What is the relationship between points and neighborhoods in topology?

[NeutronStar]

Discrete vs. Continuum

Can one have a gap between two points without assuming a coordinate system?

Yes. This kind of falls into the chicken question. Which came first, the coordinate system or the point?

The best answer that I can give you is that if there must necessarily be gaps between points, then any coordinate system that is considered to be a field of points must necessarily contain gaps. And the best way to think about those gaps is to think of them as quantum jumps. You simply can't talk about locations within these gaps. It's nonsensical to try to do so.

The problem with "pure thought" is that you can always imagine setting up finer coordinate systems within those gaps. And in "pure thought" you can. But logic has already dictated to us that those gaps must necessarily exist no matter how fine we pretend to make them.

The bottom line is this. If you want to build a "real" physical universe that exhibits a quantitative property that universe will necessarily have to contain these gaps. So to actually build one you will have to choose an actual value for the gaps. Once that's been done the universe you finally build will contain a fine structure below which it will be absolutely meaningless to talk about because those location (or points) simply don't exist.

The thing that I find so amazing about the "pure thought" or "pure logic" is that even though it can't give us an absolute number for these gaps it can tell us that they must exist. To me this is very powerful. Imagine if the mathematical community would have realized this before Max Planck discovered the grainy nature of the universe. The mathematical community could have actually predicted quantum physics. They may not have been able to predict the numerical value of Planck's Constant. But they could have at least predicted the quantum nature of the universe. Instead they decided to go down the road of pure abstraction pursing some lofty notion of a continuum. That notion simply isn't supported by the quantitative behavior of our universe.

Surely the points have to be at (or have to be conceived as being at) different coordinates? To me a location seems to be the same thing as a set of coordinates.

Well, you're just to used to thinking in terms of coordinates. I tried to make it clear earlier that the idea of a single point is really meaningless. It takes at least two distinct points before the idea of a point actually begins to make sense. But even when thinking in terms of two points our human minds tend to think in terms of a 3-D space. We are just so used to thinking in these terms not to mention that this is our everyday experience.

Humans simply aren't capable of thinking in terms of only two points. Intuitively the first thing that pops into their mind is the gap between the two points and the fact that this also can be thought of as a point. In other words, it's really hard to fight the urge to think of more than two points at a time. Yet the whole purpose of the exercise is to imagine that only two points exist. What would be the result of that? They would have to be separated by a gap otherwise they would be the same point.

That's the conclusion. Period amen. To try to claim that we can then add more points is to miss the whole exercise of thinking of only two points. The thrust of the argument is that it is simply impossible to conceive of an idea of two dimensionless points without thinking about a gap existing between them. The urge to toss a third point that represents the location between them must be suppressed. In other words, that is a forbidden location and therefore cannot be thought of as a point. Why? Because the whole premise is that only two points exist. To introduce a third point is to violate the premise.

It's a purely philosophical argument that appeals to the intuition showing that in order to conceive of an idea of at least two dimensionless points (and no more) we have no choice but to accept this gap between them that we cannot (by our premise) consider to be an additional location or point.

I can personally handle this type of intuitive comprehension and understand the logic upon which is it based. So I have no problem with the consequences of this intuitive idea. It simply means that if I want to move from one point to another I'm going to have to make a quantum jump to get there.

Now, what's the alternative?

I'm more than willing to listen to any logical or intuitive arguments concerning any ideas of a continuum that is constructed of dimensionless points.

I hold that the points must necessarily be dimensionless because to introduce a concept of points that have any breadth is to introduce discreteness right there. Any such theory would simply be moving the discreetness out of the gap and into the breadth of these so-called dimensional points.

So we need to describe a logical and intuitive idea of a continuum of two points. In other words, to dimensionless points that are not the same point and yet do not require a gap to exist between them.

I'd be more than happy to hear of any logically intuitive ideas or consequences of such an idea. I personally can't even begin to conceive of any such idea. It's a totally nonsensical concept to me.

The idea of discontinuous points with gaps that do not qualify as valid locations I can live with. This is an idea that I can imagine intuitively. I can even cheat a little bit and say to myself "Hey the gaps between the points are really there, we just can't get into them! They are forbidden to our physical existence!"

In that way, I can conceptualize the gaps in pure abstract theory while recognizing that logically they can't exist in any physical universe that might display this property that we call "quantity". They simply can't be considered even logically in any formalism that might try to model this property of our universe that we call quantity.

That's where I stand on the topic.

I would be more than happy to listen to any conceptual arguments for a continuum that is based on the concept of dimensionless points.

 

[NeutronStar]

Clarification of Points and Coordinate Systems

Yes, this is where I start to lose the plot. If a point is dimensionless what can it mean to say there can only be one point at each location? There seems to be some sleight of hand involved in defining points as imaginary and dimensionless but reifying their discrete locations.

The point is the location. The location is the point.

Does it make any sense to talk about two locations at the same location?

That would be precisely the same thing as trying to claim that there are two or more points at the same location. If they are at the same location then they are the same point.

Don't think of points as being little entities like sub-atomic particles. They are locations. Period amen. They aren't physical entities.

