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Problematic geometric concepts

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Okay, so this has occurred to me more than once but it hasn't occurred to me until today to post about it here.

I'm in a geometry class, and of course one has to deal with the concepts of the point, the line, and the plane in geometry. However, I was recently set about the task of defining what a line is, and when I went back into my geometry book, I found this: that the the concepts of the point, the line, and the plane are taken to be intuitional. Meaning, I assume, that even the writers of the geometry book were puzzled by the task of making such a definition.

When I think about it, there's an obvious problem right off the bat with trying to define these things: definitions are based on concretes, and you cannot find a concrete of any of those three things in reality. A point has no length, width, or depth, we only represent it as having the such. A line has no width or depth, we only represent it as having the such. A plane has no depth, we only represent it as having the such. This means that they do not exist in reality, which obviously poses a problem for making a definition. And yet, they're such vital parts of geometry, one could call them axiomatic, and that's what occurred to me next - that no shape can be defined without these terms, and yet there can be no "proof" for these terms themselves. You can only point and say "this has lines" or "this has planes" or "this has points".

My problem, I suppose, is this: are these three concepts axiomatic to geometry, and do they even exist?

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I might be able to help with some physics. There's a rule in physics for calculating when it becomes important not to treat space as being euclidean or flat, and that's when you're in a region very close to a very massive black hole G*M/(r*c^2) >= 1. Around the earth this comes out to around 10^-9 which is much clearly less than one. The difference between the reality we face here on earth and the geometry Euclid describes is very small. So I think its fair to say that Euclid's geometry probably wasn't developed through any kind of irrational leaps of faith.

In different frameworks lines might be defined differently, but this would add complexity to your geometry course that the lecturer probably isn't interested in. You can think of a line as a one dimensional plane. You can also define a line by its analytic properties, but that requires coordinate systems, algebra, and probably some calculus to do in a cool way. If you have the concept of a vector you can define a line as a curve upon which you can transport the vector so that the vector doesn't change direction relative to the line. Although, I do admit the line took creativity, just like the wheel. There aren't wheels in nature just as there aren't perflectly straight things, but some things such as light rays come close.

Fundamentally speaking, your ability to imagine up concepts comes from the hardware your brain evolved to measure nature. A valid concept that you can understand cannot come from outside reality. If you can visualize a line, its because your brain has the machinery to visualize a line. Although sometimes we can't visualize things, but we can translate them into things we can visualize to dumb them down.

The difference between a line and a god is that a god can take on any shape it wants whereas a line has a structure, limitations, and a proper mental image. It is a tough issue though.

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Okay, so this has occurred to me more than once but it hasn't occurred to me until today to post about it here.

I'm in a geometry class, and of course one has to deal with the concepts of the point, the line, and the plane in geometry. However, I was recently set about the task of defining what a line is, and when I went back into my geometry book, I found this: that the the concepts of the point, the line, and the plane are taken to be intuitional. Meaning, I assume, that even the writers of the geometry book were puzzled by the task of making such a definition.

When I think about it, there's an obvious problem right off the bat with trying to define these things: definitions are based on concretes, and you cannot find a concrete of any of those three things in reality. A point has no length, width, or depth, we only represent it as having the such. A line has no width or depth, we only represent it as having the such. A plane has no depth, we only represent it as having the such. This means that they do not exist in reality, which obviously poses a problem for making a definition. And yet, they're such vital parts of geometry, one could call them axiomatic, and that's what occurred to me next - that no shape can be defined without these terms, and yet there can be no "proof" for these terms themselves. You can only point and say "this has lines" or "this has planes" or "this has points".

My problem, I suppose, is this: are these three concepts axiomatic to geometry, and do they even exist?

It comes from perceptual abstraction. We see a cube. From that we can abstract a planar face from it. Where two planes come to form an edge, we abstract the line. Where three of the planes form a corner, or where two or three of the lines intersect, we are able to abstract the point.

