![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
Quaero: Principia Mathematica Animata?
So I've had this idea running through my head recently: taking the images in my head which encode my understanding of mathematics, and either writing the down or (better) animating them.
If nothing else it might provide a useful resource for students who think as visually as I do, or even for those who have difficulty decoding the static drawings and too often stilted and/or formal descriptions which are a plague upon those who have language difficulties. (It would still have a voice-over, but if it were done really well, it might not even need one.)
[a black screen, completely and utterly black.]
This is the void. An infinity of nothing. You can go for as far as you like, and you may as well have gone nowhere, because there's nothing to compare your movement to. Everywhere is identical to everywhere else. For all you know, you're travelling faster than light. But there's not even light to compare against. It is completely full of absolutely nothing.
Good thing this place doesn't exist. It's a mental model: a place which exists in our minds so that we can play in it. So in one way, this infinite nothing is very exciting, because it allows us to play with ideas. But right now, there's only nothing to play with.
Let's add something. The smallest something we can add.
[*pling!* a dot appears slightly off-center]
This is a point. It has zero dimensions: it has no height at all. It has no depth at all. It has no width at all. We've just made it look bigger, so we can see it. It is scale-invariant, which means that it doesn't matter how close or far away you are, it's still just a point. For all you know, we're zooming in on it now. There's no way to tell. But...
[dot starts to move gently around the screen]
... now we can tell if we are moving around, because we now have an anchor: a reference to measure against. Or is it the one that's moving? It doesn't matter right now, because wherever it's moving are the same, so we may as well just say that it's fixed, and we were moving.
[dot slows and stops moving.]
But still, it's not very interesting. Let's make our universe a bit more complicated.
[*pling!* another dot appears]
Now we have two dots. Already there's more information in the Universe. For a start, scale now matters. We can zoom out...
[dots move close together]
... and in...
[dots more further apart]
... and be able to tell. (Or are they moving closer and farther from each other? Again, let's assume that they're fixed in place and we're moving.)
Now, there are lots of things you can do with two dots. For a start, you can make a line between them. Find a point halfway between the two...
[*pling!*]
... then halfway between each point,...
[*pling!*]
... and again, and again, and again ...
[*pling!* *pling!* *pling!* *pling! pling! plingplingplinplinplipliplplpppppppppp* as the line fills in]
... until it's filled in all the way. Or is it? Because each of these points has no length at all, how can we fill in a line with them? It would take a infinite number of them, no matter how long that line is.
[line expands to fill the screen (ie., the ends expand off the sides.)]
Otherwise there would always be a gap between any point and the next along.
[line breaks into even spaced points again.]
Remember, this is an imaginary place, and physics doesn't apply unless we tell it to. These points are pure things, not made up of atoms. It wouldn't take long in Reality for the gap between points to be much, much smaller than the quarks which make up the particles which make up atoms. But here, there is no smallest limit, and there is always a gap. We have to make a leap to get from an infinite number of points to a line. A bit of mental effort,...
[line fills in, and zooms out to show the ends again]
and it's a continuous line.
Or,...
[line fades away, leaving the end points again]
... we could go the other way. Take the distance and direction from one point to the other, and then put a new point that far away on the other side.
[p3-p2 = p2-p1: *pling!*]
And do that again.
[*pling!*]
And again, and so on. In both directions.
[a line of evenly spaced dots, racing away to the conceptual horizon]
Now, this line also goes on forever. It doesn't matter how far you go along it, there's always room for another point.
[slowly, then faster, zooming along the line towards the horizon, then slow and stop]
And because it goes forever in both directions, there's no center. So any point at all could be the center: it doesn't matter. There's still an infinite number of points on both sides.
Let's just pick this one and call it "Zero".
[a point has "0" appear above it.]
Now we have to consider two concepts: we can count the points, or we can count the distances between them. To start with, let's consider the space between the points, so that this...
[an arrow travels from "0" to the next point, and stops with the arrowhead at the point]
we can call this "One".
["1" appears above the second point.]
But the thing is, if we have "Zero", and we have "One", then all the other integers follow. One plus One...
