I spent the better part of November writing miniature essays in this. It’s really quite addictive. My favourites:
Parallax and cepheid variables (Dead stars that flash in space)
Basic linear algebra (four-sided boxes of numbers that eat each other)
The Gold Standard (Should a bit of money be the same as a bit of sun-colored stuff that comes out of the ground?)
The Central Limit Theorem (The Middle Thing-It-Goes-To Idea-You-Can-Show-Is-True-With-Numbers—when you take lots of Middle Numbers of lots of groups, it looks like the Normal Line!)
Complex numbers (“I have just found out I can use the word ‘set’. This makes me very happy.”)
Utility, utilitarianism and the problems with interpersonal utility comparison (“If you can’t put all your wants into this order, you have Not-Ordered Wants”)
The triumvirate brain hypothesis (“when you lie down on the Mind Doctor’s couch, you are lying down next to a horse, and a green water animal with a big smile”)
Arrow’s Impossibility Theorem (“If every person making their mark on a piece of paper wants the Cat Party more than the Dog Party, then the Dog Party can’t come out higher in the order than the Cat Party.”)
The concept of “degenerate case” (“If your boyfriend or girlfriend has a different meaning for ‘box’ than you do, and you give them a line, not only will they be cross with you, but you will be wrong, and that is almost as bad”)
The word “sublimate” (“When Dry Ice goes into the air, it is beautiful, like white smoke. There is a word for this situation, and we also use that word to talk about things that are beautiful, because they are perfect, and become white smoke without being wet first”)
The Central Limit Theorem (The Middle Thing-It-Goes-To Idea-You-Can-Show-Is-True-With-Numbers—when you take lots of Middle Numbers of lots of groups, it looks like the Normal Line!)
Does it really simplify things if you replace “limit” with “thing-it-goes-to” and theorem with “idea-you-can-show-is-true-with-numbers”? IMO this is a big problem with the up-goer five style text: you can still try to use complex concepts by combining words. And because you have to describe the concept with inadequate words, it becomes actually harder to understand what you really mean.
There are two purposes of writing simple English:
writing for children
writing for non-native speakers
In both cases is “sun-colored stuff that comes out of the ground” really the way you would explain it? I would sooner say something like: “yellow is the color of the sun, it looks like . People like shiny yellow metal called gold, because there is little of it”.
I suppose the actual reason we are doing this is
artificially constrained writing is fun.
If your boyfriend or girlfriend has a different meaning for ‘box’ than you do, and you give them a line, not only will they be cross with you, but you will be wrong, and that is almost as bad
“give them a line” and “be cross with you” are expressions that make no sense with the literal interpretation of these words.
Using the most common 1,000 words is not really about simplifying or clarifying things. It’s about imposing an arbitrary restriction on something you think you’re familiar with, and seeing how you cope with it.
There are merits to doing this beyond “it’s fun”. When all your technical vernacular is removed, you can’t hide behind terms you don’t completely understand.
In fact, I’m not sure what “give them a line” means. Give them a line like this ------------- instead of a box? From context, it could also mean ‘just make something up’. (English is not my first language, in case you couldn’t tell.)
**googles**
Yes, it turns out that “give someone a line” can mean “to lead someone on; to deceive someone with false talk” (or “send a person a brief note or letter”, but that doesn’t make sense in this context).
I was quoting a single sentence of my mini-essay. “Give them a line” probably doesn’t make much sense out of context.
The original context was that a line segment is a degenerate case of a rectangle (one with zero width). You can absolutely say a line segment is a rectangle (albeit a degenerate case of one). However, if your partner really wanted a rectangle for their birthday, and you got them a line segment, they may very well be super-pissed with you, even if you’re using the same definition of “line segment” and “rectangle”.
If you’re not using the same definition, or even if you’re simply unsure whether you’re using the same definition, then when you get your rectangle-wanting partner a line segment for their birthday, not only would they be pissed with you, but you may also be factually incorrect in your assertion that the line segment is a rectangle for all salient purposes.
I don’t know why that one caught my eye, but here I go.
You’ve probably seen the number line before, a straight line from left to right (or right to left, if you like) with a point on the line for every real number. A real number, before you ask, is just that: real. You can see it in the world. If I point to a finger on my hand and ask, “how many of these do I have?”, the answer is a real number. So is the answer to “how tall am I?”, and the answer to “How much money do I have?” The answer to that last question, notice, might be less than nothing but it would still be real for all that.
