How can something be located in space but not occupy any space?

[LOVEthroughINTELLECT]

I can't be nitpicky like I usually am about providing accurate information about sources--I am too tired right now and probably do not have time to verify anything. Let's just say that I believe it was the latest issue of Philosophy Now that I was reading. I did not catch the author's name, but I got the impression that he writes a column in every issue. Here is what I understood him to be saying:

Euclidean points, the foundation of all of geometry, by definition have no dimensions, no thickness, etc. You know, stuff that you were taught the first day of 10th-grade geometry. Yet, by definition they are located in space--we use them to describe and measure space. Therefore, we have something that is supposedly part of space but does not occupy any space.

I have always kind of rolled my eyes and thought to myself, "If you say so", when mathematics at any level has been presented to me. I think that it would be a fun time to address all of the philosophical problems that mathematics and physics present. Alas, at my school the one such course that the philosophy department offers has university physics (not vanilla physics for non-majors and scientifically-illiterate people like me) as a prerequisite. Seeing how the aforementioned author seems to put things (you know, like, how can something be located in space but not occupy any space), I know that I must be missing out on a lot of fun.

Of course, it is probably more fun for the really talented people out there to make fun of scientifically-illiterate people like me asking questions like the one at the top of this thread. Have fun doing it--I won't get upset like people thinking that this thread belongs elsewhere.

 

[Keri]

tl;dr

But based on the title I feel it a, "You think too much, LTI," response.

 

[Sophrosyne]

Pretty simple actually, in able for something to occupy space it has to have volume. Since a point has no volume because by geometry volume is length x width x height, something with a 0 on all three of these dimensions has 0 volume therefore cannot take up any space. You could also consider a line takes up no space because it has length of some definable number X but no width or height so with LxWxH if you have just one 0 (zero) the volume = 0 also. You could consider a line just a bunch of points connected together that each has no volume either. It is only when you have a number other than 0 in all 3 dimensions that you can start to consider something takes up space (with some exceptions of course).

 

[r035198x]

One way is to think of a Euclidean space (a.k.a Cartesian space) as the space of all n-tuples of real numbers. It's easier to see why no space is occupied then because intuitively, numbers do not occupy space.

 

[Sophrosyne]

One way is to think of a Euclidean space (a.k.a Cartesian space) as the space of all n-tuples of real numbers. It's easier to see why no space is occupied then because intuitively, numbers do not occupy space.

It is like memory locations in computers, they don't take up any of the memory that is allocated at that location for storage, like an address doesn't take up space it is what that is located at that address that can take up space.

 

[Nanopants]

What's even more fun is thinking about how something that is exactly pi inches long will never align precisely with any measurement in base 10 no matter how precise the measurement is.

 

[acropolis]

Mathematical objects don't always have real-world equivalents. Similarly, you can define a 3d shape which, if real, could be filled with paint (finite volume) but not painted (infinite surface area). There are many instances of things like this, many of which involve the idea of infinity.

 

[LOVEthroughINTELLECT]

Pretty simple actually, in able for something to occupy space it has to have volume. Since a point has no volume because by geometry volume is length x width x height, something with a 0 on all three of these dimensions has 0 volume therefore cannot take up any space. You could also consider a line takes up no space because it has length of some definable number X but no width or height so with LxWxH if you have just one 0 (zero) the volume = 0 also. You could consider a line just a bunch of points connected together that each has no volume either. It is only when you have a number other than 0 in all 3 dimensions that you can start to consider something takes up space (with some exceptions of course).

So it has no volume, therefore it does not occupy any space, but it is part of space.

 

[LOVEthroughINTELLECT]

Mathematical objects don't always have real-world equivalents...

In other words, many of the things that we claim to know about the physical world are really logical extensions of metaphysical conceptions.

I think that such an inconsistency is what the author is trying to show. I could be wrong, though.

 

[Niels]

Coordinates can exist in space without occupying space.

 

[Boondock_Saint]

Math is where the money is.