Doesn't this assume that these dimensionless points exist at some specific location within some coordinate system?

Once again you are getting way ahead of the game here. Coordinate systems are nothing more than systematic ways of labeling locations. Once you have the concept of a coordinate system you can have all the locations you like.

The ultimate restriction is that if your coordinate system is a field of locations, and locations are nothing more than dimensionless points in that system, then you are stuck with our original philosophical conclusion that there must necessarily be gaps between these locations. In other words, you're coordinate system (if it is to display a quantitative nature) must necessarily be a quantum field.

This is just a consequence of the logic associated with the concept of having more than one location.

I submit to you that if the universe really was a continuum it would not display the quantitative nature that our universe displays. It simply isn't possible to talk about more than one location in a universe that is a continuum.

It misses the point in a way, but not exactly. If the existence of a point does not imply a coordinate system then why should millions and billions do so? That is, if the gap between two points is filled with an infinite number of dimensionless points then the gap would seem to be the sum of gaps between an infinite number of points placed into a finite space, which to me seems to be zero.

Actually that's a very profound bit of philosophy right there and you may very well be correct. Our entire universe may not take up any 'space' at all actually. In fact, on the deepest philosophical level I wouldn't be a bit surprised if that isn't the 'true' nature of our existence.

However, if you want to think that deeply then consider this. A universe that is a continuum doesn't really have any space at all between any of its points so it wouldn't require any space to exist in its entirety either! :scream:

But getting back to the logic. You keep wanting to put more points into the gap between two points. But if you go back to the discussion of the preceding quote above our logic showed us that there must necessarily be a smallest gap between two points which cannot be thought of as being divided up further. In other words, by pure logic, there necessarily must exist some smallest gap where it is simply meaningless to talk about inserting more points. That was the whole purpose of the premise of thinking about what it would mean to talk about only TWO dimensionless points that are not the same point.


Oh,… and the existence of points does imply a coordinate system. Imagine a coordinate system that contains only two points. You're either on point A or point B. That's the entirety of that coordinate system. There's simply no other place to be. If point A and point B are the only two points that exist. And they are not the same point. Then you are either on point A or point B, but it's meaningless to talk about being half-way between them.

That was really the line of reasoning that I was trying to get at with the previous premise of considering what will happen if we restrict our condition to only two points. Talking about the point in-between them is meaningless because the two points are the universe of our coordinate system. To get from A to B we must make a quantum jump.

And more to the point (not pun intended), if the points are dimensionless and they are not the same point then they must be two different locations (because that's what a point is) and they cannot be touching because they are dimensionless. So there is no other conclusion to come to except that they are separate locations with no other location permitted to exist between them. Thus the notion of a gap where no points can logically be said to exist.

 

[NeutronStar]

On Reifying Abstract Concepts

The difficulty I have is in separating the concept of a point as used in the calculus, where it is dimensionless and can be arbitrarily close to a different point, and the idea of a point as a reified entity, as it is when the calculus is used as an answer to Zeno.

I don’t see how calculus can claim to have answered Zeno's concerns. I would highly recommend studying Weierstrass's epsilon-delta definition of the limit though. Then you can come to your own conclusions. That limit definition is often taught in first year calculus courses. Unfortunately it is usually passed over relatively quickly and most of the course time is spent doing algebraic manipulations to mechanically find limits and derivatives of popular functions.

This is the heart of the issue for me. It seems a muddle. I cannot see how one can reify the gaps without reifying the points. Presumably one can fit an infinite number of points into the gap between any two points. In this case the gaps between these points must be infinitely small, and zero at the limit. Also, according to this there are no gaps between the gaps, since the points are dimensionless, so the line must be made entirely of gaps. I struggle to make sense of that.

I don't think that anyone can reify the concept of a line. A line is an abstract notion that has no physical existence. However, I do believe that the concept of number can be reified in terms of collections of things. And it is in that sense that I can also reify the concept of a line segment. However, to do so requires the use of set theory, and since current mathematical formalism has a problem in that area too this presents a problem. That damn empty set is a real pain! I certainly don't want to get into that here.

However, for what it's worth, those gaps in the line can indeed be reified intuitively in the gaps in physical quantum fields. It's tricky business though! Some have claimed to have done it and it introduced irresolvable problems. Well, that's because they did it wrong.

All of this talk about discrete space does not imply the existence of an absolute space. Realizing that space is discrete does not deny relativity. The gaps between lines (or spatial coordinates) do not need to be Newtonian in nature. They can "flex". There's nothing in our original logic that prevents them from flexing. Our logic merely told us that they must exist. It said absolutely nothing about their actual nature.

If we are moving relative with respect to each other our gaps will appear to be different sizes just like everything else. It's also not just an illusion. Our gaps really will change size relative to each other. A third observer can look at us and say, "Hey! I say observer A's points existing inside of observer B's gaps therefore we can put points inside of gaps! The universe is a continuum after all!

