Deeper than primes - Continuation

doronshadmi · Apr 18, 2013

Kage said:
No one claims that the real line is isomorphic to the natural numbers.

I also do not clime such a thing.

You still miss my argument that 1-dimesional space is not a collection of 0-dimesional spaces along it.

Kage said:
The real line is constructed from the rational numbers.

According to my non-standard framework, no amount of elements is the real-line (which is not less than a non-composed 1-dimensional space, and no collection of elements along it (call them real numbers, or what ever) has the power of the continuum of this non-composed 1-dimensional space).

Kage said:
I did not claim this.

Ok (it was a stupid mistake of me anyway).

Kage said:
1/epsilon < 1/delta implies that epsilon is greater than delta, but delta is equal to 1 - sum(k = 1 to a)9*10^(-k), which is greater than 1 - sum(k)9*10^(-k). this means that delta is simultaneously less than epsilon and greater than epsilon,

You can't conclude anything about an expression like "1 - 0.999..." by using finite indexes like k along a non-finite expression like "0.999...".

Kage said:
Have you actually used calculus, abstract algebra, or differential geometry? Math is one of the most spatially challenging subjects. You are making assumptions.

Have you actually used also your visual_spatial brain skills in order to be opened to the notion that no one of the k-dimensional spaces is composed by other k-dimensional spaces, but their possible associations provide any possible mixing of them, where the real-numbers are no more than local measurement tools of this mixing? (by your verbal_symbolic-only reasoning there is no way to distinguish between, for example, the local number 1 and the non-local number 0.999...₁₀).

Kage said:
Actually, dimensions do not have to be a natural number.

Their mixing are not measured by natural numbers.

Kage said:
This is super cool, and one of the things you get to understand when you study mathematics, specifically fractals.

The place value method (no mater what base > 1 is used) has fractal dimensions, and this is exactly the reason why the local number 1 > the non-local number 0.999...₁₀ exactly by the non-local number 0.000...1₁₀.

Kage said:
Again, why call it entropy? Are you talking about the degree of ordering in the system? Chaos? Information? What? Why use "entropy" other than it sounds cool.

I am talking about not less than Distinction Quantity and Order of the discussed subject.

doronshadmi · Apr 18, 2013

My apology to the traditional mathematician in this case.

The expressions "epsilon > delta" and "delta < epsilon" are the same.

My stupid mistake and thanks to the traditional mathematician for correcting me.

doronshadmi · Apr 18, 2013

In addition to what I wrote to you about the inability to conclude anything about "0.999..." expression, by using finite indexes, let's look at the following:

kage said:
(2)
Assume that .999.... does not equal 1
Therefore, there exists a difference, epsilon = 1 - .9999...
For each epsilon, choose a delta such that 1/epsilon is less than 1/delta ...

What do you mean by "For each epsilon, choose a delta"?

1- 0.999...₁₀ provides exactly one epsilon, which is the non-local number 0.000...1₁₀ (where the "...1" part of the the non-local number 0.000...1₁₀ is exactly the irreducibility of 1-dimensional space into 0-dimensional space).

So the part "For each epsilon, choose a delta" has nothing to do with my framework, unless different bases are used for delta and epsilon (where in both cases no finite indexes are used), but then your argument does not work.

Some examples:

By using verbal_symbolic AND visual_spatial brain skills one knows that 0.111...₂ > 0.111...₃, where both of them < 0.222...₃

By Traditional Mathematics 0.111...₂ = 0.222...₃ exactly because only verbal_symbolic brain skills are used (where both numbers are forced to represent the local number 1).

jsfisher · Apr 18, 2013

doronshadmi said:
By Traditional Mathematics 0.111...₂ = 0.222...₃ exactly because only verbal_symbolic brain skills are used (where both numbers are forced to represent the local number 1).

If they were different in value, Doron, it should then be possible to demonstrate rigorously, not with gibberish nor with doronetics' patent-pending direct perception, that they can be distinguished in some mathematical way. Visual difference in representation does not equate to mathematical difference in value.

You have flatly stated in the past, Doron, that 1/4 and 0.25 were different numbers (presumably because of representational differences), but you have made so many outrageous statements. That 2 may not be a member of the set, { 2 }, is perhaps my favorite. Your cognitive limitations, however, do not trump mathematical fact.

You got it wrong.

