Where are the irrational numbers?

[I_am_learning]

Thanx for giving me hope that i am not a total fool.

well for me, i can more easily understand 6 parts in 100 or 59 parts in 1000 or 588 parts in 10000 than i can understand 1 part in 17 because of the use of powers of ten which are easy to visualize for me.

IMHO there is nothing wrong with your visualization but if 0.05882... doesn't make much sense, i hope my way of visualizing it helps.

Well that was really silly of me to not realize that 0.05822 is just around 5.822/100 ( 5.8 part in 100) or even better, 5822 part in 100000 (don't count the 0's because, I didn't .

However, after all its just a crude guessing. I can't really find any difference in my visualization of either 5 part in 237 or 37 part in 806 or 41 part in 1000. In all case I visualize the part being a tiny fraction of the whole.
Do you have any such sharp visualization?

 

[Hurkyl]

Are all approximations that give better and better precision to sqrt(2). Just consider the concept of a fraction as an idea. What is happening here? The numerator is going to infinity, the denominator is going to infinity and because of this you get what you want, sqrt(2), an irrational number.

Ah, you conceive wrongly! It's not the concept of a fraction you're thinking of, it's the concept of a sequence of fractions. (And specifically, of a sequence that converges to something)

 

[agentredlum]

Ah, you conceive wrongly! It's not the concept of a fraction you're thinking of, it's the concept of a sequence of fractions. (And specifically, of a sequence that converges to something)

Not at all. I talked about a single fraction, you mentioned sequence. Can you think of a fraction as an abstract idea or do you have to quantify it with numbers?

In the product formula for sqrt(2) how many fractions do you see?

 

[agentredlum]

Look, you can think of it as a limit if you want. Take the limit as x and y go to infinity of x/y. This is an indeterminate form (infinity)/(infinity) which means it could be any real number, including an irrational number.

 

[agentredlum]

Well that was really silly of me to not realize that 0.05822 is just around 5.822/100 ( 5.8 part in 100) or even better, 5822 part in 100000 (don't count the 0's because, I didn't .

However, after all its just a crude guessing. I can't really find any difference in my visualization of either 5 part in 237 or 37 part in 806 or 41 part in 1000. In all case I visualize the part being a tiny fraction of the whole.
Do you have any such sharp visualization?

Not beyond the first few decimal points. I can get a rough estimation at a glance using notion of distance but then i zoom in, in my minds eye and start comparing to other objects that i am familiar with, such as cells bacteria, DNA strands, molecules, atoms, protons, electrons. Then I start thinking in terms of wavelength of light. Ultraviolet, x-ray, etc. Particularly useful is an ANGSTROM because it's 10^(-10). This is not easy and takes great effort but you get better as you practice.

Something that helped me get an understanding about decreasing quantities was a video by Arthur C. Clarke, may he rest in peace.



I'm going to watch it again now that your question reminded me of it.

Here is a great example of 'zooming in'. As far as fractals are concerned you can do this forever because they have infinite complexity.

http://www.youtube.com/watch?v=0jGaio87u3A&NR=1

 

[Hurkyl]

Not at all. I talked about a single fraction, you mentioned sequence.

Eh? You were not talking about a single fraction -- you fairly explicitly listed a family of fractions.

The inability to tell the difference between the idea of "a number" and "a family of numbers" is a rather serious liability in calculus. It is, for example, one of the major causes that lead people to insist that "0.999\ldots \neq 1" -- e.g. "0.999... is a number that tends to 1, but isn't actually 1".

I honestly can't tell if you are just in love with using "fraction" to describe "family of fractions", or if you are starting down a path that will have you coming back here in three months insisting that \sqrt{2} is a rational number.

Yes, I know you might think that last comment silly -- but there have been at least two people who have visited this forum who have insisted exactly that, using arguments that resemble what you are arguing.

In the product formula for sqrt(2) how many fractions do you see?

Explicitly, I see contained in the notation one fraction-valued expression in the variable n. Implicitly there are two related sequences of fractions: the infinite sequence of terms, and the infinite sequence of partial products.

And the product formula itself is, of course, not a fraction at all, e.g. because the outermost verb is "The infinite product of..." and not "The quotient of..."

