Infinite Infinitesimal...

[Orion1]

What IS Infinity?

Many of us might consider numbers the most sure-footed way to come within sight of infinity, even if the mathematical notion of infinity is something we'll never even remotely comprehend.

Mathematicians tell us that any infinite set?anything with an infinite number of things in it?is defined as something that we can add to without increasing its size. The same holds true for subtraction, multiplication, or division. Infinity minus 25 is still infinity; infinity times infinity is?you got it?infinity. And yet, there is always an even larger number: infinity plus 1 is not larger than infinity, but 2^infinity is.

When we're told that the decimals in certain significant numbers, like pi and the square root of two, go on forever, we can somehow accept that, especially when we learn that computers have calculated the value of pi, for one, to over a trillion places, with no final value for pi in sight. (For more on pi, see Approximating Pi.) When we're told that there are 43,252,003,274,489,856,000 possible ways to arrange the squares on the Rubik Cube's six sides, we may feel intuitively (if not rationally) that we must be on our way to the base of that loftiest of all peaks, Mt. Infinity.

One reason we may feel this way is that such numbers are as intellectually unapproachable to the mathematically challenged as infinity itself. Take a Googol. A Googol is 10100, or 1 followed by 100 zeroes, and is the largest named number in the West. The Buddhists have an even more robust number, 10140, which they know as asankhyeya. Just for fun, I'll name a larger number yet, 101000. I'll call it the "Olivian," after my daughter. Now, doesn't an Olivian get me a little closer to infinity than the Googolians or even the Buddhists can get? Nope. Infinity is just as far from an Olivian as it is from a Googol?or, for that matter, from 1.

Infinities do come in two sizes, of course?not only the infinitely large but also the infinitely small. As Jonathan Swift wrote, "So, naturalists observe, a flea/Has smaller fleas that on him prey/And these have smaller still to bite `em/And so proceed ad infinitum."

Of course, just when we think we have infinity in the palm of our hands, we watch it evaporate in the harsh light of another of those confounding paradoxes: the numerals 2 and 3 are separated by both a finite number (1) and an infinity of numbers. This conundrum spawned one of the great paradoxes of history, known as Zeno's paradox. Zeno was a Greek philosopher of the fourth century B.C. who "proved" that motion was impossible. For a runner to move from one point to another, Zeno asserted, he must first cover half the distance, then half the remaining distance, then half the remaining distance again, and so on and so on. Since this would require an infinite number of strides, he could never reach his destination, even if it lay just a few strides away.

It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.

We believe Hamlet when he says "I could be bounded in a nutshell/And count myself a king of infinite space." The shortest length physicists speak of is the Planck length, 10^-33 centimeters. But might not there be an even shorter length, say, 10^-333 centimeters, or 10-an infinite number of 3's centimeters?

as one mathematician pointed out to me, infinity is an abstract concept, appearing only in our mental images of the universe. It is not actually in the universe.

The Greeks, in fact, invented apeirophobia, fear of the infinite. (The term comes from the Greek word for infinity, apeiron, which means "without boundary.") Aristotle would only admit that the natural numbers (1, 2, 28, etc.) could be potentially infinite, because they have no greatest member. But they could not be actually infinite, because no one, he believed, could imagine the entire set of natural numbers as a finished thing. The Romans felt just as uncomfortable, with the emperor Marcus Aurelius dismissing infinity as "a fathomless gulf, into which all things vanish."

The ancients' horror infiniti held sway through the Renaissance and right up to modern times. In 1600, the Inquisitors in Italy deemed the concept so heretical that when the philosopher Giordano Bruno insisted on promulgating his thoughts on infinity, they burned him at the stake for it. Later that century, the French mathematician Blaise Pascal deemed the concept truly disturbing: "When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there ... now instead of then." Martin Buber, an Israeli philosopher who died in 1965, felt so undone by the concept of infinity that he "seriously thought of avoiding it by suicide."

...like that engendered by this gem from another anonymous sufferer of our common infirmity: "Infinity is a floorless room without walls or ceiling."