Well actually the observation would be correct, but the conclusion would be incorrect. The conclusion is based on the idea of an absolute space. There is no absolute space, therefore talking about absolute gaps is just as meaningless. We need to consider only relative gaps here. :biggirn:

Alright, I really didn't want to go there, but I think it is important to realize that we aren't talking about the nature of any absolute space here. We are talking about the quantitative nature of our universe as a whole and we already know that our universe has this relativistic property.

In short, if you are going to attempt to reify points, lines, or gaps in a quantitative way you need to do it in a way that is compatible with physical reality. After all, it is the physical universe that exhibits this quantitative nature in the first place. Trying to imagine it entirely in an abstract sense is to abandon its origins.

But I take your point about the idea of a continuum. A continuum suggests one thing, and I think it was Leibnitz who argued that something that was one thing could not have physical extension. I agree with his argument, but do not see it as showing that reality is not (ultimately) one continuous thing. But that's a different can of worms.

Well, my only concern here is that if our universe is a continuum then why does it have a consistent quantitative nature?

If you can suggest an answer to that question I'm all ears.

 

[NeutronStar]

On Reality

There's something a little odd about this if we leave mathematics for a moment and consider reality. If a finite line can contain an infinite number of points then a finite line can contain an infinite number of gaps between those points, each of which has a finite length. These gaps must be infinitely small, and, if I understand the earlier discussion about the way limits are treated in the calculus, they must therefore be considered to be equal to zero.

Precisely! And that contradicts the idea that separate points must be separated by a gap otherwise they would be the same point!

I'm with you on this one. "Infinitely small", to me can only mean one thing,… zero! Yet we have clearly shown that points that are not the same point must necessarily be separated by some non-zero gap.

A continuum always presents irresolvable paradoxes like this for me. So far I have not found an irresolvable paradox like this associated with discontinuous points. If you find one let me know, I'll think about it.

I'd be interested to see if your proof convinces others here. At the moment it seems to me that if points are dimensionless then to talk about how many can be fitted into a finite space is to make a category error.

I honestly don't expect to convince anyone of anything. Most mathematicians will defend traditional mathematical ideas to their death whether they can justify them intuitively or not. The idea that the number line is a continuum is pretty well accepted by the mathematical community. They automatically go into "defense" mode when anyone suggests otherwise.

The idea of an empty set has also been concretely accepted. To attack that idea is almost taken as a personal attack by many mathematicians. They will defend it to their death. This is because mathematicians have genuinely become "purists". They sincerely don't care whether mathematics matches up with any observed quantitative property of the universe or not. All they car about anymore is proving that things satisfy arbitrary formal definitions. This is what mathematics has become over the centuries. It's basically a meaningless axiomatic system that does indeed fall prey to Gödel's inconsistency theorem.

This bothers me because I'm interested in science and discovering the true nature of our universe. Yet modern science is becoming almost entirely mathematical and less experimental. So if mathematics doesn't reflect the true nature of quantity, and science is becoming more mathematical, then eventually science will fail to describe the universe.

In any case, I'd like to hear your thoughts on anything that I said here, and also any ideas that you might have on trying to describe the nature of two distinct points in a continuum. I can't even begin to imagine trying to intuitively describe the notion of two dimensionless points in a continuum. How would one go about conceiving of such an idea?

 

[matt grime]

I honestly don't expect to convince anyone of anything. Most mathematicians will defend traditional mathematical ideas to their death whether they can justify them intuitively or not. The idea that the number line is a continuum is pretty well accepted by the mathematical community. They automatically go into "defense" mode when anyone suggests otherwise.

The idea of an empty set has also been concretely accepted. To attack that idea is almost taken as a personal attack by many mathematicians. They will defend it to their death. This is because mathematicians have genuinely become "purists". They sincerely don't care whether mathematics matches up with any observed quantitative property of the universe or not. All they car about anymore is proving that things satisfy arbitrary formal definitions. This is what mathematics has become over the centuries. It's basically a meaningless axiomatic system that does indeed fall prey to Gödel's inconsistency theorem.

This bothers me because I'm interested in science and discovering the true nature of our universe. Yet modern science is becoming almost entirely mathematical and less experimental. So if mathematics doesn't reflect the true nature of quantity, and science is becoming more mathematical, then eventually science will fail to describe the un
In any case, I'd like to hear your thoughts on anything that I said here, and also any ideas that you might have on trying to describe the nature of two distinct points in a continuum. I can't even begin to imagine trying to intuitively describe the notion of two dimensionless points in a continuum. How would one go about conceiving of such an idea?

Then don't do mathematics if it offends you since mathematics is the manipulation of formal objects, some of which can model (quite accurately) things in the "real world".

That the real line is a continuum (an ugly phrase that few of us would ever use) is a formal consequnce of the axioms. That axiomatized system has proved good enough to put GPS satellites up there and verify Einstein's theories of relativity. It's also good enough to describe quantum states and model quantum logic gates.

If we don't have the empty set then what is the set of real solutions to the equation x^2+1=0? It is a useful tool, that is all.

If you seek things that aren't there, you may not find them.

You basically seem to be attacking mathematicians for not being physicists. Well, fortunately there are physicists doing physics (experimentally verifiable things).