 

[agentredlum]

Eh? You were not talking about a single fraction -- you fairly explicitly listed a family of fractions.

The inability to tell the difference between the idea of "a number" and "a family of numbers" is a rather serious liability in calculus. It is, for example, one of the major causes that lead people to insist that "0.999\ldots \neq 1" -- e.g. "0.999... is a number that tends to 1, but isn't actually 1".

I honestly can't tell if you are just in love with using "fraction" to describe "family of fractions", or if you are starting down a path that will have you coming back here in three months insisting that \sqrt{2} is a rational number.

Yes, I know you might think that last comment silly -- but there have been at least two people who have visited this forum who have insisted exactly that, using arguments that resemble what you are arguing.

Explicitly, I see contained in the notation one fraction-valued expression in the variable n. Implicitly there are two related sequences of fractions: the infinite sequence of terms, and the infinite sequence of partial products.

And the product formula itself is, of course, not a fraction at all, e.g. because the outermost verb is "The infinite product of..." and not "The quotient of..."

Well, i guess people see what they want to see depending on the point they want to make. Yes, i listed fractions that converge to sqrt(2), but i am not interested in ANY intermediate fractions. I am only interested in the fraction whose numerator and denominator have gone to infinity by some rule, as I_AM_LEARNING pointed out in post # 89. This is a fraction I CANNOT list in the normal sense so I am asking for a little lattitude here, and for people to use the power of their imagination. This is not a rigorous approach, i understand that, but my original observation was not intended to be unquestionable truth. I used the words 'in some sense'

When I look at the product formula for sqrt(2), I see all those things you mentioned, no question, but i also see a single fraction whose numerator and denominator have gone to infinity by some rule.

I know what a sequence is. I know what partial products are. I'm not sure if you know what 'taking a step back' and' looking at the big picture' means. That's what I'm trying to do here, in a way that makes a little sense, not perfect sense.

I think sqrt(2) is irrational, no question, but I am open to the possibility that sqrt(2) can be thought about as being rational 'in some sense'

I think .999... = 1 no question, but i am open to the possibility that this can create other problems 'in some sense'

 

[agentredlum]

How many fractions do you see when you look at 2/3 ? 'In some sense' I see an infinite product of fractions.

...(-3/-3)(-2/-2)(-1/-1)(2/3)(1/1)(2/2)(3/3)...

You can get creative.

...(-pi/-e)(-e/-pi)(2/3)(e/pi)(pi/e)...

you can fill in the dots however you like, just make sure it works.

The possibilities are only limited by a persons imagination.

 

[micromass]

How many fractions do you see when you look at 2/3 ? 'In some sense' I see an infinite product of fractions.

...(-3/-3)(-2/-2)(-1/-1)(2/3)(1/1)(2/2)(3/3)...

You can get creative.

...(-pi/-e)(-e/-pi)(2/3)(e/pi)(pi/e)...

you can fill in the dots however you like, just make sure it works.

The possibilities are only limited by a persons imagination.

This is very dangerous to do. There are many traps when dealing with infinite products, so you better say exactly what you mean with it. How do you evaluate such an infinite fraction??

 

[agentredlum]

This is very dangerous to do. There are many traps when dealing with infinite products, so you better say exactly what you mean with it. How do you evaluate such an infinite fraction??

I disregard everything and pick the number in the 'middle'.



Yeeess, I know it's dangerous but IMHO so are the subtle points of arithmetic.

 

[micromass]

I disregard everything and pick the number in the 'middle'.

OK, so what is the point in the middle?? You have an infinite product which extends to both sides, there is no middle...

Yeeess, I know it's dangerous but IMHO so are the subtle points of arithmetic.

Arithmetic is not dangerous. It's very well defined. You just need to follow the rules.

 

[agentredlum]

OK, so what is the point in the middle?? You have an infinite product which extends to both sides, there is no middle...
Arithmetic is not dangerous. It's very well defined. You just need to follow the rules.

I thought that it was obvious i was using middle sarcastially bcause i put quotes around it. Then there was the biggrin...

Just because its not easy to FIND 2/3 in that infinite product doesn't mean it isn't THERE. You can write it down first THEN surround it however you wish, just make sure it works.

