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Old 16th February 2017, 05:37 AM   #2481
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You claimed that any proof for 2X = 2(a + b + c + d +...) would be "based on symbolic-only reasoning". You were wrong.

Last edited by Hevneren; 16th February 2017 at 05:39 AM.
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Old 16th February 2017, 05:46 AM   #2482
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Originally Posted by Hevneren View Post
Evidence?
Clearly given by the diagrams if visual AND symbolic brain skills are used.

Originally Posted by Hevneren View Post
No, I don't care about your dichotomy. You're the one who keeps mentioning it. I only care about the results.
Results (mathematical or not) are comprehended by using our brain skills, and this is not my dichotomy, but it is exactly the dichotomy that is deeply involved in your brain, during your mathematical activity.
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Old 16th February 2017, 06:11 AM   #2483
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Originally Posted by MetalPig View Post
You assert that the difference is > 0, but offer no proof, or even evidence.
I offer you to use your visual AND symbolic brain skills abut the issue at hand.

Originally Posted by MetalPig View Post
Why is that difference not fixed? X has a specific value
In finite terms.
Originally Posted by MetalPig View Post
and the sum has a specific value. There really should be a specific difference.
Only in case of finite terms.

Originally Posted by MetalPig View Post
Symbolic tells me there is, and visual tells me there is. Why should that change if I use symbolic and visual?
Please simultaneously use your visual AND symbolic brain skills in order to understand the considered diagram, in order to provide the answer to your question.
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Old 16th February 2017, 06:13 AM   #2484
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Originally Posted by Hevneren View Post
You claimed that any proof for 2X = 2(a + b + c + d +...) would be "based on symbolic-only reasoning". You were wrong.
Evidence?
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Old 16th February 2017, 06:17 AM   #2485
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Originally Posted by doronshadmi View Post
Evidence?
The fridge evidence. That wasn't based on any symbols.
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Old 16th February 2017, 06:22 AM   #2486
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Originally Posted by MetalPig View Post
Why is that difference not fixed? X has a specific value and the sum has a specific value. There really should be a specific difference.
Doron probaby won't admit this, but he seems to think there's an ongoing, never-ending process of adding smaller and smaller line pieces, so the difference will get smaller for each small piece you add. I think this is his fundamental error. He can't see the X line as a whole.
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Old 16th February 2017, 06:30 AM   #2487
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Originally Posted by Hevneren View Post
The fridge evidence. That wasn't based on any symbols.
Your fridge "evidence" does not distinguish between infinite addition and finite addition.

Moreover, it artificially separates what actually have in the considered diagrams, exactly because it uses visual-only reasoning.

Furthermore, by writing "based on symbolic-only reasoning" (seen in the beginning of http://www.internationalskeptics.com...postcount=2471) I am talking about the traditional mathematical stuff about the issue at hand.

If you disagree with me then please provide traditional mathematical stuff about the considered diagrams that deduce them by using both visual AND symbolic reasoning.
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Old 16th February 2017, 06:46 AM   #2488
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Originally Posted by Hevneren View Post
Doron probaby won't admit this, but he seems to think there's an ongoing, never-ending process of adding smaller and smaller line pieces, so the difference will get smaller for each small piece you add. I think this is his fundamental error. He can't see the X line as a whole.
X line is a non-composed positive thing if taken on its own.

If it is measured by positive sub-things along it, then finitely many positive sub-things cover it, but do not have its own non-composed property.

If it is measured by ever smaller infinitely many positive sub-things along it, then infinitely many positive sub-things do not cover it AND do not have its own non-composed property.

Originally Posted by Hevneren View Post
He can't see the X line as a whole.
Your fundamental mistake is that you take X line as a collection of positive things AND you also do not distinguish between finite collections and infinite collections along the non-composed X line.

X line on its own is a positive non-composed thing, which is a notion that your collection-oriented reasoning of X, can't comprehend.

Also you avoid http://www.internationalskeptics.com...postcount=2487 and http://www.internationalskeptics.com...postcount=2466 and http://www.internationalskeptics.com...postcount=2467.

Moreover, you ignored http://www.internationalskeptics.com...postcount=2465 (and what is written there about Archimedean property (which is definitely not used in my framework)).
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Old 16th February 2017, 07:09 AM   #2489
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Originally Posted by doronshadmi View Post
Your fridge "evidence" does not distinguish between infinite addition and finite addition.

Moreover, it artificially separates what actually have in the considered diagrams, exactly because it uses visual-only reasoning.