Reference:
http://www.pbs.org/wgbh/nova/archimedes/contemplating.html
http://www.pbs.org/wgbh/nova/archimedes/infinity.html

[wuliheron]

Words and concepts only have demonstrable meaning according to their function in a given context. Outside of a given context, infinity is just as meaningless as any other concept.

Within any context, it is a notably vague term. When I talk about a chair, I can point to an example and ramble on for days about exactly why it is called a chair, pointing out each and every feature and their uses, which you are free to examine for yourself. Although people certainly ramble on for days about infinity, no one has ever been able to prove infinity exists as more than an idea.

That is not to say it is a useless fantasy. Infinity possesses the properties of both the demonstrable and undemonstrable. It is a cross between the utterly paradoxical and rational. Even the paradoxical has its uses, and being a bit less extreme infinity has even more obvious uses. However, in and of itself infinity is demonstrably useless, it is only a useful concept within the context of the finite. Here is an ancient chinese poem which expresses such relationships.

Tools

Thirty spokes meet at a nave;
Because of the hole we may use the wheel.
Clay is moulded into a vessel;
Because of the hollow we may use the cup.
Walls are built around a hearth;
Because of the doors we may use the house.
Thus tools come from what exists,
But use from what does not.

[Hurkyl]

A Googol is 10100, or 1 followed by 100 zeroes, and is the largest named number in the West.

Wee, math trivia! We actually have a number called a googolplex which is a 1 followed by a googol zeroes.

[arildno]

Besides, the Ramsey number is a LOT bigger than either..

[Smurf]

Wee, math trivia! We actually have a number called a googolplex which is a 1 followed by a googol zeroes.

I had trouble comprehending a googol, now you go and throw this at me :cry:

-Ruler of the Universe,
Smurf

[metacristi]

Wee, math trivia! We actually have a number called a googolplex which is a 1 followed by a googol zeroes.

Still not enough to win the 'who can name the bigger number' contest :-)

http://www.cs.berkeley.edu/~aaronson/bignumbers.html

[NeutronStar]

It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.

This is actually a huge misconcpetion on the part of the mathematical community. Calculus does not resolve Zeno's paradoxes. For two reasons,...

First, if we exam the defintion of the limit in detail it never actually says that any of those infinite processes actually equal the value of the limit. It only says that as some variable approaches some value then a particular function or process approaches infinity.

Well, gee, that's what Zeno was saying all along! So how is that a solution?

In fact the definition of the limit is clearly set up as to not permit the variable from actually reaching the value in question. In other words, there exists a delta greater than zero 0 such that,... In other words, the calculus limit says absolutely nothing about what happens should we decide to consider delta=0. The definition of the limit is no longer valid in that form. This prevents us from actually claiming that the value is ever actually reached. This is why good mathematicians are sure to say things like "in the limit" a value is equal to something. Because to actually say flat out that it is equal is simply wrong. It denies the very conditions of the definition of the limit. Unfortunately far too many mathematicians seem to have dropped the "in the limit" phrase and think that something is actually equal to a calculus limit if they can prove that a limit exists. It is not. That is actually a misuse of the definition of a limit.

Secondly, Zeno's real question is this,... "How can it be that an infinite number of tasks can be completed". Nowhere in calculus is it ever claimed that an infinite number of processes can be completed. In fact, to prove a limit all we need to do is prove things like boundedness and trends. If we look at all of the definitions and proofs for any limit we will clearly see that we haven't proven anywhere along the way that we have actually completed an infinite process, nor have we proven that it can be completed. All we have done is shown that no matter how long we continue the process we will continue to get closer and closer to the value that we call the limit.

Well, again, Zeno would say, "So what? That's what I've been saying all along!"

The mathematical community is absolutely incorrect to claim that it has solved Zeno's paradoxes with calculus yet this is a widely held misconception. The calculus limit really doesn't say any more than Zeno had alread pointed out. If he were alive today I am absolutely certain that he would simply say that these people just haven't truly understood the question that he is asking,... He wants to know how an infinite number of tasks ever be completed? Calculus does NOT answer that question, nor does it claim to. Yet it still claims to have solved Zeno's paradox. Clearly the mathematical community doesn't understand the question that Zeno is asking.