Consider how much trouble you had seeing your error in understanding delta < epsilon. If comprehending the simple things is that hard for you, guess what must be true for the stuff that's not so simple.

epix · Apr 19, 2013

doronshadmi said:
My apology to the traditional mathematician in this case.

The expressions "epsilon > delta" and "delta < epsilon" are the same.

My stupid mistake and thanks to the traditional mathematician for correcting me.

I show you a trick how to solve an inequality:

1/a < 1/b : a/1 ? b/1

Go to the New Testament, Revelation 22:13.

I am the Alpha and the Omega, the First and the Last, the Beginning and the End.

It immediately follows that BELLOW is to ABOVE as LEFT is to RIGHT, and you are done:

1/a < 1/b : a/1 > b/1

epix · Apr 19, 2013

doronshadmi said:
Let's look at this phrase of the traditional mathematician here:

"Doronetics is clearly meant for those situations where reality is extra flexible."

I claim that 1/epsilon < 1/delta in case that epsilon > delta.

So one can't clime that delta < epsilon and also claim that 1/epsilon < 1/delta.

(and this is exactly what kage did here:

Originally Posted by Kage
(2)
Assume that .999.... does not equal 1
Therefore, there exists a difference, epsilon = 1 - .9999...
For each epsilon, choose a delta such that 1/epsilon is less than 1/delta and that delta is a power of 10 (with Log(delta) = k).
This exists as there is no upper bound to the rational numbers.
delta is less than epsilon.However, delta is 10^(-k) = 1 - .99..9 (with k 9s).
This implies delta is greater than 1 - .99999999.... (because there are more 9s) which is a contradiction.
Therefore .9999999... is equal to 1, QED.

Click to expand...

)

According to this traditional mathematician it is true that, for example, 1/4 < 1/2 AND 4 < 2, or in other words, he always have some negative replies about what I write, no matter what.

Nice, isn't it?

Yes, it is. The highlighted text is the only sentence in the quote that starts with a lower-case letter, namely letter d. That's not an analytic mistake, but it could evolve into one, as it did... or Did... or diD... or even D<d?

: D =

doronshadmi · Apr 19, 2013

Let's see this phrase:

"Consider how much trouble you had seeing your error in understanding delta < epsilon. If comprehending the simple things is that hard for you, guess what must be true for the stuff that's not so simple."

I admit that I made a mistake about the delta < epsilon case, as written by Kage.

Does this mistake is related to my arguments in http://www.internationalskeptics.com/forums/showpost.php?p=9161741&postcount=2203 ?

Certainly not (as long as one uses his\her verbal_symbolic AND visual_spatial brain skills, as done in the link above).

doronshadmi · Apr 19, 2013

By using verbal_symbolic AND visual_spatial brain skills it is not possible to define delta < AND > epsilon if an expression like "1-0.999..." is involved, and delta is defined according to finite indexes along "0.999...".

In other words, Kage's argument does not distinguish between the non-finite expression "1-0.999..." (or "0.000...1" (please look at http://www.internationalskeptics.com/forums/showpost.php?p=9161741&postcount=2203)) and infinitely many finite expressions that are derived from finite indexes.

As for "2" and "{2}" expressions:

"2 is a member of A(={2} or {2,...})" expression does not mean that this expression holds also if only "2" expression is used.

Some notes about this phrase:

"Visual difference in representation does not equate to mathematical difference in value."

It is true if visual difference is considered only as a representation of a given mathematical value.

It is not true if visual difference is also involved in order to define a given mathematical value, for example, let's examine this phrase:

"You have flatly stated in the past, Doron, that 1/4 and 0.25 were different numbers"

If one understands the place value method only as a representation of a given mathematical value, then "1/4" and "0.25" are nothing but two representations of the same (local) mathematical value.

This is not the case if place value method is considered as a given mathematical value, and not only Locality is considered, for example:

The vertical red line is 1/4. If one understands the place value method only as a representation of a given mathematical value, then 0.25₁₀ = 1/4 = 0.01₂.

If place value method is considered as a given mathematical value, and not only Locality is considered, then 0.25₁₀ ≠ 1/4 ≠ 0.01₂ as clearly seen in

I did not expend (yet) my framework in order to deal with these differences in details, but visual_spatial brain skills and non-locality are clearly demonstrated, in addition to the verbal_symbolic brain skills.