Alright, what if you have a number line on a piece of paper, and then turn the paper around by half of half a turn, so the number line has to run from top to bottom instead of left to right? It’s still a number line, of course. But now you can draw another number line from left to right, so that it will go over the first line. Then you have not one line but two.
What if you next put a point on the paper? Because there is not one number line but two, the point can mean not just one real number but two. You can read off the first from the left-to-right line, and then a second from the top-to-bottom line. And here is a funny thing: since you still have just one point on the paper, you still have one number — at the very same time that you have two!
I recognize that this may confuse. What’s the deal? The thing to see is that the one-number-that-is-really-two is a new kind of number. It is not a real number! It’s a different kind of number which I’ll call a complete number. (That’s not the name other people use, but it is not much different and will have to do.) So there is no problem here, because a complete number is a different kind of number to a real number. A complete number is like a pair of jeans with a left leg and a right leg; each leg is a real number, and the two together make up a pair.
Why go to all this trouble for a complete number that isn’t even real? Well, sometimes when you ask a question about a real number, the answer to the question is a complete number, even if you might expect the answer to be a real number. You can get angry and shout that you don’t want an answer that’s complete, and that you only want to work with a number if it’s real, but then you’ll find many a question you just can’t answer. But you can answer them all if you’re cool with an answer that’s complete!
For what it’s worth, I dislike the term “real number” precisely because it suggests that there’s something particularly real about them. Real numbers have a consistent and unambiguous mathematical definition; so do complex numbers. Real numbers show up in the real world; so do complex numbers. If I were to tell someone about real numbers, I would immediately mention that there’s nothing that makes them any more real or fake than any other kind of number.
Unrelatedly, my favorite mathematical definition (the one that I enjoy the most, not the one I think is actually best in any sense) is essentially the opposite of Up-Goer Five: it tries to explain a concept as thoroughly as possible using as few words as possible, even if that requires using very obscure words. That definition is:
The complex numbers are the algebraic closure of the completion of the field of fractions of the initial ring.
I thought I might get some pushback on taking the word “real” in “real number” literally, because, as you say, real numbers are just as legitimate a mathematical object as anything else.
We probably differ, though, in how much we think of real & complex numbers as showing up in the real world. In practice, when I measure something quantitatively, the result’s almost always a real number. If I count things I get natural numbers. If I can also count things backwards I get the integers. If I take a reading from a digital meter I get a rational number, and (classically) if I could look arbitrarily closely at the needle on an analogue meter I could read off real numbers.
But where do complex numbers pop up? To me they really only seem to inhere in quantum mechanics (where they are, admittedly, absolutely fundamental to the whole theory), but even there you have to work rather hard to directly measure something like the wavefunction’s real & imaginary parts.
In the macroscopic world it’s not easy to physically get at whatever complex numbers comprise a system’s state. I can certainly theorize about the complex numbers embodied in a system after the fact; I learned how to use phasors in electronics, contour integration in complex analysis class, complex arguments to exponential functions to represent oscillations, and so on. But these often feel like mere computational gimmicks I deploy to simplify the mathematics, and even when using complex numbers feels completely natural in the exam room, the only numbers I see in the lab are real numbers.
As such I’m OK with informally differentiating between real numbers & complex numbers on the basis that I can point to any human-scale quantitative phenomenon, and say “real numbers are just right there”, while the same isn’t true of complex numbers. This isn’t especially rigorous, but I thought that was a worthwhile way to avoid spending several introductory paragraphs trying to pin down real numbers more formally. (And I expect the kind of person who needs or wants an up-goer five description of real numbers would still get more out of my hand-waving than they’d get out of, say, “real numbers form the unique Archimedean complete totally ordered field (R,+,·,<) up to isomorphism”.)
As far as I know, the most visible way that complex numbers show up “in the real world” is as sine waves. Sine waves of a given frequency can be thought of as complex numbers. Adding together two sine waves corresponds to adding the corresponding complex numbers. Convoluting two sine waves corresponds to multiplying the corresponding complex numbers.
Since every analog signal can be thought of as a sum or integral of sine waves of different frequencies, an analog signal can be represented as a collection of complex numbers, one corresponding to the sinusoid at each frequency. This is what the Fourier transform is. Since convolution of analog signals corresponds to multiplication of their Fourier transforms, now a lot of the stuff we know about multiplication is applicable to convolution as well.
