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Bill S. Offline OP
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Orac, in spite of your accusations, I am trying to learn. Some 10 days ago I posted the same OP on TNS.

http://www.thenakedscientists.com/forum/index.php?topic=53002.0

56 posts on, there is very little personal sniping, and quite a lot of material that I am keeping in note form. This is the sort discussion I had hoped to initiate here on SAGG so that my notes could contain a wider range of opinions.

Comments like: “I think you are being even more obtuse and you are play like the crazy pollack fool you aren't remotely interested in trying to get to the bottom of this.” Find no place in my notes. They have no learning value to me.

I would like to suggest that we draw a line under what has gone before, and make a fresh start, perhaps using Orac’s link as our starting point.

https://www.math.toronto.edu/mathnet/answers/infinity.html

Let’s consider its opening question: Is there really such a thing as "infinity"?

Any takers?


There never was nothing.
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Quote:
I propose asking a few questions, in the form of a “poll”, which may help to ameliorate the situation. Each question requires only a yes/no answer, but hopefully these would be accompanied by some thoughts.


1. Is infinity a number? No
infinity does not exist , it is a descriptive word that humans use vs other descriptive words such as forever , eternity, when describing the largest amount of time passing...

2. Is eternity a length of time? Yes
but the length of time is unknown like infinity , forever.

3. Is it possible to define Cantor’s “absolute infinity”? No
because a human could not know what absolute infinity could be.

4. If there had ever been (absolutely) nothing, could there be something now? No
something would require something , a spirit is something
even though most humans are not capable of any type of recognition / detection of a spirit through any means.
although most humans do feel spirits they do not recognize them as being spirits.

5. Could there be change without time? No
time is just the measurable part of infinity , forever , eternity , and time passes ...

number 4 might get me into some trouble here on SAGG , but does anybody really care? LOL wink

I know I don't care ...




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You have some good answers there Paul. No. 4 is the one that is really iffy. There may or may not have ever been nothing. but there is not really any way to determine if there ever was or will be nothing.

And I tend agree with your answer to No. 5. Some theoreticians have suggested that there is no such thing as time, but it appears to me that time is an integral part of our 'bookkeeping'. Without time it would be impossible to separate events because they would all occur in one single instant. But I'm not even sure how to say that, because the word 'instant' implies time. Time is deeply imbedded in our understanding of the world.

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Originally Posted By: Paul
1. Is infinity a number? No
infinity does not exist , it is a descriptive word that humans use vs other descriptive words such as forever , eternity, when describing the largest amount of time passing...

2. Is eternity a length of time? Yes
but the length of time is unknown like infinity , forever.


If one takes the “dictionary definitions” of infinity and eternity, you are spot on. I’m reluctant to accept everyday definitions as though they were accepted laws of physics. I think accepting them into our more analytical thinking can lead to trouble.

You say infinity is not a number, and justify that statement, yet you say that eternity is a length of time; although you liken it to infinity. Do you not feel there is something of a conflict here?

I have no problem with your answer to 4. Had there ever been nothing, I see no logical explanation for the obvious fact that there is something now. I cannot provide any proof of what the something that has always existed might have been, so I’m not going to try to put down anyone’s thoughts or beliefs on that subject.

In 5; I agree with your “No”, but have reservations about your reasoning. Perhaps we can come back to that.

Quote:
I know I don't care ...


I think if any of us really cared about what others on SAGG say about us, or call us, we wouldn’t still be posting. smile


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Originally Posted By: Bill
There may or may not have ever been nothing. but there is not really any way to determine if there ever was or will be nothing.


One way to establish that there could have been nothing would be to come up with a self consistent explanation for how something could emerge from nothing.


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Quote:

You say infinity is not a number, and justify that statement, yet you say
that eternity is a length of time; although you liken it to infinity.
Do you not feel there is something of a conflict here?


1. Is infinity a number?
2. Is eternity a length of time?

if you would have asked if infinity was a lenght of time my answer
would have been : Yes

if you would have asked if eternity was a number my answer
would have been : No

Quote:

5. Could there be change without time?
In 5; I agree with your “No”, but have reservations about your reasoning.
Perhaps we can come back to that.