---------------------------------

In other worlds, Traditional Mathematics is based only on Locality and verbal_symbolic brain skills when it deals with the place value method, and there can't be any agreement with a notion that defines the place value method as mathematical values (and in this case also visual_spatial brain skills and non-locality must be involved).

This disagreement is shown all along this thread, but has no impact on a mathematical work that is done by using verbal_symbolic AND visual_spatial brain skills.

laca · Apr 19, 2013

epix · Apr 19, 2013

doronshadmi said:
In addition to what I wrote to you about the inability to conclude anything about "0.999..." expression, by using finite indexes, let's look at the following:

Originally Posted by kage
(2)
Assume that .999.... does not equal 1
Therefore, there exists a difference, epsilon = 1 - .9999...
For each epsilon, choose a delta such that 1/epsilon is less than 1/delta ...

Click to expand...

What do you mean by "For each epsilon, choose a delta"?

1- 0.999...₁₀ provides exactly one epsilon, which is the non-local number 0.000...1₁₀ (where the "...1" part of the the non-local number 0.000...1₁₀ is exactly the irreducibility of 1-dimensional space into 0-dimensional space).

So the part "For each epsilon, choose a delta" has nothing to do with my framework, unless different bases are used for delta and epsilon (where in both cases no finite indexes are used), but then your argument does not work.

Different bases? I think that it is about different colors, namely yellow. Can't you see where your advanced symbolism such as 0.000...1₁₀ fits the best? Follow the yellow line...

Originally Posted by Kage
(2)
Assume that .999.... does not equal 1
Therefore, there exists a difference, epsilon = 1 - .9999...
For each epsilon, choose a delta such that 1/epsilon is less than 1/delta and that delta is a power of 10 (with Log(delta) = k).
This exists as there is no upper bound to the rational numbers.
delta is less than epsilon.
However, delta is 10^(-k) = 1 - .99..9 (with k 9s).
This implies delta is greater than 1 - .99999999.... (because there are more 9s) which is a contradiction.
Therefore .9999999... is equal to 1, QED.

The highlighted term was hastily rendered. According to the syntax of doronetics, it should look like 0.999...9₁₀, where the "...9" part is non local... you know. Take it from here.

1- 0.999...₁₀ provides exactly one epsilon

That's not really true. You create numbers by writting them down with no regard to the source that generate these numbers. You regard t = 0.9999... as a compact unit. But the magnitude of t is increasing with each step performed in the generating formula F. For each step in F, there is one indexed epsilon. The value of t is converging; the point that represents t is moving to the right and negatively accelerating.

How do you perform operations with values that are converging? Is it the same as with those represented by a fixed point, such as the one that represents an integer? Simply assert Moving = Stationary and the answer is YES.

doronshadmi · Apr 20, 2013

Let's look at this phrase:

"But the magnitude of t is increasing with each step performed in the generating formula F. For each step in F, there is one indexed epsilon."

It is wrong since an expression like "0.999...₁₀ is taken at once as exactly one infinite non-local number, where no steps (and I would say collections of finite non-local values, which are known as finite local values by Traditional Mathematics) of the forms 0.9₁₀, 0.99₁₀, 0.999₁₀, ..., etc. along it are involved, in order to define it (in terms of sets, the member of {0.999...₁₀} is not a member of {0.9₁₀, 0.99₁₀, 0.999₁₀, ...}).

In other words, "delta < epsilon" is a false term.

laca · Apr 20, 2013

doronshadmi said:
Let's look at this phrase:

"But the magnitude of t is increasing with each step performed in the generating formula F. For each step in F, there is one indexed epsilon."

It is wrong since an expression like "0.999...₁₀ is taken at once as exactly one infinite non-local number, where no steps (and I would say collections of finite non-local values, which are known as finite local values by Traditional Mathematics) of the forms 0.9₁₀, 0.99₁₀, 0.999₁₀, ..., etc. along it are involved, in order to define it (in terms of sets, the member of {0.999...₁₀} is not a member of {0.9₁₀, 0.99₁₀, 0.999₁₀, ...}).

In other words, "delta < epsilon" is a false term.

Pray tell, oh doron, which is the last member of the set {0.9₁₀, 0.99₁₀, 0.999₁₀, ...}?

In other words, you fail. Again and always. Forever and ever.

doronshadmi · Apr 20, 2013

To anyone who wonders, set {0.9₁₀, 0.99₁₀, 0.999₁₀, ...} does not have a last member, but no one of its members has an infinite amount of 9s (as found in the case of the member of set {0.999...₁₀}).