 

[micromass]

I thought that it was obvious i was using middle sarcastially bcause i put quotes around it. Then there was the biggrin...

Just because its not easy to FIND 2/3 in that infinite product doesn't mean it isn't THERE. You can write it down first THEN surround it however you wish, just make sure it works.

You can write everything you want to. But it's useless if you can't evaluate it properly...

 

[Hurkyl]

I'm not sure if you know what 'taking a step back' and' looking at the big picture' means.

It means, among other things, to stop focusing on minor details and evaluate what you're doing and why.

So let's try it. You are very fixated on the idea of trying to apply the word "fraction" to a lot of situations where it doesn't fit.

Can you explain to me your motivation for this fixation? Can you describe what you are having to do in order to follow this fixation? Is this approach really a good way to achieve whatever goals you have?



I hate to say things that way, but it seriously looks like you are being self-destructive -- you're not only resisting attempts to examine the idea in your head, but you are actively rejecting knowledge that could be useful for the purpose.

This latest post looks like you are trying to wrap your head around the fact that the rationals are dense in the reals, or possibly that the reals are the Cauchy completion of the rational numbers.

Are you going to continue crippling yourself by refusing to move beyond thoughts like "\sqrt{2} is a fraction in some sense" -- or are you going to start examining just what that "some sense" really is, and try to explain it in terms of existing mathematical ideas or even to devise new mathematical ideas invented just for this purpose, should no existing ones apply?




I'm reminded of the time I first digested the notion of "tangent vector to a manifold". I remember actively disliking the definition I had seen. But rather than just mentally rejecting the notion I had seen, I set out trying to define what I thought a tangent vector ought to be. By the time I had worked out all the difficulties and had something I was happy with, I had pretty much written down exactly what I had originally rejected.

 

[agentredlum]

It means, among other things, to stop focusing on minor details and evaluate what you're doing and why.

So let's try it. You are very fixated on the idea of trying to apply the word "fraction" to a lot of situations where it doesn't fit.

Can you explain to me your motivation for this fixation? Can you describe what you are having to do in order to follow this fixation? Is this approach really a good way to achieve whatever goals you have?
I hate to say things that way, but it seriously looks like you are being self-destructive -- you're not only resisting attempts to examine the idea in your head, but you are actively rejecting knowledge that could be useful for the purpose.

This latest post looks like you are trying to wrap your head around the fact that the rationals are dense in the reals, or possibly that the reals are the Cauchy completion of the rational numbers.

Are you going to continue crippling yourself by refusing to move beyond thoughts like "\sqrt{2} is a fraction in some sense" -- or are you going to start examining just what that "some sense" really is, and try to explain it in terms of existing mathematical ideas or even to devise new mathematical ideas invented just for this purpose, should no existing ones apply?

I'm reminded of the time I first digested the notion of "tangent vector to a manifold". I remember actively disliking the definition I had seen. But rather than just mentally rejecting the notion I had seen, I set out trying to define what I thought a tangent vector ought to be. By the time I had worked out all the difficulties and had something I was happy with, I had pretty much written down exactly what I had originally rejected.

Oh man...

I REJECT NOTHING, I QUESTION EVERYTHING. Questioning everything is not the same as rejecting anything. Wouldn't you think I was a fool if I accepted everything without question? Making my point does not mean i have to reject yours. You think it does but I am not responsible for that. I can see it your way and agree it's useful. You can't see it my way even after i ask for a little latitude. That's not fair.

 

[agentredlum]

You can write everything you want to. But it's useless if you can't evaluate it properly...

I don't agree with that. Just because a person does not see a use for it doesn't mean it isn't useful.

I am tempted to Quote Faraday here when they asked him "Of what use is electricity?" and he replied, "Of what use is a newborn baby?"

Keep an open mind, that's all I'm asking.

 

[agentredlum]

I'm reminded of the time I first digested the notion of "tangent vector to a manifold". I remember actively disliking the definition I had seen. But rather than just mentally rejecting the notion I had seen, I set out trying to define what I thought a tangent vector ought to be. By the time I had worked out all the difficulties and had something I was happy with, I had pretty much written down exactly what I had originally rejected.