Originally Posted By: paul

time is just the measurable part of infinity , forever , eternity , and time passes ...


there will always be "time" to measure change.

and changes occur as time passes... but there can not be change without "time" passing.

you haven't answered my question about the weather in the UK
in the Climate Change Forum , I would like to have the opinion
of a human and his experiences through the years vs a bunch of measuring instrument's if you don't mind.



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"1. Is infinity a number?
I would think concept, not number. Meaning not a real number but I don't understand quaternions and octonions.


2. Is eternity a length of time?
I would think an infinite length of time (if time exists). But I'm troubled that it seems photons don't experience time or distance.


3. Is it possible to define Cantor’s “absolute infinity”?
Haven't looked into that.


4. If there had ever been (absolutely) nothing, could there be something now?
I would not think so. What would anything form from?


5. Could there be change without time?"
Wouldn't think so but there's the time and photon scenario again.

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Bill S I am more than happy to discuss things in a civil detailed manner but at the moment you are not actually looking at the matter in a open investigative way and I couldn't help but laugh as you guys dug yourself into a huge hole.

Pokey bought up probably the first point as Layman you guys don't understand why complex number and greater systems exist. So if you really want to get some understanding it is worth taking a few minutes to run thru that. I will have a crack to give the ultra short condensed version as best I know which goes like this.

In the 15th century the imaginary number system was created to deal with square roots of negative numbers. In the 17th century problems to do with engineering often required the use of an imaginary axis(s) along with the normal 3D world to solve problems. A simple example would be the stress inside a 3D beam where you would end up with for every x,y,z a stress value. When you try to solve these problems you find you are dealing with a cubic equation and complex numbers came from the merge of imaginary system and real number systems to solve them.

The number system could have infinity or not depending on use and no one really took any real thought about it because it was not considered to be real. You could use infinity as a number or a concept because the number system was only used for analysis and the relevance of infinity was to the use.

Electromagnetic light waves caused the first problems here we had a waveform which was in 2D in movement (it moves in straight line so 2D not 3D) but contained 2 field strengths for each point on the 2D movement plane. So every point along the 2D space plane required X,Y,E,B as values when doing analysis on the waves. E being the electric field strength and B being the magnetic field strength. If that wasn't bad enough the totally energy between the two fields was linked. This was our first encountering of a real world situation in which 4 degrees of freedom were involved (I avoided using dimension deliberately).

The analysis of the waveforms required the use of complex numbers which was developed in the 18th century by Maxwell. At this stage you can sort of wave your hands and say the complex number mathematics is just a way of doing the mathematics we had not worked out how to do with real numbers.

In 1905 Einstein introduces GR & SR which like electromagnetism involves 4 degrees of freedom linked via equations. In SR and GR the traditional approach was to make time the imaginary component of a complex number co-ordinate of spacetime (it's mostly no longer taught that way but it was and is still valid). So now two of the cornerstones of physics required complex number system to provide solutions.

Finally QM and Heisenberg uncertainty principle create the problem that everything in the universe requires quantum wave description and that can not be done or even written in real number systems. The technical reason is usually expressed as this

Quote:
In quantum mechanics, probabilities are the only thing we can compute about the outcomes of any experiments or phenomena. And the last steps of such calculations always include the squaring of absolute values of complex probability amplitudes. Complex numbers are fundamental for all predictions in modern science.

In simple layman terms the universe is composed of things that can only be represented by complex numbers.

Once you realize the above fact worrying about the implications and stupidity of what infinity does or doesn't mean in the real number system has about as much relevance to the universe as to whether green aliens exist outside the universe.

Hopefully you might actually use some intelligence and give some more serious thought to the number system because you can't use real numbers to discuss the universe.

Well you can but don't expect me to treat you not like an idiot especially after I told you a number of times smile

There is no agreed way to handle infinity in complex numbers (largely because of it's history) and there are other similar problems which is why efforts were made to more strictly handle complex mathematics with more formal rules and that is what quaterions and octernions are.

The lesson is (and I told you it) sometimes you need to look carefully at the question and what assumptions underpin it. The problem with the question in the way that you are answering it is that you think real numbers have anything to do with real world physics wink

You could have saved yourself a lot of grief and embarrassment by a simple google search of "why does physics use complex numbers" when I pointed out that situation smile

This was one of the funniest threads I have been in for a while and I guess shows the gulf between layman and science dramatically. You just could not see how you could be wrong and I couldn't believe you didn't get the error and problem.