In other words, the member of set {0.999...₁₀} is not a member of set {0.9₁₀, 0.99₁₀, 0.999₁₀, ...}.

jsfisher · Apr 20, 2013

doronshadmi said:
... the member of set {0.999...₁₀}...

Just can't bring yourself to say just 0.999..., eh? It is almost as if you do not comprehend that 0.999... is a member (the one and only member) of { 0.999... }.

By the way, whether 0.999... be a member of {0.9, 0.99, 0.999, ...} is an open question since no precise definition for the set was given. What is probably the most natural interpretation of the notation used would be that the members all have a finite number of 9's in their decimal representation and therefore 0.999... be not a member, but that is not the only possible interpretation.

Precision has never been your forte, Doron.

doronshadmi · Apr 20, 2013

To make a long story short, given two arbitrary different k-dimensional spaces (such that k = 0 or any possible natural number) the greater dimensional space is not composed by the smaller dimensional space (where the term "dimensional space" is not limited only to metric space).

I'll be happy to know what is the reasoning of traditional mathematicians to not agree with what is written above.

doronshadmi · Apr 20, 2013

The phrase "Precision has never been your forte, Doron" was written by a traditional mathematics that in this case ignored what is written in http://www.internationalskeptics.com/forums/showpost.php?p=9166749&postcount=2211 (which clearly provides the reason of why the member of set {0.999...₁₀} is not a member of set {0.9₁₀, 0.99₁₀, 0.999₁₀, ...}).

Because of this ignorance he\she can't understand why "delta < epsilon" is a false term.

As about Precision, this traditional mathematician does not distinguish between 0.999...₁₀ expression and {0.999...₁₀} or {0.999...₁₀, ...} expressions.

jsfisher · Apr 20, 2013

Ignored? Discounted as irrelevant, illogical, gibberish, contradictory, stupid, fantastically stupid, or just plain wrong, sure. Ignored? No.

doronshadmi · Apr 20, 2013

The phrase "Ignored? Discounted as irrelevant, illogical, gibberish, contradictory, stupid, fantastically stupid, or just plain wrong, sure. Ignored? No." does not provide anything about the discussed subject.

If the traditional mathematician wishes to do it right, he is asked to provide a rigorous argument about "delta < epsilon" expression, which appears in the following quote:

Kage said:
(2)
Assume that .999.... does not equal 1
Therefore, there exists a difference, epsilon = 1 - .9999...
For each epsilon, choose a delta such that 1/epsilon is less than 1/delta and that delta is a power of 10 (with Log(delta) = k).
This exists as there is no upper bound to the rational numbers.
delta is less than epsilon.
However, delta is 10^(-k) = 1 - .99..9 (with k 9s).
This implies delta is greater than 1 - .99999999.... (because there are more 9s) which is a contradiction.
Therefore .9999999... is equal to 1, QED.

jsfisher · Apr 20, 2013

doronshadmi said:
If the traditional mathematician wishes to do it right, he is asked to provide a rigorous argument about "delta < epsilon" expression, which appears in the following quote:

What's left to provide? You've already conceded that delta being less than epsilon is a direct consequence of 1/epsilon being less than 1/delta. (Admittedly, though, it took you a bit to get there, your reading comprehension and reasoning skills being what they are and all.)

As for whether 1 is distinguishable from 0.999..., the claim is yours despite the preponderance of mathematical evidence to the contrary so the burden falls to you to show how they behave differently.

So far, though, you haven't exhibited any difference to separate the two. I wonder why that is.

doronshadmi · Apr 20, 2013

The phrase "What's left to provide? You've already conceded that delta being less than epsilon is a direct consequence of 1/epsilon being less than 1/delta." does not provide anything about the discussed subject, simply because 1/epsilon can't be less than 1/delta exactly because epsilon is exactly one and only one result of the expression 1 - 0.999...₁₀, and no delta (which is the result of "1 - .99..9 (with k 9s)" expression is smaller than this one and only one epsilon.

In other words, no one of the infinitely many expressions that uses finite values in order to provide infinitely many delta's, provides a delta that is less than this one and only one epsilon.

Kage · Apr 20, 2013

doronshadmi said:
You still miss my argument that 1-dimesional space is not a collection of 0-dimesional spaces along it.