Furthermore, by writing "based on symbolic-only reasoning" (seen in the beginning of http://www.internationalskeptics.com...postcount=2471) I am talking about the traditional mathematical stuff about the issue at hand.

If you disagree with me then please provide traditional mathematical stuff about the considered diagrams that deduce them by using both visual AND symbolic reasoning.
I could easiy do that, but I have already tried to explain this to you using visual, symbolic, AND combined lines of thought. You just keep rejecting whatever I say. It's your turn to prove your stuff now, not just repeating you false claims. You are totally out of line with mathematics here, so the burden of proof is upon you.
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Old 16th February 2017, 07:17 AM   #2490
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Originally Posted by doronshadmi View Post
X line is a non-composed positive thing if taken on its own.

If it is measured by positive sub-things along it, then finitely many positive sub-things cover it, but do not have its own non-composed property.

If it is measured by ever smaller infinitely many positive sub-things along it, then infinitely many positive sub-things do not cover it AND do not have its own non-composed property.


Your fundamental mistake is that you take X line as a collection of positive things AND you also do not distinguish between finite collections and infinite collections along the non-composed X line.
Rubbish. I take X as a line of some length, AND I know the sum of an infinite geometric series. Collections don't enter into it.

Originally Posted by doronshadmi View Post
X line on its own is a positive non-composed thing, which is a notion that your collection-oriented reasoning of X, can't comprehend.
It's in fact you that don't seem to get this.

Originally Posted by doronshadmi View Post
Also you avoid http://www.internationalskeptics.com...postcount=2487 and http://www.internationalskeptics.com...postcount=2466 and http://www.internationalskeptics.com...postcount=2467.

Moreover, you ignored http://www.internationalskeptics.com...postcount=2465 (and what is written there about Archimedean property (which is definitely not used in my framework)).
Link spamming to you own posts won't get you anywhere. You haven't explained or proved anything at all in your previous posts, so why link to them? If you have some evidence, please just provide it.
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Old 16th February 2017, 07:21 AM   #2491
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Originally Posted by Hevneren View Post
I could easiy do that
So do that, please provide traditional mathematical stuff about the considered diagrams that deduce them by using both visual AND symbolic reasoning.
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Old 16th February 2017, 07:30 AM   #2492
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Originally Posted by Hevneren View Post
Rubbish. I take X as a line of some length, AND I know the sum of an infinite geometric series. Collections don't enter into it.
Please show me an infinite geometric series that is not a collection of infinitely many things.

Originally Posted by Hevneren View Post
It's in fact you that don't seem to get this.
On the contrary, you actually have no clue what is actually X line as a non-composed thing of its own, which is independent of how we measure it (you still do not comprehend what is written at the beginning of http://www.internationalskeptics.com...postcount=2488).

EDIT: Moreover, please provide a traditional mathematical stuff which according to it X line is a non-composed thing.

Originally Posted by Hevneren View Post
Link spamming to you own posts won't get you anywhere. You haven't explained or proved anything at all in your previous posts, so why link to them? If you have some evidence, please just provide it.
You ignore the links in those links (which are not my works), exactly because they do not fit to your reasoning.
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Old 16th February 2017, 07:31 AM   #2493
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Originally Posted by doronshadmi View Post
So do that, please provide traditional mathematical stuff about the considered diagrams that deduce them by using both visual AND symbolic reasoning.
Sorry, but no It's your turn. **** or get off the pot.
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Old 16th February 2017, 07:34 AM   #2494
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Originally Posted by Hevneren View Post
Sorry, but no It's your turn. **** or get off the pot.
So, you can't do that.

Now please try http://www.internationalskeptics.com...postcount=2492.
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Old 16th February 2017, 07:58 AM   #2495
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Originally Posted by doronshadmi View Post
I offer you to use your visual AND symbolic brain skills abut the issue at hand.
Exactly. You offer no proof, or even evidence. That's what I said and youseem to confirm.


Originally Posted by doronshadmi View Post
In finite terms.
<...>
Only in case of finite terms.
Are you saying the sum of infinitely many numbers can not have a specific value?


Originally Posted by doronshadmi View Post
Please simultaneously use your visual AND symbolic brain skills in order to understand the considered diagram, in order to provide the answer to your question.
Why don't *you* give me the answer?
Symbolic tells me the sum exists, and visual tells me the sum exists. Why should that change if I use symbolic AND visual skils?


Take some length X. Make it into 2 halves. Call one of the halves a and put it aside. Take the remaining part and make it into two halves. Call one of the halves b and put it aside. Take the remaining part and make it into two halves. Call one of the halves c and put it aside.
Repeat infinitely many times (inventing new letters as you go). At no point in the process do I throw anything away, so the sum of the infinitely many bits I end up with must be X.