I spent the better part of November writing miniature essays in this. It’s really quite addictive. My favourites:
Parallax and cepheid variables (Dead stars that flash in space)
Basic linear algebra (four-sided boxes of numbers that eat each other)
The Gold Standard (Should a bit of money be the same as a bit of sun-colored stuff that comes out of the ground?)
The Central Limit Theorem (The Middle Thing-It-Goes-To Idea-You-Can-Show-Is-True-With-Numbers—when you take lots of Middle Numbers of lots of groups, it looks like the Normal Line!)
Complex numbers (“I have just found out I can use the word ‘set’. This makes me very happy.”)
Utility, utilitarianism and the problems with interpersonal utility comparison (“If you can’t put all your wants into this order, you have Not-Ordered Wants”)
The triumvirate brain hypothesis (“when you lie down on the Mind Doctor’s couch, you are lying down next to a horse, and a green water animal with a big smile”)
Arrow’s Impossibility Theorem (“If every person making their mark on a piece of paper wants the Cat Party more than the Dog Party, then the Dog Party can’t come out higher in the order than the Cat Party.”)
The concept of “degenerate case” (“If your boyfriend or girlfriend has a different meaning for ‘box’ than you do, and you give them a line, not only will they be cross with you, but you will be wrong, and that is almost as bad”)
The word “sublimate” (“When Dry Ice goes into the air, it is beautiful, like white smoke. There is a word for this situation, and we also use that word to talk about things that are beautiful, because they are perfect, and become white smoke without being wet first”)
Does it really simplify things if you replace “limit” with “thing-it-goes-to” and theorem with “idea-you-can-show-is-true-with-numbers”? IMO this is a big problem with the up-goer five style text: you can still try to use complex concepts by combining words. And because you have to describe the concept with inadequate words, it becomes actually harder to understand what you really mean.
There are two purposes of writing simple English:
writing for children
writing for non-native speakers
In both cases is “sun-colored stuff that comes out of the ground” really the way you would explain it? I would sooner say something like: “yellow is the color of the sun, it looks like . People like shiny yellow metal called gold, because there is little of it”.
I suppose the actual reason we are doing this is
artificially constrained writing is fun.
“give them a line” and “be cross with you” are expressions that make no sense with the literal interpretation of these words.
Using the most common 1,000 words is not really about simplifying or clarifying things. It’s about imposing an arbitrary restriction on something you think you’re familiar with, and seeing how you cope with it.
There are merits to doing this beyond “it’s fun”. When all your technical vernacular is removed, you can’t hide behind terms you don’t completely understand.
In fact, I’m not sure what “give them a line” means. Give them a line like this ------------- instead of a box? From context, it could also mean ‘just make something up’. (English is not my first language, in case you couldn’t tell.)
**googles**
Yes, it turns out that “give someone a line” can mean “to lead someone on; to deceive someone with false talk” (or “send a person a brief note or letter”, but that doesn’t make sense in this context).
Still can’t tell which type of line is meant.
I was quoting a single sentence of my mini-essay. “Give them a line” probably doesn’t make much sense out of context.
The original context was that a line segment is a degenerate case of a rectangle (one with zero width). You can absolutely say a line segment is a rectangle (albeit a degenerate case of one). However, if your partner really wanted a rectangle for their birthday, and you got them a line segment, they may very well be super-pissed with you, even if you’re using the same definition of “line segment” and “rectangle”.
If you’re not using the same definition, or even if you’re simply unsure whether you’re using the same definition, then when you get your rectangle-wanting partner a line segment for their birthday, not only would they be pissed with you, but you may also be factually incorrect in your assertion that the line segment is a rectangle for all salient purposes.
There’s also several meanings of “box”, such as:
a package (as might be used to hold a gift)
to punch each other for sport (as in boxing)
a computer (in hobbyist or hacker usage)
a quadrilateral shape (as in the game Dots and Boxes)
… and the various Urban Dictionary senses, too.
(Heck, if one of my partners talked about getting a box, it might mean a booster box of Magic cards.)
I don’t know why that one caught my eye, but here I go.
You’ve probably seen the number line before, a straight line from left to right (or right to left, if you like) with a point on the line for every real number. A real number, before you ask, is just that: real. You can see it in the world. If I point to a finger on my hand and ask, “how many of these do I have?”, the answer is a real number. So is the answer to “how tall am I?”, and the answer to “How much money do I have?” The answer to that last question, notice, might be less than nothing but it would still be real for all that.