Why did you stop? You should have kept going. If everything you knew up to that point made you question a definition then if i were you i would continue seeking confirmation from multiple sources before i concluded that i was wrong.

As a matter of fact, i would bring into question the very system itself that allowed me to make an erroneous assumption in the first place. I woudn't reject it, but an eyebrow would certainly be raised, and i would think long and hard about that.

 

[micromass]

I don't agree with that. Just because a person does not see a use for it doesn't mean it isn't useful.

I am tempted to Quote Faraday here when they asked him "Of what use is electricity?" and he replied, "Of what use is a newborn baby?"

Keep an open mind, that's all I'm asking.

Having an open mind in mathematics is really not a good idea...

When I'm doing research, I always like to be my own biggest critic. I criticize every step I take, and I put everything into question. Handwaving and non-rigorous arguments are ok, but they need to be formalized soon.
Once you've satisfied your own critics, only then can you present your work to somebody else. The point being that this other person criticizes your work again and shows possible flaws in your work.

In short: being skeptic in mathematics is a very good thing!

 

[micromass]

Oh man...

I REJECT NOTHING, I QUESTION EVERYTHING. Questioning everything is not the same as rejecting anything. Wouldn't you think I was a fool if I accepted everything without question? Making my point does not mean i have to reject yours. You think it does but I am not responsible for that. I can see it your way and agree it's useful. You can't see it my way even after i ask for a little latitude. That's not fair.

We do understand your point very well. But we see the possible flaws and mistakes too, and we point that out to you.
It's not because we criticize your point-of-view, that we can't see it your way...

 

[micromass]

Do check out http://en.wikipedia.org/wiki/Supernatural_numbers
This can be generalized to superrational numbers (not sure of the term), in which arbitrary infinite fractions can be studied.
However, I'm very unsure how (or if) the reals can be embedded in the superrationals...

 

[Hurkyl]

I REJECT NOTHING, I QUESTION EVERYTHING.

So, what were the results of questioning the notion of fraction you've been trying to push onto the thread?

 

[Hurkyl]

Why did you stop? You should have kept going. If everything you knew up to that point made you question a definition then if i were you i would continue seeking confirmation from multiple sources before i concluded that i was wrong.

As a matter of fact, i would bring into question the very system itself that allowed me to make an erroneous assumption in the first place. I woudn't reject it, but an eyebrow would certainly be raised, and i would think long and hard about that.

You're serious? You would reject your own reasoning in favor of clinging desperately to a first impression, even when the textbook agrees with your reasoning? And then when you finally give up your first impression, you would try and blame everyone else for filling your head with misleading thoughts?


I stopped because I was interested in learning differential geometry, and had concluded a useful exercise that shed insight onto the concept of tangent vector and demonstrated my own thoughts on the matter were already in alignment with the textbook despite my first impression. There is no expectation of utility in being contrary for the sake of being contrary.

 

[agentredlum]

We do understand your point very well. But we see the possible flaws and mistakes too, and we point that out to you.
It's not because we criticize your point-of-view, that we can't see it your way...

WOOOOOOOW! Finally a little respect, thank you, it means a LOT to me.

Oh yeah, of course there are many flaws, no question. However I do not reject any idea because of a few flaws.

I am fascinated by alternative ways of thimking. Example, in Stewart's calculus text in the section on alternating series, there are a few words about some famous mathematician proving that an infinite alternating series can be arranged to give ANY sum. I don't have the book anymore so i can't quote it verbatim but i remember the fascination i felt and the main idea.

Years ago i went to lunch with my math professor, who was an awsome teacher, and we talked and he gave an example i find fascinating even to this day.

He said a point has no length, height or width. Take a point and translate it to the right until you get a line segment, use your pencil if you like. That line segment is an infinite collection of points that have no length, wdth or height. Take the line segment and translate it up until you get a plane. Now that plane can be considered an infinite collection of line segments. Translate the plane out of the paper until you get a rectangular box. That box can be considered an infinite collection of planes. Now translate that box until it fills up all space.

