Quantum Physics Falls Apart without Imaginary Numbers
June 2, 2023 11:36 AM Subscribe
From Scientific American: Imaginary numbers—the square roots of negative numbers—are an inescapable part of quantum theory (includes the history of imaginary numbers).
That's a suspicious amount of handwaving for an article posted on April 1.
posted by flabdablet at 11:57 AM on June 2, 2023 [1 favorite]
posted by flabdablet at 11:57 AM on June 2, 2023 [1 favorite]
shurely the date is april √-1 ?
posted by lalochezia at 12:13 PM on June 2, 2023 [6 favorites]
posted by lalochezia at 12:13 PM on June 2, 2023 [6 favorites]
Imaginary numbers were one of the reasons I switched from computer to software engineering when I was in university. I thought I liked math until I started needing to solve equations involving imaginary numbers.
posted by exolstice at 12:38 PM on June 2, 2023 [1 favorite]
posted by exolstice at 12:38 PM on June 2, 2023 [1 favorite]
Mathematicians have traditionally called the square root of negative one .
Electrical engineers have traditionally called it , to avoid confusion with their customary use of to denote current (which, in turn, is probably done to avoid confusion with to denote charge... but I digress).
I've always had a sneaking suspicion that is actually negative and nobody's noticed that all the complex plane diagrams that the engineers have been drawing are upside down, but I know of no way to test this.
posted by flabdablet at 12:51 PM on June 2, 2023 [9 favorites]
Electrical engineers have traditionally called it , to avoid confusion with their customary use of to denote current (which, in turn, is probably done to avoid confusion with to denote charge... but I digress).
I've always had a sneaking suspicion that is actually negative and nobody's noticed that all the complex plane diagrams that the engineers have been drawing are upside down, but I know of no way to test this.
posted by flabdablet at 12:51 PM on June 2, 2023 [9 favorites]
Dammit! denotes capacitance, not charge. Charge is conventionally even though the unit of charge is the coulomb, for which the symbol is C. I'm old and confused.
posted by flabdablet at 12:58 PM on June 2, 2023 [2 favorites]
posted by flabdablet at 12:58 PM on June 2, 2023 [2 favorites]
Don't even get me started on the difference between µL and mL
posted by lalochezia at 1:11 PM on June 2, 2023 [2 favorites]
posted by lalochezia at 1:11 PM on June 2, 2023 [2 favorites]
"THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."
- "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960).
posted by mhoye at 1:13 PM on June 2, 2023 [5 favorites]
- "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960).
posted by mhoye at 1:13 PM on June 2, 2023 [5 favorites]
3blue 1brown did a beautiful youtube expanding & explaining that Wigner pi anecdote:
Why π is in the normal distribution (beyond integral tricks)
posted by lalochezia at 1:17 PM on June 2, 2023 [6 favorites]
Why π is in the normal distribution (beyond integral tricks)
posted by lalochezia at 1:17 PM on June 2, 2023 [6 favorites]
Imaginary numbers were one of the reasons I switched from computer to software engineering when I was in university.
Interestingly, you can use the split-complex numbers to build a dating site.
posted by lock robster at 1:43 PM on June 2, 2023 [2 favorites]
Interestingly, you can use the split-complex numbers to build a dating site.
posted by lock robster at 1:43 PM on June 2, 2023 [2 favorites]
So you can somewhat avoid thinking of imaginary numbers as "the square root of -1". But it requires some linear algebra.
Take the 2x2 matrix [[0 -1][1 0]] and call it "i", and take the 2x2 "identity" matrix [[1 0][0 1]] can call it "1". Now the complex numbers ℂ a + ib becomes the matrix [[a -b][b a]].
As [[0 -1][1 0]]^2 is [[-1 0][0 -1]] (aka i^2 is -1) this has the important properties of the complex numbers.
Now you can also do this by taking ℝ[x]/(x^2+1) in another algebraic trick. But to some people, linear algebra seems more real.
posted by NotAYakk at 1:54 PM on June 2, 2023 [2 favorites]
Take the 2x2 matrix [[0 -1][1 0]] and call it "i", and take the 2x2 "identity" matrix [[1 0][0 1]] can call it "1". Now the complex numbers ℂ a + ib becomes the matrix [[a -b][b a]].
As [[0 -1][1 0]]^2 is [[-1 0][0 -1]] (aka i^2 is -1) this has the important properties of the complex numbers.
Now you can also do this by taking ℝ[x]/(x^2+1) in another algebraic trick. But to some people, linear algebra seems more real.
posted by NotAYakk at 1:54 PM on June 2, 2023 [2 favorites]
TIL that Safari doesn't display <math> tags. Flabdablet's comment works in Firefox, but everything in a <math> tag is just a blank space in Safari.
posted by fedward at 1:58 PM on June 2, 2023 [11 favorites]
posted by fedward at 1:58 PM on June 2, 2023 [11 favorites]
MetaFilter: To some people, linear algebra seems more real.
posted by vibrotronica at 2:12 PM on June 2, 2023 [3 favorites]
posted by vibrotronica at 2:12 PM on June 2, 2023 [3 favorites]
TIL that Safari doesn't display <math> tags.
Both Safari and Chrome can display MathML, but both are a little more pedantic about MathML syntax. You have to use <math><mi>i</mi></math> instead of <math>i</math>.
For example:
Both Safari and Chrome can display MathML, but both are a little more pedantic about MathML syntax. You have to use <math><mi>i</mi></math> instead of <math>i</math>.
For example:
Mathematicians have traditionally called the square root of negative one .posted by 1970s Antihero at 2:21 PM on June 2, 2023 [10 favorites]
Electrical engineers have traditionally called it , to avoid confusion with their customary use of to denote current (which, in turn, is probably done to avoid confusion with to denote charge... but I digress).
I've always had a sneaking suspicion that is actually negative and nobody's noticed that all the complex plane diagrams that the engineers have been drawing are upside down, but I know of no way to test this.
MetaFilter: I'm old and confused.
posted by hippybear at 2:21 PM on June 2, 2023 [12 favorites]
posted by hippybear at 2:21 PM on June 2, 2023 [12 favorites]
To some people, linear algebra seems more real.
It did to me. I kind of hated calculus, probably because of how it was taught at the high school level, but when I took a high-level linear algebra class in grad school, I really got into it. You can build it up from comparatively elementary first principles (i.e. the definition of a linear space), and the way it unfolds is genuinely beautiful.
posted by mikeand1 at 3:08 PM on June 2, 2023 [4 favorites]
It did to me. I kind of hated calculus, probably because of how it was taught at the high school level, but when I took a high-level linear algebra class in grad school, I really got into it. You can build it up from comparatively elementary first principles (i.e. the definition of a linear space), and the way it unfolds is genuinely beautiful.
posted by mikeand1 at 3:08 PM on June 2, 2023 [4 favorites]
I've always had a sneaking suspicion that is actually negative and nobody's noticed that all the complex plane diagrams that the engineers have been drawing are upside down, but I know of no way to test this.
There is actually a very clear sense in which j (EE) = -i (physics)
When building out Fourier methods, time-harmonic analysis, and phasor representation, one finds that one would like to work in terms of not sine and cosine but rather in terms of complex exponentials. They come, as you might expect, as a conjugate pair ( exp(j omega t) and exp(-j omega t) in EE parlance ). There is of course no reason to prefer one over the other as to which is the primary and which the conjugate. Physics has the convention to take the primary solution as exp(-i omega t), while EE has the convention that exp(j omega t) is the primary.
Mapping the symbol j to -i then converts EE convention phasor/Fourier equations into the corresponding phasor/Fourier equations in the physics convention (and vice versa, of course).
posted by BlueDuke at 3:42 PM on June 2, 2023 [1 favorite]
There is actually a very clear sense in which j (EE) = -i (physics)
When building out Fourier methods, time-harmonic analysis, and phasor representation, one finds that one would like to work in terms of not sine and cosine but rather in terms of complex exponentials. They come, as you might expect, as a conjugate pair ( exp(j omega t) and exp(-j omega t) in EE parlance ). There is of course no reason to prefer one over the other as to which is the primary and which the conjugate. Physics has the convention to take the primary solution as exp(-i omega t), while EE has the convention that exp(j omega t) is the primary.
Mapping the symbol j to -i then converts EE convention phasor/Fourier equations into the corresponding phasor/Fourier equations in the physics convention (and vice versa, of course).
posted by BlueDuke at 3:42 PM on June 2, 2023 [1 favorite]
You just made that up.
posted by slogger at 3:56 PM on June 2, 2023 [3 favorites]
posted by slogger at 3:56 PM on June 2, 2023 [3 favorites]
As Don Knuth pointed out when discussing imaginary numbers, negative numbers are also really weird and obviously don't "exist." You can't make a pile of apples and then take away more than you have, leaving yourself with some negative quantity. It's just they arise in various useful operations unless you go out of your way to avoid them, and are relatively easy to handle consistently.
Imaginary numbers are basically the same. The type of work for which they are useful just tends to be a bit less common in everyday work than balancing your bank account.
posted by mark k at 4:06 PM on June 2, 2023 [11 favorites]
Imaginary numbers are basically the same. The type of work for which they are useful just tends to be a bit less common in everyday work than balancing your bank account.
posted by mark k at 4:06 PM on June 2, 2023 [11 favorites]
I am biased because I learned whatever I know of electromagnetism from him, but I do recommend the delightful Barton Zwiebach's gloss on how imaginary numbers are central to quantum mechanics in a way they are not essential to many other fields.
posted by range at 4:33 PM on June 2, 2023 [2 favorites]
posted by range at 4:33 PM on June 2, 2023 [2 favorites]
Aren't those the thing that does the Mandelbrot?
posted by credulous at 4:56 PM on June 2, 2023 [1 favorite]
posted by credulous at 4:56 PM on June 2, 2023 [1 favorite]
What does the B in Benoît B. Mandelbrot stand for?
Benoît B. Mandelbrot
posted by avocet at 5:28 PM on June 2, 2023 [27 favorites]
Benoît B. Mandelbrot
posted by avocet at 5:28 PM on June 2, 2023 [27 favorites]
Multiplication is just "do one thing, then the other". For example, if I ask you to zoom a picture by 2x, and then zoom it again by 3x, what's the result? It's zoomed by 6x.
Squaring is just "do the thing twice". 2x then 2x is 4x.
Now something that squares to -1 is hard to imagine at first, because how can you zoom a picture such that if you do it twice, the picture comes out upside down? You can't! The answer isn't even "turn it upside down" because if you do that twice, it's back to the right side up: -1 times -1 is 1.
No problem: just rotate the picture 90 degrees. If you do that twice, it's rotated 180 degrees, which is upside down! That's i.
posted by a car full of lions at 5:37 PM on June 2, 2023 [4 favorites]
Squaring is just "do the thing twice". 2x then 2x is 4x.