Alright, what if you have a number line on a piece of paper, and then turn the paper around by half of half a turn, so the number line has to run from top to bottom instead of left to right? It’s still a number line, of course. But now you can draw another number line from left to right, so that it will go over the first line. Then you have not one line but two.
What if you next put a point on the paper? Because there is not one number line but two, the point can mean not just one real number but two. You can read off the first from the left-to-right line, and then a second from the top-to-bottom line. And here is a funny thing: since you still have just one point on the paper, you still have one number — at the very same time that you have two!
I recognize that this may confuse. What’s the deal? The thing to see is that the one-number-that-is-really-two is a new kind of number. It is not a real number! It’s a different kind of number which I’ll call a complete number. (That’s not the name other people use, but it is not much different and will have to do.) So there is no problem here, because a complete number is a different kind of number to a real number. A complete number is like a pair of jeans with a left leg and a right leg; each leg is a real number, and the two together make up a pair.
Why go to all this trouble for a complete number that isn’t even real? Well, sometimes when you ask a question about a real number, the answer to the question is a complete number, even if you might expect the answer to be a real number. You can get angry and shout that you don’t want an answer that’s complete, and that you only want to work with a number if it’s real, but then you’ll find many a question you just can’t answer. But you can answer them all if you’re cool with an answer that’s complete!
For what it’s worth, I dislike the term “real number” precisely because it suggests that there’s something particularly real about them. Real numbers have a consistent and unambiguous mathematical definition; so do complex numbers. Real numbers show up in the real world; so do complex numbers. If I were to tell someone about real numbers, I would immediately mention that there’s nothing that makes them any more real or fake than any other kind of number.
Unrelatedly, my favorite mathematical definition (the one that I enjoy the most, not the one I think is actually best in any sense) is essentially the opposite of Up-Goer Five: it tries to explain a concept as thoroughly as possible using as few words as possible, even if that requires using very obscure words. That definition is:
I thought I might get some pushback on taking the word “real” in “real number” literally, because, as you say, real numbers are just as legitimate a mathematical object as anything else.
We probably differ, though, in how much we think of real & complex numbers as showing up in the real world. In practice, when I measure something quantitatively, the result’s almost always a real number. If I count things I get natural numbers. If I can also count things backwards I get the integers. If I take a reading from a digital meter I get a rational number, and (classically) if I could look arbitrarily closely at the needle on an analogue meter I could read off real numbers.
But where do complex numbers pop up? To me they really only seem to inhere in quantum mechanics (where they are, admittedly, absolutely fundamental to the whole theory), but even there you have to work rather hard to directly measure something like the wavefunction’s real & imaginary parts.
In the macroscopic world it’s not easy to physically get at whatever complex numbers comprise a system’s state. I can certainly theorize about the complex numbers embodied in a system after the fact; I learned how to use phasors in electronics, contour integration in complex analysis class, complex arguments to exponential functions to represent oscillations, and so on. But these often feel like mere computational gimmicks I deploy to simplify the mathematics, and even when using complex numbers feels completely natural in the exam room, the only numbers I see in the lab are real numbers.
As such I’m OK with informally differentiating between real numbers & complex numbers on the basis that I can point to any human-scale quantitative phenomenon, and say “real numbers are just right there”, while the same isn’t true of complex numbers. This isn’t especially rigorous, but I thought that was a worthwhile way to avoid spending several introductory paragraphs trying to pin down real numbers more formally. (And I expect the kind of person who needs or wants an up-goer five description of real numbers would still get more out of my hand-waving than they’d get out of, say, “real numbers form the unique Archimedean complete totally ordered field (R,+,·,<) up to isomorphism”.)
As far as I know, the most visible way that complex numbers show up “in the real world” is as sine waves. Sine waves of a given frequency can be thought of as complex numbers. Adding together two sine waves corresponds to adding the corresponding complex numbers. Convoluting two sine waves corresponds to multiplying the corresponding complex numbers.
Since every analog signal can be thought of as a sum or integral of sine waves of different frequencies, an analog signal can be represented as a collection of complex numbers, one corresponding to the sinusoid at each frequency. This is what the Fourier transform is. Since convolution of analog signals corresponds to multiplication of their Fourier transforms, now a lot of the stuff we know about multiplication is applicable to convolution as well.