Fortunately physics has answered Zeno's paradox. Time and space are not continuous, they are quantized. Zeno was probably right. If time and space were continuous we probably wouldn't be able to move. But since they are quantized only need to complete a finite number of tasks to move and finite distance because we do it in quantum jumps skiping over the supposed continuum.

After all, Zeno's paradox is only a paradox in a continuous universe. Once the universe is known to be quantized it's no longer a paradox as to why we can move. Calculus has nothing at all to do with the paradox at all.

[robert Ihnot]

Orion1: It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit.

As Neutron Star points out this is commonly supposed. It is a way for calculus buffs to indulge in self-congratulations. But, probably every generation could find some answer to Zeno.

However, I do believe that Archimedes was aware of the finite limit of the infinite sum. Indeed he used "infinite triangle sums" to discover the area of parabola.

[Hurkyl]

Fortunately physics has answered Zeno's paradox. Time and space are not continuous, they are quantized.

Physics has said no such thing.

No empirically verified theory exhibits any sort of geometric quantization. While some of the promising research topics do exhibit geometric quantiziation, they still don't exhibit quantization of position.

[NeutronStar]

Physics has said no such thing.

No empirically verified theory exhibits any sort of geometric quantization. While some of the promising research topics do exhibit geometric quantiziation, they still don't exhibit quantization of position.
I beg to differ.

It is meaningless to speak in terms of absolute position. That might be an abstract mathematical concept, but it has no meaning with respect to the universe.

In our universe any "real" particle or phenomenon exhibits some form of energetic disturbance. If it didn't how could we even claim that it exists? How could we even know of such a "invisible" entity. It would be completely undetectable in any possible way.

Most particles in the universe exhibit as least some mass which associates them with gravitational energy. Any known massless particle (like say a photon) would is associated with some other form of energy (like maybe electromagnetism)

So in any case, every particle that can possibly be associated with a position must necessarily take on a quantized position relative to other particles. In other words, it can only move in a way to take on, or give up, a quantum of energy.

Therefore, any "real" concept of position in the universe is necessarily a quantized concept due to an objects potential energy relative to other objects.

It's true that we can fool ourselves into believing that there are somehow "abstract" possible positions between these particles that aren't quantized. But isn't that putting something onto the universe that isn't really there? I mean, what good is a notion of a position that no particle can take on?

So I do hold that physics has shown that position is necessarily quantized in the real universe, and it is totally meaningless to talk about some kind of continuum when the particles in the universe do not actually behave in that fashion. Why make things up that aren't a true reflection of the properties of our universe?

It's going to take people a very long time to let go of this idea of the continuum. Our universe just isn't continuous. That's all there is to it. At the quantum level things "jump" around, They just don't move continuously. They also don't traverse the space between their jumping. There are either here or there, but never in between. That's the true nature of our quantized universe.

This isn't just true of things that are bound to energy levels within atoms. Anything that changes position in the universe is changing its relative position to something else. Therefore it is changing the relative potential energy whether it be gravitational energy or some other form. Therefore position is quantized for everything in the universe.

One could argue that a so-called "free electron", or whatever, is not restrained to quantized motions. But for the reasons I just gave an electron is never actually "free". Also, would it really even be meaningful to talk about the position of such a free particle? To speak of its position we can only do so in relative terms. Once we have set its position relative to something then in a very real sense it is bound to that reference point by mere convention and can only change its potential energy relative to that position in a quantized fashion.

So how can the idea of position ever not be quantized in our universe?

[Hurkyl]

It is meaningless to speak in terms of absolute position.

Even so, you can still speak about position being quantized.


But I should have been more precise: what I meant to say is that length is not quantized.


Energy is also not quantized. It is true that bound particles exhibit quantized energy levels, but energy, in general, can come in any quantity.


P.S. don't forget that there is no quantum theory of gravity. Furthermore, TMK, there has only been one experiment, ever, that has demonstrated quantum effects in a gravitationally bound system. (Of course there has only been one experiment that has tested it)

[anti_crank]

So in any case, every particle that can possibly be associated with a position must necessarily take on a quantized position relative to other particles. In other words, it can only move in a way to take on, or give up, a quantum of energy. How does this follow? What is this quantum of energy, and where does it come from?

