DID YOU KNOW………I didn't think so………Point
A point…..defined by Euclid as ” that which has no parts”. Playfair defines it
as “that which has position
but not magnitude”, and Legendre viagra sale says it ” is a limit terminating a line;” but none of these definitions can be called either philosophical or exact.
November 12th, 2007 at 9:19 pm
As I understand the Euclidian development, points (and lines for that matter) are not so much defined as they are described or characterized by axioms. Or, in other words, they follow from certain axioms that are developed.
November 13th, 2007 at 11:54 am
I thought that would pull you out. Now, would you like to explain it?
I kind of liked the one about “no parts”. Like Monday mornings.
November 13th, 2007 at 2:23 pm
Actually, I think I get the “axiom” bit. It would explain the question that kept me tossing in bed all last night. I was trying to think of what would be the “smallest” point possible. I gave up and fell asleep after I got to dark matter.
November 14th, 2007 at 10:53 pm
I don’t know enough pure math to be able to answer this properly, but I just remember my 9th grade geometry teacher emphasizing that points and lines did not have definitions per se.
The Euclid and the Playfair ideas make the most sense to me. I think of a point as being simply position (a_1, a_2, … , a_n) for an n-dimensional space. No quantity is associated with simple position, which is different than scalar or vector fields (or higher order beasties like the stress tensor in fluid dynamics) which are functions of position.
I can’t quite figure out the Legendre idea, “limit terminating a line.” He’s saying one has to know what lines are before points, yet this seems backwards, as a point is a more fundamental entity than a line, which contains an infinite number of points. So it seems a line derives from points rather than the other way around.
November 15th, 2007 at 1:43 am
A point and a line are fictions until there is an axiom….I guess. I didnt have to lose any sleep over the smallest point. It’s a fiction until I find an axiom.
The BIG BOOK said pretty much the same as you about Legendre. I didn’t include the whole lengthy quote which went on to question the three statements.
November 15th, 2007 at 1:57 am
Oh crap! Now I’ll be up trying figure out what would be an axiom to find the smallest point.
November 15th, 2007 at 2:18 pm
What do you mean by smallest?
November 15th, 2007 at 4:41 pm
The smallest point in the universe. Say I wanted to find a quark but even smaller. What would be the smallest point and where is it?
November 15th, 2007 at 4:51 pm
If the smallest point is an infinite then I guess we can only count our way there.
November 15th, 2007 at 5:29 pm
I guess my point is that for this to even make sense you have to define your terms. Size has many meanings (and does matter).
You began by quoting Euclid, which implies Euclidean geometry. You could define the size as the area on a Euclidean plane, for example, and a point—any point—would have area zero.
You could define the size of a set—a point, a collection of points, a line-segment, a polygon, etc.—as the number of elements it contains. By this measure, a point counts for 1, two points counts as 2, and a line counts as infinity. This is an interesting area—there turn out to be many infinities…
Both of these definitions of “size” are valuable, but they are very different.
You also talk about physics, but the problem is that nobody really knows what the underlying topology of space is. Relativity seems to assume that space is smooth—not Euclidean, but locally so. If this is what space actually looks like, then the answers for Euclidean space should be similar as well. This seems to have problems, and may not be true.
November 15th, 2007 at 6:12 pm
In over my head again. Yes, I was mixing up my “points”. You seem to have summed the problem.
November 15th, 2007 at 6:37 pm
Modern physics is, alas, way over my head. 🙁
November 15th, 2007 at 6:49 pm
Plus, I keep mixing this all up with Plato’s “Ideal World”.
November 15th, 2007 at 11:33 pm
Zook,
Huh?
November 16th, 2007 at 12:51 am
I’m not sure what part you’re “huh”ing. 🙂
We have concepts that we want to encode in the language of mathematics. We do this by creating mathematical constructs to deal with them. E.g., we can talk about the real line—all real numbers, something which really needs some development itself—and define a point in this space as a single number. We can talk about points in the plane; these are just pairs of numbers.
If you want to talk about size, you have to define what you mean by size. E.g., the “size” of a circle could be its area, its radius, its circumference, the number of points it contains, etc.
Whether the real world *actually* matches up with the constructs we’ve defined is not a mathematical question, it’s a physics question. If we look around us (without looking too hard) we may think that the surface of the earth is a plane: on the small scale it looks like it.
If we make that assumption, we find that some things in our model break down. E.g., on the plane the interior angles of a triangle add up to 180 degrees. If we take a triangle on the earth, the angles add up to more than 180: the surface of the earth is non-Euclidean.
The breakdown here isn’t in the mathematics—the math is sound if you work on a true plane—but in the geography: the earth isn’t the plane we assumed.
So, there are two distinct problems:
1) What do we mean by size?
2) If we’re concerned about the physics, what is the right mathematical model? (This is what things like string theory are trying to answer—I’ve got no clue.)
November 16th, 2007 at 9:42 am
Right, everything follows from the axioms that are initially employed. But it seems to me that the ability a specific mathematical development to be able to explain natural phenomena or make predictions grants it more validity than a development that is not able to do so (my biased physical scientist view). I realize pure mathematicians have their silly games, and this is why many history/philosophy of science people consider math to be more in the realm of philosophy than science, since the only requirement for the former is to be internally self-consistent.
November 16th, 2007 at 11:25 am
Should be an “of” after “ability” in that second sentence. Duh on me.
November 16th, 2007 at 12:28 pm
Did everyone get that? The Earth is not flat. And math is silly.
I was furtherer educated over a beer last night by a Boeing engineer. He made the same points so I’m starting to think you might be right. Great beer.
November 16th, 2007 at 12:50 pm
MR: Are you thinking like me, Ground control to Major Tom, Earth calling the orbiting thread? Besides, “The Point Is Moot” (Lincoln), Not the Prez.
November 18th, 2007 at 1:10 am
I don’t know enough about Euclidian geometry OR physics to be commenting on this post, but I think I might have accidentally learned something…I think. Thanks, doods.
I think J.J. means it’s a MOO point–a cow’s opinion ( doesn’t matter ).