What is the relationship between points and neighborhoods in topology?

[jcsd]

It's not as bad as it sounds. Physicists really wouldn't be forbidden to use these concepts, they would just realize that they aren't quantitative concepts and that they actually arise from self-referenced relationships. Having a firm understanding of this might actually help physicists better understand the nature of what they are actually attempting to describe.

Yes it is because clearly then by your defintion physical quantities are no longer 'quantitve concepts' as there is no way of avoiding solutions to funademntal physical equations where all the pararmeters are rational, but the solution (representing a physical quantity) is irrational/trancendental.

Many, though not all, physicists already know how the real numbers are constructed. You have relaize thta to physicsts maths is a tool.

After all, I'm not really suggesting that we would be forbidden to describe irrational concepts and such. I'm simply saying that they shouldn't be thought of in terms of "number". They should be seen as the relative relationships from which they arise. Thinking of them as pure number concepts that have some sort of meaning outside of these self-referenced relationships is actually not true anyway so why try to define them as such?

There's no reason not to think of them as a number as has already been pointed out your objections are more down your own misconceptions about maths than anything else.

 

[matt grime]

Don't forget we've also got the ill-defined (fuzzy) notion of "self referenced" which applies to sqrt(2) for some bizarre "collecting together" reasons, and pi because of the "self reference of the diameter to circumference of the circle", and e because e can be defined as

lim n to inf of (1+1/n)^{1/n}

quite what links all those and implies that the only way to describe these quantities is self referential is as yet unexplained. Why for instance is the fact that 1/n is defined to be the number that when multiplied by n gives 1 not a self referential definition. Is 0 self referential since it is the limit of(1/n)^n?

 

[synergy]

Calling infinity "one thing" is not qualitative at all!

It's like: A pack of cards, A bussload of people, or a single proof (which contains many concepts that were themselves proven in proofs). A single set is an "object" which may be countably or even uncountably infinite if you examine the elements of the set. That's why we can talk about the cardinality of a set being finite, infinite, or uncountably infinite. If we can construct a bijection between the natural numbers and the members of a set, than that SINGLE set has cardinality = infinity (i.e. has an infinite number of MEMBERS of the set). So we can talk about a single element as being singular, or an entire set of those elements as being singular. We can even have a SINGLE collection of collections of sets which each contain an infinite number of elements. etc. Like Matt said, look up point-set topology. There is even a smallest uncountable set (weird idea, huh?).
Aaron

 

[matt grime]

Well ordering the cardinals requires the axiom of choice - take it or leave it (Cantor took it).

 

[Canute]

Evidently there is no mathematically acceptable way to express an infinite "decimal" series that is not assumed to sum to its notational limit because, as was just pointed out, this *is* the definition of a decimal series in mathematics.

It's a bummer ain't it!

I'm genuinely astonished. I can see that it's usually useful to treat an infinite series as summing to its limit, but I didn't know that mathematicians believed that such a series really is equal to it. Isn't this to assume that the number line is a series of points and a continuum at the same time?

If we start with the concept of a dimensionless point we can show that any finite line of dimensionless points must necessarily be discontinuous. That is much easier to see. Then once we have understood the proof of this then it should be obvious that any finite line that is said to be a continuum cannot be constructed with dimensionless points.

I've agreed with almost everything you've posted so far, but this seem like the reverse of the truth. To me it seems that the only way to model a continuous line mathematically is by treating it as a series of dimensionless points. If the line is discontinuous then the points wouldn't be dimensionless. Isn't it the fact that spacetime is continuous and that the the calculus models it as such that allows the calculus to work in the first place?

So now we have this thing that is both quantitatively infinite and quantitatively One at the same time! If you actually think about this for a moment doesn't it make you wonder whether infinity equals one?

I suspect that mathematician George Spencer-Brown would agree with you. If I understand him right he regards infinities as conceptual potentia that should not be reified. (And, thinking of what you said about sets, he regards Russell as 'a fool' for his misunderstanding of empty sets).

A point is a one location. It must be dimensionless otherwise it would represent more than one location.

It seems that way to me. That's why I was struggling with your idea of a line of dimensionless points, since the line would surely be dimensionless.

After all, I'm not really suggesting that we would be forbidden to describe irrational concepts and such. I'm simply saying that they shouldn't be thought of in terms of "number". They should be seen as the relative relationships from which they arise. Thinking of them as pure number concepts that have some sort of meaning outside of these self-referenced relationships is actually not true anyway so why try to define them as such?

Most of the mathematics is well beyond me but if I understand you right then I agree. I'm not clear yet why anyone would disagree.

 

[synergy]

qualitative difference

Okay, I did say that quantitatively we can call an infinite set "one thing" but you were talking qualitatively, so I guess you could say that qualitatively an infinite set is different than one of its members. I would say it is still a single set. This is similar to the pack of cards being different than a carton of eggs. For one thing, they are different types of things, and for another they contain a different number of elements. An infinite set is both qualitatively and quantitatively different than a finite set. And it is both qual. and quan. different than one of its members (just as a carton of eggs is different than a single egg). Since we can talk this way about finite sets of eggs, with a "SINGLE" carton and a single egg, what is the real difference between a single finite collection and a single infinite collection, except the number of elements in each?
Aaron

 

[synergy]

continued

All this to say, that qualitatively an infinite, non-terminating, possibly non-repeating number is no different than a finitely represented terminating number.

 

[Hurkyl]

I'm genuinely astonished. I can see that it's usually useful to treat an infinite series as summing to its limit, but I didn't know that mathematicians believed that such a series really is equal to it. Isn't this to assume that the number line is a series of points and a continuum at the same time?

Well, when you disagree with the very notion of a number (e.g. mathematically, each number is a fixed thing, it doesn't "expand"), it shouldn't be surprising that you find more sophisticated ideas disagreeable.