[NeutronStar]

P.S. don't forget that there is no quantum theory of gravity. Furthermore, TMK, there has only been one experiment, ever, that has demonstrated quantum effects in a gravitationally bound system. (Of course there has only been one experiment that has tested it)
Oops! You're right. I forgot they didn't get there yet.

But I think it's pretty obvious that this will have to be the case. I mean, do we think that quantum theory will end up giving up its quantized nature in order to agree with GR. Or is it more likely that GR will have to give up its smooth continuum to agree with QM?

[Hurkyl]

I won't pretend to know what quantum geometry looks like. :smile: I will just comment that the amount of discretization inherent in quantum mechanics is vastly exaggerated in popular accounts.

[Canute]

NeutronStar

I thought what you said about the notion of infinitessimals and limits not solving Zeno's paradoxes was spot on. Especially:

"The mathematical community is absolutely incorrect to claim that it has solved Zeno's paradoxes with calculus yet this is a widely held misconception. The calculus limit really doesn't say any more than Zeno had alread pointed out."

However you go on to say:

"Fortunately physics has answered Zeno's paradox. Time and space are not continuous, they are quantized. Zeno was probably right. If time and space were continuous we probably wouldn't be able to move. But since they are quantized only need to complete a finite number of tasks to move and finite distance because we do it in quantum jumps skiping over the supposed continuum."

I see this as the reverse of the truth for two reasons. Firstly, I can't agree that science has shown that time and space are quantised. How has science done this?

Second, Zeno's point was that if one takes spacetime to be quantised, or, equivalently, takes the number line to be a series of points, then motion is (mathematically) paradoxical. If one takes spacetime as a continuum then the paradox disappears. This is consistent with the effectiveness of the calculus, since a 'point' in a continuum is actually an infinitely divisible range. By that I mean, the only thing that is infinitewly divisible is a point in a continuum. If spacetime (or the number line) is quantised then why do we need to use infinitessimals to calculate motion?

If you bring Zeno up to date and imagine Achilles and the Tortoise to be two particles, each being one fundamental quanta in diameter, moving in a spacetime that is quantised into fundamental quanta of time and space (discrete moments and discrete positions), then relative motion between A and T makes no sense at all. That is, not unless you allow length contraction and time dilation.

[selfAdjoint]

If you bring Zeno up to date and imagine Achilles and the Tortoise to be two particles, each being one fundamental quanta in diameter, moving in a spacetime that is quantised into fundamental quanta of time and space (discrete moments and discrete positions), then relative motion between A and T makes no sense at all. That is, not unless you allow length contraction and time dilation.

Well you could define position change; you observe particle T at time 0 and again later at time t, and get different postions, and likewise for particle A.

[NeutronStar]

Zeno's point was that if one takes spacetime to be quantized, or, equivalently, takes the number line to be a series of points, then motion is (mathematically) paradoxical.
Yes, that is true if it is also assumed that a finite line segment contains an infinite number of points. Which is what was believed in Zeno's day. And is still believed to be true today. But actually that can't possibly be the case. That situation is a logical contradiction. That is to say that such a concept is a logical inconsistency.

If one takes spacetime as a continuum then the paradox disappears. I'm afraid I don't see why it would disappear if spacetime is a continuum. How does this change Zeno's paradox or resolve it?

This is consistent with the effectiveness of the calculus, since a 'point' in a continuum is actually an infinitely divisible range. By that I mean, the only thing that is infinitely divisible is a point in a continuum. If spacetime (or the number line) is quantized then why do we need to use infinitesimals to calculate motion?
I'm afraid that you've lost me here.

How can a 'point' be an infinitely divisible range? I can see the space between points as being an infinitely divisible range. But what sense does it make to say that any given point itself is an infinitely divisible range?

And more to the point (no pun intended) where in the formalism of calculus is there any mention of a point being an infinitely divisible range, or that such a concept has anything at all to do with infinitesimals?

It's not in the formal definition of the limit I can say that much with certainty.

