What is the relationship between points and neighborhoods in topology?

[StatusX]

actually limits never invoke the concept of infinty. A limit is defined in purely finite terms. to jump from saying that the limit of a continuous function f(x) as x goes to c is L to saying that f(c)=L would be invoking the idea of infinity. In fact, a function is DEFINED to be continuous at c if this happens to be true. This captures the intuitve idea of infinity without explicitly using it. Like I said before, 0.999... is DEFINED as the limit of that sequence of sums, and infinity is never involved.

 

[NeutronStar]

Infact you can certainly define all the algebraic numbers without ever mentioning a limit, for example the set {x in Q|x>0 and x^2 > 3} defines sqrt(3) without a mention of a limit.

Well, yes. But this rests on the assumption that finite intervals are continuums. This is an assumption that I must confess that I am probably alone in not accepting. However, I have never seen a convincing proof that finite intervals are continuums. On the contrary I can show several solid logical arguments why they can't be.

actually limits never invoke the concept of infinty. A limit is defined in purely finite terms.

I absolutely agree with this.

to jump from saying that the limit of a continuous function f(x) as x goes to c is L to saying that f(c)=L would be invoking the idea of infinity. In fact, a function is DEFINED to be continuous at c if this happens to be true.

Yes, but don't confuse the word continuous with the word continuum. This state of affairs does not imply a continuum in any way. As a matter of fact, by Weierstrass's strict definition of the limit this particular definition of "continuous" necessarily implies a quantized situation. This is true because we are not permitted to consider delta to ever be equal to zero. Therefore we must necessarily view this definition of "continuous" as being quantized.

So to mathematically say that a function is continuous (by this definition) does not in any way imply that it defines a continuum. In fact, it actually implies that it must necessarily be discrete (or quantized).

 

[jcsd]

Well, yes. But this rests on the assumption that finite intervals are continuums. This is an assumption that I must confess that I am probably alone in not accepting. However, I have never seen a convincing proof that finite intervals are continuums. On the contrary I can show several solid logical arguments why they can't be.

The proof that it defines a continuum is the fact that it defines the real numbers, this is trivially true as the continuum is the set of real numbers!

Yes, but don't confuse the word continuous with the word continuum.

I think you've yuor terminology mixed up here, in the most general sense a continuum is a set with some sort of order realtion something like the real line i.e. has the cardianlity of the continuum and between any two numbers there is another number.

We're dealing with sequences which are functions of the type f:N->R so the epilson-delata defitnion is irrelvant.

This state of affairs does not imply a continuum in any way. As a matter of fact, by Weierstrass's strict definition of the limit this particular definition of "continuous" necessarily implies a quantized situation. This is true because we are not permitted to consider delta to ever be equal to zero. Therefore we must necessarily view this definition of "continuous" as being quantized.

No it does not imply it is quantized, the fact that we don't want delat ito eb equal to zero is that we are taking the limit of the function at a point, so we must ignore what is going on at that point.

So to mathematically say that a function is continuous (by this definition) does not in any way imply that it defines a continuum. In fact, it actually implies that it must necessarily be discrete (or quantized).

It does imply that the function is not discrete and as I said anyway this defitnion os irrelvant to whta is being talked about.

 

[Hurkyl]

There is no logical problem with a continuum made up of individual points, but it is true that ranges play an important part of the concept. A topology is made up of two things: points, and neighborhoods. For the real line, we can indeed take the neighborhoods to be "ranges". More specifically, the neighborhoods can be taken to be the open intervals.


.999... is a number, not a sequence of things for which you need to take a limit to get a number.

It's simply a logical contradiction in terms to claim that an infinite process can be completed. It makes no sense.

You're certainly free to have the opinion that it makes no sense, but when you call something a logical contradiction you should demonstrate it.

 

[NeutronStar]

You're certainly free to have the opinion that it makes no sense, but when you call something a logical contradiction you should demonstrate it.

Well, I must confess that I could never prove it to anyone who believes that a point is range. My entire proof requires that a point be dimensionless.

I must be getting old because I was always taught that a mathematical point is dimensionless. I wasn't aware that they've become engorged over the years. What was the purpose of that? Who engorged them?

I thought they were pretty cool concepts when they were dimensionless.

In any case, I would like to point out that these recent developments in mathematics can hardly be called "The careful work of centuries". I'm sure that Euclid would roll over in his grave if he knew that his dimensionless points had gone off their diets.