I do miss your argument. You know that there are a lot of numbers, right? Infinite numbers of significant digits?

doronshadmi said:
According to my non-standard framework, no amount of elements is the real-line (which is not less than a non-composed 1-dimensional space, and no collection of elements along it (call them real numbers, or what ever) has the power of the continuum of this non-composed 1-dimensional space).

What is this power of the continuum? Continuity at a point, by definition, means that the limit as x approaches that point exists. The real line is defined as the metric space where all Cauchy sequences converge, which by definition implies that it is continuous. What extra power have we been missing all this time?

doronshadmi said:
You can't conclude anything about an expression like "1 - 0.999..." by using finite indexes like k along a non-finite expression like "0.999...".

In mathematics, we use expressions like for all k, or from k = 1 to infinity. This is what I am doing here. K is not a finite index.

doronshadmi said:
Have you actually used also your visual_spatial brain skills in order to be opened to the notion that no one of the k-dimensional spaces is composed by other k-dimensional spaces, but their possible associations provide any possible mixing of them, where the real-numbers are no more than local measurement tools of this mixing? (by your verbal_symbolic-only reasoning there is no way to distinguish between, for example, the local number 1 and the non-local number 0.999...₁₀).

This is a pretty standard method of argument -- you don't understand because you are closed minded. I actually have considered your argument, and have been pretty reasonable here. I, like most people who studied college level mathematics, have used spatial reasoning a lot. You need to make arguments beyond telling people that they have not visualized the problem.

doronshadmi said:
The place value method (no mater what base > 1 is used) has fractal dimensions, and this is exactly the reason why the local number 1 > the non-local number 0.999...₁₀ exactly by the non-local number 0.000...1₁₀.

I will prove to you that your "number" .0000...0001 is identically zero. I use the expression "number" because this expression is not really a properly formatted number:

Assume that .0...01 is different than 0
Due to the way .0..01 is written, there should be no number between it and 0
Set epsilon equal to to the difference between .0...01 and 0
Choose delta such that 1/delta is greater than 1/epsilon
This is possible because there is no upper limit to numbers
epsilon is less than delta, which implies that there is a number between .0..01 and 0
This is a contradiction and .0...01 is identical to 0

A second proof:

.0...01 is the same as 10^-k as k grows without bound
10^-k = 1/10^k equals 1/infinity as k grows without bound
1/infinity is identical to zero by the definition of infinity (it is a symbol, not a number)

This is the problem with writing things like .0...01. This is not a real number. It is 1/infinity. Essentially you are making the argument that infinity minus one is not equal to infinity. This is not true. There are specific ways that infinity behaves differently than numbers, as infinity can be best defined as "growing without bound." Growing without bound is the same as growing without bound minus one. One divided by growing without bound is zero. In order to use things like infinity, you have to play by the rules.

Again, if you don't like this, you have to ignore calculus and all the beautiful things that calculus has made possible.

jsfisher · Apr 20, 2013

doronshadmi said:
The phrase "What's left to provide? You've already conceded that delta being less than epsilon is a direct consequence of 1/epsilon being less than 1/delta." does not provide anything about the discussed subject, simply because 1/epsilon can't be less than 1/delta exactly because epsilon is exactly one and only one result of the expression 1 - 0.999...₁₀, and no delta (which is the result of "1 - .99..9 (with k 9s)" expression is smaller than this one and only one epsilon.

In other words, no one of the infinitely many expressions that uses finite values in order to provide infinitely many delta's, provides a delta that is less than this one and only one epsilon.

It's that reading comprehension thing again, isn't it?

doronshadmi · Apr 20, 2013

Kage said:
This is the problem with writing things like .0...01. This is not a real number.

I agree with you. Real numbers have exact locations along a given 1-dimensional space.

A number like 0.000...1₁₀ is a non-local number, and it is the result of the irreducibility of 1-dimensional space into 0-dionasional space (no matter if k = 1 to infinity, where infinity is understood as an amount of members of a given collection).

Kage, you and me have no communication about the discussed subject exactly because you do not use also your visual_spatial brain skills (at least in the abstract sense) in addition to your verbal_symbolic brain skills, in order to discuss about the given subject.

Once again, as long as your mathematical paradigm is closed under the notion that (for example) 1-dimensional space is composed, there can't be any communication between us about the discussed subject.

In order to (I hope) understand better my argument, please look at http://www.internationalskeptics.com/forums/showpost.php?p=9161741&postcount=2203 .

doronshadmi · Apr 20, 2013

The phrase "It's that reading comprehension thing again, isn't it?" does not provide anything to the discussed subject.

