Do you believe in Creation?

[N0B0DY]

Just a couple questions, Roger, if I may.

"The size [ie length] of a point is zero but drawing a line means adding an infinite number of points together."

If points are zero-dimensional, wouldn't an infinite number of them be zero? Wouldn't it take either arbitrary finite increments to equal two units, or an infinite number of infinitesimal one-dimensional increments to equal two units?

Also how can one divided by zero equal infinity, if one does not equal zero multiplied by infinity?

[neutralino]

*** Multiplication is repetitive addition so that multiplying 2 x 10 say means adding 2 to itself 10 times ie 2 + 2 + ...2 = 20.

The size [ie length] of a point is zero but drawing a line means adding an infinite number of points together. So if I wish to get a line of 2 units long I must add infinity2 points together: I hope you can see this.

But adding infinity2 points together is the same as multiplying one point by infinity. That is, infinity2 x 0 = 2.

Similarly a line of 7 units long comprises an infinite amount of points of 0 length added together where the infinite amount in this case is infinity7 and so on.

And lastly if I add an infinite amounts of lines of length 7 units and thickness 0 units side by side I will get an area of 49 square units if the infinite amount is infinity7.

I think this is the extent of the explanation I can give and if I say anything more I will only be repeating myself. Therefore if you can't understand the points I have made thus far I think we will just have to agree to disagree on the issue of multiplying by infinity.

I don't understand this. I don't see the need in introducing infinitely many infinities.

*** For imaginary numbers i^2 = -1 or i = square root of (-1). Imaginary numbers obey the laws of multiplication so it is incorrect to say that multiplication is only defined over the real numbers.

This is a good point. I waved my hands a little, since I didn't know your knowledge of mathematics. However, since you do, I can be precise. What I meant to say is that multiplication is defined such that division is it's inverse only in a commutative division ring; otherwise known as a field. The real numbers and the complex numbers are both fields, so this is fine. However, there are subsets of the complex number that are not fields (under the usual operations of addition and multiplication). An example of such a set is the integers. This is not a field since it is not closed under division; that is [imath]\exists a,b\in\mathbb{Z} : a/b\notin\mathbb{Z}[/imath].

You have to be very careful when defining a new number system and presuming it is a field. For example, for the set [imath]\mathbb{R}\cup\{\infty\}[/imath] one will find that some of the field axioms are violated.

*** The general condition basically is: if F(s) is a function such that in the limit as 's' tends to zero F(s) tends to infinity then the limit of sF(s) as 's' tends to zero is finite.

Mathematicians usually have a very abstract and difficult way of saying simple things so that's why the above appears so complicated and the notation it is written in makes it even more difficult to understand.

What Laplace was actually saying in simpler terms is this: if you have a function F(s) = 1/s say. Then as 's' tends to zero [ or when 's' is zero] 1/s = 1/0 = infinity.

Laplace is saying that sF(s) which is s x 1/s when 's' is 0 is finite.

He is therefore effectively saying that 0 x 1/0 is finite which is what I have been saying to you in my earlier post about multiplying 0 x infinity.

Firstly, I still can't find the theorem anywhere: all I can find is a deMoivre-Laplace theorem with something to do with probability distributions.

Secondly, one has to be very careful when taking limits of indeterminate forms. The expression [math]\lim_{x\to 0}\frac{1}{x}[/math] is not the same as simply saying 1/0.

I'll need to look more at the specific theorem, and its restrictions on the functions F before commenting any further.

It may take an advanced mathematician to follow my argument so I will not be offended if you can't follow it.

Roger

Please, use as much mathematics as you wish.

[ThirdWorld]

If points are zero-dimensional, wouldn't an infinite number of them be zero?

*** No, an infinite number of zero sized points added (in one direction) will give a line of finite length.

Wouldn't it take either arbitrary finite increments to equal two units, or an infinite number of infinitesimal one-dimensional increments to equal two units?