 

[StatusX]

The fact is, continuity in the sense I was referring to is only a valid concept for functions from R to R. Discrete functions cannot possibly be continuous, because it would be impossible for the limit to exist. That's the point. You can pick ANY e>0, and if the function is continuous, there will be a d>0 such that |f(c-d)-f(c)|<e. This is not possible if the function is quantized, because if e<|f(c-q)-f(c)|, where q is the size of the quanitization, then there will be no d that will work. So I wasn't confusing continuum with continuous, but they are more closely related than you seem to think.

Actually, I'm not even sure what were talking about. Whether 0.999... is 1? Whether the real numbers exist? If infinite tasks are impossible? Which is it?

 

[Canute]

Seeing as how it was my offhand laymans remark about .99... equaling 1 that set this off I'll clarify what I meant about points and ranges. I'm well aware that I'm not a mathematician, but for me this is more about meta-mathematics than mathematics per se. I expect some of this is incorrect but I'll say my piece and you can tear it apart.

A mathematical point is by definition dimensionless. How then can such a point be divisible? If one models motion as taking place in a medium made up of points which are each infinitely divisible then this in fact models motion in a continuum, not quantised motion. Thus, for practical purposes, the calculus gets around Zeno's objections to motion in quantised spacetime by un-quantising it. However it is a fudge, since the calculation of these points uses the concept of limits. It defines points in spacetime as infinitely divisible but then treats them as if they are not.

When I said .999...=1 I should have said 'in the limit'. I was suggesting that in reality .999... does not equal 1, it equals .999... Obviously this number, as it expands, approaches 1. However it never becomes 1, precisely because if points are infinitely divisible then there is always a number between .999... and 1. To round off .999... to 1 is to assume that points are not infinitely divisible.

I was suggesting that the concept of infinitessimals does not solve Zeno's paradoxes because in the calculus one has ones cake and eats it. Points are considered to infinitely divisible but they are not infinitely divided. In the end isn't the whole purpose of the calculus the avoidance of infinite divisions?

Thus the concept of infinitessimals allows us to model motion mathematically, but does not answer the question of how motion is possible if spacetime is quantised.

To put it another way, infinitessimals are conceptual things, mathematical tools, not things that exist. If spacetime is quantised then its fundamental quanta have physical extension. As such they are not points but ranges, the extent of the range determined by the diameter of the point. When we divide it again and again the range is reduced, but it cannot be reduced to nothing except conceptually. (The Dedekind Cut seems relevant here but I won't risk saying anything about about that).

That's not very clear but best I can do at the moment.

 

[matt grime]

It is incorrect to say that "in the limit 0.999...=1" as a piece of English, since 0.999... is itself a limit point so you're being tautogical.

It is correct to say that the limit as n tends to infinity of the n'th partial sum is 1, it also correct to say 0.999...=1 since we have implicitly constructed it as a limit, and they are in the same equivalence class of cauchy convergent sequences and hence in the space of reals they are equal. (Better to say equivalent perhaps, but equal is the norm).

You are saying that we must always say "in the limit the limit of the partial usm of 0.99... is 1" which is completely unnecessary.

 

[NeutronStar]

Constructing Mathematical Objects from Limits?

It is correct to say that the limit as n tends to infinity of the n'th partial sum is 1, it also correct to say 0.999...=1 since we have implicitly constructed it as a limit.

Constructed it as a limit?

This is the kind of talk that just blows me away.

Where's the justification for claiming that anything has been "constructed"?

This isn't a personal attack on you Matt. I realize that many mathematicians have been taught this. But who's teaching this and why? Who decided that a limit can be said to have "constructed" something? Where did that idea come from? Can we point to a famous mathematician that came up with the theory of "constructions" by limits? Or is this just something that kind of crept into mathematics on the sly?

I mean, I'd really like to know just who it was that justified the idea that Weierstrass's limit definition can be used to claim that a mathematical object has been constructed in its entirety.

This isn't merely a philosophical question. This is a problem with logic. As I mentioned in an earlier post Weierstrass included in his limit definition the condition that delta cannot equal zero,… i.e.:

$$\exists\delta\ni\delta>0$$

This little piece of information is not trivial. In fact, it's there because without it the rest of the definition would fall apart. It's crucial to the definition that delta not be allowed to become zero. This a very important part of the logic that insures that conclusions of the definition are meaningful. Take that little piece of the definition out and the resulting conclusions have no foundation in logic.