[StatusX]

Why can't we complete an infinite number of tasks? This is an assumption. If it is taken true, and a "task" is defined as anything that you can describe in words, such as "going half the distance left," then Zeno has a point. If not, then his argument is meaningless. What reasons do you have for believing it?

[Canute]

Well you could define position change; you observe particle T at time 0 and again later at time t, and get different postions, and likewise for particle A.
True. But try working out the relative motion one instant at a time. Particle A can go no faster than one quanta of distance per instant (otherwise it would be in more than one place at the same time) and particle B can go no slower than one quanta of distance per instant, (otherwise it would be stationary). The only solution is to allow A to be at more than one place at a time (changing its length) or for the instants of A to be longer than the instants of B (changing its clock). I can't see a way around this problem except to say that spacetime is not quantised. This doesn't seem to disagree with any evidence as far as I can tell. (We seem to having this discussion in two threads so sorry if I'm repeating stuff).

[Canute]

Yes, that is true if it is also assumed that a finite line segment contains an infinite number of points. Which is what was believed in Zeno's day. And is still believed to be true today. But actually that can't possibly be the case. That situation is a logical contradiction. That is to say that such a concept is a logical inconsistency.
I agree. However the problem does not arise if the number line is conceived of as a continuum (as Charles Sanders Peirce argued it should be).

I'm afraid I don't see why it would disappear if spacetime is a continuum. How does this change Zeno's paradox or resolve it?
It allows points to be treated as if they are infinitely divisible ranges, and thus they can be treated non-paradoxically by mathematics. An infinitely divisible point is by definition not a point but a range. If it was a mathematical point it would not be divisible.

How can a 'point' be an infinitely divisible range? I can see the space between points as being an infinitely divisible range. But what sense does it make to say that any given point itself is an infinitely divisible range?
You're right, a point cannot be a range. However an infinitely divisible point (or rather, the concept of an infinitely divisible point) is not a point.

And more to the point (no pun intended) where in the formalism of calculus is there any mention of a point being an infinitely divisible range, or that such a concept has anything at all to do with infinitesimals?
There's no mention of it because it's taken for granted that such points have extension (conceptually at least). If they did not then how could they be divided?

It's not in the formal definition of the limit I can say that much with certainty.
A limit is defined according to some specified tolerance of error. That is, at the limit .9999... equals 1. But only because we decide to stop calculating and accept some innacuracy.

[matt grime]

A limit is defined according to some specified tolerance of error. That is, at the limit .9999... equals 1. But only because we decide to stop calculating accept some innacuracy.

Oh, dear, I cannot begin to explain the inaccuracies there, they all tumble to get out of my head at the same time.

[Canute]

Oh, dear, I cannot begin to explain the inaccuracies there, they all tumble to get out of my head at the same time.
Yes, I knew I was pushing my luck it a bit there. But it's not very helpful to me if you just throw up your hands. What was wrong with what I said?

[matt grime]

What are the real numbers? They are by definition the complete totally ordered field, or the set of dedekind cuts of the rationals or the set of equivalence classes of convergent cauchy sequences of rationals or a model of one of these, or possibly many other equivalent things. In any case 0.99... =1 is a simple consequence of the axioms. It has nothing to do with approximating, or adding things up and approaching and so on. These are formal mathematical objects, this is not a constructive universe, what you wrote just isn't mathematical.

If I gave in it's simply because this is such a common problem that I can't believe you haven't come across many of the threads on it. All objections are based upon the fact that the objector doesn't know the mathematics of the real numbers presuming them to be something they are not.

[NeutronStar]

The limit is defined according to some specified tolerance of error.
Gee, the formal definition of the calculus limit that I was taught has nothing in it at all about any specified tolerance of error. That would be the formal Weierstrass delta-epsilon definition. There's nothing in that definition about any tolerance of error.


That is, at the limit .9999... equals 1. But only because we decide to stop calculating and accept some innacuracy.
This is perfectly true. "In the limit" 0.999999.... = 1. But that is not the same thing as saying that 0.9999.... = 1.