The value of an infinite sum is defined to be the limit of the partial sums because it's the most convenient way to define it, and it's in line with many mathematicians' intuition.


There are other ways one could go about defining the value of an infinite sum, but since they turn out to yield the same value as the limit definition, there's no gain.

 

[NeutronStar]

Evidently there is no mathematically acceptable way to express an infinite "decimal" series that is not assumed to sum to its notational limit because, as was just pointed out, this *is* the definition of a decimal series in mathematics.

It's a bummer ain't it!

I'm genuinely astonished. I can see that it's usually useful to treat an infinite series as summing to its limit, but I didn't know that mathematicians believed that such a series really is equal to it. Isn't this to assume that the number line is a series of points and a continuum at the same time?

It's quite refreshing to meet someone who also sees the logical incompatibility between the concept of a series of points and the concept of a continuum. Most mathematicians don't seem to appreciate the logical inconsistency associated with these two entirely different concepts.

If we start with the concept of a dimensionless point we can show that any finite line of dimensionless points must necessarily be discontinuous. That is much easier to see. Then once we have understood the proof of this then it should be obvious that any finite line that is said to be a continuum cannot be constructed with dimensionless points.

I've agreed with almost everything you've posted so far, but this seem like the reverse of the truth. To me it seems that the only way to model a continuous line mathematically is by treating it as a series of dimensionless points. If the line is discontinuous then the points wouldn't be dimensionless. Isn't it the fact that spacetime is continuous and that the calculus models it as such that allows the calculus to work in the first place?

Calculus doesn't actually model a continuum at all. If you exam all of the calculus definitions with great care you will actually see how they avoid the concept of a continuum altogether. In fact, all of modern calculus is ultimately based on Karl Weierstrass's definition of a limit. And Weierstrass was very clever in avoiding any direct confrontation with the idea of a continuum. So calculus doesn't actually model a continuum at all. It does however address a mathematical notion of continuity which doesn't actually imply a continuum at all. This is really quite clear if you simply inspect the formal definition in great detail.

But forget about the calculus for now. There are much simpler ways to approach the concept. In your first quote above you seem to recognize intuitively that there is a logical inconsistency between the idea of a continuum and the idea of a series of points. Yet in your second quote you seem to be indicating that if a line is thought of as being made up of a series of points that these points should maybe have some dimension to them. Actually, that isn't the case. A line that is made up of dimensionless points cannot possibly form a continuum. I'll try to explain this here, but please bear with me because it's some heavy logic and this is a short post.

The Dimensionless Point
Ok, we begin with the postulate that points are dimensionless. They have no "physical" existence in the sense that they don't take up any space. In fact they aren't actually entities. Don't think of them as entities. Think of a point as nothing more than a location period amen. A point is a location.

That's the foundational idea.

Now,… what does it mean to have a single point? In truth such a concept is totally meaningless. Why? Because a point is a location. If all we have is a location and nothing else then the whole concept of location has absolutely no meaning. Something can only be located relative to something else. So the idea of having a single point is absurd. Unless of course you already imagine that you have a 3-D space or coordinate system with which you can use to refer to your point. But that's really cheating because in that case your imaginary space is already full of possible locations and therefore it is full of points.

So trust me on this. The concept of a single location in an otherwise empty universe is a meaningless idea. Fortunately for us we never have to think about such lonely points.

A New Premise
Let there exists a second location which is not the same as the first location. Ok, now we have a meaningful relative concept. We have two locations which are not the same location. We still don't need a coordinate system or anything. All we need to know is that we have two points that are not the same point. They can't be at the same location because, by definition (i.e. a point is a location) they would be the same point if they were at the same location. So it follows logically that any two points that are not the same point must necessarily be separated by some gap. If this wasn't the case they'd be the same location, and thus they'd be the same point.

So we can clearly see from this line of reasoning that any two points, that are not the same point, must necessarily be separated by a gap. There may be an urge by the reader to want to start shoving more points in between these two primitive points. But I seriously ask, "What's the point in doing that?". The point's already been made. No matter how many points we imagine tossing in the gap there will always be one basic naked truth left,..

"Any two points that are not the same point must necessarily exist at different locations, and since they are dimensionless points this means that they must necessarily be separated by some gap otherwise they'd be the at same location and therefore they'd be the same point."

There's just no getting around this fundamental truth about the nature of dimensionless points. The mere fact that they are indeed dimensionless is what gives rise to this natural property. If they had any breath at all, we could claim that they were "touching" even though their centers were at different locations thus qualifying them as different points. But this isn't the case. Dimensionless points have no breath. Therefore for any two dimensionless points to be considered to be not the same point they must necessarily be separated by some gap.

Now if you start thinking about wanting to break this gap down into smaller and smaller gaps by considering more and more locations between these two original locations then you've really missed the whole concept. The whole purpose to this discussion is to simply address the fundamental nature that must exist between any two point that are not the same point. So considering millions and billions of points existing between our original two is to simply miss the point altogether actually. And there's seriously no pun intended here.

The Ultimate Conclusion
While it may be true that a single point can be thought of as being dimensionless, it is quite impossible to imagine two points (which are not the same point) to be dimensionless. In other words, there is necessarily a dimension (or width) associated with every two points that are not the same point.

This is really an extremely important observation when it comes time to talk about the number of points that can exist in a finite line. We might be able to ignore the dimension of the individual points, but we simply can't ignore the necessary dimension that must exists between every two of these dimensionless points.

A point is a one location. It must be dimensionless otherwise it would represent more than one location

It seems that way to me. That's why I was struggling with your idea of a line of dimensionless points, since the line would surely be dimensionless.