Well,… ignoring that little piece of the definition is precisely what we must do if we want to claim that we have completed the limit process and constructed a complete mathematical object in its entirety. Yet this is precisely what mathematicians are doing when they claim to have constructed a mathematical objecting using the definition of the limit.

Therefore I hold that those constructions are logically flawed and therefore they cannot be depended upon to be logically meaningful in any way.

To say that 0.9999…. = 1 in an absolute way is to say that we have taken delta to equal zero in Weierstrass's limit definition and that's a violation of logic. We are simply logically incorrect to make such a claim. We have no logical justification for doing so.

This is much more than merely a philosophical opinion. This is a serious logical issue.

 

[matt grime]

If the reals are not constructed as either dedekind cuts or cauchy sequences, what are they? as a model of a complete totally ordered field I mean.

Things are constructed in mathematics all the time. Arguably everything in mathematics is a construction in some sense. Are you positioning yourself as a platonist and claiming that there is a physical object that is the real numbers? If so what is it? Maths generally isn't done like that. It is merely a formal thing we play around with. If it can be usefully used to model the real world so much the better, I suppose, but no one should actually think that the things we use in maths have any existential form. It is not necessary, and frequently not useful.

 

[StatusX]

Constructed it as a limit?

This is the kind of talk that just blows me away.

Where's the justification for claiming that anything has been "constructed"?

...

This isn't merely a philosophical question. This is a problem with logic. As I mentioned in an earlier post Weierstrass included in his limit definition the condition that delta cannot equal zero,… i.e.:

$$\exists\delta\ni\delta>0$$

This little piece of information is not trivial. In fact, it's there because without it the rest of the definition would fall apart. It's crucial to the definition that delta not be allowed to become zero. This a very important part of the logic that insures that conclusions of the definition are meaningful. Take that little piece of the definition out and the resulting conclusions have no foundation in logic.

Well,… ignoring that little piece of the definition is precisely what we must do if we want to claim that we have completed the limit process and constructed a complete mathematical object in its entirety. Yet this is precisely what mathematicians are doing when they claim to have constructed a mathematical objecting using the definition of the limit.

Therefore I hold that those constructions are logically flawed and therefore they cannot be depended upon to be logically meaningful in any way.

To say that 0.9999…. = 1 in an absolute way is to say that we have taken delta to equal zero in Weierstrass's limit definition and that's a violation of logic. We are simply logically incorrect to make such a claim. We have no logical justification for doing so.

This is much more than merely a philosophical opinion. This is a serious logical issue.

I understand what your saying, but youre missing something important. All I want you to do is tell me which step has the mistake:

1.
$$0.999... \equiv \lim_{n \rightarrow \infty} \sum_{k=1}^n 9 \cdot 10^{-k}$$

That triple equal sign means "is defined as."

2.
$$\lim_{n \rightarrow \infty} \sum_{k=1}^n 9 \cdot 10^{-k} = L$$

If there is a number L such that for any e>0, you can find an n such that |L - S(n)| < e, where S(n) is the partial sum of the first n terms.

3.
In this case, it is easy to prove that n = floor(2 - log(e)) will satisfy the condition |1 - S(n)| < e for any e>0, and so L=1.

4.
So, by the transitivity of equality:

$$0.999... = 1$$

I know this 0.999... thing is getting tiring, but this will help you understand why your wrong. And please get back to reality, you are not discovering some hidden flaw no one has seen before. You are misunderstanding basic concepts about real numbers, the way they were defined.

 

[NeutronStar]

I understand what your saying, but youre missing something important. All I want you to do is tell me which step has the mistake:

Well obviously your very first step is logically incorrect.

rewrite it as follows and you'll see why,…

$$f(c) \equiv \lim_{x \rightarrow c} f(x)$$

You simply have no logical basis for defining something as its limit. f(x) may not have a value at f(c). You can't use the definition of the limit to define the value f(c). There's nothing in the definition of the limit that supports this.

Just because 0.999… has a limit defined at infinity doesn’t' mean that it is equal to that limit.

There is nothing in the definition of a limit that allows you to claim that any process actually equals its limit.

On the contrary, the part of the limit definition that says,…

$$\exists\delta\ni\delta>0$$

actually forbids you from claiming that equality using the defintion of the limit alone.

So not only are you incorrect in doing this, but you are actually forbidden by definition to do it.