 

[Hurkyl]

I can't answer that because I can't even being to conceive the idea of a continuum.

If you cannot even conceive of the idea of a continuum, how can you possibly find it paradoxical?

In fact, if you look at the entire situation from a bird's-eye view you'll clearly see that Weierstrass's limit definition actually forbids the conclusion of a continuum.

A bold statement from someone who can't even conceive the idea of a continuum.


Incidentally, the usual usage of the term "infinitessimal" means something that has a size smaller than any rational number. (Yes, 0 is infinitessimal) Though, in algebraic geometry, it's used (I believe) to refer to a quantity x that satisfies x^n=0 for some positive integer n.

The idea of a continuum simply has too many logical inconsistencies associated with it for me.

When you can start from an accepted definition of "continuum" and derive a contradiction with rigorous logic, then you can make this statement.

The most profound of which is the idea of two dimensionless points, which are not the same point, but are also not separated by any gap. That is simply an irresolvable paradox for me. It's clearly a logical contradiction that cannot be resolved.

You're right, it is. (more or less)

And that's fine, because aside from yourself, no mathematician or physicist has ever claimed that's what "continuum" means. I'll repeat my example again, maybe you'll see it this time.

There is no gap between the point 0 and the collection of points {1, 1/2, 1/3, 1/4, ...}. Yes, it is true that there is a gap between 0 and each individual point of that collection, but there is no gap between 0 and that collection when taken as a whole.

But it does have great implications with respect to an idea associated with set theory, and that is the idea of an empty set. I don't want to get into that here because it would appear to be a side-track.

If by "great implications" you mean "little to no impact", then yes. If you don't want to get into it, then you probably shouldn't have said it.

Just as one final note, this whole continuum vs. discrete issue does related directly to set theory.

Wrong again. One can talk about "continua" and discrete topologies without using any set theory.

Yes. This kind of falls into the chicken question. Which came first, the coordinate system or the point?

Well, if you knew any history at all, it's fairly obvious the point came first. Furthermore, mathematicians are fully capable of speaking about points without using coordinates at all -- we don't even need to have a notion of distance!

If you want to build a "real" physical universe that exhibits a quantitative property

The most important prerequisite to "building" a real physical universe that exhibits a "quantitative" property is, well, for there to be a real physical universe that exhibits a quantitative property. Once you have empirical evidence that your "method" (if it can be said to even be a method) gives better results than what people do now, then you'd have an argument.

Imagine if the mathematical community would have realized this before Max Planck discovered the grainy nature of the universe.

That would be tough, because, as far as all experiments have shown, the universe does not have a grainy nature. But of course you've heard us tell you this before, and just ignore it. The quantization in quantum mechanics is more analogous to the vibration of a violin string than the pixels on a computer screen. It moves through a "continuum", but it can only move in some combination of a discrete set of ways.

Humans simply aren't capable of thinking in terms of only two points.

Wow, so I'm not human?

Does it make any sense to talk about two locations at the same location?

Actually, there are situations where it makes sense to talk about a point that is "made up of" other points. (This is related to what I meant earlier about technical details)

All of this talk about discrete space

(You're the only one talking about discrete space)

They automatically go into "defense" mode when anyone suggests otherwise.

Is that what it's called when people who show little understanding of the subject speak like they know better than all the experts, and the experts step into refute the plethora of mistakes, rhetoric, and self-aggrandizement, not because they think it will convice this person, but for the sake of others who actually want to learn about the subject?

 

[Canute]

Hurkyl

Thanks for trying but even that little bit of mathematics you posted to me earlier was beyond me. I think I'd better retire from this one. To me the nature of the number line, or our concept of the number line, is an epistemilogical or meta-mathematical issue, and I can't accept that it is this or that just because it has been defined for formal reasons as being like this or that. Not that I've got anything against defining it formally, but only if I agree with the definition.

I wish I could talk about it more mathematically. I've got nothing against doing it, but I sat next to a fantastic girl in a mini skirt all through an important year of mathematics classes, and by the end of it I'd forgotten even what I'd managed to learn in the previous year. My mathematics never recovered.

 

[Canute]

I can't answer that because I can't even being to conceive the idea of a continuum. The very idea is paradoxical to me. So whether calculus could be used to model such an idea is not even a sensible question to me.

I'm with Hurykl on this one. There seems no reason that your inability to conceive of a continuum should have any bearing on whether or not the calculus can be used to model one. The question seemed sensible to me, and I thought the answer would be well known. If you don't know the answer then on what basis are you arguing that the number line cannot be considered as a continuum? Because you can't imagine it?

I can say that there is absolutely nothing in any of the calculus definitions that supports any idea of a continuum. On the contrary the calculus definitions clearly depend on things being discrete.

Yes, this is what mathematicians generally say. I disagree. I might well be wrong, I know that, but I have yet to be convinced that the calculus, with its notion of infinititessimals, differentials, fluxions, or whatever we call them, would have to be altered in any way if we wanted to use it to model the mathematics of a continuum.