In other words, detailed arguments that supports it, must be provided in order to really contribute to the discussed subject.

doronshadmi · Apr 20, 2013

Kage said:
Essentially you are making the argument that infinity minus one is not equal to infinity.

Essentially my argument is that no collection of infinity many objects on a given dimensional space > 0 (where the given dimensional space is not necessarily a metric space), has the power of infinity (I do not use here the term continuum, because you will get it by using the paradigm that gets it in terms of collections) of that space.

Kage · Apr 20, 2013

doronshadmi said:
In addition to what I wrote to you about the inability to conclude anything about "0.999..." expression, by using finite indexes, let's look at the following:

It is not a finite index. Hard to imagine there being that many numbers, but there are.

doronshadmi said:
What do you mean by "For each epsilon, choose a delta"?

This is actually a pretty standard way of proving things in the field of analysis (fancy talk for calculus). To prove that an expression converges on a limit, the proof will say, assume that the expression never gets delta away from the limit. The prover will then construct an epsilon such that epsilon is derived from the expression and that epsilon is smaller than delta. This concludes the proof. The proof shows that no matter how close the expression is to the limit it will be closer, and therefor the limit exists and the expression converges to it.

BTW I think I have been writing my proofs backwards according to the convention, but again, numbers do not care what you call them.

doronshadmi said:
1- 0.999...₁₀ provides exactly one epsilon, which is the non-local number 0.000...1₁₀ (where the "...1" part of the the non-local number 0.000...1₁₀ is exactly the irreducibility of 1-dimensional space into 0-dimensional space).

Ok, so numbers don't care what you call them, but I kinda do. I think you need to either be more or less specific, and not use jargon of your own making. I don't see what you are trying to say.

doronshadmi said:
So the part "For each epsilon, choose a delta" has nothing to do with my framework, unless different bases are used for delta and epsilon (where in both cases no finite indexes are used), but then your argument does not work.

What is wrong with using letters from a dead language to mean arbitrary numbers? My entire degree is based on using letters from dead languages.

Again, numbers don't care what you call them. Different bases do not make numbers different. Primes are prime in every base. Rational numbers are rational in every base. Crazy, but there you go.

doronshadmi said:
Some examples:

I LOVE examples!

doronshadmi said:
By using verbal_symbolic AND visual_spatial brain skills one knows that 0.111...₂ > 0.111...₃, where both of them < 0.222...₃

[qimg]http://farm5.staticflickr.com/4103/5096227808_e362e07fe9_b.jpg[/qimg]

By Traditional Mathematics 0.111...₂ = 0.222...₃ exactly because only verbal_symbolic brain skills are used (where both numbers are forced to represent the local number 1).

Ok, some quick things:

The red line in the second graph corresponding to .111... is equal to 1/2 in base three. Congratulations, you have proven that 1/2 does not equal 1. While numbers don't care what you call them, you have to be careful when you change bases.

So back to your argument that .111... in base two is smaller than .222... in base three:

.111... in base 2 is sum_{k is 1 to infinity} 2^-k.111... = lim_{k -- infinity} (2^k-1)/(2^k)
.111... = lim_{k -- infinity} 1 - 2^-k = 1 - lim_{k -- infinity} 2^-kbut 2^(-k) as k grows without bound is 1/infinity is 0, and the limit is 1

.222... in base 3 is sum_{k is 1 to infinity} 2 * 3^-k.222... = lim_{k -- infinity} (3^k-1)/3^k.222... = lim_{k -- infinity} 1 - 3^-k= 1 - lim_{k -- infinity} 3^-kand same as above.

ok, so .222... in base 3 minus 1111 in base 2:
sum_{k is 1 to infinity} 2 * 3^-k - sum_{k is 1 to infinity} 2^-klim_{k -- infinity} (3^k-1)/3^k - (2^k-1)/2^klim_{k -- infinity} 2^-k - 3^-k = lim_{k -- infinity} 6^-kwhich is zero. .222... in base 3 gets off to a head start, but .111... catches up given enough time...

doronshadmi · Apr 20, 2013

Kage, please reply to http://www.internationalskeptics.com/forums/showpost.php?p=9167113&postcount=2215 .

doronshadmi · Apr 20, 2013

Kage said:
Again, numbers don't care what you call them. Different bases do not make numbers different. Primes are prime in every base. Rational numbers are rational in every base. Crazy, but there you go.