If mathematicians are doing this on a regular basis then all they are really doing is ignoring the details of Weierstrass's definition.

 

[CrankFan]

Constructed it as a limit?

This is the kind of talk that just blows me away.

Where's the justification for claiming that anything has been
"constructed"?

There are many equivalent ways to define real numbers, one method that's
used a lot is the Cauchy construction. A real number can be thought of
as an equivalence class of Cauchy sequences of rational numbers.

What's a Cauchy sequence, you say? I'm glad you asked!

A Cauchy sequence of rational numbers is a sequence $$x$$ of
rationals such that for every positive rational number $$\epsilon$$
there exists a positive integer $$N$$ such that for
every $$m, n > N$$ we have:

$$\vert x_n - x_m \vert < \epsilon$$

In English, for any epsilon (no matter how small!) there is some point
in the sequence, after which the difference of any two terms is less
than that epsilon.

In this construction reals aren't actually Cauchy sequences of
rationals, but equivalence classes of Cauchy sequences of
rationals, and this is how we get to .999... = 1

1, 1, 1, 1, 1, 1, ... is a Cauchy sequence of rationals. You can
confirm this with the definition provided above.

9/10, 99/100, 999/1000, ... is also a Cauchy sequence of rationals,
again you can confirm it based on the definition above.

So the question you might be thinking about at this point is how is the
equivalence relation on Cauchy sequences of rationals defined? I'm glad
you asked!

We say that two Cauchy sequences x and y are equivalent iff
for every positive rational number $$\epsilon$$ there is an
integer $$N$$ such that for all $$n > N$$ we have:

$$|x_n - y_n | < \epsilon$$

In other words, for every epsilon greater than zero (no matter how small!)
there is a point in both sequences after which the difference between any
two terms is less than that epsilon.

Recall the two Cauchy sequences in question:

$$x = 1, 1, 1, 1, 1, 1, ...$$
$$y = 9/10, 99/100, 999/1000, ...$$

To prove they are equivalent we must show that for every $$\epsilon$$
there is an $$N$$ such that $$n > N$$ implies $$|x_n - y_n | < \epsilon$$

$$|x_n - y_n | = 1^n - { (10^n - 1) \over 10^n } = { 1 \over 10^n }$$

If we choose an $$N$$ such that $$10^N > { 1 \over \epsilon }$$ then we have

$$|x_n - y_n | <= { 1 \over 10^n }$$

$$|x_n - y_n | < { 1 \over 10^N }$$

$$|x_n - y_n | < \epsilon$$

And we're done. (Uh, I think... pending any corrections from mentors )

 

[Hurkyl]

I'm sure that Euclid would roll over in his grave if he knew that his dimensionless points had gone off their diets.

I'm not sure whose comment you were addressing, but it wasn't mine. I was addressing your comment that "It's simply a logical contradiction in terms to claim that an infinite process can be completed."

How then can such a point be divisible?

Nobody says a point is divisible. Divisibility refers to space.

I was suggesting that in reality .999... does not equal 1, it equals .999..

Just like 1/2 doesn't equal 2/4, I suppose.

Well obviously your very first step is logically incorrect.

rewrite it as follows and you'll see why,?

$$f(c) \equiv \lim_{x \rightarrow c} f(x)$$

This is exactly your misconception, because this is exactly what StatusX was not saying.

 

[StatusX]

You simply have no logical basis for defining something as its limit.
...
Just because 0.999… has a limit defined at infinity doesn’t' mean that it is equal to that limit.
...
There is nothing in the definition of a limit that allows you to claim that any process actually equals its limit.

I'll try to be extremely careful and thorough here so if there are any errors they can be easily spotted.

Decimal notation is defined in terms of limits. A decimal expansion consists of an infinite series of integers between 0 and 9:

$${d_N, d_{N-1}, ..., d_1, d_0, d_-1,...}$$

In general, this starts at some integer N and goes to $$-\infty$$. The real number r represented by this series is defined as:

$$r \equiv \sum_{n=-\infty}^{N} d_n \cdot 10^n$$

If you want to get really technical, you can rewrite this (yea, this is just a rewrite) as:

$$r \equiv \lim_{m \rightarrow \infty} \sum_{n=-m}^{N} d_n \cdot 10^n$$

This is a definition. ok? If you dispute the truth of this statement, then you arent talking about the same decimal notation as the rest of us. If you don't think this definition is logically sound, I'll address that below.