Now that is mind boggling! You have just used something that has no length, width, height to (IN SOME SENSE) construct all 3-space. Is it rigorous, absolutely not. Is this thought experiment interesting, imho absolutely yes!

Later in a Linear Algebra course i learned the most amazing thing, the first day of class, from the same professor.

0x + 0y + 0z = 0

This is the equation of all 3-space. EVERY SINGLE POINT OF 3-space satisfies this equation.

Now, that is mind boggling and something Anton's Linalg book did not mention. Apparrantly, out of NOTHING you get EVERYTHING. Is it rigorous? no. Is it fascinating? Definitely yes!

So you see why i am not too eager to reject ideas?

 

[agentredlum]

So, what were the results of questioning the notion of fraction you've been trying to push onto the thread?

That there is more going on here than simple definitions could account for.

I'm not trying to 'push' anything on anybody. I would be hypocrite if i didn't accept scrutiny of my opinions. I'm just 'floating' it out there sort of like a colorful balloon with the word WARNING! on it.

 

[agentredlum]

You're serious? You would reject your own reasoning in favor of clinging desperately to a first impression, even when the textbook agrees with your reasoning? And then when you finally give up your first impression, you would try and blame everyone else for filling your head with misleading thoughts?I stopped because I was interested in learning differential geometry, and had concluded a useful exercise that shed insight onto the concept of tangent vector and demonstrated my own thoughts on the matter were already in alignment with the textbook despite my first impression. There is no expectation of utility in being contrary for the sake of being contrary.

If you convinced yourself that your first impression was superficial then i don't blame you for stopping, i would have done the same.

 

[Hurkyl]

Example, in Stewart's calculus text in the section on alternating series, there are a few words about some famous mathematician proving that an infinite alternating series can be arranged to give ANY sum. I don't have the book anymore so i can't quote it verbatim but i remember the fascination i felt and the main idea.

If, for a given infinite sequence of real numbers:

• The positive terms converge to zero
• The positive terms add to +\infty
• The negative terms converge to zero
• The positive terms add to -\infty

Then for any extended real number a, there exists a permutation of the sequence whose infinite sum converges to a.

And there is a related theorem: if

• The positive terms add to a finite number
• The negative terms add to a finite number

(this case is called "absolute convergence")

Then all permutations of the sequence sum to the same number.



This is rather important, since people like to rearrange sums arbitrarily, and these two facts not only tell you either a sum behaves 'perfectly' under rearrangement or it is capable of misbehaving in the worst way possible, but they also give you a very, very good way to tell which is which.

('perfect' is, of course, subject to the situation. Sometimes you want a sum that behaves badly under rearrangement)




One particular example of rearranging having actual practical importance (rather than just being a neat example) is double summations -- it is really, really, really convenient to think of it as just having a set of numbers to add up without having to pay attention to how they're arranged and in what order they are being summed. You can only get away with it in the case of absolute convergence.

(e.g. the sum might be given as adding up the rows first, then adding the results -- but it might be easier to instead add up the columns first, or sometimes adding up along diagonals is the way to go)

 

[agentredlum]

Check out bullet #4 of your post. What do you think about my powers of observation now?

 

[cb174503]

If the volume is equal to the numerical value of the Planck length then i am interested to know the dimensions of this geometrical solid.

An example: Suppose volume is a cube with side Planck length. Then the numerical value you get is about 10^-105. This is a lot smaller than the quantized value. Problems like this arise with any kind of geometrical object, including marshmellows, you can't even make a square with side Planck length without violating quantization.

Lets make the problem simpler so i can make my point without confusion. Pretend PL is .5 and you have a quantized system that does not allow smaller values. Then someone asks,'what happens if you square PL?'
You get .25 which is a smaller value than PL. 'No problem' you say, 'I'll just put .25 into my system as an exception.' What if you square the exception? then you get a smaller number still. You can repeat this process ad infinitem. You always get a number smaller than the previous and greater than zero. So you are rapidly creating an INFINITE number of exceptions approaching zero in your quantized system that was designed to have a FINITE number of points between zero and one. This is one of the reasons IMHO i believe quantization is dead.