Well, from my argument above perhaps you can see now why just the opposite is true. If all of the dimensionless points where 'touching' then the line would surely be dimensionless. It would have to be because, since the points are dimensionless, and they are all 'touching' then they would have to be the very same point (i.e. the same location). It would be impossible to build a line from dimensionless points if they were 'touching'. Yet this is precisely what they would need to do if they were considered to be a continuum! In order to have a continuum the points would need to have dimension so that they can touch and not be the same point as their neighbor. But we really don't need to be bothered with that because there really isn't any need to try to build a continuum. The whole idea of a continuum is paradoxical and probably can't even exist in nature.

Now there is an age-old proof that a finite line contains an infinite number of points. The proof is really quite simple and it goes like this:

Say we have a finite line made up of a finite set of points. Well, we can always stick some more points between those existing points right? I mean, points are dimensionless! They don't take up any space. There's nothing stopping us from putting even more points between those points, and so on, ad infinity. Therefore we can clearly stick an infinite number of dimensionless points in a finite line. End of argument. Who could possibly argue with that?

Well, actually I can show that while this reasoning appears to be sound it is actually quite flawed. The bottom line is that the argument does not support the conclusion. I can actually prove this mathematically if anyone is interested in seeing the proof.

In fact, I can then go on and use calculus to prove why a finite line cannot possibly contain an infinite number of points.

It's ironic to me that mathematics can actually be used to conclusively prove that a finite line cannot contain an infinite number of dimensionless points, yet the mathematical community continues to insist that it can contain them.

 

[StatusX]

place a point on a line. now I assert there is a point a distance e from this point for every e>0. Is there a gap between the original point and the point closest to it?

you say yes, but I say there is no such point. For any point you pick out, you can find another one closer to the original. there is an ordering of these points, but there is no way to list them in this order. this is the idea behind continuity, and it defies common sense.

 

[Hurkyl]

I don't feel like a very long response today, so I'll keep it short:

Now if you start thinking about wanting to break this gap down into smaller and smaller gaps by considering more and more locations between these two original locations then you've really missed the whole concept. The whole purpose to this discussion is to simply address the fundamental nature that must exist between any two point that are not the same point. So considering millions and billions of points existing between our original two is to simply miss the point altogether actually.

millions, billions, centillions, googolplexes, so what? Those are all finite. And aside from some technical details and the philosophical problem that you can't speak of a "gap" between two points without already having some a priori notion of places between them, I agree with you.

But, you haven't addressed infinitely many points at all. While any two individual points may be isolated, you've said absolutely nothing about the whole.

Well, actually I can show that while this reasoning appears to be sound it is actually quite flawed. The bottom line is that the argument does not support the conclusion. I can actually prove this mathematically if anyone is interested in seeing the proof.

In fact, I can then go on and use calculus to prove why a finite line cannot possibly contain an infinite number of points.

Sure. I'll get you started by posting the alledged proof that a line segment has infinitely many points:



notation: A*B*C means "B lies between A and C". By definition, it means that A, B, and C are distinct collinear points, and this relation satisfies the axioms of betweenness.

A point X is said to lie on the line segment YZ if and only if X=Y, X=Z, or Y*X*Z.


Lemma: Let A and B be distinct points. Then, there exists a point, C, such that A*C*B.
(I can prove this one too, if you need it)



Theorem: Let AB be any line segment. For every positive integer n, I can construct points C1, C2, ..., Cn such that
(1) they all lie on AB
(2) A*Cx*Cy if 1 <= y < x <= n.

Proof:

By the lemma, there exists a point X such that A*X*B. Choose C1 to be any such point. Condition (1) is satisfied by definition of line segment, and condition (2) is vacuously true. Thus, the theorem has been proven for n = 1.

Suppose the theorem is true for n = k. Then, by the lemma, there exists a point X such that A*X*Ck. Choose C(k+1) to be any such point.

We already know that Cx lies on AB if 1 <= x <= k. Because A*C(k+1)*Ck and A*Ck*B, we have A*C(k+1)*B (axioms of betweenness, or maybe it was a theorem), so C(k+1) lies on the line segment AB.

We alraedy know that A*Cx*Cy if 1 <= y < x <= k. Now, suppose x = k+1.
case 1: y = k. C(k+1) was constructed such that A*C(k+1)*Ck, so this case is proven.
case 2: y < k. Both A*C(k+1)*Ck and A*Ck*Cy, so we have that A*C(k+1)*Cy.
So, we see that A*Cx*Cy if 1 <= y < x <= k

So, we see that conditions (1) and (2) are both true for n = k+1. By the principle of mathematical induction, the theorem is proven for every n.


Corollary: Any line segment has infinitely many points lying on it.

Proof: suppose otherwise: that there exists a line segment AB that doesn't have infinitely many points lying on it. Let n be the number of points lying on AB.

By the theorem, we can construct points C1, C2, ..., C(n+1) such that
(1) they all lie on AB
(2) A*Cx*Cy if 1 <= y < x <= n+1.

Now, if p and q are different integers with 1 <= p, q <= n+1, we have either p < q or q < p. So, either A*Cq*Cp or A*Cp*Cq. Either way, this means Cp and Cq are distinct.

Therefore, each of the Ck are distinct, and there n+1 of them, contradicting the fact that AB has only n points lying on it.

Therefore, the initial assumption was wrong, so we conclude that every line segment has infinitely many points lying on it.

QED

 

[NeutronStar]

place a point on a line. now I assert there is a point a distance e from this point for every e>0. Is there a gap between the original point and the point closest to it?

you say yes, but I say there is no such point. For any point you pick out, you can find another one closer to the original. there is an ordering of these points, but there is no way to list them in this order. this is the idea behind continuity, and it defies common sense.

You seem to require that I need to pick out some particular point to which no point can be closer. Yet I never claimed anywhere that I could do any such thing!

All I've shown is that no matter how close any point that you chose is to the original point there is necessarily a gap between your point and the original point. Assuming, of course, that your point is not the same point as the original point!

So in other words, no matter how small you imagine making your distance e you still end up with a gap between your points.