Please read the relevant part in http://www.internationalskeptics.com/forums/showpost.php?p=9164030&postcount=2208 .

Kage · Apr 20, 2013

doronshadmi said:
I agree with you. Real numbers have exact locations along a given 1-dimensional space.

Hooray.

doronshadmi said:
A number like 0.000...1₁₀ is a non-local number, and it is the result of the irreducibility of 1-dimensional space into 0-dionasional space (no matter if k = 1 to infinity, where infinity is understood as an amount of members of a given collection).

Is "non-local" a fancy way of saying "made up? I get that you are trying to make some sort of "glue" to hold the number line together, but it is quite firmly constructed out of the rationals via cuts.

doronshadmi said:
Kage, you and me have no communication about the discussed subject exactly because you do not use also your visual_spatial brain skills (at least in the abstract sense) in addition to your verbal_symbolic brain skills, in order to discuss about the given subject.

So it is my fault that you cannot win an argument? The only picture you have sent with regards to .999... being different then 1 proved that 1/2 is not identical to 1. It was convincing.

doronshadmi said:
Once again, as long as your mathematical paradigm is closed under the notion that (for example) 1-dimensional space is composed, there can't be any communication between us about the discussed subject.

Prove that it is not. It is that simple. My mind is quite open to things that are intuitively incorrect. This has happened many times in studying math. The beauty of established mathematics is that everything is backed up with reasoned argument.

For example, between any two rational numbers there is an irrational number. Between any two irrational numbers there is a rational number. However, rational numbers are countably infinite and irrational numbers are un-countably infinite.

jsfisher · Apr 20, 2013

doronshadmi said:
The phrase "It's that reading comprehension thing again, isn't it?" does not provide anything to the discussed subject.

It does, but it would require action on your part to be fruitful.

In other words, detailed arguments that supports it, must be provided in order to really contribute to the discussed subject.

What would be helpful for this subject you feign so much interest would be for you to actually do what you claim no one else does in response. For the case at hand, show that 1 and 0.999... behave differently in some circumstance. If they be different as you so erroneously claim, then the difference should be demonstrable.

Instead, you wave your hands and stomp your feet like a child. Histrionics are no substitute for mathematics.

Kage · Apr 20, 2013

doronshadmi said:
To make a long story short, given two arbitrary different k-dimensional spaces (such that k = 0 or any possible natural number) the greater dimensional space is not composed by the smaller dimensional space (where the term "dimensional space" is not limited only to metric space).

I'll be happy to know what is the reasoning of traditional mathematicians to not agree with what is written above.

Ummm, you mean that ordered pairs are not made out of their fields?

R² is simply the collection of ordered pairs (a, b) such that a, b are elements of R. C is the collection of numbers x such that x = a + b*i, a, b are elements of R.

How are these two spaces NOT composed of their subfields?

Also, have you considered infinite dimensional space, such as the set of all polynomials? Mind-blowing, but that is how many degrees of freedom are in there.

And that is just it, dimension is an expression of the degrees of freedom. R² is not C, but they have the same dimension.

You make these claims about spaces being "non-composed" or whatever, but they are assumptions and lead to conclusions that are either irrelevant or wrong.

doronshadmi · Apr 20, 2013

Kage said:
but 2^(-k) as k grows without bound is 1/infinity is 0,

Only if you understand infinity only in terms of collection.

By using also your visual_spatial brain skills of the discussed subject, you are able to understand the power of infinity in terms of the irreducibility of higher dimensional spaces into lower dimensional spaces.

Kage · Apr 20, 2013

doronshadmi said:
By using verbal_symbolic AND visual_spatial brain skills it is not possible to define delta < AND > epsilon if an expression like "1-0.999..." is involved, and delta is defined according to finite indexes along "0.999...".

In other words, Kage's argument does not distinguish between the non-finite expression "1-0.999..." (or "0.000...1" (please look at http://www.internationalskeptics.com/forums/showpost.php?p=9161741&postcount=2203)) and infinitely many finite expressions that are derived from finite indexes.

As for "2" and "{2}" expressions:

"2 is a member of A(={2} or {2,...})" expression does not mean that this expression holds also if only "2" expression is used.

Some notes about this phrase:

"Visual difference in representation does not equate to mathematical difference in value."

