Register to reply

Infinite Infinitesimal...

by Orion1
Tags: infinite, infinitesimal
Share this thread:
Orion1
#1
Sep14-04, 10:19 PM
Orion1's Avatar
P: 989
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/archime...emplating.html
http://www.pbs.org/wgbh/nova/archimedes/infinity.html
Phys.Org News Partner Science news on Phys.org
Fungus deadly to AIDS patients found to grow on trees
Canola genome sequence reveals evolutionary 'love triangle'
Scientists uncover clues to role of magnetism in iron-based superconductors
wuliheron
#2
Sep15-04, 02:13 AM
P: 1,967
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
#3
Sep15-04, 03:37 AM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,091
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
#4
Sep15-04, 03:44 AM
Sci Advisor
HW Helper
PF Gold
P: 12,016
Infinite Infinitesimal...

Besides, the Ramsey number is a LOT bigger than either..
Smurf
#5
Sep16-04, 06:42 AM
P: 2,891
Quote Quote by Hurkyl
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

-Ruler of the Universe,
Smurf
metacristi
#6
Sep18-04, 12:42 PM
P: 261
Quote Quote by Hurkyl
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
#7
Nov2-04, 11:51 AM
P: 418
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
#8
Nov4-04, 09:50 PM
PF Gold
P: 1,059
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
#9
Nov5-04, 05:19 PM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,091
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
#10
Nov5-04, 06:10 PM
P: 418
Quote Quote by Hurkyl
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
#11
Nov5-04, 06:53 PM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,091
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
#12
Nov5-04, 06:54 PM
Sci Advisor
P: 131
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
#13
Nov5-04, 07:24 PM
P: 418
Quote Quote by Hurkyl
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
#14
Nov6-04, 08:07 AM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,091
I won't pretend to know what quantum geometry looks like. I will just comment that the amount of discretization inherent in quantum mechanics is vastly exaggerated in popular accounts.
Canute
#15
Nov6-04, 08:50 AM
P: 1,499
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
#16
Nov6-04, 10:16 AM
Emeritus
PF Gold
P: 8,147
Quote Quote by Canute
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
#17
Nov6-04, 06:30 PM
P: 418
Quote Quote by Canute
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
#18
Nov9-04, 01:11 AM
HW Helper
P: 2,567
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?


Register to reply

Related Discussions
Infinitesimal generator Set Theory, Logic, Probability, Statistics 8
Sum of an infinitesimal variable Calculus 5
The ultimate physical paradox: infinite from infinitesimal cosmos Special & General Relativity 0
Help on infinitesimal calculation General Math 6
Infinite Dimensional Infinitesimal General Physics 1