Now, you might argue that irrational numbers are poorly defined in this system, but I'm ignoring them for now. Any repeating decimal can be rewritten as an infinite sum, or again, if you are fussy, as the limit of a sequence of partial sums. For example, for 0.333...(where the dots just imply that $$d_m=3$$ for any arbitrarily large negative integer m):

$$\lim_{m \rightarrow \infty} \sum_{n=-m}^{-1} 3 \cdot 10^n$$

$$= \lim_{m \rightarrow \infty} \sum_{n=1}^{m} 3 \cdot 10^{-n}$$

$$= 1/3$$

I could show this using the epsilon delta defintion, but I hope you don't need me to. So, just to reiterate: 0.333... is a mathematical symbol, just like an integral sign or a radical. It is defined as the value of a limit, which is in turn defined by the epsilon delta method.


Now your problem, which is demonstrated nicely in the quotes above, is that you are confusing the value of a limit with the process of taking partial sums. These are not the same. There are no variables in 0.999...: it is a constant, and it is meaningless to take a limit of it.

What you are implicitly assuming is that the number has to be written out completely to have a value. If you sat down with a pen and paper and started writing "0." followed by as many nines as you could, you would be performing a process. The number you write down would never equal 1, no matter how many nines you write (note that infinity isn't a number, not to mention the universe is finite). However, this is NOT what 0.999... means in ANY sense. The abstract mathematical symbol 0.999... is defined as above, and is equal to 1.

to be clear:
A zero, followed by a decimal point, followed by three nines, followed by three dots is an abstract symbol which is meant to reperesent the value of an infinite sum, or more precisely, the limit of a sequence of partial sums, which in this case turns out to be equal to 1.

It is very important you understand the difference between the process of listing numbers and taking a limit. When you say something that I can only interpret as "limits can never exactly equal their limit, they just approoach it," you are talking nonsense. Specifically, by the first "limits," you mean the partial sums, or the values of a function as x gets closer and closer to c. It is true, these never equal the limit value, but they are completely separate entities from this value. The limit is the number L as defined in the epsilon delta method. By this definition, none of the partial sums or close values are equal to it. But these are only used in calculating the limit. L is a real number, and it is not changing in any sense.


One final point. You mentioned that I can't say that:

$$\lim_{x \rightarrow c} f(x) = f(c)$$

This is true in general. However, in this case, c is infinity, and in this case, that statement is true. In fact, it's how infinity is defined!

$$f(\infty) = \lim_{x \rightarrow \infty} f(x)$$

There is no number infinity, so it must be treated as special, as in this example. Also, infinite limits are different in ordinary limits because instead of getting closer and closer to some value, x is alowed to get bigger and bigger without bound. Again, infinity is only definined in the context of limits.

 

[NeutronStar]

$$r \equiv \lim_{m \rightarrow \infty} \sum_{n=-m}^{N} d_n \cdot 10^n$$

This is a definition. ok? If you dispute the truth of this statement, then you arent talking about the same decimal notation as the rest of us. If you don't think this definition is logically sound, I'll address that below.

Alright,... I see what you are saying. It's an arbitrary definition. It's not intended as a logical proof.

Well, all I can say is that this is a terrible shame because by defining the real numbers in this way it forces the field of real numbers to be a continuum. Yet that idea is logically incompatible with the idea of dimensionless points, and with the idea of quantitative individuality, both of which are important to correctly model the quantitative nature of the universe.

I'm just surprised that the scientific community accepts this. This is simply an incorrect picture of the quantitative property of our universe. With physical theories becoming more and more dependent on abstract mathematics this is not good.

How can an arbitrary definition ever be disproved? You either accept it or you don't. It's kind of like religion. Mathematics has become a faith-based discipline I suppose. I honestly didn't realize that until just now.

I guess these forums are a good learning tool even though I'm not real happy with what I have learned here.

 

[StatusX]

I should just note that what I meant to do was define decimal notation. Although this is probably a valid definiton of the reals if the dn are allowed to take on any values between 0 and 9, its usually done the other way around. That is, real numbers are defined in some way, and then a decimal expansion of a real number is defined as the series which makes that statement true. You could also recursively define the decimal expansion from the real number using floor and mod 10 functions. The reason I did it this way is because the argument about 0.999... isn't one of real numbers, it is one of notation, and a rigorous definition of decimal notation is needed to prove that it equals 1.

