Caveat: Nothing deep here just some rambling thoughts. No actual research was involved in the making of this blog post.
I was reading William Rowan Hamilton’s original “On Quaternions” lecture. It’s my cup of tea… he takes 15 pages to define what a “Vector” is. I love it.
It’s on Google Books… a scan of an original from Harvard College science library.
But then here he goes…
“For my own part I cannot conceal that I hold it to be of great and even fundamental importance , to regard Pure Mathematics as being primarily the science of ORDER (in Time and Space), and not primarily the science of MAGNITUDE”
Compare this with the introductory chapters to Thorne, Wheeler, and Misner’s “Gravitation” (Which, if you saw the movie Interstellar, was the big black book on the table at the NASA headquarters discussion room).
They basically say a similar thing. They are trying to explain Einstein’s General Relativity, the one where Space is curved by Mass. To badly summarize, they begin by asking the reader to forget, for a moment, about the idea of Distance. Just look at the relationship between Events in Space-Time… their ordering between each other.
Now pile on top of this another idea.. that measurement itself is a bit of a conundrum, because it typically involves Real Numbers, which we can’t actually do arithmetic on, nor store in a computer, nor write down. This is my bad paraphrase and interpretation of Dr Norman Wildberger’s articles like this one on “uncomputable decimals”.
Sometimes I wonder… then…
Was Renee Descartes invention, the idea of assigning numbers to points in space, a kind of leap of incredible faith that we take a bit for granted? After all, the point 1,1 appears to be quite simple, but if viewed from a polar coordinate system, 45 degrees off zero, it’s distance from origin, being square root of 2, contains an infinite amount of information. It is not an infinite distance… but it is an infinitely complex distance. Every book, every film, every person’s DNA, every particle position of every planet, every star of every galaxy, everything, is contained in that little square root of 2 (well, with a bit of invented steganography perhaps). An infinite number of digits can be imagined as representing, literally, anything. So… this thing we call Distance… is it not something strange and wonderful that we don’t really understand at all?
But can we still say, at least, say that a point at 1,1 comes “after” a point at 1,0 … depending on our definition of before and after? I supposed it depends on how you define distance… or rather… should I say… it depends on how you define ordering? If you say the ordering is such that
is compared to
For two points, and , I guess you could say that is a kind of ordering… based on the concept of the circle or the ‘Blue Quadrance’.
So in the example… say and
Then you’d have the ordering, . So is “after” .
But the fun part, the clever bit, is that if point and point are both finite size rational numbers, then your ordering is always a Finite Rational Number, so you can always, always do arithmetic on it. There is no ambiguity, ever. Even if you stick it in a computer program (assuming you use Rational Number types and not floating point).
Why? Well… There was this dude named Gauss, and these things called Algebraic Numbers, Constructible numbers… etc… but that’s another story for another time (better told at http://www.cut-the-knot.com/arithmetic/rational.shtml )
But back to this story. What if you take a different definition of order, say
compared to . That’s kind of based on the idea of a Hyperbola… or another way to say it… the Red Quadrance.
So the order of the two points is reversed! x,y is “before” x1,y1. But we are still all Finite Rational here. So we can still do the calculations precisely.
So there are multiple ways to order points… depending on how you want to do it.
Which reminds me of this other idea. Is the idea of infinitely complex distance… relative to the observer, in some way? Our observer at 0,0 who can only use polar coordinates, has this issue of 1,1 being an infinitely complex distance, that they will never be able to calculate or store or write. But the observer at 0,0 using rectangular coordinates, 1,1, does not have this problem. But, in theory, they are both “Seeing” the same point! This is what I mean by how I take Descarte’s ideas for granted. To really look at it, maybe is to see a wonderworld of strange projections of impossible objects.
Even as the Perspective artist can draw a one-point perspective, that point being considered “at infinity”…. maybe the simplest square could be argued from the observer stuck with only a Polar Coordinate system, to contain it’s own kind of infinity….but one that we can imagine as finite by the magic of “Projection”, down to one dimension, so that the Polar Coordinate person can imagine their own “fake” version of rectangular coordinates…. the point at 45 degrees, root 2 distance, is transmuted into the point standing at 0 degrees and 1 distance step, followed by a left turn of 90 degrees and another 1 distance step. Then there are four numbers (0,1,90,1) to describe what is in theory a two dimensional object…. but at least now the Polar Coordinate person has Finite Rational Numbers to work with!
As I said. Nothing too deep. I found it fun to ponder, though. And I did almost no research and have no real knowledge of the subject. But even a child can ask, meaningfully, what is distance? What is measure? How can you order things in space and/or space time? And is it kind of funny that William Hamilton is saying something a little bit similar to Thorne, Wheeler, and Misner, trying to describe General Relativity 130+ years later?