To say that the 0.9999.... = 1 outside of the context of the limit is absurd. And anyone who fully understands the Weierstrass delta-epsilon definition of the limit knows that delta can never be said to be zero. Therefore to say that 0.9999...=1 in any asbsolute sense is to simply deny the very defintion of the limit. Weierstrass was very careful to include in his definition:

\exists\delta\ni\delta>0

Yet so many mathematicians seem to claim that deta can actually be thought of as having been taken completely to zero. They're simply wrong. They are ignoring Weierstrass's definition. They are jumping to conclusions that simply aren't in the formal definition.

By the way, Weierstrass didn't put that condition into his definition of the limit just for fun. He realized that the whole definition would break down if that condition isn't included! It's an extremely important concept to the whole idea of the limit. Yet mathematicians consistently try to ignore it! It just goes to show that they don't fully grasp the concept I suppose.

Any mathematician who tries to claim that 0.9999,...=1 outside of the concept of the limit should be totally embarrassed. There's just no justification for it. It's not a mathematical idea.

More to the point, 0.9999,.... does not equal 1. However, the limit of 0.9999... does equal 1. See what I mean? L=1 The limit equals 1.

0.9999.... doesn't equal 1. Its limit equals 1. That's a whole different concept than saying that 0.9999.... = 1. In fact, to say that latter is simply mathematically incorrect. There's no basis for it. It's simply incorrect.

So I actually agree with you Canute. We do decide to stop calculating. In fact all we really need to do to satisfy Weierstrass's limit definition is to prove trends actually. All that the limit really says is that 0.999999... is approaching 1. It never claims in any way that it ever actually equals 1. That is an incorrect conclusion. I simpy can't undestand why so many mathematicians insist on jumping to this conclusion. It really makes me wonder who they had as a teacher?

[matt grime]

They had mathematicians as teachers, Fields Medal winning mathematicians. So, Neutron, what is the definition of Real Numbers you are using and what is the definition of 0.99... you are using?

0.999... is approcahing 1? 0.999... is a fixed real number, it is not itself approaching anything in this active sense.

Two real numbers x and y are equal iff, for all d>0 |x-y| < d.

Proof: Suppose not, if two real numbers are not equal that x-y=r some r. Then d=r/2 will suffice. The reverse implication is trivially left as an exercise for the reader.

By the definition of the reals, I need only consider d in the rationals.

So 1- 0.999... < 1/10^n for all n clearly (it is larger than all its partial sums) hence 1=0.999....

(the reals are the space where all convegent cauchy sequences of rationals have a limit).

[NeutronStar]

Why can't we complete an infinite number of tasks? This is an assumption.
Unless you believe that an infinite number is finite how could it ever be completed?

To claim that an infinite number of tasks can be completed is to simply deny the meaning of "infinite".

All you would be saying is that infinity is finite. If you can live with that more power to you. It doesn't make any sense at all to me .

Intuitively that would be like saying that an endless process can come to an end. Well, if it can come to an end then why did we call it an "endless" process to begin with? We must have been wrong in the first place!

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

[NeutronStar]

(the reals are the space where all convegent cauchy sequences of rationals have a limit).

Well, there you go. You just proved that I'm correct.

The very space that you are calling the "reals" is defined on the concept of the Weierstrass delta-epsilon definition of the calculus limit via the Cauchy sequences of rationals.

Therefore anything that any mathematician ever says about the real numbers should be preceded by the phrase, "In the limit".

That's all I ask because I fully understand the concept of the calculus limit and as long as you precede all of your statements about real numbers with the phrase "In the limit" I have no problem because I know what that means.

So as far as I can see you've just inadvertently agreed with me.

[StatusX]

Unless you believe that an infinite number is finite how could it ever be completed?

To claim that an infinite number of tasks can be completed is to simply deny the meaning of "infinite".

All you would be saying is that infinity is finite. If you can live with that more power to you. It doesn't make any sense at all to me.


What kind of argument is this? You are still assuming an infinite number of "tasks" can never be completed. Your saying you cant complete an infinite number of tasks because to complete a number of tasks, that number must be finite, and so that would mean infinity is finite, which is a contradiction. That is absurd reasoning. If the tasks take less and less time, such that the infinite sum of the times of the tasks converge, then what possible reason is there to believe you couldn't do them?