Now something that squares to -1 is hard to imagine at first, because how can you zoom a picture such that if you do it twice, the picture comes out upside down? You can't! The answer isn't even "turn it upside down" because if you do that twice, it's back to the right side up: -1 times -1 is 1.
No problem: just rotate the picture 90 degrees. If you do that twice, it's rotated 180 degrees, which is upside down! That's i.
posted by a car full of lions at 5:37 PM on June 2, 2023 [4 favorites]
All numbers are made up.
posted by interogative mood at 6:06 PM on June 2, 2023 [5 favorites]
posted by interogative mood at 6:06 PM on June 2, 2023 [5 favorites]
All right, my searching is failing me, but I recall one complex formulation of the uncertainty principle that was really fascinating. Basically, reality is non-communicative and complex, and AB - BA equals i times h-bar or somesuch. Leading to physics based upon non-communicative geometry, which gets odd very quickly.
posted by indexy at 6:09 PM on June 2, 2023
posted by indexy at 6:09 PM on June 2, 2023
I do recommend the delightful Barton Zwiebach's gloss on how imaginary numbers are central to quantum mechanics
Just a derail but don't run onto someone you know in theses discussions, and Barton is just the sweetest man. Had not noticed he had a class on mit courseware.
posted by sammyo at 7:48 PM on June 2, 2023
Just a derail but don't run onto someone you know in theses discussions, and Barton is just the sweetest man. Had not noticed he had a class on mit courseware.
posted by sammyo at 7:48 PM on June 2, 2023
I think it was here on Metafilter that I read that Gauss or someone similar thought negative numbers should be called "adverse numbers," since they go "against" positive numbers. I like that, because it also suggests a great alternative name for imaginary numbers: their actions describes a counterclockwise, or leftward, rotation, so they could be called "sinister numbers."
posted by biogeo at 8:14 PM on June 2, 2023 [11 favorites]
posted by biogeo at 8:14 PM on June 2, 2023 [11 favorites]
I haven't dug through the paper in detail yet, but I think the game here goes like this:
In typical, complex valued quantum mechanics, if I have two independent systems and I want to talk about them as a compound system, I use an operation called a tensor product to combine their individual descriptions. A tensor product is pretty simple. Say I have the position of one classical particle as (x, y, z) and a second as (p, q, r). The tensor product of their positions is the vector of length six (x, y, z, p, q, r).
To get a real valued quantum theory that satisfies Bell inequalities, they had to make combining the two systems more complicated. So the actual physics that gets captured in tensor products over states (entanglement, basically) gets captured by a more complex way of describing compound systems. And they got something that works for all the Bell inequalities...which describe a system with a single source of particles.
When you go to more than one source, then the modification of the tensor product to let real valued theories work doesn't have enough structure to handle it. But what they hope they've shown is that no such modification can handle multiple sources.
posted by madhadron at 10:34 PM on June 2, 2023 [2 favorites]
In typical, complex valued quantum mechanics, if I have two independent systems and I want to talk about them as a compound system, I use an operation called a tensor product to combine their individual descriptions. A tensor product is pretty simple. Say I have the position of one classical particle as (x, y, z) and a second as (p, q, r). The tensor product of their positions is the vector of length six (x, y, z, p, q, r).
To get a real valued quantum theory that satisfies Bell inequalities, they had to make combining the two systems more complicated. So the actual physics that gets captured in tensor products over states (entanglement, basically) gets captured by a more complex way of describing compound systems. And they got something that works for all the Bell inequalities...which describe a system with a single source of particles.
When you go to more than one source, then the modification of the tensor product to let real valued theories work doesn't have enough structure to handle it. But what they hope they've shown is that no such modification can handle multiple sources.
posted by madhadron at 10:34 PM on June 2, 2023 [2 favorites]
> As Don Knuth pointed out when discussing imaginary numbers, negative numbers are also really weird and obviously don't "exist."
> All numbers are made up.
Yeah, sorry, this is one of my hobby horses, but I will ride it as often as I like.
Literally ALL numbers are made up. As in, they are products of the mind and of logic, not concrete objects that exist out in the real world.
Like you didn't wake up on morning to find a giant throbbing 1 looming over your bed - in that same way you might have seen a giant tree, or a bear, or a moose, or a vulture.
Trees, bears, mooses, and vultures are real concrete objects that exist in the objective world.
Numbers are not.
Literally all numbers are logical constructs that exist in our minds. In that sense, they are all imaginary.
1, 2, -1, 1/2, 3/4, 12.43, and so on - all the numbers we use in daily life and that, for example, make it possible for me to send this message to your over the internet - ALL of them are simply agreed upon logical constructs that we have found to be tremendously useful.
The point is, "imaginary numbers" just happened to get a bad rap because of the name they happen to have.
In fact, they are not any more or less "imaginary" than any other number.
But once people learn the numbers are imaginary they just cannot let go of that idea that, somehow, they don't really exist.
"They don't exist - they are just IMAGINARY. It says it right there in the very name." Endless philosophical discussions ensue.
Literally the exact same philosophical discussions could ensue about natural numbers, or integers, or rational numbers, or (especially!) real numbers.
But they don't. Not because the numbers themselves are any more "natural" or "rational" or "real" but simply because they have a better name.
So please, I beg you: Forget completely and entirely about the name imaginary numbers. Just think of them as, let's say, friendly numbers.
That's as good a name as any.
I guarantee, as soon as you spend a few years thinking of them as friendly numbers, all your doubts and trouble with them will vanish in an instant.
And they really, really are very easy in their essence. Now yes, complexity can arise in application, just as they can with any other type of number. But in their essence, "friendly numbers" are super, super easy.
For instance: +1 we can think of as moving 1 unit rightwards on a number line. -1 is moving 1 unit leftwards.
Multiplying by 1 leaves everything on the number line unchanged. Multiplying by -1 reverses the number line - so the numbers that were on the left switch to the right and those that were on the right switch to the left.
And (as car full of lions mentions above) multiplying by our special "friendly number" f rotates the number line by 90 degrees.
That's it. That's all there is to it.
I'm sorry, so sorry, if the educators in your life somehow made you think friendly numbers were super difficult or impossible to understand. They're really not.
They are super friendly.
Just keep repeating that to yourself, and soon it will be true.
posted by flug at 11:13 PM on June 2, 2023 [14 favorites]
> All numbers are made up.
Yeah, sorry, this is one of my hobby horses, but I will ride it as often as I like.
Literally ALL numbers are made up. As in, they are products of the mind and of logic, not concrete objects that exist out in the real world.
Like you didn't wake up on morning to find a giant throbbing 1 looming over your bed - in that same way you might have seen a giant tree, or a bear, or a moose, or a vulture.
Trees, bears, mooses, and vultures are real concrete objects that exist in the objective world.
Numbers are not.
Literally all numbers are logical constructs that exist in our minds. In that sense, they are all imaginary.
1, 2, -1, 1/2, 3/4, 12.43, and so on - all the numbers we use in daily life and that, for example, make it possible for me to send this message to your over the internet - ALL of them are simply agreed upon logical constructs that we have found to be tremendously useful.
The point is, "imaginary numbers" just happened to get a bad rap because of the name they happen to have.
In fact, they are not any more or less "imaginary" than any other number.
But once people learn the numbers are imaginary they just cannot let go of that idea that, somehow, they don't really exist.
"They don't exist - they are just IMAGINARY. It says it right there in the very name." Endless philosophical discussions ensue.
Literally the exact same philosophical discussions could ensue about natural numbers, or integers, or rational numbers, or (especially!) real numbers.
But they don't. Not because the numbers themselves are any more "natural" or "rational" or "real" but simply because they have a better name.
So please, I beg you: Forget completely and entirely about the name imaginary numbers. Just think of them as, let's say, friendly numbers.
That's as good a name as any.
I guarantee, as soon as you spend a few years thinking of them as friendly numbers, all your doubts and trouble with them will vanish in an instant.
And they really, really are very easy in their essence. Now yes, complexity can arise in application, just as they can with any other type of number. But in their essence, "friendly numbers" are super, super easy.
For instance: +1 we can think of as moving 1 unit rightwards on a number line. -1 is moving 1 unit leftwards.
Multiplying by 1 leaves everything on the number line unchanged. Multiplying by -1 reverses the number line - so the numbers that were on the left switch to the right and those that were on the right switch to the left.
And (as car full of lions mentions above) multiplying by our special "friendly number" f rotates the number line by 90 degrees.
That's it. That's all there is to it.
I'm sorry, so sorry, if the educators in your life somehow made you think friendly numbers were super difficult or impossible to understand. They're really not.
They are super friendly.
Just keep repeating that to yourself, and soon it will be true.
posted by flug at 11:13 PM on June 2, 2023 [14 favorites]
Calvin: Here's another math problem I can't figure out. What's 9 + 4?
Hobbes: Ooh, that's a tricky one. You have to use calculus and imaginary numbers for this.
Calvin: Imaginary numbers?!
Hobbes: You know, eleventeen, thirty-twelve, and all those ... it's a little confusing at first.
Calvin: How did you learn all this? You've never gone to school!
Hobbes: Instinct, tigers are born with it.
Link to comic.
posted by Kattullus at 11:19 PM on June 2, 2023 [9 favorites]
Hobbes: Ooh, that's a tricky one. You have to use calculus and imaginary numbers for this.
Calvin: Imaginary numbers?!
Hobbes: You know, eleventeen, thirty-twelve, and all those ... it's a little confusing at first.
Calvin: How did you learn all this? You've never gone to school!
Hobbes: Instinct, tigers are born with it.
Link to comic.
posted by Kattullus at 11:19 PM on June 2, 2023 [9 favorites]
mikeand1, you might enjoy this brief video that YouTube's algorithm coincidentally served up to me today, in which Gil Strang talks about the advantages of linear algebra over calculus: "It's all about flat things."
posted by mpark at 11:43 PM on June 2, 2023 [1 favorite]
posted by mpark at 11:43 PM on June 2, 2023 [1 favorite]
That's what unfortunately set linear algebra off on the wrong foot for for me before I even got started --- I thought linear equations of the form y=ax+b were boringly flat, and that linear algebra would only bring me much more of that boring, in surround-sound. I wish I'd done a flug-like trick and gave it a cooler-sounding name in my head.
That said, we already have friendly numbers --- they're sets of "two or more natural numbers with a common abundancy index", don't you know. "Not to be confused with amicable numbers" of course. Those number theorists use up all the names!
posted by Chef Flamboyardee at 12:52 AM on June 3, 2023 [2 favorites]
That said, we already have friendly numbers --- they're sets of "two or more natural numbers with a common abundancy index", don't you know. "Not to be confused with amicable numbers" of course. Those number theorists use up all the names!
posted by Chef Flamboyardee at 12:52 AM on June 3, 2023 [2 favorites]
When little ms flabdablet went to primary school, she was taught that friendly numbers are the pairs of digits that sum to 10: 1 and 9, 2 and 8, 3 and 7, 4 and 6, and 5 and 5. Yes, 5 is its own friend. I think there's a lesson in that for all of us.
posted by flabdablet at 2:00 AM on June 3, 2023 [2 favorites]
posted by flabdablet at 2:00 AM on June 3, 2023 [2 favorites]
Trees, bears, mooses, and vultures are real concrete objects that exist in the objective world.