You can't make your e = 0 because to do that you would simply be to select the original point and not have two points after all. Therefore your e must be some distance (no matter how arbitrarily small).

So in essence your whole methodology supports my claim that there must be some no-zero gap (i.e. e>0) between all your points.

So I see your post as being in agreement with what I've already been saying.

 

[Hurkyl]

I always forget how much explicit examples help.

Tell me, what is the gap between the point labelled

0

and the points labelled

1, 1/2, 1/3, 1/4, 1/5, 1/6, ...


Certainly between any two individual points there is a gap, but what is the gap between these two groups of points?

 

[NeutronStar]

millions, billions, centillions, googolplexes, so what? Those are all finite.

Oh boy! I'm carving this quote in STONE!

I know that we'll be coming back to this concept very shortly so I am very glad that you have made your stance on this concept quite clear.

If you really believe what you have stated above (as do I) then you should end up agreeing with me when all is said and done. I'm putting this quote on file.

{sniped out long proof},...Therefore, the initial assumption was wrong, so we conclude that every line segment has infinitely many points lying on it.

I agree that you have proven that your initial assumption was wrong.

What I disagree with is your conclusion. In other words, I'm saying that I can demonstrate why there is absolutely no connection between your initial assumption and your conclusion.

That fact that you've disproven your initial assumption does not lead to your conclusion. I can show this conclusively. And ironicly, my demonstration is based on the idea stated in your quote at the top of this very post!

"millions, billions, centillions, googolplexes, so what? Those are all finite."

I'll type in my demonstration of why your conclusion does not follow from having proven your initial assumption to be wrong.

Unfortunately I don't have time to type it in anytime soon.

But I'll be back!

 

[Hurkyl]

I agree that you have proven that your initial assumption was wrong.

What I disagree with is your conclusion.

The initial assumption was:

"There exists a line segment that doesn't have infinitely many points lying on it".

You agree that this was wrong, however the negation of this statement is:

"Every line segment has infinitely many points lying on it."

So I'm understandably confused when you say you don't agree with my conclusion that every line segment has infinitely many points lying on it.


I guess I'll have to expect you to come back describing an alternative logical system that doesn't have the law of contradiction:

(~P --> false) --> P

Or maybe where it's impossible to say the phrase "infinitely many" (but then, it would also be impossible to say the phrase "finitely many" because if you could say "finitely many" then you can say "not finitely many" which is the definition of "infinitely many")

 

[StatusX]

All I've shown is that no matter how close any point that you chose is to the original point there is necessarily a gap between your point and the original point. Assuming, of course, that your point is not the same point as the original point!

So in other words, no matter how small you imagine making your distance e you still end up with a gap between your points.

You can't make your e = 0 because to do that you would simply be to select the original point and not have two points after all. Therefore your e must be some distance (no matter how arbitrarily small).

So in essence your whole methodology supports my claim that there must be some no-zero gap (i.e. e>0) between all your points.

So I see your post as being in agreement with what I've already been saying.

What youre missing is that there are still an infinite number of points between any two points. there is a point halfway between them. There are points halfway between the endpoints and the midpoint. And so on. All the distances between these points are greater than 0. You can find a point ARBITRARILY close to a given point, and so there is no discreteness. The key here is ANY e>0. You can't name a distance such that there are no two points closer than this.

 

[NeutronStar]

The initial assumption was:

"There exists a line segment that doesn't have infinitely many points lying on it".

Careful,... this isn't what you assumed or proved,...

Therefore, each of the Ck are distinct, and there n+1 of them, contradicting the fact that AB has only n points lying on it.

This is what you proved!

You proved that AB has n+1 points on it, and not n points.

In other words, you proved that the situation is unbound, but you didn't prove that it has the property of being infinite. There's a huge difference between these two concepts that the mathematical community actually uses every day. Yet for some reason they don't seem to see it in the case of the number of points in a line.

I'll see if I can type this up fairly quickly. I just want it to be complete and error free before I actually post it. I'll also be using some Latex so please be patient. I'm not real quick with Latex.

 

[NeutronStar]

The Proof
We can prove using various mathematical methods and intuitive reasoning that we can always insert more points in a finite line. In other words, we can show clearly that there is no total number of points n that we will eventually reach thus preventing us from adding anymore.

The Conclusion
Having clearly demonstrated that no definite number exits that prevents us from adding more points to a finite line segment we have every reason to conclude that the line must therefore contain an infinite number of points.

The Fallacy in the Logic

Let us begin with a finite line segment that contains only two points (the end points). Not much of a line to be sure, but it's a good reference place to start. What we will do is start adding points between these points and see just how far we can go and what the ultimate consequences will be.

Now before we begin adding more points to our line let's create a set to keep track of the number of points in our line. Let this set be called P

Let each element of this set hold the number of points contained in the line with each successive addition of points. So in the case of our original line we have: P=\{2\}

This shows that our line starts off with only two points. Well adding a point between these two points we get a line that contains 3 points. So 3 becomes the next element in our set.

P=\{2, 3\}

Adding points between those 3 points we get 5. So P=\{2, 3, 5\}

Adding points between those 5 points gives us 9 points. So P=\{2, 3, 5, 9\}

If we keep this up we get P = \{2, 3, 5 , 9, 17, 33, 65, 129, 257, 513, 1025, 2049, ...\}

The set continues to grow without bound. There can be no doubt that the set P is an infinite set because there is absolutely nothing stopping us from continuing to add points between existing points forever.

So we conclude that a finite line segment contains an infinite number of points!

Was this a Logical Conclusion?

The answer to that question is no, it was not!

