![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
Today I went out and bought Coraline and read it, because that’s what I do. It was a very good book, except for the hour when it vanished on me, eventually turning up on my desk, where I’d left it. (I expect it was off conversing with my copy of Milton, and I hope they became quick enough friends that I can lure it back with threats of a hostage situation.)
There are times when you’re walking in the fog and you can’t see your destination, and you’re given to wonder if maybe the world has forgotten that it’s supposed to be there. I don’t know if that was the feeling Neil Gaiman was going for; in any case, it was excellent.
But this is not news, or even what I wanted to say.
While I was reading, I became very bemused by a thought: it’s like math. Which it was, of course, because I had hit the part where they walked off in a straight line and ended up where they started. It was like trying to walk in a line on a sphere – there were too many dimensions.
It’s like math, I said to myself. This was somewhat abstract, so I clarified. Math is *scary*.
Well, yes, replied my brain. I was there when you took it. I remember.
Oh, I didn’t mean it like that, I said. I mean, it makes you think.
…, said my brain, not amused (and, I fear, vaguely insulted).
What I mean is this: the best fantasy, the stuff that gives me goosebumps, is never just about people with wings or mindreading or spacecraft, but about the nature of reality. And what I mean, as I tried to explain to myself, is this: whatever little thrill we suffer by suspecting the universe isn’t quite what it should be, mathematics can match.
And I mean not only that staring into the abyss of infinity can drive a man mad.
The first problem with math is probably fairly obvious - draw a picture of 4. (And I don’t mean the symbol ‘4’ or the letters ‘four.’ That’s cheating.)
Chances are, you drew four little circles, or four little squares, or balloons or trees or neckties. To illustrate 4 to a child, we have to do it indirectly, by having many sets of other things and suggesting that ‘4’ is what they all have in common. A number like 4 seems as solid to us as anything in this world, but all the same, it isn’t real - it’s an idea. You can’t hold 4 or describe 4 in it’s own right, and it’s much too concrete an idea to become a god. Even mathematicians worship more exotic numbers.
But you know what 4 is, and what it means, and for all we know of the universe, you seem to be right.
The opposite of four is, of course, negative four. Or -4, if you prefer. So draw a picture of -4? This one’s a little more tricky, because now the whole point is that the squares or circles or neckties aren’t there. Not only are they not there, they’re actively missing, so one might imagine four necktie-shaped black holes in the universe, or four white holes on a white paper, or four anti-matter neckties which explode when they hit the real things and leave nothing. But we know what -4 is too: it’s just 4 in the opposite direction.
The negative numbers seem, at the simplest level, to behave just like their positive counterpart only backwards. There are some differences, but these come into play in more subtle, or at least more complicated, ways. (And who hasn’t heard this fairy tale, in words? What should one find on the other side of the mirror? Exposed in the negatives of a roll of film?)
Now, we know that the square root of 16 must be 4, on account of 4 times 4 being 16, but we become slightly concerned when asked what the square root of -16 should be, because -4 times itself equals 16 as well. So, negative numbers may seem the reflection – flat and colourless in comparison to the real, whole numbers.
Except, as math becomes deeper, the lack of a square root for the negative numbers becomes more difficult to work around. So we imagine there must be another number-line, comprised of numbers which, when multiplied together, result in negative products. These would be, then, the imaginary numbers, denoted by i. (And a world analogous to the imaginary number-line, one imagines, would be very interesting indeed.)
So 4 is only real in the sense that we think it is, and -4 is only real in the sense that we think it isn’t, and 4i is only real in the sense that we need it to be. But this is interesting, not frightening.
And the frightening part isn’t zero, which can also be considered the opposite of four, because it’s the opposite of everything. 0 is the pivot point of math, the origin, the point where the positive and negative number-lines join, the reason that Romans would have never invented computers. It defines all math by its nothingness. (And this is a story too, because what meaning has something without the possibility of nothing?)