Numbers are not.
Literally all numbers are logical constructs that exist in our minds. In that sense, they are all imaginary.
Although I agree with your wider point, I feel I should point out that trees bears mooses and vultures are also logical constructs that exist only in your mind in the very exact same sense as numbers.
posted by nzero at 2:56 AM on June 3, 2023 [12 favorites]
Numbers are not.
Literally all numbers are logical constructs that exist in our minds. In that sense, they are all imaginary.
Although I agree with your wider point, I feel I should point out that trees bears mooses and vultures are also logical constructs that exist only in your mind in the very exact same sense as numbers.
posted by nzero at 2:56 AM on June 3, 2023 [12 favorites]
I feel some bears at least exist in their own minds as well.
posted by a car full of lions at 3:28 AM on June 3, 2023 [6 favorites]
posted by a car full of lions at 3:28 AM on June 3, 2023 [6 favorites]
trees bears mooses and vultures are also logical constructs that exist only in your mind in the very exact same sense as numbers.
Seems to me that there is a coherent and useful distinction that can be made between those kinds of logical construct, though.
Trees, bears, mooses and vultures are logical constructs, sure; but it's easy to imagine Alice and Bob developing those constructs independently of one another via similarity of experience, and being able to have useful conversations employing them, without Bob ever having had a use for numbers.
Inventing counting seems like the kind of thing that nobody would do until they already had some things to count. Numbering things emerges from categorizing them. There's nothing to be gained from counting until one of your categories contains at least two things, and very little to be gained until some categories contain more things than others.
In other words, the logical construct of numbers is a rider to constructs like trees and bears and mooses in a way that a construct like vultures is not.
I feel some bears at least exist in their own minds as well.
Seems to me that any logical construct within a bear's mind that represents that bear would have more features in common with the logical construct in my mind that represents me than with the one that represents bears.
posted by flabdablet at 4:07 AM on June 3, 2023 [2 favorites]
Seems to me that there is a coherent and useful distinction that can be made between those kinds of logical construct, though.
Trees, bears, mooses and vultures are logical constructs, sure; but it's easy to imagine Alice and Bob developing those constructs independently of one another via similarity of experience, and being able to have useful conversations employing them, without Bob ever having had a use for numbers.
Inventing counting seems like the kind of thing that nobody would do until they already had some things to count. Numbering things emerges from categorizing them. There's nothing to be gained from counting until one of your categories contains at least two things, and very little to be gained until some categories contain more things than others.
In other words, the logical construct of numbers is a rider to constructs like trees and bears and mooses in a way that a construct like vultures is not.
I feel some bears at least exist in their own minds as well.
Seems to me that any logical construct within a bear's mind that represents that bear would have more features in common with the logical construct in my mind that represents me than with the one that represents bears.
posted by flabdablet at 4:07 AM on June 3, 2023 [2 favorites]
Academic philosophers have of course pushed these ideas much further than I personally will ever find useful. It's nice to see them happy, I guess.
posted by flabdablet at 4:11 AM on June 3, 2023 [1 favorite]
posted by flabdablet at 4:11 AM on June 3, 2023 [1 favorite]
When I was a smart-Alec kid, I asked high school math teachers, what is the square root of i? Hint 1: Like 16 has two square roots, 4 & -4, the square root of i has two opposite square roots. Hint 2: They're irrational. Sigh. And sigh again. This question was more fun before one could Google a solution.
posted by gregoreo at 5:14 AM on June 3, 2023 [1 favorite]
posted by gregoreo at 5:14 AM on June 3, 2023 [1 favorite]
There's nothing to be gained from counting until one of your categories contains at least two things[..]
I have a turtle, which in theory has a name, but mostly I think of it as "the turtle". If I had two turtles, then I would need names.
Or not. I would probably have "big turtle" and "small turtle", or "turtle with a spot" and "turtle without a spot". But at some multiple of turtles, simpler labels would become useful.
posted by rochrobbb at 5:41 AM on June 3, 2023
I have a turtle, which in theory has a name, but mostly I think of it as "the turtle". If I had two turtles, then I would need names.
Or not. I would probably have "big turtle" and "small turtle", or "turtle with a spot" and "turtle without a spot". But at some multiple of turtles, simpler labels would become useful.
posted by rochrobbb at 5:41 AM on June 3, 2023
When I was a smart-Alec kid, I asked high school math teachers, what is the square root of i? ... This question was more fun before one could Google a solution.
Lemme try that from first principles without reference to Google...
I'm thinking the square root of i would have to be a complex number. So I'm looking for a + bi such that (a + bi)(a + bi) = i.
Expanding: a2 + 2abi + b2i2 = i
Collecting: a2 - b2 + 2abi = i
Real and imaginary parts on both sides must match, so a2 - b2 = 0 and 2ab = 1. The fact that 2ab is positive means a and b have the same sign; together with a2 - b2 = 0, that means they're equal. So now we have 2a2 = 1, therefore a2 = ½, therefore a = either √½ or -√½ and b is the same, which means the square roots we're after are
±√½(1 + i)
QED.
posted by flabdablet at 5:46 AM on June 3, 2023 [1 favorite]
Lemme try that from first principles without reference to Google...
That was fun!
I'm thinking the square root of i would have to be a complex number. So I'm looking for a + bi such that (a + bi)(a + bi) = i.
Expanding: a2 + 2abi + b2i2 = i
Collecting: a2 - b2 + 2abi = i
Real and imaginary parts on both sides must match, so a2 - b2 = 0 and 2ab = 1. The fact that 2ab is positive means a and b have the same sign; together with a2 - b2 = 0, that means they're equal. So now we have 2a2 = 1, therefore a2 = ½, therefore a = either √½ or -√½ and b is the same, which means the square roots we're after are
±√½(1 + i)
QED.
posted by flabdablet at 5:46 AM on June 3, 2023 [1 favorite]
I am not in a position to do the real math because I’m waiting for a doc appointment, but I went with the rotation analogy, thinking that it would represent a 45 degree angle on a circle, because that would be the halfway point such that two rotations of that angle from 1,0 would get you to 1,0.
posted by notoriety public at 5:51 AM on June 3, 2023 [3 favorites]
posted by notoriety public at 5:51 AM on June 3, 2023 [3 favorites]
sin(45°) and cos(45°) are both √½, so that works. But so does going ⅝ of the way around, which gets us the other root.
posted by flabdablet at 5:56 AM on June 3, 2023
posted by flabdablet at 5:56 AM on June 3, 2023
To show that "imaginary" numbers are just as "real" as any other you can get to them purely from geometry.
The short version is that there's a system called Geometric Algebra which defines a way to multiply vectors together via a geometric product. When you multiply two vectors you get a little patch of area defined by taking copies of your vectors and forming them into a parallelogram. This "patch of area" is formally called a bivector. And bivectors basically are "imaginary" numbers*. In particular, when you square the unit bivector using the geometric product it equals -1. There's a bit more to it of course. Bivectors are oriented areas, for instance. Just like vectors have a direction so do bivectors. But for bivectors it's a rotational direction sort of like a ratchet. It can go one direction or the other but not both. Imaginary numbers aren't super weird (but convenient) mathematical objects they can be thought of as just tiny patches of area on a plane with a "desire" to rotate either counter-clockwise (if it's positive) or clockwise (if it's negative.)
In more detail in case anyone is interested:
Geometric algebra defines the geometric product like this for vectors v and w**:
vw = v . w + v^w
Read the right side as "v dot w plus v wedge w." The dot product takes the two vectors and produces a scalar (aka: just a number) and the wedge product produces a bivector. If you take copies of v and w and form them into a parallelogram, the wedge product calculates the area of that parallelogram.
In 2D given:
v=(x1, y1) and w = (x2, y2)
v^w = x1y2 - x2y1
Which, if you're familiar with linear algebra, you'll recognize as the formula for calculating the determinant of a 2x2 matrix.
There are a couple things interesting about the wedge product. First, since it's calculating an area any vector wedged with itself equals 0 because straight lines don't have area. And, as mentioned above, bivectors are oriented. If you reverse v and w in the above definition you get the same absolute value but opposite sign (anti-commutativity.) That is:
v^w = -w^v
Geometrically that tells you which direction you have to rotate to get from v to w. If v^w is positive you need to rotate counter-clockwise and if it's negative you rotate clockwise. So the anti-commutativity makes sense because if going from v to w (v^w) means rotating counter-clockwise then obviously going from w to v (w^v) means you need to rotate clockwise and hence the change of sign.
Due to the anti-commutativity of the wedge product, the geometric product is also anti-commutative:
vw = -wv
For geometric algebra to make sense the set it operates on has to be closed meaning multiplying vectors together should give us other vectors. But scalars and bivectors aren't vectors. To get around this we just expand our idea of "vector." Geometric algebra works on things called multi-vectors which, in 2D, contain four components: 1 scalar component, 2 vector components (our x and y directions) and 1 bivector component (which again can be though of as just a patch of area with a clockwise or counter-clockwise orientation.)
So if we take x and y to be the unit basis vectors of our 2D space then 2D geometric algebra has multi-vectors of the form:
a + bx + cy + dxy
Where a, b, c, and d are all just numbers, x and y are our basis vectors and xy denotes the bivector. Okay so let's see what happens when we square each of our basis entities.
The basis for scalars is 1 (think of a as 1 times a) and 1 times 1 = 1 which is not terribly interesting. How about x?
Using the geometric product:
xx = x.x + x^x
xx = x.x + 0 (because any vector wedged with itself equals 0)
xx = |x||x| (by definition of dot product)
xx = 1 (because x is a unit vector |x| = 1)
And the exact same is true for y so:
yy = 1
Now what about that bivector?
(xy)(xy) = (-yx)(xy) (because of anti-commutativity of geometric product)
(xy)(xy) = -yxxy
(xy)(xy) = -yy (because xx = 1)
(xy)(xy) = -1 (because yy = 1)
So there you have it! The unit bivector, a purely geometric entity representing an oriented area—and therefore just as "real" as anything else—squares to -1. Also, from the definition of the geometric product above if both v and w are vectors we get both a scalar and a bivector as a result. In other words something of the form:
a + bxy
Which is really just (isomorphic to):
a + bi
Which is to say, a complex number. And specifically a complex number representing a rotation (in the direction of the bivector) by the angle between v and w. Multiplying two vectors in geometric algebra gives you a rotation and multiplying that rotation by another vector rotates the vector.***
Bonus Higher Dimensions and Quaternions!