Last edited by Orac; 11/30/14 04:46 PM.

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And all of this talk still has nothing to do with whether infinity is a number. Some calculations in both GR and QM wind up with infinities in the answer. In QM this is frequently corrected by renormalization, but it doesn't affect the fact that infinities occur in complex number calculations, just as they do in real number calculations.

Example:
let A = 10 + 10i
let B = 0 + 0i
then:
A/B = infinity, if you ignore the prohibition against dividing by zero.

So all of your blather about complex numbers still doesn't give us an answer as to whether you think infinity is a number.

And Paul's answer to the questions that Bill S. asked at the start of this thread are still pretty good answers.

Bill Gill


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Thanks for the interesting history, Orac.

Quote:
The problem with the question in the way that you are answering it is that you think real numbers have anything to do with real world physics


A large part of the problem is that you are reaching that sort of conclusion without any justification. I make no such assumption about the Universe, and if you were able to take my questions and reasoning at face value, instead of trying to force them into a confused and confusing mathematical mould, you would understand that. Interestingly, in this forum you are urging me to study complex numbers in order to solve what you perceive as my "problem"; and in TNS, Pete, who you will remember from his brief sojourn in SAGG, is urging me to study calculus for the same reason.

Possibly the key question from the OP is:

If there had ever been (absolutely) nothing, could there be something now?

Pokey's answer: "I would not think so. What would anything form from?" is concise, reasonable and lacking in extraneous maths. It also anticipates the question that would naturally follow an affirmative answer.

I accept willingly that mathematics is the best language that humans have yet found to probe the mysteries of the cosmos, but I don't necessarily accept that there is some grand mathematical model on which some preexisting intelligence designed the cosmos, and that we cannot even think about the nature of existence unless we discover the mind of some mathematical God.


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Originally Posted By: Bill
So all of your blather about complex numbers still doesn't give us an answer as to whether you think infinity is a number.

I have told you the answer a number of times if you bothered to actually read the problem instead or reacting because it's me that said it.

There is no formal way to answer the question in the complex number system. The way you define the complex number system will either allow you to use infinity as a digit in both answer and operator or not. Due to the history of complex numbers and every complex number containing "imaginary" terms the idea of them being or meaning something real is sort of foreign and weird. Perhaps you could argue all complex numbers are concepts because they are all like infinity, but that leads to interesting conclusions about the world smile

Hopefully you see the problem trying to distinguish any number as being "real" as opposed to "concept" in complex numbers is well interesting by any standard. I would really like to have a mathematician make a comment on it perhaps I am missing something.

Personally there is no selection criteria I could give you as to whether infinity is a "digit representation" or "concept" in complex numbers other than can you use it as a normal "digit" in operations. I really see no other criteria I could use there is nothing special about it compared to any other complex number.

That is why I wanted to talk about operators which you didn't like and instead wanted to make it about "counting". I didn't jibe you about it but had a good chuckle because counting is the operator of "addition" as a heads up.

There is no "authority" that can settle or answer that question you will need to work a basis to make the selection of the answer.

In other words it requires something you refuse to do, which Bill S has sort of come around to which is discuss what is the question really asking and think about it.

Last edited by Orac; 12/01/14 01:33 AM.

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Originally Posted By: Bill S.
If there had ever been (absolutely) nothing, could there be something now?

Again Bill S there is a huge problem with that question.

1.) You could try to answer that from your experience.

The problem is we live and have experience in one tiny part of a very big universe that may actually be nothing like what we see around us. As humans we once believed the earth was flat, we were the centre of the universe, the sun revolved around the earth by using exactly that sort of reasoning.

Do you really want to go back to the dark ages?


2.) You could try answering the question from the point of view of physics. I know you asked this on a number of science forums and they gave you the same answer I will.

Our current understanding and mathematics of physics does not exclude the possibility that you can create something from nothing, regardless of the fact that probably every scientist hates the idea.

3.) You could try answering the question from what you personally prefer as a cult (although that has really negative context and I don't mean to imply it, my english I can't find a better wording)

Originally Posted By: Bill S
Pokey's answer: "I would not think so. What would anything form from?" is concise, reasonable and lacking in extraneous maths. It also anticipates the question that would naturally follow an affirmative answer.