[a ghost of the arrow from "0" to "1" moves to between "1" and the next point. "2" appears above that point]
... equals "Two". Two plus One equals Three. And so on.
[Arrow moves to create "3". Numbers appear above the points: "4", "5", "6" ... until you can't read them any more.]
But wait... there's a complete infinity still waiting, in the other direction.
[pan to the points on the other side of "0"]
If you can move in one direction, you can move in the other direction. There must be a number which, when you add One, you get Zero. It's like One, but it's its mirror. Let's call the numbers on this side "negative", and mark them thus:
[the arrow from "0" to "1" ghosts and shifts until its tip is resting on "0". Above the other end appears "-1"]
And we can replicate this in the other direction.
["-2", "-3", "-4", ... and so on.]
We've just invented the Integers. And there's an infinity of them. Start from Zero and start going, and you will never get to the end. There is no "last" integer. You can always add One.
But wait, there are all those gaps. Didn't we fill them?
[the line between "0" and "1" reappears.]
So now we have this continuous line. And because the difference between the space between Zero and One and and other point and the next is because we arbitrarily pointed to one and called it "Zero" and nothing else, then we can fill in every other space as well. All of them at once.
[Line fills in, with blips for the points, and numbers remain above]
This is the Real number line.
Now think about this: the space between any integer and the next is filled with an infinite number of points: whatever number you're thinking of, it's more. and there are an infinite number of these spaces -- as there are an infinite number of integers. So the number of points on the Real number line must be an infinity of infinities... it's an infinity which is bigger than the infinity we first thought of.
We started from absolutely nothing, and created two levels of infinity, just by inventing Numbers.
...
So, we start from the first couple of Euclid's Axioms, work our way through Peano, and take a large helping of Cantor on the way. And we're still in one dimension. I haven't started talking about circles yet. In this first step, I'd continue by talking about the difference between ordinals (the number as a vector) and cardinals (the count of points), and how Zero is different, because every other number is a vector, except for Zero, which is the point from which all vectors are measured: which is how all numbers have a magnitude except for Zero. Then extrapolating from addition to multiplication/division as a function of constructive ratios (ie., 4.5 * 8 => 1:4.5 :: 8:?), and by extension why division by Zero makes no sense.
Sure, it skips and jumps around thousands of years of mathematical history, but I'm trying to lay out the concepts in an order which logically follows, one idea from another. It's difficult, because it's all really a net of interconnected ideas, and making any given path is to some extent arbitrary.
I've run through some of these ideas with Miss A, and her head didn't spontaneously burst into flame... although I'm not sure how much of it she understood.
Do people think this is worth putting effort into?
If nothing else it might provide a useful resource for students who think as visually as I do, or even for those who have difficulty decoding the static drawings and too often stilted and/or formal descriptions which are a plague upon those who have language difficulties. (It would still have a voice-over, but if it were done really well, it might not even need one.)
[a black screen, completely and utterly black.]
This is the void. An infinity of nothing. You can go for as far as you like, and you may as well have gone nowhere, because there's nothing to compare your movement to. Everywhere is identical to everywhere else. For all you know, you're travelling faster than light. But there's not even light to compare against. It is completely full of absolutely nothing.
Good thing this place doesn't exist. It's a mental model: a place which exists in our minds so that we can play in it. So in one way, this infinite nothing is very exciting, because it allows us to play with ideas. But right now, there's only nothing to play with.
Let's add something. The smallest something we can add.
[*pling!* a dot appears slightly off-center]
This is a point. It has zero dimensions: it has no height at all. It has no depth at all. It has no width at all. We've just made it look bigger, so we can see it. It is scale-invariant, which means that it doesn't matter how close or far away you are, it's still just a point. For all you know, we're zooming in on it now. There's no way to tell. But...
[dot starts to move gently around the screen]
... now we can tell if we are moving around, because we now have an anchor: a reference to measure against. Or is it the one that's moving? It doesn't matter right now, because wherever it's moving are the same, so we may as well just say that it's fixed, and we were moving.
[dot slows and stops moving.]