A 2D geometric algebra as multi-vectors with four components and, in general, an n-D geometric algebra has 2^n (two to the power n, not 2 wedge n) component multi-vectors. In 3D we add trivectors which you get by wedging three vectors together. Then quadvectors in 4D and so on. And the distribution of those components is determined by the binomial expansion (or Pascals Triangle).
So in 2D we have: 1 scalar, 2 vectors, 1 bivector.
In 3D we have: 1 scalar, 3 vectors, 3 bivectors, 1 trivector.
Now, just like the single bivector in 2D squares to -1, so do all three bivector components in 3D. Each of them representing one of the primary planes in 3D space: xy-plane, xz-plane, yz-plane. And just like taking the scalar and bivector parts by themselves in 2D gave us the complex numbers, it hopefully doesn't come as a surprise that the scalar and bivector parts taken by themselves in 3D are isomorphic to quaternions.
Quaternions confuse people. A lot. They're this weird, quasi 4-dimensional vector-like thing which somehow acts on 3D vectors to get them to rotate...somehow? But just like we could say that the imaginary unit i is really just an oriented patch of area in 2D we can do the same with the quaternions i, j and k. We can think of each one as not a coordinate of this 4D vector-thing but instead as the primary planes in 3D. When you take a linear combination of those planes what you get is another plane. Only instead of being oriented along the main axes of our system it's a plane oriented arbitrarily in 3D space. And just like any other plane (bivector) is has a rotational direction. It's this plane that objects rotate about in 3D just like they rotate in the xy plane in 2D when multiplying by complex numbers. You can think of the scalar as literally that; a scaling factor to keep objects from growing or shrinking as they rotate. Quaternions aren't vectors and they're not terribly confusing from this perspective. A quaternion just defines a plane of rotation and stuff rotates in that plane. Easy.
* Technically I should probably say that the bivectors are isomorphic to the imaginary numbers but meh.
** Actually this definition of the geometric product is valid only if both v and w are vectors. You can define both the dot and wedge products in a more general way so that you can, for instance, take the dot product of a vector and a bivector but it's not quite the same as what most people know as the dot product.
*** This isn't quite true—at least not in more than 2 dimensions—but it's close enough for my purposes.
posted by Mister_Sleight_of_Hand at 6:30 AM on June 3, 2023 [15 favorites]
The short version is that there's a system called Geometric Algebra which defines a way to multiply vectors together via a geometric product. When you multiply two vectors you get a little patch of area defined by taking copies of your vectors and forming them into a parallelogram. This "patch of area" is formally called a bivector. And bivectors basically are "imaginary" numbers*. In particular, when you square the unit bivector using the geometric product it equals -1. There's a bit more to it of course. Bivectors are oriented areas, for instance. Just like vectors have a direction so do bivectors. But for bivectors it's a rotational direction sort of like a ratchet. It can go one direction or the other but not both. Imaginary numbers aren't super weird (but convenient) mathematical objects they can be thought of as just tiny patches of area on a plane with a "desire" to rotate either counter-clockwise (if it's positive) or clockwise (if it's negative.)
In more detail in case anyone is interested:
Geometric algebra defines the geometric product like this for vectors v and w**:
vw = v . w + v^w
Read the right side as "v dot w plus v wedge w." The dot product takes the two vectors and produces a scalar (aka: just a number) and the wedge product produces a bivector. If you take copies of v and w and form them into a parallelogram, the wedge product calculates the area of that parallelogram.
In 2D given:
v=(x1, y1) and w = (x2, y2)
v^w = x1y2 - x2y1
Which, if you're familiar with linear algebra, you'll recognize as the formula for calculating the determinant of a 2x2 matrix.
There are a couple things interesting about the wedge product. First, since it's calculating an area any vector wedged with itself equals 0 because straight lines don't have area. And, as mentioned above, bivectors are oriented. If you reverse v and w in the above definition you get the same absolute value but opposite sign (anti-commutativity.) That is:
v^w = -w^v
Geometrically that tells you which direction you have to rotate to get from v to w. If v^w is positive you need to rotate counter-clockwise and if it's negative you rotate clockwise. So the anti-commutativity makes sense because if going from v to w (v^w) means rotating counter-clockwise then obviously going from w to v (w^v) means you need to rotate clockwise and hence the change of sign.
Due to the anti-commutativity of the wedge product, the geometric product is also anti-commutative:
vw = -wv
For geometric algebra to make sense the set it operates on has to be closed meaning multiplying vectors together should give us other vectors. But scalars and bivectors aren't vectors. To get around this we just expand our idea of "vector." Geometric algebra works on things called multi-vectors which, in 2D, contain four components: 1 scalar component, 2 vector components (our x and y directions) and 1 bivector component (which again can be though of as just a patch of area with a clockwise or counter-clockwise orientation.)
So if we take x and y to be the unit basis vectors of our 2D space then 2D geometric algebra has multi-vectors of the form:
a + bx + cy + dxy
Where a, b, c, and d are all just numbers, x and y are our basis vectors and xy denotes the bivector. Okay so let's see what happens when we square each of our basis entities.
The basis for scalars is 1 (think of a as 1 times a) and 1 times 1 = 1 which is not terribly interesting. How about x?
Using the geometric product:
xx = x.x + x^x
xx = x.x + 0 (because any vector wedged with itself equals 0)
xx = |x||x| (by definition of dot product)
xx = 1 (because x is a unit vector |x| = 1)
And the exact same is true for y so:
yy = 1
Now what about that bivector?
(xy)(xy) = (-yx)(xy) (because of anti-commutativity of geometric product)
(xy)(xy) = -yxxy
(xy)(xy) = -yy (because xx = 1)
(xy)(xy) = -1 (because yy = 1)
So there you have it! The unit bivector, a purely geometric entity representing an oriented area—and therefore just as "real" as anything else—squares to -1. Also, from the definition of the geometric product above if both v and w are vectors we get both a scalar and a bivector as a result. In other words something of the form:
a + bxy
Which is really just (isomorphic to):
a + bi
Which is to say, a complex number. And specifically a complex number representing a rotation (in the direction of the bivector) by the angle between v and w. Multiplying two vectors in geometric algebra gives you a rotation and multiplying that rotation by another vector rotates the vector.***
Bonus Higher Dimensions and Quaternions!
A 2D geometric algebra as multi-vectors with four components and, in general, an n-D geometric algebra has 2^n (two to the power n, not 2 wedge n) component multi-vectors. In 3D we add trivectors which you get by wedging three vectors together. Then quadvectors in 4D and so on. And the distribution of those components is determined by the binomial expansion (or Pascals Triangle).
So in 2D we have: 1 scalar, 2 vectors, 1 bivector.
In 3D we have: 1 scalar, 3 vectors, 3 bivectors, 1 trivector.
Now, just like the single bivector in 2D squares to -1, so do all three bivector components in 3D. Each of them representing one of the primary planes in 3D space: xy-plane, xz-plane, yz-plane. And just like taking the scalar and bivector parts by themselves in 2D gave us the complex numbers, it hopefully doesn't come as a surprise that the scalar and bivector parts taken by themselves in 3D are isomorphic to quaternions.
Quaternions confuse people. A lot. They're this weird, quasi 4-dimensional vector-like thing which somehow acts on 3D vectors to get them to rotate...somehow? But just like we could say that the imaginary unit i is really just an oriented patch of area in 2D we can do the same with the quaternions i, j and k. We can think of each one as not a coordinate of this 4D vector-thing but instead as the primary planes in 3D. When you take a linear combination of those planes what you get is another plane. Only instead of being oriented along the main axes of our system it's a plane oriented arbitrarily in 3D space. And just like any other plane (bivector) is has a rotational direction. It's this plane that objects rotate about in 3D just like they rotate in the xy plane in 2D when multiplying by complex numbers. You can think of the scalar as literally that; a scaling factor to keep objects from growing or shrinking as they rotate. Quaternions aren't vectors and they're not terribly confusing from this perspective. A quaternion just defines a plane of rotation and stuff rotates in that plane. Easy.
* Technically I should probably say that the bivectors are isomorphic to the imaginary numbers but meh.
** Actually this definition of the geometric product is valid only if both v and w are vectors. You can define both the dot and wedge products in a more general way so that you can, for instance, take the dot product of a vector and a bivector but it's not quite the same as what most people know as the dot product.
*** This isn't quite true—at least not in more than 2 dimensions—but it's close enough for my purposes.
posted by Mister_Sleight_of_Hand at 6:30 AM on June 3, 2023 [15 favorites]
But so does going ⅝ of the way around, which gets us the other root.
I mean, that goes without saying! Or would have, until you said it.
“We’re sorry. The number you have reached is imaginary. Please rotate your phone 90 degrees, and dial again.”
posted by notoriety public at 7:03 AM on June 3, 2023 [4 favorites]
I mean, that goes without saying! Or would have, until you said it.
“We’re sorry. The number you have reached is imaginary. Please rotate your phone 90 degrees, and dial again.”
posted by notoriety public at 7:03 AM on June 3, 2023 [4 favorites]
To show that "imaginary" numbers are just as "real" as any other you can get to them purely from geometry.
To show that any mathematical construct is in some way "real" involves finding features of reality to which the abstractions and operations codified by that construct can be usefully applied.
Abstraction is kind of a weird skill that mainly consists of developing sound intuitions about what can safely be ignored in order to make useful the ascribing of some kind of essence to a thing. Done right, it leads to all kinds of useful discoveries about novel ways to categorize and manipulate things that are not at all obviously related otherwise.
Done wrong, it leads down bifurcating and progressively narrower conceptual rabbit holes that ultimately feed into a deep cave sparkling with ideal forms, where mathematics exists in some unspecifiable way prior to reality and reality is a consequence of mathematics rather than the other way around.
The unfortunate denizens of these burrows become sorely afflicted by questions such as what the Universe "really is", or what it might be that "breathes fire into the equations", whose obvious answers they've long since persuaded themselves to disregard.
posted by flabdablet at 7:45 AM on June 3, 2023
To show that any mathematical construct is in some way "real" involves finding features of reality to which the abstractions and operations codified by that construct can be usefully applied.
Abstraction is kind of a weird skill that mainly consists of developing sound intuitions about what can safely be ignored in order to make useful the ascribing of some kind of essence to a thing. Done right, it leads to all kinds of useful discoveries about novel ways to categorize and manipulate things that are not at all obviously related otherwise.
Done wrong, it leads down bifurcating and progressively narrower conceptual rabbit holes that ultimately feed into a deep cave sparkling with ideal forms, where mathematics exists in some unspecifiable way prior to reality and reality is a consequence of mathematics rather than the other way around.
The unfortunate denizens of these burrows become sorely afflicted by questions such as what the Universe "really is", or what it might be that "breathes fire into the equations", whose obvious answers they've long since persuaded themselves to disregard.
posted by flabdablet at 7:45 AM on June 3, 2023
Please rotate your phone 90 degrees, and dial again.
No, not about that axis. No, not that one either. Or that one.