If, by some fat chance, the mathematical community would decide to accept the notion that a finite line must be constructed of finite points, this decision would not affect calculus at all. Nor would it have any affect at all on Weierstrass's definition of a limit. Weierstrass's definition simply doesn't not require a continuum to work.

I agree with you that there is something odd about the way mathematicians define lines and points. But if a point is defined as infinitely small then by definition there must be an infinity of them on a finite line.

So in answer to your question,… "infinitesimals" (or "differentials" as are they are more commonly referred to by mathematicians) do not overcome any awkward infinities. It is the entire Weierstrass delta-epsilon definition of a mathematical limit that overcomes these problems. And it actually does this partially by requiring that delta must be greater than zero (i.e. it forces us to address the problem as a discrete problem rather than as a continuum)

If this is the case then I have misunderstood something. I thought the notion of infinitessimals entailed the notion of limits. I don't quite see how we could have one without the other.

 

[Canute]

Yes. This kind of falls into the chicken question. Which came first, the coordinate system or the point?

The best answer that I can give you is that if there must necessarily be gaps between points, then any coordinate system that is considered to be a field of points must necessarily contain gaps. And the best way to think about those gaps is to think of them as quantum jumps. You simply can't talk about locations within these gaps. It's nonsensical to try to do so.

That seems a rather ad hoc solution. Of course there must be gaps between points if we define points as necessarily having gaps between them. The question is, is this conceptual picture of gaps and points logically coherent.

The problem with "pure thought" is that you can always imagine setting up finer coordinate systems within those gaps. And in "pure thought" you can. But logic has already dictated to us that those gaps must necessarily exist no matter how fine we pretend to make them.

But you didn't use logic, you took it as axiomatic that those gaps exist.

The bottom line is this. If you want to build a "real" physical universe that exhibits a quantitative property that universe will necessarily have to contain these gaps. So to actually build one you will have to choose an actual value for the gaps. Once that's been done the universe you finally build will contain a fine structure below which it will be absolutely meaningless to talk about because those location (or points) simply don't exist.

If I read this right I agree, especially since you put "real" in quote marks.

The thing that I find so amazing about the "pure thought" or "pure logic" is that even though it can't give us an absolute number for these gaps it can tell us that they must exist.

I disagree. You say they exist, but logic does not tell us that they exist. Logically those gaps give rise to paradoxes.

To me this is very powerful. Imagine if the mathematical community would have realized this before Max Planck discovered the grainy nature of the universe.

I don't think he discovered that.

The mathematical community could have actually predicted quantum physics. They may not have been able to predict the numerical value of Planck's Constant. But they could have at least predicted the quantum nature of the universe. Instead they decided to go down the road of pure abstraction pursing some lofty notion of a continuum. That notion simply isn't supported by the quantitative behavior of our universe.

You keep saying this, but providing no argument. It's not obvious to everyone that the universe, or 'the fabric of reality', can be represented as being quantised without contradiction. I gather that Charles Sanders Peirce also argued that the number line was better represented as being a continuum, although I haven't got around to reading him yet.

Well, you're just to used to thinking in terms of coordinates. I tried to make it clear earlier that the idea of a single point is really meaningless. It takes at least two distinct points before the idea of a point actually begins to make sense.

I'd say it takes at least two points and a gap.

Humans simply aren't capable of thinking in terms of only two points. Intuitively the first thing that pops into their mind is the gap between the two points and the fact that this also can be thought of as a point. In other words, it's really hard to fight the urge to think of more than two points at a time. Yet the whole purpose of the exercise is to imagine that only two points exist. What would be the result of that? They would have to be separated by a gap otherwise they would be the same point.

I agree. But our inability to conceptualise two points without a gap between them into which other points could be fitted doesn't mean, contrary to intuition or common sense, that 'really' it makes sense to say that two points can exist without a gap between them. It may just be that it makes no sense think that. If we define two points as being different they must be different in some way. If the only difference between them is their location then they must be at different locations.

That's the conclusion. Period amen. To try to claim that we can then add more points is to miss the whole exercise of thinking of only two points. The thrust of the argument is that it is simply impossible to conceive of an idea of two dimensionless points without thinking about a gap existing between them. The urge to toss a third point that represents the location between them must be suppressed. In other words, that is a forbidden location and therefore cannot be thought of as a point. Why? Because the whole premise is that only two points exist. To introduce a third point is to violate the premise.

But you're saying no more than that we mustn't think about gaps because you say so.

It's a purely philosophical argument that appeals to the intuition showing that in order to conceive of an idea of at least two dimensionless points (and no more) we have no choice but to accept this gap between them that we cannot (by our premise) consider to be an additional location or point.

I can personally handle this type of intuitive comprehension and understand the logic upon which is it based. So I have no problem with the consequences of this intuitive idea. It simply means that if I want to move from one point to another I'm going to have to make a quantum jump to get there.

Yes, this where Zeno becomes relevant. Where will you be when you are not at a point, during these quantum jumps? These jumps would have to take no time, otherwise you be late arriving at the next point.