You can visualize the process or put it into symbols or both. The sum will always be X.
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Old 16th February 2017, 10:34 AM   #2496
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Originally Posted by MetalPig View Post
Exactly. You offer no proof, or even evidence. That's what I said and youseem to confirm.
Once again you do not use your visual AND symbolic brain skills in order to realize that in the considered diagram the fact that 2X>X√2 is inseparable of the fact that 2X>2(a+b+d+d+...). The diagram is the proof of this inseparability only if you are actually simultaneously using both your visual AND symbolic brain skills during its observation.

Originally Posted by MetalPig View Post
Are you saying the sum of infinitely many numbers can not have a specific value?
What I say is that there is no sum to the added positive values in the considered diagram.

Originally Posted by MetalPig View Post
Why don't *you* give me the answer?
Because the right answer about the considered diagram is deeply related of how *you* are actually simultaneously using both your visual AND symbolic brain skills during the considered diagram observation.

Originally Posted by MetalPig View Post
Symbolic tells me the sum exists, and visual tells me the sum exists. Why should that change if I use symbolic AND visual skils?
Because this is the whole point, *your* brain can't comprehend the diagram unless *you* simultaneously observe it by using *your* visual AND symbolic brain skills.

As long as *you* search for a string of symbols that is observed by *you* as a mathematical proof about the considered diagram, *you* are using *your* symbolic-only brain skills, which actually prevent the actual understanding of the considered diagram.

Originally Posted by MetalPig View Post
Take some length X. Make it into 2 halves. Call one of the halves a and put it aside. Take the remaining part and make it into two halves. Call one of the halves b and put it aside. Take the remaining part and make it into two halves. Call one of the halves c and put it aside.
Repeat infinitely many times (inventing new letters as you go). At no point in the process do I throw anything away, so the sum of the infinitely many bits I end up with must be X.

You can visualize the process or put it into symbols or both. The sum will always be X.
Look how *you* actually observe X separately of how it is actually visually AND symbolically defined in the following diagram:



FACT 1: No infinitely many staircases with constant value 2X are equal to X√2 (written as 2X>X√2).

FACT 2: a+b+c+d+... is the result of the intersections of the diagonal black lines on the peaks of the infinitely many staircases, with the 2X sides of the diagram.

FACT 3: The interactions along the 2X sides are inseparable of FACT 1.

FACT 4: By FACT 3, one realizes that no infinitely many 2(a+b+c+d+...) are equal to 2X ( written as 2X>2(a+b+c+d+...) ).

FACT 5: No infinitely many 2(a+b+c+d+...) have a fixed sum, exactly because 2X>X√2 AND 2X>2(a+b+c+d+...) are inseparable in the considered diagram.

FACT 6: FACTS 1 to 5 are known only if one actually simultaneously using both his\her visual AND symbolic brain skills during his\her observation of the considered diagram.
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Old 16th February 2017, 11:06 AM   #2497
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Exclamation

Originally Posted by doronshadmi View Post
FACT 1: No infinitely many staircases with constant value 2X are equal to X√2 (written as 2X>X√2).
True. Nobody disagrees.

Originally Posted by doronshadmi View Post
FACT 2: a+b+c+d+... is the result of the intersections of the diagonal black lines on the picks of the infinitely many staircases, with the 2X sides of the diagram.
OK.

Originally Posted by doronshadmi View Post
FACT 3: The interactions along the 2X sides are inseparable of FACT 1.
What do you mean by inseparable? This isn't clear.

Originally Posted by doronshadmi View Post
FACT 4: By FACT 3, one realizes that no infinitely many 2(a+b+c+d+...) are equal to 2X ( written as 2X>2(a+b+c+d+...) ).
Maybe "one" does, but then "one" is wrong. This has been expained to you many times.

Originally Posted by doronshadmi View Post
FACT 5: No infinitely many 2(a+b+c+d+...) have a fixed sum, exactly because 2X>X√2 AND 2X>2(a+b+c+d+...) are inseparable in the considered diagram.
False. The sum is 2X.

Originally Posted by doronshadmi View Post
FACT 6: FACTS 1 to 6 are known only if one actually simultaneously using both his\her visual AND symbolic brain skills during its observation of the considered diagram.
It doesn't matter what skills were used when the results are wrong.

I think you may have observed that the black diagonals approach length X√2 as the staircase steps grow smaller, and it may look like some paradox that the staircase has constant length 2X while it looks more and more like a straight line as the steps get smaller. Geometry is a fascinating subject, but your answers are still wrong. Now try to disprove the following (it should be easy if you are right):

S = 1/2 + 1/4 + 1/8 + 1/16 + ...
2S = 1 + 1/2 + 1/4 + 1/8 + 1/16 ...