What do you mean, run out of axes? You must be holding it wrong.
posted by flabdablet at 7:49 AM on June 3, 2023 [4 favorites]
No, not about that axis. No, not that one either. Or that one.
What do you mean, run out of axes? You must be holding it wrong.
posted by flabdablet at 7:49 AM on June 3, 2023 [4 favorites]
Scientists are not poets, a phrase or metaphor is not their primary skill. Like the "God particle" or the Rutherford diagram of an atom that looks like the solar system, it's just not well imagined for noobs. Worked for their peers that needed an idea tweak. The most imaginary number ever is zero. Hmm is it even a number?
posted by sammyo at 7:52 AM on June 3, 2023
posted by sammyo at 7:52 AM on June 3, 2023
i is not √(-1).
The square root function is a function, and hence, is defined to always return a positive number to avoid a function returning two numbers. Example: √4 is +2, despite the fact that (-2)2 = 4.
i has the property that i2 = -1. That is distinct from saying that √(-1) = i.
(</pedantry>)
posted by saeculorum at 8:22 AM on June 3, 2023 [5 favorites]
The square root function is a function, and hence, is defined to always return a positive number to avoid a function returning two numbers. Example: √4 is +2, despite the fact that (-2)2 = 4.
i has the property that i2 = -1. That is distinct from saying that √(-1) = i.
(</pedantry>)
posted by saeculorum at 8:22 AM on June 3, 2023 [5 favorites]
To show that any mathematical construct is in some way "real" involves finding features of reality to which the abstractions and operations codified by that construct can be usefully applied.
Fair point. Perhaps, "to show that 'imaginary' numbers are no more weird or scary than 'real' numbers...." would have been a better way to phrase it. Or perhaps not. Regardless, I like math, I've been on a geometric algebra kick lately and here we are.
posted by Mister_Sleight_of_Hand at 8:26 AM on June 3, 2023 [1 favorite]
Fair point. Perhaps, "to show that 'imaginary' numbers are no more weird or scary than 'real' numbers...." would have been a better way to phrase it. Or perhaps not. Regardless, I like math, I've been on a geometric algebra kick lately and here we are.
posted by Mister_Sleight_of_Hand at 8:26 AM on June 3, 2023 [1 favorite]
I remember somewhere that number was defined as the cardinality of a set, i.e. how many members it has. The cardinality of the set of square circles is 0, so 0 must be a number.
posted by njohnson23 at 9:26 AM on June 3, 2023
posted by njohnson23 at 9:26 AM on June 3, 2023
The square root function is a function, and hence, is defined to always return a positive number to avoid a function returning two numbers.
For any complex number z (including all the values that also happen to be real numbers, i.e. those with an imaginary part of zero) there are two values that square to z. That doesn't make √z somehow not a function of z. When z is -1, those values are √-1 = i and -√-1 = -i.
posted by flabdablet at 10:15 AM on June 3, 2023 [1 favorite]
For any complex number z (including all the values that also happen to be real numbers, i.e. those with an imaginary part of zero) there are two values that square to z. That doesn't make √z somehow not a function of z. When z is -1, those values are √-1 = i and -√-1 = -i.
posted by flabdablet at 10:15 AM on June 3, 2023 [1 favorite]
I feel like no discussion involving imaginary numbers is complete without a mention of Euler's identity, one of the most amazing bits of mathematical trivia that exists: e^(i pi)+1=0
posted by Slothrup at 10:35 AM on June 3, 2023 [3 favorites]
posted by Slothrup at 10:35 AM on June 3, 2023 [3 favorites]
"I’ve heard of Gödel," Waterhouse put in helpfully. "But could we back up just a sec?"
"Of course Lawrence."
"Why bother? Why did Russell do it? Was there something wrong with math? I mean, two plus two
equals four, right?"
Alan picked up two bottlecaps and set them down on the ground. "Two. One-two. Plus—" He set down
two more. "Another two. One-two. Equals four. One-two-three-four."
"What’s so bad about that?" Lawrence said.
"But Lawrence—when you really do math, in an abstract way, you’re not counting bottlecaps, are
you?"
"I’m not counting anything."
Rudy broke the following news: "Zat is a very modern position for you to take."
"It is?"
Alan said, "There was this implicit belief, for a long time, that math was a sort of physics of bottlecaps.
That any mathematical operation you could do on paper, no matter how complicated, could be reduced
—in theory, anyway—to messing about with actual physical counters, such as bottlecaps, in the real
world."
"But you can’t have two point one bottlecaps."
"All right, all right, say we use bottlecaps for integers, and for real numbers like two point one, we use
physical measurements, like the length of this stick." Alan tossed the stick down next to the bottlecaps.
"Well what about pi, then? You can’t have a stick that’s exactly pi inches long."
"Pi is from geometry—ze same story," Rudy put in.
"Yes, it was believed that Euclid’s geometry was really a kind of physics, that his lines and so on
represented properties of the physical world. But—you know Einstein?"
"I’m not very good with names."
"That white-haired chap with the big mustache?"
"Oh, yeah," Lawrence said dimly, "I tried to ask him my sprocket question. He claimed he was late for
an appointment or something."
"That fellow has come up with a general relativity theory, which is sort of a practical application, not of
Euclid’s, but of Riemann’s geometry—"
"The same Riemann of your zeta function?"
"Same Riemann, different subject. Now let’s not get sidetracked here Lawrence—"
"Riemann showed you could have many many different geometries that were not the geometry of
Euclid but that still made sense internally," Rudy explained.
"All right, so back to P.M. then," Lawrence said.
"Yes! Russell and Whitehead. It’s like this: when mathematicians began fooling around with things like
the square root of negative one, and quaternions, then they were no longer dealing with things that you
could translate into sticks and bottlecaps. And yet they were still getting sound results."
"Or at least internally consistent results," Rudy said.
"Okay. Meaning that math was more than a physics of bottlecaps."
"It appeared that way, Lawrence, but this raised the question of was mathematics really true or was it
just a game played with symbols? In other words—are we discovering Truth, or just wanking?"
--Neal Stephenson, Cryptonomicon
posted by indexy at 10:57 AM on June 3, 2023 [1 favorite]
"Of course Lawrence."
"Why bother? Why did Russell do it? Was there something wrong with math? I mean, two plus two
equals four, right?"
Alan picked up two bottlecaps and set them down on the ground. "Two. One-two. Plus—" He set down
two more. "Another two. One-two. Equals four. One-two-three-four."
"What’s so bad about that?" Lawrence said.
"But Lawrence—when you really do math, in an abstract way, you’re not counting bottlecaps, are
you?"
"I’m not counting anything."
Rudy broke the following news: "Zat is a very modern position for you to take."
"It is?"
Alan said, "There was this implicit belief, for a long time, that math was a sort of physics of bottlecaps.
That any mathematical operation you could do on paper, no matter how complicated, could be reduced
—in theory, anyway—to messing about with actual physical counters, such as bottlecaps, in the real
world."
"But you can’t have two point one bottlecaps."
"All right, all right, say we use bottlecaps for integers, and for real numbers like two point one, we use
physical measurements, like the length of this stick." Alan tossed the stick down next to the bottlecaps.
"Well what about pi, then? You can’t have a stick that’s exactly pi inches long."
"Pi is from geometry—ze same story," Rudy put in.
"Yes, it was believed that Euclid’s geometry was really a kind of physics, that his lines and so on
represented properties of the physical world. But—you know Einstein?"
"I’m not very good with names."
"That white-haired chap with the big mustache?"
"Oh, yeah," Lawrence said dimly, "I tried to ask him my sprocket question. He claimed he was late for
an appointment or something."
"That fellow has come up with a general relativity theory, which is sort of a practical application, not of
Euclid’s, but of Riemann’s geometry—"
"The same Riemann of your zeta function?"
"Same Riemann, different subject. Now let’s not get sidetracked here Lawrence—"
"Riemann showed you could have many many different geometries that were not the geometry of
Euclid but that still made sense internally," Rudy explained.
"All right, so back to P.M. then," Lawrence said.
"Yes! Russell and Whitehead. It’s like this: when mathematicians began fooling around with things like
the square root of negative one, and quaternions, then they were no longer dealing with things that you
could translate into sticks and bottlecaps. And yet they were still getting sound results."
"Or at least internally consistent results," Rudy said.
"Okay. Meaning that math was more than a physics of bottlecaps."
"It appeared that way, Lawrence, but this raised the question of was mathematics really true or was it
just a game played with symbols? In other words—are we discovering Truth, or just wanking?"
--Neal Stephenson, Cryptonomicon
posted by indexy at 10:57 AM on June 3, 2023 [1 favorite]
'imaginary' numbers are no more weird or scary than 'real' numbers
The only thing that ever really bothered me about the whole business of complex numbers is the convention of using the addition operator to construct them. I had difficulty letting go of the idea that adding something to something else carried an implication that ultimately both of those somethings had to be things of the same kind.
Even algebraic expressions, where you're constantly writing expressions like 5x + 2y + 7 and have to be scrupulous not to mix up the x and y terms and the bare numbers, I saw as legitimate because both 5x and 2y are going to end up having the same kind of value as 7, at which point the addition just gets done and disappears.
I'd held on tightly to this mental model all the way through primary school, and it helped me not get as confused as many of my peers when working out stuff like ⅕ + ⅜. I wasn't even tempted to add the 1 and the 3 because it was clear to me that the two things either side of the + were not yet the same kind of thing and that the first order of business was resolving that conflict. But you can never evaluate away the + in a + bi, and that bothered me. As did calling the b term "imaginary". I'd decided very early that all this stuff was really imaginary.
So I would have been much happier if the standard convention for complex numbers had always involved treating them as two-component tuples with some tidy purpose-built notation like (a, b) instead of what I saw as this cheap hack where the good nature of the addition operator is abused by kludging a non-commensurable-by-definition factor onto the right-hand term and just leaving it there. Ugh.
It wasn't until much later that I appreciated the elegance of the idea of having addition and multiplication operators defined implicitly and only by the legitimate ways in which they and their associated identities can be manipulated symbolically, thereby leaving the meaning of those operators completely open to interpretation by the user. There's a lot of useful mathematics accessible via that kind of approach to operators.
posted by flabdablet at 10:57 AM on June 3, 2023 [1 favorite]
The only thing that ever really bothered me about the whole business of complex numbers is the convention of using the addition operator to construct them. I had difficulty letting go of the idea that adding something to something else carried an implication that ultimately both of those somethings had to be things of the same kind.
Even algebraic expressions, where you're constantly writing expressions like 5x + 2y + 7 and have to be scrupulous not to mix up the x and y terms and the bare numbers, I saw as legitimate because both 5x and 2y are going to end up having the same kind of value as 7, at which point the addition just gets done and disappears.