I'm more than willing to listen to any logical or intuitive arguments concerning any ideas of a continuum that is constructed of dimensionless points.

One way to conceive of a continuum is that it is constructed of an infinity of dimensionless points, the other is as one undifferentiated thing. This is the only choice we have, and it seems a paradoxical one. However there are ways around the paradoxes.

So we need to describe a logical and intuitive idea of a continuum of two points. In other words, two dimensionless points that are not the same point and yet do not require a gap to exist between them.

I'd be more than happy to hear of any logically intuitive ideas or consequences of such an idea. I personally can't even begin to conceive of any such idea. It's a totally nonsensical concept to me.

It does seem that way. I don't want to disrupt the discussion, but I should mention that this apparent paradox is resolved in Buddhist cosmology.

I would be more than happy to listen to any conceptual arguments for a continuum that is based on the concept of dimensionless points.

I'd be happy to attempt one, but this isn't really the place. I'm going to retire from this thread because my mathematics isn't up to it. But I'll carry on under metaphysics if you want.

 

[NeutronStar]

That the real line is a continuum (an ugly phrase that few of us would ever use) is a formal consequnce of the axioms. That axiomatized system has proved good enough to put GPS satellites up there and verify Einstein's theories of relativity. It's also good enough to describe quantum states and model quantum logic gates.

This is ridicules. None of the things that you've mentioned here depend on the idea of a continuum or on the idea of an empty set. If I believed that my ideas were going to change basic classical physics or even special relativity I would have big problems with my idea. Not to mention the fact that I would also drop the idea in a heartbeat. What do you think I am? A crackpot?

Everything that we are talking about here has to do with the nature of infinities. These ideas won't have any affect on classical physics or special relativity. What they very well may have an affect on, however, is various aspects of quantum theory and/or general relativity. Unfortunately I'm not well educated enough in the mathematics of those fields to know where those affects will show up. I wish I did know because that would give me great insight into the problem one way or the other.

May I ask you the following questions?

Right now it seems to me that many mathematicians are confused about the actual property of cardinality between the set of real numbers and the set of natural numbers so,…

Can you tell me in clear intuitive terms what this difference is?

Does the set of real numbers actually contain "more" elements than the set of natural numbers?

Or is the cardinal difference between these two sets based on a different quality other than quantity?

If you answered the previous question "yes:, then just what is this other quality? What makes these sets cardinally different from each other if one does not contain more elements than the other?

 

[Canute]

The point is the location. The location is the point.

Does it make any sense to talk about two locations at the same location?

No, two locations are clearly different locations if they have been defined as such. I don't mind whether we call them points or locations. We have defined them as being the same thing.

That would be precisely the same thing as trying to claim that there are two or more points at the same location. If they are at the same location then they are the same point.

Ok. And if they are the same point then they are at the same location.

Don't think of points as being little entities like sub-atomic particles. They are locations. Period amen. They aren't physical entities.

I understand that. These are points located in our imagination.

Once again you are getting way ahead of the game here. Coordinate systems are nothing more than systematic ways of labeling locations. Once you have the concept of a coordinate system you can have all the locations you like.

But only if your coordinate system is infinitely finely grained.

The ultimate restriction is that if your coordinate system is a field of locations, and locations are nothing more than dimensionless points in that system, then you are stuck with our original philosophical conclusion that there must necessarily be gaps between these locations. In other words, you're coordinate system (if it is to display a quantitative nature) must necessarily be a quantum field.

I didn't mean to say anything much about coordinate systems. I was just pointing out that two locations imply a coordinate system.

I submit to you that if the universe really was a continuum it would not display the quantitative nature that our universe displays. It simply isn't possible to talk about more than one location in a universe that is a continuum.

That was Parmeneides' and Zeno's point, and many others. The question is perhaps, what meaning can points and locations have outside of the coordinate system we call spacetime. As far as we can tell spacetime, our universe anyway, has not always existed, but exploded into being just as if the BB happened at every point in it at once.

Actually that's a very profound bit of philosophy right there and you may very well be correct. Our entire universe may not take up any 'space' at all actually. In fact, on the deepest philosophical level I wouldn't be a bit surprised if that isn't the 'true' nature of our existence.

Yes, this is the fundamental issue. Really we're talking about the nature of the one and the many, and back with Plato et al.

However, if you want to think that deeply then consider this. A universe that is a continuum doesn't really have any space at all between any of its points so it wouldn't require any space to exist in its entirety either! :scream:

Exactly. What could it mean to say that the universe takes up space? The idea makes no sense.

But getting back to the logic. You keep wanting to put more points into the gap between two points. [/quote}
I don't want to put them in. It just follows from the fact that points are defined only by their location that there must be points between different points. It's just a consequence of the definition.

But if you go back to the discussion of the preceding quote above our logic showed us that there must necessarily be a smallest gap between two points which cannot be thought of as being divided up further. In other words, by pure logic, there necessarily must exist some smallest gap where it is simply meaningless to talk about inserting more points. That was the whole purpose of the premise of thinking about what it would mean to talk about only TWO dimensionless points that are not the same point.