But still, it's not very interesting. Let's make our universe a bit more complicated.
[*pling!* another dot appears]
Now we have two dots. Already there's more information in the Universe. For a start, scale now matters. We can zoom out...
[dots move close together]
... and in...
[dots more further apart]
... and be able to tell. (Or are they moving closer and farther from each other? Again, let's assume that they're fixed in place and we're moving.)
Now, there are lots of things you can do with two dots. For a start, you can make a line between them. Find a point halfway between the two...
[*pling!*]
... then halfway between each point,...
[*pling!*]
... and again, and again, and again ...
[*pling!* *pling!* *pling!* *pling! pling! plingplingplinplinplipliplplpppppppppp* as the line fills in]
... until it's filled in all the way. Or is it? Because each of these points has no length at all, how can we fill in a line with them? It would take a infinite number of them, no matter how long that line is.
[line expands to fill the screen (ie., the ends expand off the sides.)]
Otherwise there would always be a gap between any point and the next along.
[line breaks into even spaced points again.]
Remember, this is an imaginary place, and physics doesn't apply unless we tell it to. These points are pure things, not made up of atoms. It wouldn't take long in Reality for the gap between points to be much, much smaller than the quarks which make up the particles which make up atoms. But here, there is no smallest limit, and there is always a gap. We have to make a leap to get from an infinite number of points to a line. A bit of mental effort,...
[line fills in, and zooms out to show the ends again]
and it's a continuous line.
Or,...
[line fades away, leaving the end points again]
... we could go the other way. Take the distance and direction from one point to the other, and then put a new point that far away on the other side.
[p3-p2 = p2-p1: *pling!*]
And do that again.
[*pling!*]
And again, and so on. In both directions.
[a line of evenly spaced dots, racing away to the conceptual horizon]
Now, this line also goes on forever. It doesn't matter how far you go along it, there's always room for another point.
[slowly, then faster, zooming along the line towards the horizon, then slow and stop]
And because it goes forever in both directions, there's no center. So any point at all could be the center: it doesn't matter. There's still an infinite number of points on both sides.
Let's just pick this one and call it "Zero".
[a point has "0" appear above it.]
Now we have to consider two concepts: we can count the points, or we can count the distances between them. To start with, let's consider the space between the points, so that this...
[an arrow travels from "0" to the next point, and stops with the arrowhead at the point]
we can call this "One".
["1" appears above the second point.]
But the thing is, if we have "Zero", and we have "One", then all the other integers follow. One plus One...
[a ghost of the arrow from "0" to "1" moves to between "1" and the next point. "2" appears above that point]
... equals "Two". Two plus One equals Three. And so on.
[Arrow moves to create "3". Numbers appear above the points: "4", "5", "6" ... until you can't read them any more.]
But wait... there's a complete infinity still waiting, in the other direction.
[pan to the points on the other side of "0"]
If you can move in one direction, you can move in the other direction. There must be a number which, when you add One, you get Zero. It's like One, but it's its mirror. Let's call the numbers on this side "negative", and mark them thus:
[the arrow from "0" to "1" ghosts and shifts until its tip is resting on "0". Above the other end appears "-1"]
And we can replicate this in the other direction.
["-2", "-3", "-4", ... and so on.]
We've just invented the Integers. And there's an infinity of them. Start from Zero and start going, and you will never get to the end. There is no "last" integer. You can always add One.
But wait, there are all those gaps. Didn't we fill them?
[the line between "0" and "1" reappears.]
So now we have this continuous line. And because the difference between the space between Zero and One and and other point and the next is because we arbitrarily pointed to one and called it "Zero" and nothing else, then we can fill in every other space as well. All of them at once.
[Line fills in, with blips for the points, and numbers remain above]
This is the Real number line.
Now think about this: the space between any integer and the next is filled with an infinite number of points: whatever number you're thinking of, it's more. and there are an infinite number of these spaces -- as there are an infinite number of integers. So the number of points on the Real number line must be an infinity of infinities... it's an infinity which is bigger than the infinity we first thought of.
We started from absolutely nothing, and created two levels of infinity, just by inventing Numbers.