I'd held on tightly to this mental model all the way through primary school, and it helped me not get as confused as many of my peers when working out stuff like ⅕ + ⅜. I wasn't even tempted to add the 1 and the 3 because it was clear to me that the two things either side of the + were not yet the same kind of thing and that the first order of business was resolving that conflict. But you can never evaluate away the + in a + bi, and that bothered me. As did calling the b term "imaginary". I'd decided very early that all this stuff was really imaginary.
So I would have been much happier if the standard convention for complex numbers had always involved treating them as two-component tuples with some tidy purpose-built notation like (a, b) instead of what I saw as this cheap hack where the good nature of the addition operator is abused by kludging a non-commensurable-by-definition factor onto the right-hand term and just leaving it there. Ugh.
It wasn't until much later that I appreciated the elegance of the idea of having addition and multiplication operators defined implicitly and only by the legitimate ways in which they and their associated identities can be manipulated symbolically, thereby leaving the meaning of those operators completely open to interpretation by the user. There's a lot of useful mathematics accessible via that kind of approach to operators.
posted by flabdablet at 10:57 AM on June 3, 2023 [1 favorite]
are we discovering Truth, or just wanking?
REJECT ALL FALSE DICHOTOMIES
posted by flabdablet at 11:03 AM on June 3, 2023 [6 favorites]
REJECT ALL FALSE DICHOTOMIES
posted by flabdablet at 11:03 AM on June 3, 2023 [6 favorites]
flug: Kronecker said that the natural numbers came from God, and all the rest from man ;)
god creates natural numbers
god creates dinosaurs
god destroys dinosaurs
god creates man
man destroys god
man creates real and imaginary numbers
posted by dismas at 12:07 PM on June 3, 2023 [4 favorites]
god creates natural numbers
god creates dinosaurs
god destroys dinosaurs
god creates man
man destroys god
man creates real and imaginary numbers
posted by dismas at 12:07 PM on June 3, 2023 [4 favorites]
Back in my calculator days, the square root of -1 was flashing ERROR.
posted by njohnson23 at 1:37 PM on June 3, 2023 [3 favorites]
posted by njohnson23 at 1:37 PM on June 3, 2023 [3 favorites]
The claim that the square root "function" only outputs "positive numbers" trivially restricts the function to the positive real line, because there is no such thing as a "positive" complex number.
But we can ask how to extend the square root function into the complex plane. The most common approach is to rewrite the complex number z = x + iy = r eiθ, where
Note that if you use the convention where a full turn is 2π radians, rather than 360º, a phase angle of a half-turn gives you the Euler equation eiπ = –1 (or eiπ + 1 = 0, which I've always thought was prettier). The part of your mathematical education where this relationship goes from baffling and impossible to beautiful is a really delightful transition. If it hasn't happened to you yet, you're in for a treat; there are a number of sources.
One definition of the square root function on positive real numbers is to raise that real number to the one-half power. If we choose this definition to expand into the complex plane, we can define the square root of a complex number as √z = z1/2 = r1/2 eiθ/2. So for any complex number z, we can find its square root by taking the root of its magnitude and then cutting in half the angle between z and the real axis. The square root function maps the complex plane onto the complex half-plane.
The ambiguity between the positive and negative square roots is related to the ambiguity in the definition of the phase angle θ. If you have a phase angle of a third of a turn (120º = 2π/3 radians), that's the same as a phase angle of two-thirds of a turn in the opposite direction, –240º = –2π • 2/3 radians. These two choices correspond to the choice between the "positive" and "negative" square roots of z. You can see this if you extend the phase ambiguity to the positive real axis itself. The line between the origin and a real number x can make a zero-degree angle with the real axis; but it can also make an angle of 360º = 2π radian with the real axis. The square root of z = x ei•2π is z1/2 = x1/2 eiπ = –x1/2.
As you keep going, you get more roots. There are four complex numbers for which z4 = 1: they are ±1 and ±i. These are known in the business as "the fourth roots of unity." There are five "fifth roots of unity," which are 1, ±ei•2π/5, and ±ei•2π•2/5. The way you generate these extra roots is to keep multiplying your favorite representation of z by ei•2π = 1.
posted by fantabulous timewaster at 2:15 PM on June 3, 2023 [4 favorites]
But we can ask how to extend the square root function into the complex plane. The most common approach is to rewrite the complex number z = x + iy = r eiθ, where
- the Euler number e is the limit of the sum of the inverses of the factorials of the natural numbers, e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + … = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + … ≈ 2.7183;
- the "magnitude" obeys r2 = x2 + y2;
- and the "phase angle" obeys cos θ = x and sin θ = y.
Note that if you use the convention where a full turn is 2π radians, rather than 360º, a phase angle of a half-turn gives you the Euler equation eiπ = –1 (or eiπ + 1 = 0, which I've always thought was prettier). The part of your mathematical education where this relationship goes from baffling and impossible to beautiful is a really delightful transition. If it hasn't happened to you yet, you're in for a treat; there are a number of sources.
One definition of the square root function on positive real numbers is to raise that real number to the one-half power. If we choose this definition to expand into the complex plane, we can define the square root of a complex number as √z = z1/2 = r1/2 eiθ/2. So for any complex number z, we can find its square root by taking the root of its magnitude and then cutting in half the angle between z and the real axis. The square root function maps the complex plane onto the complex half-plane.
The ambiguity between the positive and negative square roots is related to the ambiguity in the definition of the phase angle θ. If you have a phase angle of a third of a turn (120º = 2π/3 radians), that's the same as a phase angle of two-thirds of a turn in the opposite direction, –240º = –2π • 2/3 radians. These two choices correspond to the choice between the "positive" and "negative" square roots of z. You can see this if you extend the phase ambiguity to the positive real axis itself. The line between the origin and a real number x can make a zero-degree angle with the real axis; but it can also make an angle of 360º = 2π radian with the real axis. The square root of z = x ei•2π is z1/2 = x1/2 eiπ = –x1/2.
As you keep going, you get more roots. There are four complex numbers for which z4 = 1: they are ±1 and ±i. These are known in the business as "the fourth roots of unity." There are five "fifth roots of unity," which are 1, ±ei•2π/5, and ±ei•2π•2/5. The way you generate these extra roots is to keep multiplying your favorite representation of z by ei•2π = 1.
posted by fantabulous timewaster at 2:15 PM on June 3, 2023 [4 favorites]
> flug: Kronecker said that the natural numbers came from God, and all the rest from man ;)
Oh, I am very well aware of that.
Kronecker was completely nuts.
(On that particular topic, anyway.)
posted by flug at 4:34 PM on June 3, 2023 [1 favorite]
Oh, I am very well aware of that.
Kronecker was completely nuts.
(On that particular topic, anyway.)
posted by flug at 4:34 PM on June 3, 2023 [1 favorite]
The only thing that ever really bothered me about the whole business of complex numbers is the convention of using the addition operator to construct them. I had difficulty letting go of the idea that adding something to something else carried an implication that ultimately both of those somethings had to be things of the same kind.
Oh yeah. If i were to put my mathematician hat on 2, 7 and i are already complex numbers so adding and multiplying them like 2 + 7i is fine, but to my programmer brain there are implicit type coercions involved to make 2 + 7i typecheck.
posted by mscibing at 5:11 PM on June 3, 2023
Oh yeah. If i were to put my mathematician hat on 2, 7 and i are already complex numbers so adding and multiplying them like 2 + 7i is fine, but to my programmer brain there are implicit type coercions involved to make 2 + 7i typecheck.
posted by mscibing at 5:11 PM on June 3, 2023
mpark "Gil Strang talks about the advantages of linear algebra over calculus"
Thanks, I like it!
posted by mikeand1 at 5:24 PM on June 3, 2023
Thanks, I like it!
posted by mikeand1 at 5:24 PM on June 3, 2023
> The most imaginary number ever is zero. Hmm is it even a number?
Oh, don't even get me started on that one - another of my favorite hobby horses.
As an undergrad I wrote like a 25 page paper about the history of zero. The upshot is, much like people get hung up on the idea that "imaginary numbers can't really exist," they also get super hung up on the issue of "how can you have something that is nothing?"
Or as Robert Kaplan - who went a little beyond me to write a whole 240-page book about the history of zero - put it: It is all about "the human race's philosophical struggle with the idea of nothingness."
End result, we have examples of the use of what we would call the "digit zero" going back about as far as our earliest examples of numerical systems - somewhere around 5000 B.C.E. with the Babylonians, for example.
But even though you'll find various examples of using the equivalent of zero as a placeholder or in various other specific situations many times throughout the centuries, the recognition of zero as an actual number, with properties like other numbers, and belonging together in equality with other numbers like 1, 2, and 3, came surprisingly late.
Just for example, pretty much the first time we see an explanation for addition with the number zero (as opposed to rules about how to use it as a placeholder, etc) is in Brahmagupta's Brahmasputha Siddhanta , which dates to the 7th century. Just for example, it states that the sum of zero with itself as zero.
This was, literally, a breakthrough. It is hard to explain to what degree it was a breakthrough, because we have zero inculcated into our mathematical thinking from the earliest ages. Earlier systems, going back many thousands of years in different parts of the world, used a symbol for zero or a placeholder in some ways that made practical sense. But absolutely no one - to our knowledge - conceptualized 0 as a number on equal footing with the other, "natural" numbers.
To make that leap took, literally, a different philosophy of what "nothing" or zero was. And even with that in place, it was centuries more before the concept became a routine part of the number systems and calculations used in most places around the world.
The centuries, and philosophical struggles, it took to get a simple concept like zero to a place where everyone just uses it and accepts it as a normal, everyday part of life is really quite amazing.
I am now hopefully imagining a time, perhaps 5000 years in the future, where our newly renamed friendly numbers are considered everyday and perfect normal, and taught to every kid starting in preschool - just like zero is today.
posted by flug at 6:09 PM on June 3, 2023 [1 favorite]
Oh, don't even get me started on that one - another of my favorite hobby horses.
As an undergrad I wrote like a 25 page paper about the history of zero. The upshot is, much like people get hung up on the idea that "imaginary numbers can't really exist," they also get super hung up on the issue of "how can you have something that is nothing?"
Or as Robert Kaplan - who went a little beyond me to write a whole 240-page book about the history of zero - put it: It is all about "the human race's philosophical struggle with the idea of nothingness."
End result, we have examples of the use of what we would call the "digit zero" going back about as far as our earliest examples of numerical systems - somewhere around 5000 B.C.E. with the Babylonians, for example.
But even though you'll find various examples of using the equivalent of zero as a placeholder or in various other specific situations many times throughout the centuries, the recognition of zero as an actual number, with properties like other numbers, and belonging together in equality with other numbers like 1, 2, and 3, came surprisingly late.
Just for example, pretty much the first time we see an explanation for addition with the number zero (as opposed to rules about how to use it as a placeholder, etc) is in Brahmagupta's Brahmasputha Siddhanta , which dates to the 7th century. Just for example, it states that the sum of zero with itself as zero.