I'm sorry but I cannot conceive of a gap so small that an infinitessimal wouldn't fit into it. It's possible to define gaps in such a way as to stop me from doing this, for practical or formal reasons, but you can't reify a definition.

Oh,… and the existence of points does imply a coordinate system. Imagine a coordinate system that contains only two points. You're either on point A or point B. That's the entirety of that coordinate system. There's simply no other place to be. If point A and point B are the only two points that exist. And they are not the same point. Then you are either on point A or point B, but it's meaningless to talk about being half-way between them.

That seems self-contradictory, but I may be misreading it.

And more to the point (not pun intended), if the points are dimensionless and they are not the same point then they must be two different locations (because that's what a point is) and they cannot be touching because they are dimensionless. So there is no other conclusion to come to except that they are separate locations with no other location permitted to exist between them. Thus the notion of a gap where no points can logically be said to exist.

Well, there is at least one other conclusion, and that is that your definition of points, locations and gaps is incoherent. Btw, I'm not trying to defend some particular theory here, I simply can't see how you arrive at your conclusions.

 

[NeutronStar]

It's not a personal limitation. It's a logical contradiction.

I'm with Hurykl on this one. There seems no reason that your inability to conceive of a continuum should have any bearing on whether or not the calculus can be used to model one. The question seemed sensible to me, and I thought the answer would be well known. If you don't know the answer then on what basis are you arguing that the number line cannot be considered as a continuum? Because you can't imagine it?

Ok, this was due to poor phrasing on my part. When I say that I can't imagine it I mean that I can't logically justify it. I don't mean to imply that I have a limited imagination.

In other words, here is what we have to imagine in order to "justify" the idea of a continuum.

We begin with the fact that points are dimensionless. Remember, if we claim to have points that have dimension then we have merely shifted the discrete gap into the points and we haven't really solved the problem.

So, the points must be dimensionless.

Ok. So far so good. I can conceive the idea of a dimensionless point as simply the idea of a location that has not breadth.

Now, we need to consider what it means to have two distinctly different points that are not the same point. (in other words, two distinct locations that are not the same location)

But wait, we're not done! We have to maintain the concept of a continuum which is the idea there there are no gaps between these distinctly different dimensionless points.

Well, how are you going to envision that? In other words, how are you going to logically justify that concept?

When I say that I can't envision it, I simply mean that it is a logical contradiction. I maintain that it cannot be logically justified. And in that sense I cannot conceive it as a meaningful idea. It's illogical.

The idea of two distinctly different locations that are located at the very same location (which they must be if these locations have no breadth, i.e. are dimensionless points) is simply a logical contradiction plain and simple.

So I probably shouldn't just feebly claim that I can't envision it. I should boldly claim that it is a logical contradiction and therefore it is nonsensical.

How can anyone claim to have an idea that cannot be conceived?

The idea of a continuum is a logical contradiction pure and simple.

This has absolutely nothing at all to do with my own personal inability to imagine it. I can clearly see what it would take to try to imagine it and I see that it is a direct contradiction in logic. We simply can't claim that two distinct dimensionless points are not the same point. It is a logical contradiction. Its an unworkable idea!

This has absolutely nothing at all to do with my own personal abilities to comprehend anything. I claim that anyone who believe that they can comprehend this idea if necessarily fooling themselves.

I would be more than happy to hear arguments to the contrary. But so far no one has offered any logically consistent picture of how a continuum can logically be said to exist even as an idea.

Yet, I have offered a logically consistent picture of what it means to have two discrete points. So I see this position as being more meaningful.

 

[matt grime]

What do you think I am? A crackpot?

Do you really want an answer to that?

Right now it seems to me that many mathematicians are confused about the actual property of cardinality between the set of real numbers and the set of natural numbers so,…

We are confused? Pray tell what the correct definition is? Simply that they are infinite? Well, that's a very old fashioned view that we can *refine*.

Can you tell me in clear intuitive terms what this difference is?

Why must it necessarily be intuitive? In the category of SET they lie in different isomorphism classes. That's all.

Does the set of real numbers actually contain "more" elements than the set of natural numbers?

that's up to you to state what you mean by "more" isn't it? By *analogy* with the case of finite sets, we could say Y has strictly more elements than X if there is an injection from X to Y, but no bijection, that is X is in 1-1 correspondence with a proper subset, but never the whole of Y. That seems a reasonable generalization of "more" doesn't it, I suppose.

With it, we can say seemingly natural statements such as there are real numbers that are not algebraic, since there are strictly more real numbers than algebraic ones. However, that is obscuring the simple fact that algebraic numbers are countable and Reals not.

Or is the cardinal difference between these two sets based on a different quality other than quantity?

no it is to do with the isomorphism class in SET, nothing more nor less.

If you answered the previous question "yes:, then just what is this other quality? What makes these sets cardinally different from each other if one does not contain more elements than the other?

i didn't answer "yes", or perhaps you think I did. Whatever, the point is that the only person who appears not to know what cardinals are is you.

