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.
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.
In the above I followed the map/territory meme, where you can find sheep in the territory, but the numbers in the map.
Yes, but the numbers also constitute a territory. This is similar to the way that a physical road map corresponds to a territory, but is also in the territory in the sense of being a physical object.
However, I am interested if you outline or link to the arguments you mention.
I’m not going to be able to formulate it as elegantly as Eliezer (and I don’t remember which of his posts and/or comments this is from), but the basic idea is that even if our sense data come from nothing, I want to know why the nothing seems to be so lawful. Basically the physical world exists because it displays consistent patterns in a lawful manner, the same obviously applies to mathematics.
Yes, but the numbers also constitute a territory. This is similar to the way that a physical road map corresponds to a territory, but is also in the territory in the sense of being a physical object.
Can’t say I agree that numbers are like physical representation of maps, but that would be a discussion for another day.
Basically the physical world exists because it displays consistent patterns in a lawful manner, the same obviously applies to mathematics.
I do not agree that this is an accurate map. Consider a random collection of dots: human eye can find patterns in it. Chances are, what we think of as physical laws are just the patterns we distill from the utter randomness.
Not working for me. Try imagebin or something. Also note that the compression can be very lossy. The real test is how well one can extrapolate from a small patch outward.
The issue isn’t where they’re stored. The explicit links I gave work, but they work only intermittently from within Markdown. If I load the permalink for my comment they don’t show. But if I open the two explicit links in separate windows, then reload the permalink, the Markdown links work.
It doesn’t work if you just click the link, but if you copy the link address and paste it in a browser then it works. (Because there isn’t a referrer header anymore.)
Your web server is giving a 403
Forbidden response to requests that
include a referer, which includes
(in most web browsers) requests for images in
elements. This can
probably be configured somewhere on your web server. You may currently have
something like
this
enabled and you’ll need to disable it.
(A reason someone might want this behavior is to prevent
hotlinking, but in this case,
you want to hotlink your own image.)
The reason that the behavior is intermittent is presumably caching on your
browser. When I tested it from the command
line the behavior seemed quite consistent.
I blame browser caching. In the process of testing I made several dozen requests without a Referer and all of them were 200 OK (and in the case of a couple dozen of those I double-checked that the file contents were identical with each other).
Edit: Actually, you can test this if your Apache install is set to log Referers to your access log (this is a common default). If so, you’ll probably find that the 403 lines have Referers and the 200 lines don’t. (You may also see some 304 Not Modified responses there, which would probably be either all without Referers or maybe a mixture.)
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 :)
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.
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″.
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 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.
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 :-)
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.
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.
Yes, but the numbers also constitute a territory. This is similar to the way that a physical road map corresponds to a territory, but is also in the territory in the sense of being a physical object.
I’m not going to be able to formulate it as elegantly as Eliezer (and I don’t remember which of his posts and/or comments this is from), but the basic idea is that even if our sense data come from nothing, I want to know why the nothing seems to be so lawful. Basically the physical world exists because it displays consistent patterns in a lawful manner, the same obviously applies to mathematics.
Can’t say I agree that numbers are like physical representation of maps, but that would be a discussion for another day.
I do not agree that this is an accurate map. Consider a random collection of dots: human eye can find patterns in it. Chances are, what we think of as physical laws are just the patterns we distill from the utter randomness.
Not enough to compress it substantially.
Random collection of dots:
Not a random collection of dots:
ETA: Well, those links weren’t working, then they were working, currently they aren’t. The actual URLs are http://kennaway.org.uk/images/noise.jpg and http://kennaway.org.uk/images/notnoise.jpg
ETA2: And now they’re working, or not, at random.
Not working for me. Try imagebin or something. Also note that the compression can be very lossy. The real test is how well one can extrapolate from a small patch outward.
The issue isn’t where they’re stored. The explicit links I gave work, but they work only intermittently from within Markdown. If I load the permalink for my comment they don’t show. But if I open the two explicit links in separate windows, then reload the permalink, the Markdown links work.
If I click on the links it just says “Forbidden”. Perhaps they work for you because you have access permissions that we don’t?
The site doesn’t know who I am when I access them. They work for me from anywhere. Is there anyone else they do work for?
The explicit links weren’t working for me.
It doesn’t work if you just click the link, but if you copy the link address and paste it in a browser then it works. (Because there isn’t a referrer header anymore.)
Your web server is giving a 403 Forbidden response to requests that include a referer, which includes (in most web browsers) requests for images in
(A reason someone might want this behavior is to prevent hotlinking, but in this case, you want to hotlink your own image.)
The reason that the behavior is intermittent is presumably caching on your browser. When I tested it from the command line the behavior seemed quite consistent.
That is probably it. It doesn’t explain the direct links failing though.
I blame browser caching. In the process of testing I made several dozen requests without a Referer and all of them were 200 OK (and in the case of a couple dozen of those I double-checked that the file contents were identical with each other).
Edit: Actually, you can test this if your Apache install is set to log Referers to your access log (this is a common default). If so, you’ll probably find that the 403 lines have Referers and the 200 lines don’t. (You may also see some 304 Not Modified responses there, which would probably be either all without Referers or maybe a mixture.)
Well, clicking on the links and then reloading doesn’t work.
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 :)
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”.
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.
s/negative//