No, the unnerving moments in math come later, when you look at the constants such as e and π and realize that these are the important things, and numbers such as 4 and 10 or 16 are the arbitrary ones. You can have an infinite line that encloses a finite area on a graph, or a shape that no matter how impossibly close you examine it seems to be a circle – but isn’t, because the circumference is drawn with triangles that keep breaking. The inside and the outside of a Mobius strip are the same side, and that suggests worrying structure possibilities if one considers that we live in more than three dimensions. There are functions that become angry and jagged when their derivatives are found, and functions that smooth to planes, and a couple that aren’t affected at all – and most of the time you can almost get them back, though scarred with constants.
And you learn that, for a continuous situation, the probability of any particular thing happening is somewhere around exactly zero… and you glance at the angle of a particular leaf in a particular tree or notice the colour of the car parked beside you or remember the precise amount of sugar you put in your coffee that morning – and suddenly the floor of reality doesn’t seem so solid.
Because it’s math, they say, that ties the universe together. And every math major in the world is doing their best to find those knots and unravel them to see where they go. (One suspects that if they ever do succeed, we will have much more reason to be afraid.)
But mathematicians have a secret, and it’s this: no matter how good they might be at manipulating numbers and inventing formulas and doing partial integration, they don’t understand any of it. They can tell you the rules, but they don’t know why they’re there. Try it. The next time you’re in math class, or calculus study or a lecture on matrices, ask your professor the most simple question in the world.
“…When n is negative, x to the n should be rewritten with respect to cosine,” your Professor will say. “Just a moment, we have a question?”
“Yes,” you will reply. “Why is two plus two four?”
And your Professor will glare balefully at you for a minute, and then make jokes at your expense and perhaps redirect you to the philosophy classroom. But you may feel better to know that all this is really just a front for their ignorance. It just is! is not a very satisfying answer, and your professor knows this (but probably, you should be warned, doesn’t care unless it’s in regard to proofs).
I suppose that any way of looking at reality can be unnerving, really. It’s just that literature and physics get most of the attention, and I think math deserves some too. There is a romance in math, wherein the numbers become more real than the world, that is akin to the romance in fine art or books, but less people tend to look for it.
There are times when you’re walking in the fog and you can’t see your destination, and you’re given to wonder if maybe the world has forgotten that it’s supposed to be there. I don’t know if that was the feeling Neil Gaiman was going for; in any case, it was excellent.
But this is not news, or even what I wanted to say.
While I was reading, I became very bemused by a thought: it’s like math. Which it was, of course, because I had hit the part where they walked off in a straight line and ended up where they started. It was like trying to walk in a line on a sphere – there were too many dimensions.
It’s like math, I said to myself. This was somewhat abstract, so I clarified. Math is *scary*.
Well, yes, replied my brain. I was there when you took it. I remember.
Oh, I didn’t mean it like that, I said. I mean, it makes you think.
…, said my brain, not amused (and, I fear, vaguely insulted).
What I mean is this: the best fantasy, the stuff that gives me goosebumps, is never just about people with wings or mindreading or spacecraft, but about the nature of reality. And what I mean, as I tried to explain to myself, is this: whatever little thrill we suffer by suspecting the universe isn’t quite what it should be, mathematics can match.
And I mean not only that staring into the abyss of infinity can drive a man mad.
The first problem with math is probably fairly obvious - draw a picture of 4. (And I don’t mean the symbol ‘4’ or the letters ‘four.’ That’s cheating.)
Chances are, you drew four little circles, or four little squares, or balloons or trees or neckties. To illustrate 4 to a child, we have to do it indirectly, by having many sets of other things and suggesting that ‘4’ is what they all have in common. A number like 4 seems as solid to us as anything in this world, but all the same, it isn’t real - it’s an idea. You can’t hold 4 or describe 4 in it’s own right, and it’s much too concrete an idea to become a god. Even mathematicians worship more exotic numbers.
But you know what 4 is, and what it means, and for all we know of the universe, you seem to be right.
The opposite of four is, of course, negative four. Or -4, if you prefer. So draw a picture of -4? This one’s a little more tricky, because now the whole point is that the squares or circles or neckties aren’t there. Not only are they not there, they’re actively missing, so one might imagine four necktie-shaped black holes in the universe, or four white holes on a white paper, or four anti-matter neckties which explode when they hit the real things and leave nothing. But we know what -4 is too: it’s just 4 in the opposite direction.