What we have shown is that the set P is clearly infinite. But that set doesn't represent the number of points in our line! That set contains elements, each of which represent the number of points in a line. And while its true that we have shown that it must contain an infinite number of elements, we have NOT shown that any of those elements have the property of being infinite. On the contrary, using the method outlined about we can clearly show why none of the elements within this set can be infinite. They all must be finite. We have started with a finite number of points and continually added finite quantities of points each time we added more points. In fact in this particular scenario we are actually restricted to adding a finite number of points with each successive addition.

There is absolutely no logic in mathematics that permits us to automatically transfer the quantitative property of a set onto the elements contained within that set. In fact, there are actually reasons that prevent us from doing this.

Consider the set of natural numbers. N = \{1, 2, 3, 4, 5,...\}.

We know that this set has the property of being infinite, yet we have absolutely no problem at all understanding why it is that none of its elements can be infinite. For if anyone of its elements were infinite, then that element would have to be the LAST element in the set instantly making the set finite. Infinity is not a member of the natural numbers for good reason.

The Real Conclusion
When we prove that we can continually add more and more points to a finite line segment we haven't really proven anything at all about how many points the line can actually hold. On the contrary. Using the reasoning outlined above we have no choice but to conclude that a finite line can only contain a finite number of points.

How many points would that be? You might ask. Pick any number you like. The only requirement is whatever number you choose to use it must have the property of being a finite number.

millions, billions, centillions, googolplexes, so what? Those are all finite.

Precisely!

There's absolutely no limit to the number of points that you can put into a finite line segment providing that the number you choose has the property of being finite.

The number of points is unbounded, but finite just like the natural numbers which are the elements of the set of natural numbers. The points must be finite in number because of the fact that the points are dimensionless. There simply must be some non-zero gap between the points. It's an unavoidable logical consequence of the very nature of the dimensionless points themselves. If the points are to be dimensionless there can only be a finite number of them in a finite line. They are unbounded, but finite, just like the individual natural numbers.

This is just like the set of Natural Numbers. There is no largest Natural Number. The SET of natural numbers has the property of being infinite, yet no single element (Natural Number) within that set can be infinite. Those elements are unbounded but finite. There is no end to the largeness that you may assign to a Natural Number, yet it must always have the property of being finite. This is really the only restriction to a natural number, and this same idea applies to the number of points within a finite line segment.

The SET containing the possible combinations of points that you can put into a finite line is infinite. But just like the elments of the SET of Natural Numbers, the actual number of points that you can claim to have in a finite line is actually finite.

So the conclusion that a finite line segment contains an infinite number of points is simply incorrect logic. It's simply not supported by mathematical reasoning.

People who want to claim that 0.999… is not equal to 1 are trying to recognize this necessary gap between the points. They are trying to say, "Hey, 0.999… is a different point than 1". It's not the same LOCATION! To try to remove that gap by claiming that 0.999... = 1 in an attempt to make the line a continuum is a direct logical contradiction to the idea of a dimensionless point.

These two concepts, a continuum, and a dimensionless point, simply aren't compatible ideas.

Calculus can be used to reinforce this very same conclusion using a completely different argument.

 

[NeutronStar]

All the distances between these points are greater than 0. You can find a point ARBITRARILY close to a given point, and so there is no discreteness. The key here is ANY e>0. You can't name a distance such that there are no two points closer than this.

How does this result in the conclusion that there is no discreteness?

You clearly agree that the distance between any two points must be non-zero, yet you continue to argue that there is no discreteness.

No matter how arbitrarily close you allow, you are still restricted to that arbitrarily non-zero quantum jump.

There's just no getting around it. As long as the distance between dimensionless points is restricted to being non-zero (which it must be) then the line is necessarily discrete. There's just no getting around it.

In other words, you can't move away from any location without moving some non-zero distance. And that requires a discrete jump. You can be at the second position and not have left the first position! To do that you'd still be at the same position.

You have to take the quantum jump. There's just no other way to do it.

This falls directly out of the logic mandated by the idea of dimensionless points, (to change locations you have to move a discrete distance). No matter how arbitrarily small you make it, it must necessarily be non-zero and therefore discrete.

It's really the idea of a continuum that fails here.

Using pure logic we can actually deduce the quantum nature of our universe. It's a shame that the mathematical community didn't discover this before Max Planck discovered it experimentally. Imagine the triumph of pure mathematics had it done so. Unfortunately the mathematical community is a bit late. In fact, they still don't seem to get it. They seem to be obsessed with the idea of a continuum. Buy why? Where did this arbitrary obsession come from?

The whole idea of a continuum just plain fails. It simply can't work. It's logically inconsistent. A change in position necessarily must be quantized at some level. It's the only logical conclusion.

 

[matt grime]

That's a pretty compelling demonstration that you don't really understand any of this.

Unbounded but finite at the same time? You do realize that we are not claiming that any natural number is infinite, but that the set of them is. (ie it is not a finite set). And since we don't ever claim that any natural number describes the cardinality of the natural numbers we're ok.

Mathematics never would discover the quantum nature of the universe, since it is not about experimentally validated ideas which may or may not be true. Also, that quantized thing you think we need to change position to? Yep, well, it wouldn't be about if it weren't for the real numbers (after all, how are you going to define Planck's constant numerically?). Note that you're only talking about bound states being quantized (demonstrating you don't really know about quantization - is time quantized?), and for that matter nor about maths: there are lots of quantum objects in mathematics (quantum binomial coefficients). All quantization is essentially is the introduciton of a variable q that indicates the failure to commute.

I'd be interested to see how using pure logic we can prove the universe is quantized. As far as I know no one has shown time to be quantized.


Discrete in mathematics in this sense means topologically distinct points, ie that given any point there is an *open* nbd of it containing no other points, The metric topology is not discrete on R. You are taking the options in the wrong order:

Fix x, fix e>0, then there is some point y not equal to x such that |x-y|<e, e was arbitrary. You now appear to want to change e so that |x-y| is not less than e, well, that isn't how the mathematics of it works. As Hurkyl as already shown you don't know how to negate universally quantified statements.