This was, literally, a breakthrough. It is hard to explain to what degree it was a breakthrough, because we have zero inculcated into our mathematical thinking from the earliest ages. Earlier systems, going back many thousands of years in different parts of the world, used a symbol for zero or a placeholder in some ways that made practical sense. But absolutely no one - to our knowledge - conceptualized 0 as a number on equal footing with the other, "natural" numbers.
To make that leap took, literally, a different philosophy of what "nothing" or zero was. And even with that in place, it was centuries more before the concept became a routine part of the number systems and calculations used in most places around the world.
The centuries, and philosophical struggles, it took to get a simple concept like zero to a place where everyone just uses it and accepts it as a normal, everyday part of life is really quite amazing.
I am now hopefully imagining a time, perhaps 5000 years in the future, where our newly renamed friendly numbers are considered everyday and perfect normal, and taught to every kid starting in preschool - just like zero is today.
posted by flug at 6:09 PM on June 3, 2023 [1 favorite]
> That said, we already have friendly numbers
Oh now, perhaps you don't know that every single everyday word is re-used at least 25 different times to name mathematical concepts of varying degrees of arcanity - preferably each one of them completely unrelated to all the others. In order to create maximum confusion.
Regardless, in a very short period of time, the new definition of friendly number we have developed here will be used a thousand times more commonly than any of the others. I have faith.
posted by flug at 6:18 PM on June 3, 2023 [1 favorite]
Oh now, perhaps you don't know that every single everyday word is re-used at least 25 different times to name mathematical concepts of varying degrees of arcanity - preferably each one of them completely unrelated to all the others. In order to create maximum confusion.
Regardless, in a very short period of time, the new definition of friendly number we have developed here will be used a thousand times more commonly than any of the others. I have faith.
posted by flug at 6:18 PM on June 3, 2023 [1 favorite]
Just to be clear, is a plate of beans one plate of beans, or is it a complex construct of a*plates + b*beans?
posted by Riki tiki at 6:21 PM on June 3, 2023
posted by Riki tiki at 6:21 PM on June 3, 2023
You want plates of imaginary beans? I'll give you plates of imaginary beans.
The centuries, and philosophical struggles, it took to get a simple concept like zero to a place where everyone just uses it and accepts it as a normal, everyday part of life is really quite amazing.
That would be because zero is not a simple concept.
The fundamental idea underpinning all of semantics is that of the referent: the thing to which something such as a word or idea refers. Computer programming has the related idea of a reference, usefully analogous here.
From our point of view as autonomous beings building our own understandings of an arbitrarily structured world, the referent of some expression that we use could potentially be anything (literally any thing). A computer processor's world is much more restricted: all of the things available to it are stored in some regular way in a machine-accessible memory, and a reference is about where some data value can be found as well as what kind of data it is.
Most references in computer programs take the form of names, which the programmer uses much as a writer of English uses nouns, and the issue of exactly where the processor is supposed to find the data referred to by a name is dealt with as part of the way the language is implemented. But some programming languages also include the idea of a pointer, where the data referred to by the pointer's name is itself a reference to some other piece of data elsewhere, data which usually never gets an explicit name of its own.
In order to work with the data referred to by a pointer, the pointer must first be dereferenced, a specific operation where its value gets used to locate the data that it points to. Dereferencing also has to happen in order to work with data identified by names, but there's usually no explicit syntax for that so it's usually not thought of as such.
The reason I bring this up is that the analogous operation in natural language is even more easily overlooked than the implicit dereferencing of names in computer programs. Many online discussions, for example, go completely off the rails when participants insist on behaving as if words and their referents were interchangeable.
But they're not. In fact a word in any human language works rather like the name of a pointer in a computer program, in that identifying the referent of a word involves two dereference operations: first from the word to an idea inside the mind of the person doing the identifying, and then from that idea to some identifiable thing in the world.
Natural languages work because we train each other to use the same words to refer to near-enough-to-the-same ideas. Failure to comprehend an unfamiliar language, or even a jargon, happens when we haven't shared enough of that training, so the first dereference fails often enough to make meaning irrecoverable. But we can't do anything about the fact that each of us has a unique library of ideas built up over years of unique experience. Sometimes the first dereference works just fine, but we end up talking past each other anyway when near-enough turns out not to be, and our second dereferences don't identify the same things.
Computer languages that support pointers almost always also allow the possibility of a null pointer: a pointer containing a value chosen in such a way that any attempt to dereference it will fail by design. Tony Hoare called this idea his billion dollar mistake but I've always rather liked it.
Dereferencing a null pointer is always a mistake and always yields invalid results. Just using a null pointer, though, is usually not and usually doesn't. The typical use of a null pointer is as a placeholder: a thing to fill in a spot where a valid pointer would normally go, but for which the data that such a pointer would otherwise reference doesn't exist.
For me, the word "nothing" is akin to the name of a null pointer. The idea that appears in my mind when I do the first dereference of "nothing" is a special one whose entire point is not to have a further referent of its own. The only purpose of this null idea is to flag attempts to dereference it as errors. The word "nothing", then, becomes a purely syntactical placeholder, a thing to drop into a spot where a noun would usually go in the service of expressing some larger idea.
This piece of interior design has stopped all rumination about the nature of nothing in and of itself dead in its tracks, replacing it with far more fruitful contemplations on the nature of assorted kinds of absences.
The thing that distinguishes absence from nothing is that any absence is inherently an absence of something. Any attempt to generalize the idea of absence in such a way as to remove that restriction would require it to refer to the absence of everything; to posit such an absence is to deny the very act of doing so, which makes the generalization unachievable. "Nothing" can serve as a useful syntactical placeholder for an absence, provided that the specific something that the absence is of can be inferred by context.
For example: it's been observed that a ham sandwich is better than complete happiness, on the basis that nothing is better than complete happiness and a ham sandwich is better than nothing. If "nothing" actually did have a referent, then this argument would be difficult to refute. But it doesn't; in the first clause it's being used as as placeholder for an absence of complete happiness, and in the second as a placeholder for an absence of food. They're not the same absence.
And at last he arrives at his point:
"0", like "nothing", is a placeholder. If it acquires any referent at all, it's doing so from context by standing in for the name of some specific absence, an idea which the reader's mind is then using to perform the second dereference.
This could be the absence of a result: if a - b = 0, then the difference between a and b is absent i.e. there isn't one. Could be things counted or measured using numbers: if I have zero apples, then I don't have any apples; I have an absence of apples.
In the specific case of positional notation for numerals, the things being counted by numbers are themselves numbers. 409 represents the sum of nine ones and four hundreds; the zero between the nine and the four denotes an absence of tens, which needs to be made explicit because the positional notation requires some digit to occupy the tens column to let the hundreds column exist. The zero is literally holding the tens' place because there aren't any tens.
But counting and measuring are not the only uses for numbers. They're also used as labels, whose salient properties are not their meanings but only their uniqueness and orderability. In that context, 0 can be as much a number as any other because anything can be a label without needing to be dereferenced.
Regardless of all of that, though, the best way to settle a mathematical dispute about whether zero is or is not a number is just to define it as one, or don't, and explore the consequences. Which is how, for example, we get a huge pile of useful theory pertaining to natural numbers, which don't include zero, and whole numbers, which do.
See? Nothing to it.
posted by flabdablet at 8:22 AM on June 4, 2023 [4 favorites]
The centuries, and philosophical struggles, it took to get a simple concept like zero to a place where everyone just uses it and accepts it as a normal, everyday part of life is really quite amazing.
That would be because zero is not a simple concept.
The fundamental idea underpinning all of semantics is that of the referent: the thing to which something such as a word or idea refers. Computer programming has the related idea of a reference, usefully analogous here.
From our point of view as autonomous beings building our own understandings of an arbitrarily structured world, the referent of some expression that we use could potentially be anything (literally any thing). A computer processor's world is much more restricted: all of the things available to it are stored in some regular way in a machine-accessible memory, and a reference is about where some data value can be found as well as what kind of data it is.
Most references in computer programs take the form of names, which the programmer uses much as a writer of English uses nouns, and the issue of exactly where the processor is supposed to find the data referred to by a name is dealt with as part of the way the language is implemented. But some programming languages also include the idea of a pointer, where the data referred to by the pointer's name is itself a reference to some other piece of data elsewhere, data which usually never gets an explicit name of its own.
In order to work with the data referred to by a pointer, the pointer must first be dereferenced, a specific operation where its value gets used to locate the data that it points to. Dereferencing also has to happen in order to work with data identified by names, but there's usually no explicit syntax for that so it's usually not thought of as such.
The reason I bring this up is that the analogous operation in natural language is even more easily overlooked than the implicit dereferencing of names in computer programs. Many online discussions, for example, go completely off the rails when participants insist on behaving as if words and their referents were interchangeable.
But they're not. In fact a word in any human language works rather like the name of a pointer in a computer program, in that identifying the referent of a word involves two dereference operations: first from the word to an idea inside the mind of the person doing the identifying, and then from that idea to some identifiable thing in the world.
Natural languages work because we train each other to use the same words to refer to near-enough-to-the-same ideas. Failure to comprehend an unfamiliar language, or even a jargon, happens when we haven't shared enough of that training, so the first dereference fails often enough to make meaning irrecoverable. But we can't do anything about the fact that each of us has a unique library of ideas built up over years of unique experience. Sometimes the first dereference works just fine, but we end up talking past each other anyway when near-enough turns out not to be, and our second dereferences don't identify the same things.
Computer languages that support pointers almost always also allow the possibility of a null pointer: a pointer containing a value chosen in such a way that any attempt to dereference it will fail by design. Tony Hoare called this idea his billion dollar mistake but I've always rather liked it.
Dereferencing a null pointer is always a mistake and always yields invalid results. Just using a null pointer, though, is usually not and usually doesn't. The typical use of a null pointer is as a placeholder: a thing to fill in a spot where a valid pointer would normally go, but for which the data that such a pointer would otherwise reference doesn't exist.
For me, the word "nothing" is akin to the name of a null pointer. The idea that appears in my mind when I do the first dereference of "nothing" is a special one whose entire point is not to have a further referent of its own. The only purpose of this null idea is to flag attempts to dereference it as errors. The word "nothing", then, becomes a purely syntactical placeholder, a thing to drop into a spot where a noun would usually go in the service of expressing some larger idea.
This piece of interior design has stopped all rumination about the nature of nothing in and of itself dead in its tracks, replacing it with far more fruitful contemplations on the nature of assorted kinds of absences.
The thing that distinguishes absence from nothing is that any absence is inherently an absence of something. Any attempt to generalize the idea of absence in such a way as to remove that restriction would require it to refer to the absence of everything; to posit such an absence is to deny the very act of doing so, which makes the generalization unachievable. "Nothing" can serve as a useful syntactical placeholder for an absence, provided that the specific something that the absence is of can be inferred by context.