The negative numbers seem, at the simplest level, to behave just like their positive counterpart only backwards. There are some differences, but these come into play in more subtle, or at least more complicated, ways. (And who hasn’t heard this fairy tale, in words? What should one find on the other side of the mirror? Exposed in the negatives of a roll of film?)
Now, we know that the square root of 16 must be 4, on account of 4 times 4 being 16, but we become slightly concerned when asked what the square root of -16 should be, because -4 times itself equals 16 as well. So, negative numbers may seem the reflection – flat and colourless in comparison to the real, whole numbers.
Except, as math becomes deeper, the lack of a square root for the negative numbers becomes more difficult to work around. So we imagine there must be another number-line, comprised of numbers which, when multiplied together, result in negative products. These would be, then, the imaginary numbers, denoted by i. (And a world analogous to the imaginary number-line, one imagines, would be very interesting indeed.)
So 4 is only real in the sense that we think it is, and -4 is only real in the sense that we think it isn’t, and 4i is only real in the sense that we need it to be. But this is interesting, not frightening.
And the frightening part isn’t zero, which can also be considered the opposite of four, because it’s the opposite of everything. 0 is the pivot point of math, the origin, the point where the positive and negative number-lines join, the reason that Romans would have never invented computers. It defines all math by its nothingness. (And this is a story too, because what meaning has something without the possibility of nothing?)
No, the unnerving moments in math come later, when you look at the constants such as e and π and realize that these are the important things, and numbers such as 4 and 10 or 16 are the arbitrary ones. You can have an infinite line that encloses a finite area on a graph, or a shape that no matter how impossibly close you examine it seems to be a circle – but isn’t, because the circumference is drawn with triangles that keep breaking. The inside and the outside of a Mobius strip are the same side, and that suggests worrying structure possibilities if one considers that we live in more than three dimensions. There are functions that become angry and jagged when their derivatives are found, and functions that smooth to planes, and a couple that aren’t affected at all – and most of the time you can almost get them back, though scarred with constants.
And you learn that, for a continuous situation, the probability of any particular thing happening is somewhere around exactly zero… and you glance at the angle of a particular leaf in a particular tree or notice the colour of the car parked beside you or remember the precise amount of sugar you put in your coffee that morning – and suddenly the floor of reality doesn’t seem so solid.
Because it’s math, they say, that ties the universe together. And every math major in the world is doing their best to find those knots and unravel them to see where they go. (One suspects that if they ever do succeed, we will have much more reason to be afraid.)
But mathematicians have a secret, and it’s this: no matter how good they might be at manipulating numbers and inventing formulas and doing partial integration, they don’t understand any of it. They can tell you the rules, but they don’t know why they’re there. Try it. The next time you’re in math class, or calculus study or a lecture on matrices, ask your professor the most simple question in the world.
“…When n is negative, x to the n should be rewritten with respect to cosine,” your Professor will say. “Just a moment, we have a question?”
“Yes,” you will reply. “Why is two plus two four?”
And your Professor will glare balefully at you for a minute, and then make jokes at your expense and perhaps redirect you to the philosophy classroom. But you may feel better to know that all this is really just a front for their ignorance. It just is! is not a very satisfying answer, and your professor knows this (but probably, you should be warned, doesn’t care unless it’s in regard to proofs).
I suppose that any way of looking at reality can be unnerving, really. It’s just that literature and physics get most of the attention, and I think math deserves some too. There is a romance in math, wherein the numbers become more real than the world, that is akin to the romance in fine art or books, but less people tend to look for it.
Tags:
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
From:
no subject
I really enjoyed reading that, and suspect I shall be mulling over some of your points for the next couple of days.
From:
no subject
Shrugs. And, hey, you're a mathematician, eh?
From:
no subject
And I'm just pretending to be a mathematician until someone finds me out. :)
From:
no subject
That, I think, would be really interesting to see... or at least to get someone to explain the just of.
But, I don't know. (And it's possible I really don't know what I'm talking about.) It's always seemed to me that they can show that 2 and 2 is 4, in a variety of creative ways, but that the whole why issue is caught up in the way the universe works.
And who says the universe has to be coherent, when you get down to it, except for us?