What is the relationship between points and neighborhoods in topology?

[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)

 

[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.