--------------------------
Re: Agentredlum's example about size and dimensionality above:
I hope everyone will forgive me more of my words... I did not expect a geometric challenge to where those doggone irrational numbers are. At any rate:

1.616252*10-35 Planck-distance, in meters,
or “amount of line” covered by a Planck-distance, otherwise known as a Planck-length

(1.616252*10-35)*(1.616252*10-35) = 2.61227*10-70 = 1planck-area, in square meters,
or “amount of surface” covered by a Planck-area

(1.616252*10-35)*(1.616252*10-35)*(1.616252*10-35) = 4.22208*10-105 = 1planck-volume, in cubic meters, or “amount of space” contained in a Planck-volume – aka, a Planck-point as a 3-d quantized value based on the same 1-d quantized value as a Planck-length.

There isn't a reason to proceed with the calculation to higher dimensions because there is no evidence that there are any. However, if there were, the same logic would apply.

Math based on a number line with a finite number of points where those points are defined based on a Planck-length yields dimensional sizes which are comparably finite. Progressively larger, but not infinitely so. Only 70 orders of magnitude larger from a “line” to a volume… The difference with Agentredlum’s example is merely a difference in point of view, but an important difference.

So...the three standard issue multiplications above are different dimensional measurements using the same value, a Planck-length. Regardless of how it’s sliced, a Planck-volume is a Planck-length from each of its corners (planck-volume would be the smallest unit volume possible, mathematically analogous to a dimensionless point, but still one Planck-length per edge, visualized as a cube). There is no conflict in measurement from the “line” to the volume; it is but the same value measured in different dimensions. The dimensions being measured, however, are vastly different. Depending on the number of dimensions you wish to discuss, the value appears to get smaller according to the exponent, but in reality the dimensions are becoming larger. The same progression occurs whether the end volumes are cubes, spheres or marshmallows or universes – or, for that matter, however many dimensions you wish to consider. Also, with no infinite outcomes, which is unlike using math where infinite values exist in a line segment.

Intriguingly, 1.616252*10 to the +35th p-v’s laid one by one next to each other would make a line of p-v’s one meter long - a finite number of Planck-points. Using p-l’s and p-v’s suggests a means of distinguishing size between dimensions – where using traditional math to try and measure size differences between dimensions doesn’t work well, if at all. It is possible the CERN machine (LHC) may find real world evidence related to the question.

Speaking of dimensions, I’ve read that some consider time to be a 4th dimension. Usually it’s referred to as an extension of the third dimension, similar to the third as an extension of the second. I don’t think that’s the case… rather, time is an extra linear dimension similar to the single dimension but not as an extension of the three we experience. It has already been suggested that time is discrete in structure at a scale of about 1*10-43 seconds. Others have considered this question in relation the Zeno paradox and have come to the solution that time isn’t something we’re “in” that can be visualized statically like “in the present instant”. They assert time is something we and our world are passing through dynamically with no stop actions in the discrete instants. To me that is unsatisfying as it seems to mix discrete and continuous structure - but hard to refute.
cb

 

[agentredlum]

If the volume is equal to the numerical value of the Planck length then i am interested to know the dimensions of this geometrical solid.

An example: Suppose volume is a cube with side Planck length. Then the numerical value you get is about 10^-105. This is a lot smaller than the quantized value. Problems like this arise with any kind of geometrical object, including marshmellows, you can't even make a square with side Planck length without violating quantization.

Lets make the problem simpler so i can make my point without confusion. Pretend PL is .5 and you have a quantized system that does not allow smaller values. Then someone asks,'what happens if you square PL?'
You get .25 which is a smaller value than PL. 'No problem' you say, 'I'll just put .25 into my system as an exception.' What if you square the exception? then you get a smaller number still. You can repeat this process ad infinitem. You always get a number smaller than the previous and greater than zero. So you are rapidly creating an INFINITE number of exceptions approaching zero in your quantized system that was designed to have a FINITE number of points between zero and one. This is one of the reasons IMHO i believe quantization is dead.