 

[Hurkyl]

That's right, you cannot disprove a definition. This is a fundamental point about mathematics that many nonmathematicians have difficulty grasping.

Mathematics is, for the most part, not empirical -- it is about definitions (and axioms), and their logical consequences.


The application of mathematics to the "real world", though, is empirical. Physics isn't done over a continuum because mathematicians "like" continua (which is entirely untrue -- many mathematicians prefer more discrete or algebraic subjects) -- physics is done over a continuum because it works.

 

[matt grime]

Why does the fact that R is a continuum imply points do not have dimension 0? What the buggery flip does quantitative individuality even mean?

Working with the real numbers as their tool physicists have managed to prove many things.

Also, what is a point, and why must it be dimensionless - ie what evidence are your (NeutronStar's) postulates based on, what empircal things do you claim to know here?

 

[Canute]

Aha. I see what I got wrong anyway. I gather that .999... is defined as being 1 by mathematicians. I was therefore wrong to use the notation '.999...' to mean some number less than 1. What is the correct way to express an infinite decimal series that is not assumed to sum to its notional limit?

 

[jcsd]

Imagine if we weren't allowed to use the number e in physics, or if we were forbideen from using transcendental functions, that just about excludes us from using quantum physics and much more besides. I can see no way that we can avoid using the real numbers in physics

 

[matt grime]

Since a decimal series does, by definition, equal the limit of the partial sums, then you are simply not talking about decimal (as real numbers).

You are more than welcome to talk about strings of digits (x_1,x_2,x_2..) with each x_i between 0 and 9 and two strings are equal if and only if they are equal componentwise, but you ain't doing anything in a model of the reals.

 

[synergy]

As was said, you can't disprove a definition. However, many useful branches of mathematics have been developed by CHOOSING different definitions. It's all about consistency. If you want to define something differently and then follow that definition to its logical implications, and if all of these implications are consistent (don't contradict each other) then you might have developed something useful. This is what happened with non-euclidean geometry, for example. Parallel lines were defined in a way that allowed zero lines through a point P to be parallel to line L (when P is not on L) and another form of non-euclidean geom has an infinite number of lines through P parallel to L. Each geometry has applications when the space in question is not flat. I've often wondered if there could be a "calculus" of discrete space rather than continuous space, where the points of the space were too small to treat en masse as we would macroscopic measures, but being discrete we couldn't take limits of the infinitely small as such a thing wouldn't exist. Maybe this doesn't quite make sense, or maybe this is an idea of quantum mechanics applied to space or something. Whatever, this IS the philosophy section so I guess it only has to make sense to me and those as crazy as I am? Anyway, maybe you'll come up with your own theory - good luck.
Aaron

 

[NeutronStar]

Hurkyl

That's right, you cannot disprove a definition. This is a fundamental point about mathematics that many nonmathematicians have difficulty grasping.

Actually I have no problem with definitions. I just wasn't aware that the concept of number had been defined in two different ways. I was much happier with the definition of number as it was based on set theory. Although I must confess that I do have some concerns about logical inconsistencies even with that definition. But I also see how they can be fixed up while maintaining the basic idea of sets. More importantly I understand the historical development of that idea and I see how it arose from empirical observations.

I honestly wasn't aware that the idea of number was defined twice within the formalism. While I agree with you that we cannot disprove a definition I do believe that it is possible to prove that a particular definition is logically inconsistent and must therefore be abandoned on that ground. After all, mathematics is based on logic and if we can show why some particular mathematical definition is logically inconsistent then that definition should be abandoned.

I believe that I can show that this limit definition of the reals is logically inconsistent. However, to do so would require the acceptance that points must be dimensionless. From what I gather I would have a problem gaining general acceptance with that idea.

Mathematics is, for the most part, not empirical -- it is about definitions (and axioms), and their logical consequences.

I agree. But I also hold that if the definitions in mathematics are based on logical inconsistencies then any logical consequences that are arrived at will necessarily also contain logically inconsistencies.

The application of mathematics to the "real world", though, is empirical. Physics isn't done over a continuum because mathematicians "like" continua (which is entirely untrue -- many mathematicians prefer more discrete or algebraic subjects) -- physics is done over a continuum because it works.

Well if that's true then mathematicians should love my ideas.

Aha. I see what I got wrong anyway. I gather that .999... is defined as being 1 by mathematicians. I was therefore wrong to use the notation '.999...' to mean some number less than 1. What is the correct way to express an infinite decimal series that is not assumed to sum to its notional limit?