 

[Hurkyl]

Careful,... this isn't what you assumed or proved,...

Funny, since you can find that exact wording in my proof.

This is what you proved!

You proved that AB has n+1 points on it, and not n points.

Yes, but n was defined to be the number of points on AB! Since n is the number of points on AB, and n+1 is the number of points on AB, then n = n+1, and that's a contradiction.

There's a huge difference between these two concepts

Yes, there is, but you seem to have overcompensated, and are making the reverse mistake from most people.

"Finite, but unbounded" only applies when you have some (necessarily infinite) class of things. For any particular bound, I can always find something bigger than that bound, but each object is finite.

It's even true that the collections of points produced by my theorem are finite, but unbounded.

But there's one piece to the puzzle you're missing: all of those collections are part of some whole -- the entire collection of points on AB contains each of the constructions made by my theorem.

But I didn't use this to prove my corollary because the logic I did use is very clear.

Assume AB has only finitely many points on it.
Define n to be the number of points on AB.
(thus, AB has exactly n points on it, no more, no less)
Apply the theorem to find n+1 distinct points on AB.
Because n+1 > n, this is a contradiction.
Therefore, AB has infinitely many points on it.


I have to go to work, so I haven't had a chance to address your argument.

 

[Canute]

NeutronStar

Thanks for your explanation. I want to pick through it to find out where (or if) I'm going wrong.

Calculus doesn't actually model a continuum at all. If you exam all of the calculus definitions with great care you will actually see how they avoid the concept of a continuum altogether. In fact, all of modern calculus is ultimately based on Karl Weierstrass's definition of a limit. And Weierstrass was very clever in avoiding any direct confrontation with the idea of a continuum. So calculus doesn't actually model a continuum at all. It does however address a mathematical notion of continuity which doesn't actually imply a continuum at all. This is really quite clear if you simply inspect the formal definition in great detail.

Ok, a question. If one was to treat spacetime as continuous then would the calculus fail as a mathematical way of modelling or calculating motion in this medium? If it would not fail then this would show that the calculus models spacetime as a continuum. (To be honest I thought the very purpose of infinitessimals was to overcome the awkward infinitities that arise when modelling continuous change in a continuous medium, or against a continuous scale of measurement).

Btw I'm not arguing that the number line is a continuum. Rather I'm suggesting that the number line, like spacetime, can be seen as either a series of points or a continuum, and that to treat it exclusively as one or the other gives rise to paradoxes. I suppose this is as much metaphysics as mathematics, but then I regard the nature of the number line as a sort of meta-mathematical metaphysical question, as undecidable as any other metaphysical question. (I rather suspect that Goedel's theorem has something to do with all this).

The Dimensionless Point
Ok, we begin with the postulate that points are dimensionless. They have no "physical" existence in the sense that they don't take up any space. In fact they aren't actually entities. Don't think of them as entities.

OK.

Think of a point as nothing more than a location period amen. A point is a location. Now,… what does it mean to have a single point? In truth such a concept is totally meaningless. Why? Because a point is a location. If all we have is a location and nothing else then the whole concept of location has absolutely no meaning.

Ok. I'm fine with the idea that a dimensionless point has no location.

A New Premise
Let there exists a second location which is not the same as the first location. Ok, now we have a meaningful relative concept. We have two locations which are not the same location. We still don't need a coordinate system or anything. All we need to know is that we have two points that are not the same point. They can't be at the same location because, by definition (i.e. a point is a location) they would be the same point if they were at the same location. So it follows logically that any two points that are not the same point must necessarily be separated by some gap. If this wasn't the case they'd be the same location, and thus they'd be the same point.

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

So we can clearly see from this line of reasoning that any two points, that are not the same point, must necessarily be separated by a gap. There may be an urge by the reader to want to start shoving more points in between these two primitive points. But I seriously ask, "What's the point in doing that?". The point's already been made. No matter how many points we imagine tossing in the gap there will always be one basic naked truth left,..

"Any two points that are not the same point must necessarily exist at different locations, and since they are dimensionless points this means that they must necessarily be separated by some gap otherwise they'd be the at same location and therefore they'd be the same point."

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.

There's just no getting around this fundamental truth about the nature of dimensionless points. The mere fact that they are indeed dimensionless is what gives rise to this natural property. If they had any breath at all, we could claim that they were "touching" even though their centers were at different locations thus qualifying them as different points. But this isn't the case. Dimensionless points have no breath. Therefore for any two dimensionless points to be considered to be not the same point they must necessarily be separated by some gap.

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

Now if you start thinking about wanting to break this gap down into smaller and smaller gaps by considering more and more locations between these two original locations then you've really missed the whole concept. The whole purpose to this discussion is to simply address the fundamental nature that must exist between any two point that are not the same point. So considering millions and billions of points existing between our original two is to simply miss the point altogether actually. And there's seriously no pun intended here.

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.

The Ultimate Conclusion
While it may be true that a single point can be thought of as being dimensionless, it is quite impossible to imagine two points (which are not the same point) to be dimensionless. In other words, there is necessarily a dimension (or width) associated with every two points that are not the same point.

That makes more sense to me. It's what I meant earlier when I said clumsily that a point is a 'range', (a range between the point +.000...1 and the point -.000...1 , i.e. a not quite dimensionless location).

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.

This is really an extremely important observation when it comes time to talk about the number of points that can exist in a finite line. We might be able to ignore the dimension of the individual points, but we simply can't ignore the necessary dimension that must exists between every two of these dimensionless points.

Well, from my argument above perhaps you can see now why just the opposite is true. If all of the dimensionless points where 'touching' then the line would surely be dimensionless. It would have to be because, since the points are dimensionless, and they are all 'touching' then they would have to be the very same point (i.e. the same location). It would be impossible to build a line from dimensionless points if they were 'touching'. Yet this is precisely what they would need to do if they were considered to be a continuum! In order to have a continuum the points would need to have dimension so that they can touch and not be the same point as their neighbor. But we really don't need to be bothered with that because there really isn't any need to try to build a continuum. The whole idea of a continuum is paradoxical and probably can't even exist in nature.