This goes down that path and science caress little what Pokey, you me or anybody else "thinks" that is the basis for a cult not science.

Originally Posted By: Bill S
I accept willingly that mathematics is the best language that humans have yet found to probe the mysteries of the cosmos, but I don't necessarily accept that there is some grand mathematical model on which some preexisting intelligence designed the cosmos, and that we cannot even think about the nature of existence unless we discover the mind of some mathematical God.

Again science does not care what you think. All it cares about is do you have an idea that makes predictions that can be tested and are those answers verified.

Last edited by Orac; 11/30/14 11:54 PM.

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Originally Posted By: Orac
There is no formal way to answer the question in the complex number system. The way you define the complex number system will either allow you to use infinity as a digit in both answer and operator or not.

Then please give an example of a complex number where infinity is a number and tell us what that number is.

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Originally Posted By: Orac
That is why I wanted to talk about operators which you didn't like and instead wanted to make it about "counting". I didn't jibe you about it but had a good chuckle because counting is the operator of "addition" as a heads up.

You are partly correct, but you have it backwards. Addition is a short hand for counting. 5,000 to 6,000 years ago when a farmer brought in his offering to the temple the priests counted how much he brought in and put it in storage. Then when another farmer brought in his they counted that and put it in storage. But inevitably the high priest would want to know how much there was. Then they had to go count it all. But then they invented a short cut, addition. Now all they had to do was to count the income from each farmer and then add it up to figure how much they had. And that was the start of all mathematics, including complex mathematics. All mathematics is a short hand for counting.

Complex math may have a way to handle infinity, but the result of any calculation is unusable it it returns an infinity. So infinity is not a number.

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Originally Posted By: Bill
Then please give an example of a complex number where infinity is a number and tell us what that number is.

Dozens of them but a fairly common one related to spacetime is Riemann sphere

http://en.wikipedia.org/wiki/Riemann_sphere

It's has a lot of uses in physics where you potentially have a real infinity and want to avoid using higher maths schemes.

Ask Dr Maths gives you the background in the best way I could ever express for a layman
http://mathforum.org/library/drmath/view/63359.html

Quote:
There's a uniform way to handle infinity, too. There is just one infinity, and the term "as z approaches infinity" simply means that z gets farther and farther away from zero in any direction, even by spiraling out or any other path you can think of.

There is an easy mapping of the complex plane plus the single "point at infinity" to something called the "Riemann sphere." Imagine a sphere sitting on the complex plane at the origin, so its "south pole" touches the origin. To do the mapping, draw a straight line from the north pole to the point on the complex plane, and wherever that line punctures the sphere is the corresponding point on the sphere. As points on the complex plane get farther and farther from the origin (in any direction), that puncture point gets closer and closer to the north pole, so the north pole is the image of infinity on the Riemann sphere.

This all works incredibly nicely and gives a completely uniform way to talk about all the complex numbers, including infinity.

Is that clear enough that infinity is handled exactly the same way as any other complex number under the system and that is why you do it.

As to what number infinity is well perhaps lets write a few complex numbers and you tell me what number they are so I can understand what you mean

3 + 2i = what number to Bill
69 + 3i = what number to Bill
7 + 0i = what number to Bill (you might be able to do)

So can you tell me what those numbers are and I might understand a basis to give you a number for infinity.

Last edited by Orac; 12/01/14 04:03 PM.

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Originally Posted By: Bill
Complex math may have a way to handle infinity, but the result of any calculation is unusable it it returns an infinity. So infinity is not a number.

Complex numbers calculations return complex numberssmile

The usability of infinity is a function of the complex number system used as discussed in Reimann Sphere smile

That was actually my point in many forms of complex number systems, quaternions and octonions you can use operations on infinity. It does not differ from any other number under these schemes because each complex number is nothing like a number in the terms you are thinking of.

So I need you to write me a complex number and tell me what number it represents so I can try and understand you

Here is the general form of a complex number

a + bi
(a is the real number part, b is the imaginary number part)

I want you to explain how any of those numbers so written are different to writing infinity.