 

[LOVEthroughINTELLECT]

Coordinates can exist in space without occupying space.

Coordinates do not exist in space. They are conceptions in our minds.

I suppose that one could say that coordinates exist in the space of our brains.

 

[LOVEthroughINTELLECT]

Math is where the money is.

And picking it apart is where the fun is.

 

[r035198x]

So it has no volume, therefore it does not occupy any space, but it is part of space.

The concept of space is only added to give a visual real word analogy. The formal definition is all about numbers. It's just numbers being translated into other numbers using basic mathematical operations. Numbers do not take up any space.
It's just helpful to think of differences between some numbers as spaces in order to visualize the differences between those numbers.

 

[Nanopants]

In other words, many of the things that we claim to know about the physical world are really logical extensions of metaphysical conceptions.

I would disagree with that. Math is an invention, relying on frames of reference which we have imagined, but it's quite useful in making meaningful comparisons of real relationships.

Take any ratio for example, say, 3/4. The quantities, the symbols, the syntax, etc were defined by us, but there is a real relationship that is expressed here. Math is just a language that helps us to make sense of reality.

 

[Sophrosyne]

So it has no volume, therefore it does not occupy any space, but it is part of space.

Who said it was a "part" of space? It is more like a postal address, just a pointer to a location. In other words you really cannot see it because it doesn't take up space required to be seen. If you were to go to an address you would see something at the location but the number (address) would not exist (physically) without someone actually putting up something with the number on it.

 

[Niels]

Coordinates do not exist in space. They are conceptions in our minds.

I suppose that one could say that coordinates exist in the space of our brains.

I understand what you're saying, but I think you've missed my point. What coordinates refer to exists independently of what we think. My current location existed before latitude and longitude were devised, the halfway point between the Earth and the Sun also exists, yet neither takes up space.

The only exceptions that come to mind involve fiction. For example, a coordinate in a fictional universe only exists in the imagination.

 

[LOVEthroughINTELLECT]

The concept of space is only added to give a visual real word analogy. The formal definition is all about numbers. It's just numbers being translated into other numbers using basic mathematical operations. Numbers do not take up any space.
It's just helpful to think of differences between some numbers as spaces in order to visualize the differences between those numbers.

Who said it was a "part" of space? It is more like a postal address, just a pointer to a location. In other words you really cannot see it because it doesn't take up space required to be seen. If you were to go to an address you would see something at the location but the number (address) would not exist (physically) without someone actually putting up something with the number on it.

So if space did not exist the mathematical relationships that we have thought of would continue to exist?

It seems to me that if there were no space there would be no relationships to express.

And I did not know that a point is simply a representation of a location. I thought that points are the thing being analyzed/described, not a representation of the thing being analyzed/described. In other words, if points did not exist, geometry would not exist.

 

[LOVEthroughINTELLECT]

What coordinates refer to exists independently of what we think. My current location existed before latitude and longitude were devised, the halfway point between the Earth and the Sun also exists, yet neither takes up space...

So a point can be a location where there is nothing.

Therefore, if a circle is defined as the set of all points in a plane that are a given distance from a given point then that means that a circle could be all of the points in a plane that are a given distance from nothing.

 

[LOVEthroughINTELLECT]

Before anybody thinks that I am being difficult or argumentative, I am not.

Just having a conversation about the seeming implications of an article that I read.

If you want to know about my personal experience with the kind of problems being discussed:

I remember being taught that the definition of a circle is the set of all points in a plane that are a give distance from a given point. Points, I was taught, have no dimensions, no thickness, etc. Yet, I was then taught a certain mathematical formula to determine the area of a circle. Well...

First, the circle, by definition, has no dimensions or thickness, so how can it have an area? More importantly, the forumla given makes calculations of space that is, by definition, not part of the circle.

If by "the area of a circle" it was really meant "the area encompassed by a circle", that distinction should have been--and should be--made clear with language.

Is that being nitpicky? No, it is being the critical thinker often called a philosopher.