...
So, we start from the first couple of Euclid's Axioms, work our way through Peano, and take a large helping of Cantor on the way. And we're still in one dimension. I haven't started talking about circles yet. In this first step, I'd continue by talking about the difference between ordinals (the number as a vector) and cardinals (the count of points), and how Zero is different, because every other number is a vector, except for Zero, which is the point from which all vectors are measured: which is how all numbers have a magnitude except for Zero. Then extrapolating from addition to multiplication/division as a function of constructive ratios (ie., 4.5 * 8 => 1:4.5 :: 8:?), and by extension why division by Zero makes no sense.
Sure, it skips and jumps around thousands of years of mathematical history, but I'm trying to lay out the concepts in an order which logically follows, one idea from another. It's difficult, because it's all really a net of interconnected ideas, and making any given path is to some extent arbitrary.
I've run through some of these ideas with Miss A, and her head didn't spontaneously burst into flame... although I'm not sure how much of it she understood.
Do people think this is worth putting effort into?
no subject
It is a little difficult to work out who it is pitched at, though, as it is covering very basic concepts, but begins using specialised terminology without adequate denotation that the meanings of ordinary words like "real", "point" and "fixed", which have different meanings and uses in this context. If I were using something like this, I would want to have access to call-outs which explained new words, or new word usages.
Darned interesting idea, though. And as someone whose (undergoing assessment for AS) 9 year old is freaking out when confronted with new maths concepts in the classroom, but flies along once she can calm down and integrate the new information into her own non-standard understanding system, I would suggest that it would quite probably prove extremely useful.
no subject
These technical terms are quite important, and it does help a lot to have exposure to the lingo even if you don't quite understand it yet, but I thought while I was writing the above that those technical terms (Integer, Real, Infinity) could appear in the corner or the screen: appearing and fading away to mark this as an Important Term.
I included the stuff about infinity because: 1. when working through some of this with Miss A, it was a natural tangent to talk about, and 2. these were exactly some of the concepts which got me stuck when I was learning all this the first time. Most kids just did the mechanical things with fractions and memorised how to manipulate bounded and unbounded intervals, but I was hung up on figuring out what all that meant, and getting lost in implications which I was completely unprepared to follow up on. If I was really lucky I was able to ask a teacher who was able to give a meaningful explanation... when I was arguing with a teacher on the interpretation of x⁰ comes to mind. (I insisted that it must mean x times itself Zero times, which was Zero, but he explained the use of xⁿ in positional number systems, and how it made sense to describe 1,234 as 1*1000 + 2*100 + 3*10 + 4*1 = 1*10³ + 2*10² + 3*10¹ + 4*10⁰. Now I understand x⁰ as a Null operation, which leaves the equation unchanged. In an additive context, the Null operation is ±0, in the multiplicative context it's *1.)
Ahem. I digress. Anyway, this would be but the first part. The idea which got me started was what would be the second part, dropping back to the two points and using them to construct a circle, then starting on trigonometry from first principles, and proving a few interesting properties. In fact, it was from playing in this mental space which led me to come up with some proofs of these properties myself, from first principles. (Such as: the angle between a tangent to a circle and the line from when the tangent and circle meet and the center is always 90˚, and I can prove that two different ways, both of them geometric.)
Um. Yeah. In any case, I have never done any animation, and so I would need to talk to someone who could at least help me get started with this. Or else I would need to rework this concept into something which could be done in prose and illustrations. Which isn't impossible, I suppose, but some of the concepts really would lose much impact when presented statically. I visualise these operations dynamically, as interacting movement of lines and points and distance and angle.
Although another way to do this would be interactively. There is an interactive implementation of Euclid's Elements which is nothing short of genius: allowing you to play with the relationships being described and gain a deeper kinesthetic understanding of them. (Indeed, you might find it useful with your own kids: a self-paced, interactive, and visual introduction to the foundations upon which a lot of geometry is based.)
I'm just going on and on, now. I think it has the potential to be a really neat idea, but I have no idea where to go with it, or how to proceed (even if I knew where I was proceeding to...)