As for what you said about 0.999... only equalling 1 in the limit, I have to ask you, the limit of what? I assume you mean the limit as n goes to infinity of:

\sum_{k=1}^{n} 9 \cdot 10^{-k}

But what you are mistakingly assuming is that 0.999... represents, in some sense, the process of taking this sum. This is unclear, and I don't know how you could prove anything about such a poorly defined concept. What you have to realize is that the definition of 0.999... is:

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

I'm pretty sure you already conceded this is 1. If you disagree with this definition, then you are not using the same notation as the rest of us, plain and simple.

Be careful before you assume you know something that professional mathematicians don't. You call their careful work of centuries "simply incorrect" because you don't understand it, and its ironic that you think theyre the ones who should be "totally embarassed."

[NeutronStar]

You call their careful work of centuries "simply incorrect"

Whoa!!!

I never said any such thing!

I believe that most serious professional mathematicians will agree that any conclusions that are based on the definition of the limit should be preceded by the phase "in the limit".

Any mathematician worth his salt will quickly point out that saying that the limit of f(x) at c is equal to L does NOT automatically imply that f(c)=L.

That *is* mathematics! To claim otherwise is "simply incorrect" not according to me, but according to mathematical formalism period amen.

Now is there any mathematician who really wants to argue with that?

[StatusX]

0.9999.... doesn't equal 1. Its limit equals 1. That's a whole different concept than saying that 0.9999.... = 1. In fact, to say that latter is simply mathematically incorrect. There's no basis for it. It's simply incorrect.
...
All that the limit really says is that 0.999999... is approaching 1. It never claims in any way that it ever actually equals 1. That is an incorrect conclusion. I simpy can't undestand why so many mathematicians insist on jumping to this conclusion. It really makes me wonder who they had as a teacher?

What you called incorrect is actually correct. You said some pretty derogatory things about people who understand something that you don't.

Any mathematician worth his salt will quickly point out that saying that the limit of f(x) at c is equal to L does NOT automatically imply that f(c)=L.

No doubt about this, but that's not what you said.

[synergy]

I just realized I only read page 1.
So,
continuum => infinitely divisible => an infinite number of tasks and
discrete => A and B always are the same speed or stationary. This is, at least, what I think Canute was saying:
Quote: Canute said:
try working out the relative motion one instant at a time. Particle A can go no faster than one quanta of distance per instant (otherwise it would be in more than one place at the same time) and particle B can go no slower than one quanta of distance per instant, (otherwise it would be stationary). The only solution is to allow A to be at more than one place at a time (changing its length) or for the instants of A to be longer than the instants of B (changing its clock). I can't see a way around this problem except to say that spacetime is not quantised. This doesn't seem to disagree with any evidence as far as I can tell.
end of quote.
It still sounds like a paradox to me, perhaps a juxtaposition of states could lead to a loophole? Space is both discrete and continuous at the same "time"?
Aaron

[StatusX]

The individual particles can only "travel" at one speed, but patterns of them are more flexible. For example, the game of life is a sort of quantized universe, and this page mentions how different "speeds" are possible in this setting. They suggestively call the maximum speed of 1 cell per unit time "c."

http://www.ericweisstein.com/encyclopedias/life/Spaceship.html

[jcsd]

Whoa!!!

I never said any such thing!

I believe that most serious professional mathematicians will agree that any conclusions that are based on the definition of the limit should be preceded by the phase "in the limit".

Any mathematician worth his salt will quickly point out that saying that the limit of f(x) at c is equal to L does NOT automatically imply that f(c)=L.

That *is* mathematics! To claim otherwise is "simply incorrect" not according to me, but according to mathematical formalism period amen.

Now is there any mathematician who really wants to argue with that?

We're talking about the limit of sequences in the rationals,, that limit is not guranteed to be rational, however it is guraenteed to correspond to a real number, by the defitnion of a real number! Your point is irrelvean as we're tlaking boaut the defintion of the reals so we have no need of the preface 'in the limit' as we know that the limit will exists as we use that limit to define the reals.

[NeutronStar]

What you called incorrect is actually correct. You said some pretty derogatory things about people who understand something that you don't.
Well, it would only be taken as derogatory by those to which it is applicable.