--------------------------
Re: Agentredlum's example about size and dimensionality above:
I hope everyone will forgive me more of my words... I did not expect a geometric challenge to where those doggone irrational numbers are. At any rate:

1.616252*10-35 Planck-distance, in meters,
or “amount of line” covered by a Planck-distance, otherwise known as a Planck-length

(1.616252*10-35)*(1.616252*10-35) = 2.61227*10-70 = 1planck-area, in square meters,
or “amount of surface” covered by a Planck-area

(1.616252*10-35)*(1.616252*10-35)*(1.616252*10-35) = 4.22208*10-105 = 1planck-volume, in cubic meters, or “amount of space” contained in a Planck-volume – aka, a Planck-point as a 3-d quantized value based on the same 1-d quantized value as a Planck-length.

There isn't a reason to proceed with the calculation to higher dimensions because there is no evidence that there are any. However, if there were, the same logic would apply.

Math based on a number line with a finite number of points where those points are defined based on a Planck-length yields dimensional sizes which are comparably finite. Progressively larger, but not infinitely so. Only 70 orders of magnitude larger from a “line” to a volume… The difference with Agentredlum’s example is merely a difference in point of view, but an important difference.

So...the three standard issue multiplications above are different dimensional measurements using the same value, a Planck-length. Regardless of how it’s sliced, a Planck-volume is a Planck-length from each of its corners (planck-volume would be the smallest unit volume possible, mathematically analogous to a dimensionless point, but still one Planck-length per edge, visualized as a cube). There is no conflict in measurement from the “line” to the volume; it is but the same value measured in different dimensions. The dimensions being measured, however, are vastly different. Depending on the number of dimensions you wish to discuss, the value appears to get smaller according to the exponent, but in reality the dimensions are becoming larger. The same progression occurs whether the end volumes are cubes, spheres or marshmallows or universes – or, for that matter, however many dimensions you wish to consider. Also, with no infinite outcomes, which is unlike using math where infinite values exist in a line segment.

Intriguingly, 1.616252*10 to the +35th p-v’s laid one by one next to each other would make a line of p-v’s one meter long - a finite number of Planck-points. Using p-l’s and p-v’s suggests a means of distinguishing size between dimensions – where using traditional math to try and measure size differences between dimensions doesn’t work well, if at all. It is possible the CERN machine (LHC) may find real world evidence related to the question.

Speaking of dimensions, I’ve read that some consider time to be a 4th dimension. Usually it’s referred to as an extension of the third dimension, similar to the third as an extension of the second. I don’t think that’s the case… rather, time is an extra linear dimension similar to the single dimension but not as an extension of the three we experience. It has already been suggested that time is discrete in structure at a scale of about 1*10-43 seconds. Others have considered this question in relation the Zeno paradox and have come to the solution that time isn’t something we’re “in” that can be visualized statically like “in the present instant”. They assert time is something we and our world are passing through dynamically with no stop actions in the discrete instants. To me that is unsatisfying as it seems to mix discrete and continuous structure - but hard to refute.
cb

Very good point that even though the numbers are decreasing, the DIMENSIONS are getting larger in the sense you can do more with them and on an intuitive level.

I also agree with you that it is disquieting to mix discrete and continuous, and brings into question the motivation behind such an endeavor. Are they using facts to fit the theory?, or are they using the theory to change the facts? Or are they doing both whenever it suits them? Or maybe it's a misunderstanding and they're doing neither?

Like I said before, I hope you succeed in your attempt to quantize, many are still working on this so you could too.

If you call your Planck Length 'one' then squaring, cubing, etc. don't present the problem I mentioned. Your meter would have about 10^35 PL. like you mention above. After all the standard length of 1m is comepletely arbitrary. Why not define PL as 'one meter'?

Then the distance of my face to the monitor is 10^35 meters...

 

[cant_count]

Thank you for suggesting Cantor - very interesting. Another quick question: there are integers, rational numbers, irrationals (which can be expressed using polynomials), transcendentals (which can't, but still can be defined, e.g. pi and e). What about numbers that cannot be defined in any way whatsoever? Does such a category have a name? And are there more of them than anything else? Are they ever used in mathematics?
(If they are still transcendentals, they'd need to be distinguished from the definable ones, I'd be thinking.) Perhaps they can't be discussed because they're undetectable or unprovable by definition? But there should be more of them than anything else, right? (Like dark matter ..!)
- Just wondering