nature.
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?
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".
All numbers are abstractions and are therefore in the map. Positive integers have no more claim for existence than quaternions or what have you.
Disagree. Mathematical objects exist in the same way physical objects do at the very least, i.e., the standard anti-solipsist arguments for the existence of the physical world apply equally well to mathematical objects.
I sort of agree, but probably not in the way you mean. In the above I followed the map/territory meme, where you can find sheep in the territory, but the numbers in the map. However, I am interested if you outline or link to the arguments you mention.