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.

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.

Say we have a finite line made up of a finite set of points. Well, we can always stick some more points between those existing points right? I mean, points are dimensionless! They don't take up any space. There's nothing stopping us from putting even more points between those points, and so on, ad infinity. Therefore we can clearly stick an infinite number of dimensionless points in a finite line. End of argument. Who could possibly argue with that?

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.

Well, actually I can show that while this reasoning appears to be sound it is actually quite flawed. The bottom line is that the argument does not support the conclusion. I can actually prove this mathematically if anyone is interested in seeing the proof.

In fact, I can then go on and use calculus to prove why a finite line cannot possibly contain an infinite number of points.

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.

(Pardon the late edit - I spotted a mistake)

 

[matt grime]

Of course it doesn't convince us. All these ideas have nothing to do with the mathematics of the real numbers, or the definitions of discrete and so on as used correctly.

Here, consider the line segment [0,1] in R. it contains points1/n for all n. There is not a finite number of them (if there were then there would be a smallest one, and there isn't) hence by the very definition of the word infinite, we conclude that there are an infinite number (ie not a finite number) of points in that interval.

The objections and "counter arguments" arise purely from misunderstanding mathematics.

I mean, that the heck is a finite line anyway? And what does calculus, and analytic tool, necessaryil have to say about geometry?

 

[StatusX]

construct the natural numbers one at a time. the first set has 1 element, the second has 2, etc. But every term in the series:

{1,2,3,4,...}

is finite. I just proved there are a finite number of natural numbers. If only people as smart as me had been running things back in the day, we'd have been able to get so much farther than we have with this false idea that there are an infinite number of natural numbers.

 

[NeutronStar]

construct the natural numbers one at a time. the first set has 1 element, the second has 2, etc. But every term in the series:

{1,2,3,4,...}

is finite. I just proved there are a finite number of natural numbers. If only people as smart as me had been running things back in the day, we'd have been able to get so much farther than we have with this false idea that there are an infinite number of natural numbers.

What was that all about?

I certainly never claimed that there are only a finite number of natural numbers. On the contrary I completely agree that the set of natural numbers is infinite. I merely stated that no member of that set has the property of being infinite. And as far as I'm aware this is the currently accepted picture.

I was merely pointing out the irony in the fact that while the mathematical community accepts this situation they reject the idea that the number of points in a line segment must be finite. Yet it's basically the very same situation that they already accept for the set of natural numbers!

NeutronStar

Thanks for your explanation. I want to pick through it to find out where (or if) I'm going wrong.

Thank you for considering these ideas. I've read your entire post and would like to comment on your concerns, but it will take me a while to respond.

I too, have considered many of the concerns that you have mentioned so I can share with you just how it is that I have come to grips with these concerns. I will be interested in hearing your ideas on these issues as well. I'll be back later to address the concerns that you've mentioned in your previous post.

 

[StatusX]

Let each element of this set hold the number of points contained in the line with each successive addition of points. So in the case of our original line we have: P=\{2\}

This shows that our line starts off with only two points. Well adding a point between these two points we get a line that contains 3 points. So 3 becomes the next element in our set.

P=\{2, 3\}

Adding points between those 3 points we get 5. So P=\{2, 3, 5\}

Adding points between those 5 points gives us 9 points. So P=\{2, 3, 5, 9\}

If we keep this up we get P = \{2, 3, 5 , 9, 17, 33, 65, 129, 257, 513, 1025, 2049, ...\}

The set continues to grow without bound. There can be no doubt that the set P is an infinite set because there is absolutely nothing stopping us from continuing to add points between existing points forever.

So we conclude that a finite line segment contains an infinite number of points!

Was this a Logical Conclusion?

The answer to that question is no, it was not!

What we have shown is that the set P is clearly infinite. But that set doesn't represent the number of points in our line! That set contains elements, each of which represent the number of points in a line. And while its true that we have shown that it must contain an infinite number of elements, we have NOT shown that any of those elements have the property of being infinite. On the contrary, using the method outlined about we can clearly show why none of the elements within this set can be infinite. They all must be finite. We have started with a finite number of points and continually added finite quantities of points each time we added more points. In fact in this particular scenario we are actually restricted to adding a finite number of points with each successive addition.

My point was to show how that logic is wrong. I used the exact same method of "proof" as you to show there are a finite number of natural numbers, which you yourself realize is false.

 

[Hurkyl]

Was this a Logical Conclusion?

The answer to that question is no, it was not!

And, by golly, I agree with this too! The "argument" you present is, in fact, invalid.

But, you'll notice, that I did not use that argument. Instead of saying "I can find as many points as I want, so the line segment must have infinitely many points! Wee!", I carefully derived a contradiction from the assumption that there was a line segment without infinitely many points.

Just because some people make a mistake doesn't mean everybody will make that mistake.

And, more importantly, just because some people make a mistake doesn't mean their conclusion is wrong.


Now, to comment on your argument.

Let us begin with a finite line segment that contains only two points (the end points).

A set of two points is not a line segment. At least, it doesn't resemble any concept of line segment I've ever seen, and it certainly doesn't resemble the geometric definition of a line segment.

As I mentioned, the geometric definition is that a point X lies on the line segment AB if and only if X = A, X = B, or A*X*B. A more set theoretic approach to geometry simply defines AB = {A, B} U {X | A*X*B}.

So, because there exists a point C such that A*C*B, it follows that {A, B} is not a line segment (because it doesn't contain C).


Anyways, I'm not sure precisely what you mean by "line segment", but you don't appear to have proven that all "line segment"s have finitely many points, just that this particular kind of "line segment" does.