Originally Posted By: Bill
And that was the start of all mathematics, including complex mathematics. All mathematics is a short hand for counting

Once you get thru that I want you to show me "counting" on a complex number. Oh boy this is going to be a Bill G classic. To give you a heads up you might find a little problem determining which direction to "count" in and I am dying to know how you decide that. Perhaps read Ask Doctor maths answer again because he talks about directions laugh

So when you are ready 23 + 7i what is the next count of that complex number please I am dying to know.

Bill can I suggest the hole is getting deeper stop digging smile

Last edited by Orac; 12/01/14 04:05 PM.

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Originally Posted By: Orac
So when you are ready 23 + 7i what is the next count of that complex number please I am dying to know.

Ok, let's take a look at that.

let A = 23 + 7i Determine A + 1
let x = 23
let y = 7
let r = length of the vector described by A
let T = the angle of the vector described by A
T = atan(y/x) = 16.927 degrees
r = x/cos(T) = 24.041
let r' = r + 1 = 25.041
x' = r' cos(T) = 23.956
y' = r' sin(T) = 7.291

Then

A' = 23.956 + 7.291i

A has thus been incremented by 1.

Did you forget that complex numbers are basically vectors?

I did take a little longer than it should have to figure that out. I don't work with the trig functions much so I had to do some review.

By the way I didn't see any infinities in that.

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Originally Posted By: Bill

A' = 23.956 + 7.291i

A has thus been incremented by 1.

Did you forget that complex numbers are basically vectors?

So if I work out the number of "bill counts" from 0 + 0i I guess that must be the bill count number thing you keep asking for and you have no infinity smile

So lets do it

0 + 0i = 0
0.956 + 0.291i = 1
1.912 + 0.582i = 2
2.868 + 0.873i = 3
....
22.944 + 6.984i = 24

hmmm not quite even so I guess

23 + 7i = 24.058577405857740585774058577406

Are you happy with Bill, I think I have followed your logic correctly?

Anyone else care to comment?

Hints:
Bill count to 23 - 7i, -23 + 7i & -23 - 7i. Those are the obvious easy ones no maths required.
Clever ones will be able to mark all the co-ordinates for a bill count of 24.058577405857740585774058577406 and how many is that smile
If you haven't got it think circle centred on 0,0smile


So hopefully you all caught up so Bill says 24.058577405857740585774058577406 bill counts from the complex origin all you have to do is work out which of the infinite points on the circle of radius 24.058577405857740585774058577406 centred on 0,0 he means smile

In fact for any point you pick you can draw a circle 1 unit in size and those are the infinite number of values 1 bill count away depending on which of the infinite directions you decide to bill count. And you have no idea which way to count until your bill count is given to you as a proper complex number so you can get its direction from 0,0. This is dramatically different from real numbers where any number on has two numbers 1 bill count away.

So every complex number has infinite other complex numbers 1 bill count away and that doesn't even include all the complex numbers because we can see the circle doesn't pass thru every complex number. Good lord there is something therefore bigger than and more tricky than infinity then what shall we call it smile

Bill can write his bill count number using digits rather than a symbol is about the only difference I can see but I am sure Bill will clear this all up for me.

Last edited by Orac; 12/02/14 04:11 AM.

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This should help



Background read the full detail
http://www.clarku.edu/~djoyce/complex/abs.html

And you note
Quote:
As you might expect, there are infinitely many of them.


No infinities were harmed in the making of this production of the infinite number of infinities.

Last edited by Orac; 12/02/14 12:40 PM.

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I'm glad to see that you can still draw completely erroneous conclusions from my simple demonstrations. In your example that you wanted me to work out you asked me to add 1 to a vector. Since one is not a vector I assumed that you meant me to add the quantity one. That means that you wanted me to add 1 to the length of the vector. So I converted the vector to polar coordinates and added 1, then converted back to XY coordinates. If I had realized you wanted the 1 to be a vector then I would have expressed it as a vector that is, 1 + 0i, and done vector addition. That is a lot simpler than doing the conversions back and forth between XY and polar coordinates. In that case the answer would be 24 + 7i.

As far as an infinite number of directions from the end of the first vector is concerned it is true, but to access those points you have to define a vector that points to the one you want. You did not give me a vector, you gave me a number. So you are once again resorting to obfuscation to hide the fact that:

You still haven't found any place where infinity is a number.

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