So let me understand you. You are saying that modern mathematical formalism says that an infinite number of tasks can be completed without referencing Weierstrass's definition of the limit which clearly does not support this conclusion.

I'm open to that. I've just never heard of that part of the formalism. Clearly any reference to the real numbers automatically references Weierstrass's limit definition since the reals are defined upon that concept so it can't have anything to do with real numbers.

[jcsd]

We're not saying that we're carrying an infinite amount of tasks we're just defining the reals in terms of limits, 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.

[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. :smile:


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. :tongue2:

[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. :tongue2:

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. :cool:

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. :surprised

[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]

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 :smile: )

[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. :frown:

[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]

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]

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

[Hurkyl]

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

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


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


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

[NeutronStar]

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

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

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


If we start with the concept of a dimensionless point we can show that any finite line of dimensionless points must necessarily be discontinuous. That is much easier to see. Then once we have understood the proof of this then it should be obvious that any finite line that is said to be a continuum cannot be constructed with dimensionless points.I've agreed with almost everything you've posted so far, but this seem like the reverse of the truth. To me it seems that the only way to model a continuous line mathematically is by treating it as a series of dimensionless points. If the line is discontinuous then the points wouldn't be dimensionless. Isn't it the fact that spacetime is continuous and that the calculus models it as such that allows the calculus to work in the first place?
Calculus doesn't actually model a continuum at all. If you exam all of the calculus definitions with great care you will actually see how they avoid the concept of a continuum altogether. In fact, all of modern calculus is ultimately based on Karl Weierstrass's definition of a limit. And Weierstrass was very clever in avoiding any direct confrontation with the idea of a continuum. So calculus doesn't actually model a continuum at all. It does however address a mathematical notion of continuity which doesn't actually imply a continuum at all. This is really quite clear if you simply inspect the formal definition in great detail.

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

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

That's the foundational idea.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[StatusX]

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

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

[Hurkyl]

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


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

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

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


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

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

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



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

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


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



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

Proof:

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

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

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

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

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


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

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

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

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

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

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

QED

[NeutronStar]

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

you say yes, but I say there is no such point. For any point you pick out, you can find another one closer to the original. there is an ordering of these points, but there is no way to list them in this order. this is the idea behind continuity, and it defies common sense.
You seem to require that I need to pick out some particular point to which no point can be closer. Yet I never claimed anywhere that I could do any such thing!

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

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

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

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

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

[Hurkyl]

I always forget how much explicit examples help.

Tell me, what is the gap between the point labelled

0

and the points labelled

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


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

[NeutronStar]

millions, billions, centillions, googolplexes, so what? Those are all finite.
Oh boy! I'm carving this quote in STONE!!!!

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

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

{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. :frown:

But I'll be back! :devil:

[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. :tongue:


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 any one 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.
:confused: :confused: :confused:

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. :smile:

[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]

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]

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]

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]

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. :biggrin:

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]

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. :smile:

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. :approve:

[NeutronStar]

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. :tongue:


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. :tongue:


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? :frown:


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 in to 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. :biggrin:
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.

[quote] 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]

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. :biggrin:

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. :approve:

[matt grime]

What do you think I am? A crackpot???


Do you really want an answer to that?

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

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

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

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

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

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

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

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

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

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

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

[Canute]

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

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

[Canute]

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

Ok. So far so good. I can conceive the idea of a dimensionless point as simply the idea of a location that has not breadth.
It's the only sort of dimensionless point there is.

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

But wait, we're not done! We have to maintain the concept of a continuum which is the idea there there are no gaps between these distinctly different dimensionless points.
Why? Your points are in your imagination, you won't find any out there in reality.

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

The idea of a continuum is a logical contradiction pure and simple.
I more or less agree with that also. However I don't derive the same conclusions from it.

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

But so far no one has offered any logically consistent picture of how a continuum can logically be said to exist even as an idea.
That's a fair point, but I won't respond here.

[StatusX]

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

[Hurkyl]

A brief introduction to topology.

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

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

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


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


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


Back to the mathematics.

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

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

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

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


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


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

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


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

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

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


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