You clearly agree that the distance between any two points must be non-zero, yet you continue to argue that there is no discreteness.

No matter how arbitrarily close you allow, you are still restricted to that arbitrarily non-zero quantum jump.

And that's because you keep missing a very important point. The statement:"that the distance between any two points must be non-zero" applies to any pair of points in a metric space.

However, the statement "there are no gaps in the line" is a statement about a whole collection of points. I gave an explicit example:

Let the first collection consist of a single point, 0.

Let the second collection consist of all of these points: {1, 1/2, 1/3, 1/4, ...}

So while there is a nonzero gap between any point in the first collection and the second collection, there is no gap between the two collections when viewed as a whole.

Using pure logic

It's not pure logic because you've made assumptions about the nature of reality. (such as what a "line segment" is, and that "line segment"s have a bearing on reality)

 

[Hurkyl]

(this one is mostly adderssing Canute)

One way to try and deal with confusion about a nebulous, intuitive concept is to try and devise a "working definition": tentatively come up with a criterion that seems to describe the nebulous concept, yet can be manipulated more rigorously.


Mathematicians generally use a notion called "completeness", but for the case of the real line, it's equivalent to "connectedness" which can be described as follows:


A topological space is connected iff the following is true:

If you take all of your points and split them into two sets, then you can find some point, call it X, such that every neighborhood of X ("range" containing X) contains points in both sets.


The mathematical definition of the real line is connected. The rationals, for example, are not connected, because, for instance, you can split the rationals into these two sets:

A = {x | x <= 0} U {x | 0 < x and x^2 < 2}
B = {x | 0 < x and 2 <= x^2}

And you can prove that for any point X you choose, there is a range containing X that lies entirely in one of these sets. (However, in the real numbers, you can choose X to be the positive square root of 2)

 

[NeutronStar]

On the Infintesimals

Ok, a question. If one was to treat spacetime as continuous then would the calculus fail as a mathematical way of modeling or calculating motion in this medium? If it would not fail then this would show that the calculus models spacetime as a continuum.

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 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.

(To be honest I thought the very purpose of infinitesimals was to overcome the awkward infinities that arise when modeling continuous change in a continuous medium, or against a continuous scale of measurement).

I don't know where you got that idea, I never heard of any formal proclamation of that. Although I can see where people might have gotten such an idea inadvertently.

An "infinitesimal" is actually an older name for the "differential" (i.e. dy, dx, etc.) The differential is clearly defined in calculus based on Weierstrass's epsilon-delta definition of the limit (i.e. via the formal definition of the derivative).

Now Weierstrass's epsilon-delta definition of the limit does overcome the awkward infinities that arise when modeling an instantaneous rate of change. In your quote above you used the words "continuous change in a continuous medium". But you need to be careful here. In mathematics the words "continuous" and "continuity" have very formal definitions. These definitions are also based on the Weierstrass definition of the limit. In fact, they actually refer back to it and depend on it entirely for their meaning.

The mathematical terms "continuous" and "continuity" do not have the same intuitive meaning that most laymen would assign to them. In other words, the mathematical terms "continuous" and "continuity" do not necessarily imply a continuum. 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. So it's much better to think of the definition of the limit as addressing instantaneous change rather than addressing some time of continuum. A continuum is simply not necessary for Weierstrass's definition to work. And the mathematical terms "continuous" and "continuity" both rely back on Weierstrass's definition so it should be quite clear that nether of those mathematical terms implies a continuum either.

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.

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)

I would strongly recommend studying Weierstrass's delta-epsilon definition of the limit. This definition is the foundation of all of modern calculus.

 

[NeutronStar]

On Logical Inconsistencies

Btw I'm not arguing that the number line is a continuum. Rather I'm suggesting that the number line, like spacetime, can be seen as either a series of points or a continuum, and that to treat it exclusively as one or the other gives rise to paradoxes. I suppose this is as much metaphysics as mathematics, but then I regard the nature of the number line as a sort of meta-mathematical metaphysical question, as undecidable as any other metaphysical question. (I rather suspect that Goedel's theorem has something to do with all this).

Unlike you, I do take the stance that a line must be a discrete series of points. The idea of a continuum simply has too many logical inconsistencies associated with it for me. So far I have been able to resolve all of the apparent paradoxes associates with a discrete series of points. I have not been able to resolve the paradoxes associated with a continuum. Moreover, I have discovered logical contradictions associated with a continuum that I am completely convinced of and therefore I cannot imagine them being resolved. 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.

I have yet to find any such irresolvable paradox associated with a discrete series of points. Therefore I simply must take the more logically sound road.

On Gödel's Incompleteness Theorem

Since you've mentioned Gödel's inconsistency theorem I'd like to make some comments on that as well. Actually Gödel's work has nothing at all to do with whether things are discrete or continuous. 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. Also, I would like to be quick add that Gödel's theorem in no way references the empty set specifically. However, Gödel's work is directly related to that concept in very important ways. In fact, if the concept of the empty set were to be removed from mathematics Gödel's inconsistency theorem would no longer even apply to mathematics!

That's a very long story! I would not want to have to try to explain that on an Internet forum board.

Just as one final note, this whole continuum vs. discrete issue does related directly to set theory. It is intimately connected with the concept of an empty set. A theory which permits the concept of an empty set is one that supports a continuum. A theory that denies the concept of an empty set support a discrete nature of quantity. Obviously I firmly reject the concept of an empty set.

But again, this appears to be a completely different topic. It's actually quite intimately related to the idea of whether the universe is discrete or a continuum. Unfortunately this relationship between set theory and the quantitative nature of the universe has been widely ignored by the mathematical community. The major historical events associated with can be found in the history of Georg Cantor, and the other famous mathematicians who lived at that same time period. (only a couple of centuries ago)