 

[Canute]

This bothers me because I'm interested in science and discovering the true nature of our universe. Yet modern science is becoming almost entirely mathematical and less experimental. So if mathematics doesn't reflect the true nature of quantity, and science is becoming more mathematical, then eventually science will fail to describe the universe.

I agree with your diagnosis, but not with the cure.

In any case, I'd like to hear your thoughts on anything that I said here, and also any ideas that you might have on trying to describe the nature of two distinct points in a continuum. I can't even begin to imagine trying to intuitively describe the notion of two dimensionless points in a continuum. How would one go about conceiving of such an idea?

I also agree that this is an inconsisistent idea. The only points there can be in a continuum are conceptual ones.

 

[Canute]

We begin with the fact that points are dimensionless...(snip) ...So, the points must be dimensionless.

Hmm.

Ok. So far so good. I can conceive the idea of a dimensionless point as simply the idea of a location that has not breadth.

It's the only sort of dimensionless point there is.

Now, we need to consider what it means to have two distinctly different points that are not the same point. (in other words, two distinct locations that are not the same location)

But wait, we're not done! We have to maintain the concept of a continuum which is the idea there there are no gaps between these distinctly different dimensionless points.

Why? Your points are in your imagination, you won't find any out there in reality.

The idea of two distinctly different locations that are located at the very same location (which they must be if these locations have no breadth, i.e. are dimensionless points) is simply a logical contradiction plain and simple.

I agree.

The idea of a continuum is a logical contradiction pure and simple.

I more or less agree with that also. However I don't derive the same conclusions from it.

This has absolutely nothing at all to do with my own personal inability to imagine it. I can clearly see what it would take to try to imagine it and I see that it is a direct contradiction in logic. We simply can't claim that two distinct dimensionless points are not the same point. It is a logical contradiction. Its an unworkable idea!

I don't think anyone has claimed that.

But so far no one has offered any logically consistent picture of how a continuum can logically be said to exist even as an idea.

That's a fair point, but I won't respond here.

 

[StatusX]

You seem to be saying that because the distance between any two points is finite, there can't be a continuum because there would have to be a distance between consecutive points. The flaw in this argument is that there arent any consecutive points in a continuum. This is counter-intuitive, but not illogical. Between any two points you pick there are still an infinite number of points, regardless of how close they are. There is no next number after 1. The open set (0,1) has no greatest or least element.

 

[Hurkyl]

A brief introduction to topology.

A topology consists of two kinds of things:
(1) points
(2) neighborhoods

The basic relationship between points and neighborhoods is that neighborhoods contain points. In fact, in the set theoretic approach to topology, neighborhoods are defined to be the set of all points they contain.

Furthermore:
Each point is contained in at least one neighborhood.
If two neighborhoods overlap (that is, have a point in common), then there is an entire neighborhood contained in both.


One example of a topology is the real line. The points of the real line are simply real numbers. The neighborhoods of the real line are the open intervals: that is, sets of the form {x | a < x < b}. for some a and b.


Non-mathematical aside: The points, by themselves, tell you very little. The neighborhoods are the "soul" of topology -- they are what describes how the points relate to each other, they describe "texture" of the topological space. As we see with the example above, the neighborhoods of the real line are precisely the neighborhoods Canute mentioned. I don't think that's a coincidence: Canute wasn't the first person to realize that these ranges are important to describing a "continuum".


Back to the mathematics.

Another type of example of a topology is a discrete space:

The points can be anything (but, IIRC, there's supposed to be at least 1).
Then, for each point, there is a neighborhood that consists of that point and nothing else.

Each point in a discrete space is isolated: for each point there is a neighborhood that contains that point and nothing else.

Contrast this with the real line: every neighborhood of a point contains many other points.


Next, I'd like to mention the notion of nearness. If you have a point (let's call it P), and you have some set of other points, (let's call it A), then the phrase P is near A means that every neighborhood of P contains a point in A.


Let's use the real line again as an example. Let's let P be the point 0, and let A be the set {1, 1/2, 1/3, 1/4, ...}. Then, P is near A.

Proof: Let (a, b) be any neighborhood of P. That means a < 0 < b. However, there exists some integer n such that 1/n < b, which means that 1/n is in the neighborhood (a, b). QED


Note that the intuitive notion of a "gap" can now be described in terms of nearness -- no need to have any concept of there being some other locations that make up the gap. We can say there's a gap between a point P and a set of points A if P is not near A.

So, we can see that in the discrete space, there is a gap between a point and any set not containing that point! However in the real line, there is no gap betwen 0 and {1, 1/2, 1/3, 1/4, ...}. But, of course, there is a gap between -1 and {1, 1/2, 1/3, 1/4, ...} (because the neighborhood (-1.5, -.5) doesn't contain any element of the set)

And, just as we'd expect, there is a gap between any two points on the real line: for instance, there's a gap between 0 and {1} because the neighborhood (-0.5, 0.5) doesn't contain any element of {1}.


There's obviously a lot more to say. I haven't even gotten far enough that we could start speaking about what it means to be a "continuum". But, I was just trying to give a taste about how one can speak of a space being made up of individual points without them being necessarily isolated.