It is true if visual difference is considered only as a representation of a given mathematical value.

It is not true if visual difference is also involved in order to define a given mathematical value, for example, let's examine this phrase:

"You have flatly stated in the past, Doron, that 1/4 and 0.25 were different numbers"

If one understands the place value method only as a representation of a given mathematical value, then "1/4" and "0.25" are nothing but two representations of the same (local) mathematical value.

This is not the case if place value method is considered as a given mathematical value, and not only Locality is considered, for example:

The vertical red line is 1/4. If one understands the place value method only as a representation of a given mathematical value, then 0.25₁₀ = 1/4 = 0.01₂.

If place value method is considered as a given mathematical value, and not only Locality is considered, then 0.25₁₀ ≠ 1/4 ≠ 0.01₂ as clearly seen in

[qimg]http://farm7.staticflickr.com/6123/5968116238_e17a4e6f90_b.jpg[/qimg]

I did not expend (yet) my framework in order to deal with these differences in details, but visual_spatial brain skills and non-locality are clearly demonstrated, in addition to the verbal_symbolic brain skills.

---------------------------------

In other worlds, Traditional Mathematics is based only on Locality and verbal_symbolic brain skills when it deals with the place value method, and there can't be any agreement with a notion that defines the place value method as mathematical values (and in this case also visual_spatial brain skills and non-locality must be involved).

This disagreement is shown all along this thread, but has no impact on a mathematical work that is done by using verbal_symbolic AND visual_spatial brain skills.

I didn't reply to this originally, but since you insist...

So you are saying because the graphs are labelled differently, then number is different? So, if I am driving my car 50 MPH, BUT the speedometer also indicates that I am going 80 KPH, what happens? Are they the same speeds, or different? How is my car going two speeds if these numbers are not references to the same thing?

Do the japanese have different numbers because they call one "ichi?"

What is a number?

Kage · Apr 20, 2013

doronshadmi said:
Only if you understand infinity only in terms of collection.

How do you understand infinity?

doronshadmi said:
By using also your visual_spatial brain skills of the discussed subject, you are able to understand understand the power of infinity in terms of the irreducibility of higher dimensional spaces into lower dimensional spaces.

Is this where you try to seduce me to the dark side of the force? Infinity is plenty powerful. I have yet to see one interesting or useful thing based on your misuse of infinity.

doronshadmi · Apr 20, 2013

Kage said:
And that is just it, dimension is an expression of the degrees of freedom.

Once again you get the concept of dimension only in terms of collections (degrees of freedom (finite or not) in this case).

Kage said:
You make these claims about spaces being "non-composed" or whatever, but they are assumptions and lead to conclusions that are either irrelevant or wrong.

This is indeed what you can get if you understand dimensional spaces > 0 only in terms of collections.

The question is this:

Can you make the needed paradigm-shift in your mind, in order to get a given dimensional space > 0 not only in terms of collection of lower dimensional spaces?

doronshadmi · Apr 20, 2013

Kage said:
What is a number?

A measurement tool that its structure is not necessarily ignored.

jsfisher · Apr 20, 2013

doronshadmi said:
A measurement tool that its structure is not ignored.

This does not parse.

doronshadmi · Apr 20, 2013

Kage said:
How do you understand infinity?

Infinity is the power of a given dimensional space > 0 w.r.t lower dimensional spaces on it, such that no amount of these dimensional spaces has the power of that dimensional space.

One of the main results of this notion, is the irreducibility of such dimensional space into a lower dimensional space (and again, do not get "dimensional space" only in terms of metric space).

doronshadmi · Apr 20, 2013

I hope that the phrase "This does not parse." is not the detailed reply to http://www.internationalskeptics.com/forums/showpost.php?p=9167315&postcount=2224 ( http://www.internationalskeptics.com/forums/showpost.php?p=9167235&postcount=2220 ) .

jsfisher · Apr 20, 2013

doronshadmi said:
I hope that the phrase "This does not parse." is not the detailed reply to http://www.internationalskeptics.com/forums/showpost.php?p=9167315&postcount=2224.

Tis' a real poser, isn't it? What could the comment, "this does not parse", possibly have referred to? Maybe we can find a clue in the full post:

jsfisher said:
doronshadmi said:

A measurement tool that its structure is not ignored.

Click to expand...

This does not parse.

Nope, no help there. If only I had included the text and pointer to the post I was replying to, then it might have been clearer.

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