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You'll want to read Rand's ItOE where she describes such things as length, width, weight, point, etc. as cognitive "units" which serve as a bridge not only between percepts and concepts but also between mathematics and concept formation.

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It comes from perceptual abstraction. We see a cube. From that we can abstract a planar face from it. Where two planes come to form an edge, we abstract the line. Where three of the planes form a corner, or where two or three of the lines intersect, we are able to abstract the point.

I don't quite understand what you're saying here. I do understand how one abstracts, what I don't understand is how one defines such things as the line, the point, and the plane. If we define a point as a place at which lines intersect, then what of points that aren't connected to lines? And why do they still have no width, length, or depth? How can you conceive of a line or a plane when neither exist in reality. You can conceive of a very thin sheet or a very thin string or a very small ball, but not of the other things. I don't see how one conceives of something that has no length or no width or no depth.

" If you can visualize a line, its because your brain has the machinery to visualize a line."

But that's the thing - one can't visualize an object that's missing one of the dimensions. It has no parallel in reality.

The only way I can think of defining these things is in the context of an actual, 3-D shape. A plane is the flat face of a shape, a line is the series of points at which planes meet, and a point is where lines meet. But those are not the definitions or the contexts given for these ideas in geometry, at least, not the geometry I'm currently learning.

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A modern axiomatic mathematical theory of Euclidean geometry is given by Hilbert's axioms. These formal axioms are chosen so that they correspond to our more intuitive notion of geometry, just as Peano's axioms are a formalization of our intuitive notions of natural numbers.

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And why do they still have no width, length, or depth? How can you conceive of a line or a plane when neither exist in reality.

The key is understanding the role units and measurement omission play in concept formation. You perceive things: a flag pole, a piece of string, the edge of a table. You abstract a similarity i.e. that they all have "length". The specific length is un-important in developing the concept of length. Units such as length, depth, weight are mental tools and as such do not exist independent of existents. There is no such THING as length. Things HAVE length as one of their attributes.

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I don't quite understand what you're saying here. I do understand how one abstracts, what I don't understand is how one defines such things as the line, the point, and the plane. If we define a point as a place at which lines intersect, then what of points that aren't connected to lines? And why do they still have no width, length, or depth? How can you conceive of a line or a plane when neither exist in reality. You can conceive of a very thin sheet or a very thin string or a very small ball, but not of the other things. I don't see how one conceives of something that has no length or no width or no depth.

" If you can visualize a line, its because your brain has the machinery to visualize a line."

But that's the thing - one can't visualize an object that's missing one of the dimensions. It has no parallel in reality.

The only way I can think of defining these things is in the context of an actual, 3-D shape. A plane is the flat face of a shape, a line is the series of points at which planes meet, and a point is where lines meet. But those are not the definitions or the contexts given for these ideas in geometry, at least, not the geometry I'm currently learning.

Try visualizing a very thin sheet, string or ball where you omit the dimension.

In the case of a plane, the thickness must be something (hence the visualization of the very thin sheet), but can be anything.

In the case of a line, the thickness must be something, but can be anything.

In the case of a point, yes, it must have a dimesion, but again, it can be anything.

The plane might also be described as where the 'air' contacts a 'face of the cube' - neither the 'air' nor the 'cube', rather a description of the 'extracted shape' where the two are in contact with one another. How much distance is there between the air molecules and where it touches the surface of the cube?

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• 9 months later...

I agree with what New Buddha is saying. You'll want to read Rand's introduction to epistemology, if you already haven't. It is the same thing with the attribute of color. Color isn't an axiomatic concept, but it is a concept that you can make just by looking around. Entities have color just like they have shapes and sizes. You look around and you figure out that one thing is red and long, another purple and short. I recommend Rand's book on epistemology. It is the most valuable (to me) book of hers. I own a copy with the appendices. The appendices are great too. I love the book.

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Double Post

There are long objects, and shorter objects, right? A ruler is an interesting example. It is a man-made object which we use as a standard by which to compare various objects' length. We put a ruler next to a bookcase, and we say ah, two rulers long. Next to a doorway, ah 3 rulers long. The doorway is more rulers long than the bookcase. When we are very young, we perceive that some objects have more length than others. Eventually we speak about length and make complex studies of it.