*** "Infinitesimal one-dimensional increments" are "finite increments". A sufficiently large number of infinitesimal one-dimensional increments will yield two units but an 'infinite' amount of these infinitesimal increments will give infinity and not two units.

Also how can one divided by zero equal infinity, if one does not equal zero multiplied by infinity?

*** One IS equal to zero multiplied by infinity (ie if the infinity is of the correct magnitude as I explained above).

Roger

[ThirdWorld]

I don't understand this. I don't see the need in introducing infinitely many infinities.

*** It's an infinite amount of zero size points (or 'nothing') I am introducing not an infinite amount of infinities.

What I meant to say is that multiplication is defined such that division is it's inverse only in a commutative division ring; otherwise known as a field.

*** Well I am alleging that infinite numbers constitute a field just as there is a field of finite numbers.

The real numbers and the complex numbers are both fields, so this is fine. However, there are subsets of the complex number that are not fields (under the usual operations of addition and multiplication). An example of such a set is the integers. This is not a field since it is not closed under division; that is .

*** I think I follow the point you are making but again I say that infinity7/ infinity2 say = infinity3.5. The set of infinite numbers does not consist only of integers so there exist elements of Z for all quotients of a and b. The set of infinite numbers is therefore a field.

You have to be very careful when defining a new number system and presuming it is a field. For example, for the set one will find that some of the field axioms are violated.

*** Which field axiom is violated? That sounds to me like saying that for X- Y- coordinates the set X Union Y violates some of the field axioms.

You will need to elaborate on your statement for me to see the point you are making.

Firstly, I still can't find the theorem anywhere: all I can find is a deMoivre-Laplace theorem with something to do with probability distributions.

*** I have not looked it up myself so I don't know if you will find it on the internet. I referred to it as the Laplace Theorem maybe you will find something under 'The Laplace Transform'.

Secondly, one has to be very careful when taking limits of indeterminate forms. The expression is not the same as simply saying 1/0.

*** You certainly seem to be much more versed in math than I thought.

To the pure mathematician the limit is thought of as different from the actual value 1/0 only because he does not understand whether 1/0 actually exists and what it is if it does, in my view. The mathematician therefore shies away from the definitive answer of saying that 1/0 is infinity and chooses rather only to approach (or tend to) zero but never to reach it.

To my mind as a theoretical physicist, however, infinity as dimensions of reality apart from the three finite spacial, one time and one mass (or energy) dimensions is a reality easy for me to think in terms of.

There is a philosophical statement made by a former Ethiopian ruler that has a part that says "...will remain but a fleeting illusion to be pursued but never attained". He wasn't talking of infinity in his speech but what he said summarizes quite nicely what the mathematician thinks of infinity: you will always pursue it but never reach it.

I'll need to look more at the specific theorem, and its restrictions on the functions F before commenting any further

*** I wish you could find it and study it.

I do not have the facility/ software to write in all the math notation that you are using so although the language I use is no where near in elegance to mathematic notation I think you can still understand and appreciate the points I make even though you may not agree with them.

Roger

[Profpat]

Hi Roger

I think you can place an infinite number of points onto a line smaller than a planck's length. I'm not even sure you even need length.

Best to you

Pat

[neutralino]

Roger,

Firstly, I don't think I understand your number system properly. I don't understand what all you infinities do. What does the sequence 1,2,3,4,..... tend to in your number system, for instance?

It's up to you to check the field axioms; there's about 8 of them.

I'll talk about the Laplace transform later; although I'm not sure you should be relying on a theorem that you haven't read!

[ThirdWorld]

I think you can place an infinite number of points onto a line smaller than a planck's length. I'm not even sure you even need length.

*** True. If the PLank length is 10^-22 units say then the number of points to make up this length would be infinity10^-22.

According to my perception of infinity, the infinite amount of points that will produce no length will be infinity0.


Roger

[ThirdWorld]

Firstly, I don't think I understand your number system properly. I don't understand what all you infinities do.

*** Well, the most pragmatic use of this numbering system [which effectively is only an extension or extrapolation of the existing system] is that it solves the problem of the "divide zero error" that we usually get with calculators and other computerized computations.