For example: it's been observed that a ham sandwich is better than complete happiness, on the basis that nothing is better than complete happiness and a ham sandwich is better than nothing. If "nothing" actually did have a referent, then this argument would be difficult to refute. But it doesn't; in the first clause it's being used as as placeholder for an absence of complete happiness, and in the second as a placeholder for an absence of food. They're not the same absence.
And at last he arrives at his point:
"0", like "nothing", is a placeholder. If it acquires any referent at all, it's doing so from context by standing in for the name of some specific absence, an idea which the reader's mind is then using to perform the second dereference.
This could be the absence of a result: if a - b = 0, then the difference between a and b is absent i.e. there isn't one. Could be things counted or measured using numbers: if I have zero apples, then I don't have any apples; I have an absence of apples.
In the specific case of positional notation for numerals, the things being counted by numbers are themselves numbers. 409 represents the sum of nine ones and four hundreds; the zero between the nine and the four denotes an absence of tens, which needs to be made explicit because the positional notation requires some digit to occupy the tens column to let the hundreds column exist. The zero is literally holding the tens' place because there aren't any tens.
But counting and measuring are not the only uses for numbers. They're also used as labels, whose salient properties are not their meanings but only their uniqueness and orderability. In that context, 0 can be as much a number as any other because anything can be a label without needing to be dereferenced.
Regardless of all of that, though, the best way to settle a mathematical dispute about whether zero is or is not a number is just to define it as one, or don't, and explore the consequences. Which is how, for example, we get a huge pile of useful theory pertaining to natural numbers, which don't include zero, and whole numbers, which do.
See? Nothing to it.
posted by flabdablet at 8:22 AM on June 4, 2023 [4 favorites]
...one of the most amazing bits of mathematical trivia that exists: e^(i pi)+1=0
Even better,
eⁱⁿ + 2ϕ = √5
where phi is the golden ratio
posted by thatwhichfalls at 9:17 AM on June 4, 2023
Even better,
eⁱⁿ + 2ϕ = √5
where phi is the golden ratio
posted by thatwhichfalls at 9:17 AM on June 4, 2023
I guarantee, as soon as you spend a few years thinking of them as friendly numbers, all your doubts and trouble with them will vanish in an instant.
C’mon.
posted by mhoye at 1:03 PM on June 4, 2023
C’mon.
posted by mhoye at 1:03 PM on June 4, 2023
This piece of interior design has stopped all rumination about the nature of nothing in and of itself dead in its tracks
Martin Heidegger rolls in grave, shouts "The nothing noths!"
posted by Pyrogenesis at 11:42 PM on June 4, 2023
Martin Heidegger rolls in grave, shouts "The nothing noths!"
posted by Pyrogenesis at 11:42 PM on June 4, 2023
The thing about Heidegger and me, though, is that he's dead and I'm not. So if I were to run this line of argument by him, he'd find nothing to disagree with.
posted by flabdablet at 4:19 AM on June 5, 2023
posted by flabdablet at 4:19 AM on June 5, 2023
Oh now, perhaps you don't know that every single everyday word is re-used at least 25 different times to name mathematical concepts of varying degrees of arcanity
My favorite usage of the term described ("normal") is in a pair of adjectival phrases, one from differential geometry ("unit normal") and one from linear algebra ("orthonormal"). Those two phrases both have, broadly speaking, the same meaning ("length one and perpendicular to something"), but they arrive at that near-identical meaning through different meanings for the term they have in common. In differential geometry, "unit" is short for "of unit length" (i.e., length one), and "normal" means "perpendicular". In linear algebra, something is "normal" when its magnitude (length, or similar concepts for non-linear structures) has been scaled down to the standard size of 1, and the "ortho-" prefix, a truncation of "orthogonal", signifies being at a right angle to something.
But even though you'll find various examples of using the equivalent of zero as a placeholder or in various other specific situations many times throughout the centuries, the recognition of zero as an actual number, with properties like other numbers, and belonging together in equality with other numbers like 1, 2, and 3, came surprisingly late.
Yeah, there is a pragmatic issue and a philosophical issue. In a place-value system (which is awfully useful for arithmetic), there's obviously a need to align numbers so that it's clear whether a particular occurrence of the digit "4" is four hundreds, or tens*, or ones, or what. A lot of early calculation systems just wrote the numbers in grids and left a blank space. Only when place-value numerals were communicated inline among text instead of being worked out in a grid did it become apparent that, hey, you need something here so readers know which digit is which. But even then it felt like, and continues to feel like, a convention of presentation rather than an important idea in its own right. You read 427 as "four hundreds, two tens, and seven ones" easily enough, but it's more natural to think of "407" as "four hundreds and seven ones" than as "four hundreds, no tens, and seven ones", because the things that are not present are generally regarded as unimportant: yeah, 407 doesn't have any tens, but it doesn't have any thousands or aardvarks either, and nobody feels the need to go out of their way to clarify that matter when reading the number out. Enumerating all the things not present in the number 407 doesn't seem like it helps you understand what 407 is. So for that reason, introducing "0" as a numeral didn't really incline anyone to think of zero as a number, because you'd have to get over that philosophical hump of thinking of absence as significant.
*Obviously, this is taking a modern, base-ten approach to the notion here, which were notably not used by many of the place-value systems preceding the Indo-Arabic one. Mesopotamian systems were 60-based, Mesoamerican 20-based. The placeholder problem still holds, though.
posted by jackbishop at 6:11 AM on June 5, 2023 [3 favorites]
My favorite usage of the term described ("normal") is in a pair of adjectival phrases, one from differential geometry ("unit normal") and one from linear algebra ("orthonormal"). Those two phrases both have, broadly speaking, the same meaning ("length one and perpendicular to something"), but they arrive at that near-identical meaning through different meanings for the term they have in common. In differential geometry, "unit" is short for "of unit length" (i.e., length one), and "normal" means "perpendicular". In linear algebra, something is "normal" when its magnitude (length, or similar concepts for non-linear structures) has been scaled down to the standard size of 1, and the "ortho-" prefix, a truncation of "orthogonal", signifies being at a right angle to something.
But even though you'll find various examples of using the equivalent of zero as a placeholder or in various other specific situations many times throughout the centuries, the recognition of zero as an actual number, with properties like other numbers, and belonging together in equality with other numbers like 1, 2, and 3, came surprisingly late.
Yeah, there is a pragmatic issue and a philosophical issue. In a place-value system (which is awfully useful for arithmetic), there's obviously a need to align numbers so that it's clear whether a particular occurrence of the digit "4" is four hundreds, or tens*, or ones, or what. A lot of early calculation systems just wrote the numbers in grids and left a blank space. Only when place-value numerals were communicated inline among text instead of being worked out in a grid did it become apparent that, hey, you need something here so readers know which digit is which. But even then it felt like, and continues to feel like, a convention of presentation rather than an important idea in its own right. You read 427 as "four hundreds, two tens, and seven ones" easily enough, but it's more natural to think of "407" as "four hundreds and seven ones" than as "four hundreds, no tens, and seven ones", because the things that are not present are generally regarded as unimportant: yeah, 407 doesn't have any tens, but it doesn't have any thousands or aardvarks either, and nobody feels the need to go out of their way to clarify that matter when reading the number out. Enumerating all the things not present in the number 407 doesn't seem like it helps you understand what 407 is. So for that reason, introducing "0" as a numeral didn't really incline anyone to think of zero as a number, because you'd have to get over that philosophical hump of thinking of absence as significant.
*Obviously, this is taking a modern, base-ten approach to the notion here, which were notably not used by many of the place-value systems preceding the Indo-Arabic one. Mesopotamian systems were 60-based, Mesoamerican 20-based. The placeholder problem still holds, though.
posted by jackbishop at 6:11 AM on June 5, 2023 [3 favorites]
I think it's quite arguable that treating zero as a number is exactly what leads to the frequently expressed confusion about what value is really represented by constructions like 1/0 or (worse still) 0/0.
The standard answer that these things are undefined strikes many as some kind of cop-out. People insist on seeking "solutions" that justify some definition or other, based on procedures like evaluating 1/x or x/x as x gets smaller, as if 0 really were a quantity that some other quantity could get "sufficiently close to" in some presumed-absolute sense.
As a matter of practical experience, the distinction between an imperceptible amount of $THING and an absence of $THING is indeed inconsequential most of the time, so this kind of thinking should come as no surprise. But division and multiplication are all about scale, and scale is always relative; there is no absolute scale to the real numbers, as a matter of deliberate design.
There may well be something like an absolute scale to the actual world, if Planck is to be believed. Not to the measuring numbers we've designed to model that world with, though, which are every bit as capable of endlessly subdividing the representation of measurements on the Planck scale as on any other.
So even if zero is thought of as a number, I'm only ever going put it "on an equal footing with" others when it's used purely as a label. In any other case, considering zero to be the absence of a number makes much more sense to me.
It also eliminates all those pesky aardvarks.
posted by flabdablet at 7:09 AM on June 5, 2023 [1 favorite]
The standard answer that these things are undefined strikes many as some kind of cop-out. People insist on seeking "solutions" that justify some definition or other, based on procedures like evaluating 1/x or x/x as x gets smaller, as if 0 really were a quantity that some other quantity could get "sufficiently close to" in some presumed-absolute sense.
As a matter of practical experience, the distinction between an imperceptible amount of $THING and an absence of $THING is indeed inconsequential most of the time, so this kind of thinking should come as no surprise. But division and multiplication are all about scale, and scale is always relative; there is no absolute scale to the real numbers, as a matter of deliberate design.
There may well be something like an absolute scale to the actual world, if Planck is to be believed. Not to the measuring numbers we've designed to model that world with, though, which are every bit as capable of endlessly subdividing the representation of measurements on the Planck scale as on any other.
So even if zero is thought of as a number, I'm only ever going put it "on an equal footing with" others when it's used purely as a label. In any other case, considering zero to be the absence of a number makes much more sense to me.
It also eliminates all those pesky aardvarks.
posted by flabdablet at 7:09 AM on June 5, 2023 [1 favorite]
they arrive at that near-identical meaning through different meanings for the term they have in common.
I'd like to take a moment to acknowledge how absurdly pleasing I find that fact.
Thanks, jackbishop, for pointing it out!
posted by flabdablet at 7:13 AM on June 5, 2023
I'd like to take a moment to acknowledge how absurdly pleasing I find that fact.
Thanks, jackbishop, for pointing it out!
posted by flabdablet at 7:13 AM on June 5, 2023
Trees, bears, mooses, and vultures are real concrete objects that exist in the objective world.
Chairs, however, don't exist. Actually this may have wider implications..
posted by FatherDagon at 12:51 PM on June 6, 2023
Chairs, however, don't exist. Actually this may have wider implications..
posted by FatherDagon at 12:51 PM on June 6, 2023
« Older It's Donut Day! Which is your region? WaPo says... | None More Pink Newer »
This thread has been archived and is closed to new comments
For more detail on the history in a digestible video format.
posted by lock robster at 11:43 AM on June 2, 2023 [2 favorites]