"[The defenders of conceptual knowledge] were unable to offer a solution to the 'problem of universals,' that is: to define the nature and source of abstractions, to determine the relationship of concepts to perceptual data - and to prove the validity of scientific induction ..."

-Rand, Introduction to Objectivist Epistemology, pg 3 (which is the forward)in my copy

As far as points go, lengths end in points, right? Pointy objects end in points, at any rate. Lengthy sentences end in points. Um, when you argue, you try to make a point. That is a different kind of point I suppose. Or is it? Pointy and sharp can be used interchangeably sometimes. I'd say the word point in mathematics classes usually refers to some part of an imaginary line. Oh, and basketball players score points by ending plays. That seems like an equivocation. Some people think it is rude to point fingers. I kind of see what they are saying, but as you can see I'm a big fan of pointing in general.

Edited by Brian9
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These concepts are tools of measurment made explicit by great thinkers over time. They are by no means intuitive.

I remember being very suprised to hear the something like this " __________ " is not a line. It is a line segment, a line I was told goes on in both direction towards infinity. This concept is much different than the original " a line is an edge in one direction ". In fact, when I was a kid, I called all sorts of thing "lines" that I should have called "edges". I called all sorts of things points, which are actually "tips", "ends", or "dots". I never called anything a plane... if I had to I would probably have called it a "slice" or something.

I think that there is a relationship between these simple ideas and the geometrical ideas.

For instance we can take the idea of an "edge", and say " a line is an edge without a shape, with no width, that never curves, and has an infinite length". This to me just seems like the most extreme form of measurment dropping. A point is a "dot that is of infintesimal size". A plane is a "slice of space of which one dimension is of infinitesmial value".

To give a really precise definition of these sorts of concepts you would most likely have to use calculus, which really weirds me out because that would mean conceptually calculus is more fundmental than geometry, I didn't know that.

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My understanding of Ayn's use of the idea "unit" does NOT encompass that which can not be experienced or imagined, at least in principle.

Ergo, infinitesimal ANYTHING is impossible -- including but not limited to infinitesimal points, lines, and planes. It is incorrect to assume the limit necessarily exists. The attempt by geometers to divorce their wares from the field of reality is traditional, but like any other attempt to vivisect mind from body, it only leads to a false dichotomy.

Real things have finite extent in every observable property, at least as far as my experience goes -- have yet to sense or imagine something that is infinitesimal, i.e., has no measurable value for some unitizable property. Go figure ...

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Lines are closer to concepts of methods. I agree with the posters who say that lines are abstracted units of length. They are idealized units of length whose only property is their length (and direction), since those are the only properties we care about in the context of plane geometry.

As for not being able to physically realize a line, that's right, we can't. But just as one needn't produce -2i cows in a pen to use complex numbers, we need not produce an infinitesimally thin ruler in order to apply plane geometry. The fact that objects can subtend some (straight) length and be a reasonable approximation to a line in some context is all that we need to abstract to the idea of a line.

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"The mathematicians feel that they can do anything they want with their abstraction because they donâ€™t relate it to reality. And, of course, they can really do anything they want with their abstractions, even though, like masturbation, it is irrelevant to the propagation of life." -- R. Buckminster Fuller

The mathematical notion of line attempts to do more than abstract the notion of relationship (and the related concept of physical extent) -- it creates a concept at least partly divorced from reality: in reality, relationships between things (including the related parts of a body which determine its extent in any given direction) are in general neither linear, nor continuous, nor static ... and they have "diameter" in the sense of physical (or conceptual!) interference effects. In reality, relationships cannot be established except by exchanging information, at minimum in the form of little energy packets called photons. Their are no real lines, only interactions among related parts of larger wholes.

Or so it appears from my perspective, consistent with the logic of my experience.

Cheers.

- ico

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