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

It's a bummer ain't it!

Why does the fact that R is a continuum imply points do not have dimension 0?

Well actually this is easier to see the other way around.

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

What the buggery flip does quantitative individuality even mean?

Whenever we call something One from a quantitative point of view, we are claiming that it has a property of quantitative individuality. I mean, why else would we want to call it "One"?

Unfortunately, this property of quantitative individuality has basically been swept under the carpet by the acceptance of the concept of an empty set. By sweeping this extremely important concept under the carpet we have actually permitted people to call things "One" that don't really have a quantitative property of individuality.

Take the set of natural numbers for example. It has a quantitative property of being infinite. Right? I mean, the set itself contains an infinity of elements, therefore it's cardinal property is infinity. Yet everyone wants to treat these set as though it has a property of being "One". In other words, we say that it is one set, and therefore we feel justified in treating it as though it is one thing. Believe it or not, it is actually the acceptance of the empty set that permits us to logically do this.

So now we have this thing that is both quantitatively infinite and quantitatively One at the same time! If you actually think about this for a moment doesn't it make you wonder whether infinity equals one? I mean, here we have an object that qualifies mathematically as being representative of both infinity and one.

I hold that what is actually going on here is that when we treat the set of natural numbers as 1 thing we are doing so in a qualitative manner that is not quantitative. In other words, we are recognizing that it can be viewed qualitatively as 1 thing if we ignore its quantitative nature.

Well, that really does introduce extreme logical inconsistencies into a formalism that is supposedly based on the idea of quantity.

Yes, I know! Everyone doesn't agree that mathematics should be restrained to be about ideas of quantity. But this is where it came from historically. Mathematics, and the concept of number, came to be because humans recognized that the universe displays a quantitative nature. This was the root of the whole idea for mathematics.

So, all I have to say is that if mathematics is not going to embrace non-quantitative concepts, the least it could do is recognize what these other concepts are and when they are being used. Up to this point in my life I have never seen any formal recognition of these non-quantitative concepts.

Working with the real numbers as their tool physicists have managed to prove many things.

I would have to ask for an example here. What have physicists proven that DEPENDS on the mathematical definition of the real numbers?

Just one example will suffice.

Also, what is a point, and why must it be dimensionless - ie what evidence are your (NeutronStar's) postulates based on, what empircal things do you claim to know here?

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

Obviously to fully understand this you would need to understand the meaning of quantitative individuality (or One). It wouldn't make much sense to talk about One point if we didn't have a definition for the meaning of One.

Imagine if we weren't allowed to use the number e in physics, or if we were forbideen from using transcendental functions, that just about excludes us from using quantum physics and much more besides. I can see no way that we can avoid using the real numbers in physics

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

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

 

[matt grime]

Calculus in the sense of limits can be "defined" for any topological space, and spaces far more complicated than just the reals. There is a notion of Moore smith convergence that is important in functional analysis where by the "sequences" are indexed by things more complicated than the Naturals. Making things simpler than R in this vague notion though often leads to trivial theories - places where no limits exist or all sequences converge to every point. Look up point set topology. One can even do integration over bizarre objects too. And whose to say that the metric we complete the rationals in is the "correct one". There are non-archimidean norms on the rationals (p-adic valuations) that lead to other number systems in the completion - the p-adics which have their own analytic results and are treated in many courses at univeristies. There is the interesting effect that working in base p, only finitely long p-adic expansions (after the decimal point) converge. Ie, 0.22... base 3 does not make sense in the 3-adics, but the infinitely long "natural" number ...2222222 does exist, and infact is equal to -1! Don't believe me then add 1 to it - what happens, the left most 2 becomes a 3, so that's a zero carry one to the left, and so on, and this converges in the 3-adic norm to 0, hence it is indeed -1.

 

[jcsd]

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

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

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

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

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

 

[matt grime]

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

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

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

 

[synergy]

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

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

 

[matt grime]

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

 

[Canute]

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

It's a bummer ain't it!

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

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

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

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

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

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

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

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

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

 

[synergy]

qualitative difference

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

 

[synergy]

continued

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

 

[Hurkyl]

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

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


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


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

 

[NeutronStar]

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

It's a bummer ain't it!

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

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

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

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

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

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

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

That's the foundational idea.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[StatusX]

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

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