With that system we will be able to divide by zero, get an answer which though not useful/ meaningful in a finite universe, will enable us to continue the computation to solve practical problems in math and physics.

An example of a physics problem that needs this math I think is Einstein's time dilation physics problems. Although I disagree with the physics of his theory the math in it can be made workable. I can't recall the equation for lamda right now however in that equation if the particle is a photon then v = C and we end up with a 0 in the denominator making Lamda infinity.

With the numbering system that I propose one will be still able to compute a meaningful time for this lamda = infinity that can be further used with velocity to determine distance between sun and earth say.

What does the sequence 1,2,3,4,..... tend to in your number system, for instance?

*** I don't hink I want to use the expression 'tend to' for the finiite number system. I haven't yet said that I think that 0 is also a field of numbers and this affects the 'points of intersection' between the finite and infinite axes.

To give the simple answer to your question however I will say that the axis of finite numbers intersect an orthogonal axis of infinite numbers at a point infinity0.

It's up to you to check the field axioms; there's about 8 of them.

*** That sounds dismissive.

I'll talk about the Laplace transform later; although I'm not sure you should be relying on a theorem that you haven't read!

*** I read it and learnt it while I was doing math as a student at university but I can't say that I understant it thoroughly or its application at present.

Roger

[swoarg]

hi there
thats all very interesting unfortunatly for me i am not a mathmatician but i would realy like to know how it realates to creation (please)

thanks Swoarg

[neutralino]

*** Well, the most pragmatic use of this numbering system [which effectively is only an extension or extrapolation of the existing system] is that it solves the problem of the "divide zero error" that we usually get with calculators and other computerized computations.

This sounds a lot like the number system that a professor at reading university came up with including the "number" that he defined as nullity. (I don't have a reference, but I'm sure a google search will return a plethora of sites).

With that system we will be able to divide by zero, get an answer which though not useful/ meaningful in a finite universe, will enable us to continue the computation to solve practical problems in math and physics.

Who said the universe was finite?>

An example of a physics problem that needs this math I think is Einstein's time dilation physics problems. Although I disagree with the physics of his theory the math in it can be made workable. I can't recall the equation for lamda right now however in that equation if the particle is a photon then v = C and we end up with a 0 in the denominator making Lamda infinity.

Well, it is precisely the physics that says you can't do that! The special theory of relativity holds for inertial frames. There is no inertial frame that can travel at the speed of light, so the limit in the equations is precisely that: a limit that cannot be breached. FYI [imath]\gamma=\left(1-v^2/c^2)\right^{-1/2}[/imath].

*** I don't hink I want to use the expression 'tend to' for the finiite number system. I haven't yet said that I think that 0 is also a field of numbers and this affects the 'points of intersection' between the finite and infinite axes.

Ok then, think of the sequence [imath](x_n)[/imath] where [imath]x_{n+1}=1+x_n[/imath].

To give the simple answer to your question however I will say that the axis of finite numbers intersect an orthogonal axis of infinite numbers at a point infinity0.

Ok.

*** That sounds dismissive.

I'm not being dismissive; but if you come up with something new, then it is upon you to prove it; not me to prove/disprove it.

*** I read it and learnt it while I was doing math as a student at university but I can't say that I understant it thoroughly or its application at present.

Roger

Ok, so the Laplace transform existence theorem goes something like this:
If [imath]f(t)[/imath] is piecewise continuous on every finite interval in [imath][0,\infty)[/imath] satisfying [imath]|f(t)|\leq m e^{at}, \forall x \in \[0,\infty)[/imath], then the laplace transform, [imath]\mathcal{L}[f(t)](s)[/imath] exists [imath]\forall s>a[/imath]. (From here). How does this help your case?

Finally, I have no special software for producing the maths; simply use [ math] and [ /math] tags (without the spaces) and type the equations in TeX. There is a thread in the "news and announcements" forum on this, if you are interested.