2S-S= 1 + 1/2 + 1/4 + 1/8 + 1/16 ...
- 1/2 - 1/4 - 1/8 - 1/16 + ...

S=1

Yes, this is symbolic only, but since you master both kinds of reasoning so superbly, it can't be too difficult.
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Old 16th February 2017, 11:15 AM   #2498
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Originally Posted by Hevneren View Post
Yes, this is symbolic only, but since you master both kinds of reasoning so superbly, it can't be too difficult.
http://www.internationalskeptics.com...postcount=2496 simply evolved beyond the artificial limitations of *your* symbolic-only | visual-only dichotomy.

Originally Posted by Hevneren View Post
and it may look like some paradox that the staircase has constant length 2X while it looks more and more like a straight line as the steps get smaller.
There is no paradox of any kind here, and exactly because there is not paradox FACTS 1 to 6 hold.
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Old 16th February 2017, 11:21 AM   #2499
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Originally Posted by doronshadmi View Post
http://www.internationalskeptics.com...postcount=2496 simply evolved beyond the artificial limitations of *your* symbolic-only | visual-only dichotomy.
I guess it did. Any comments on my proof?

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Old 16th February 2017, 02:36 PM   #2500
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Originally Posted by Hevneren View Post
What do you mean by inseparable? This isn't clear.
It is clear as a noon sun.

No infinitely many staircases are "settled down" into X√2, and since 2(a+b+c+d+...) is the result of the intersections of the diagonal black lines on the peaks of these infinitely many staircases with the 2X sides of the diagram, 2(a+b+c+d+...) can't be "added up" into 2X.
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Old 16th February 2017, 02:48 PM   #2501
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Originally Posted by Hevneren View Post
Any comments on my proof?
Originally Posted by Hevneren View Post
S = 1/2 + 1/4 + 1/8 + 1/16 + ...
2S = 1 + 1/2 + 1/4 + 1/8 + 1/16 ...

2S-S= 1 + 1/2 + 1/4 + 1/8 + 1/16 ...
- 1/2 - 1/4 - 1/8 - 1/16 + ...

S=1
Exactly as MetalPig, you observe it separately of the considered diagram, and this is the bast of what your symbolic-only | visual-only dichotomous reasoning actually achieves.

So the answer is: you have no proof whatsoever.

Originally Posted by Hevneren View Post
I just wanted to stir the pot a bit myself, ...
Still in deep sleep.

Originally Posted by Hevneren View Post
What about my fridge?
Still in deep freeze.
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Old 16th February 2017, 05:32 PM   #2502
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Originally Posted by doronshadmi View Post
Exactly as MetalPig, you observe it separately of the considered diagram, and this is the bast of what your symbolic-only | visual-only dichotomous reasoning actually achieves.
That doesn't make it wrong. The proof stands. Mathematical rigor does not yield to your misinterpretation of simple diagrams nor to your self-proclaimed superior reasoning methods.

The mathematical fact that 1/2 + 1/4 + 1/8 + ... is 1 follows directly from definitions; your insistence otherwise has no impact.

Mathematics doesn't care if you don't understand.
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Old 17th February 2017, 12:14 AM   #2503
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Originally Posted by doronshadmi View Post
No infinitely many staircases are "settled down" into X√2,
Nobody disputes this. 2 > √2 , we know.

Originally Posted by doronshadmi View Post
and since 2(a+b+c+d+...) is the result of the intersections of the diagonal black lines on the peaks of these infinitely many staircases with the 2X sides of the diagram, 2(a+b+c+d+...) can't be "added up" into 2X.
Maybe you can't add them up, but it can be proven symbolically, and it's visually obvious from your diagram that they add up to 2X.

Somehow doing it "symbolically and visually" prevents you from getting to the correct answer. Sad.
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Old 17th February 2017, 04:57 AM   #2504
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jsfisher, MetalPig and Hevneren,

http://www.internationalskeptics.com...postcount=2500 is exactly the right result about the considered diagram.

No reasoning that is based on the artificial dichotomy between *your* natural abilities to combine *your* visual AND symbolic brain skills, is going to change that right result.

Moreover, ignoring http://www.internationalskeptics.com...postcount=2466, http://www.internationalskeptics.com...postcount=2467 and http://www.internationalskeptics.com...postcount=2465 (including the rejection of the Archimedean property) actually put you in an obsolete state with respect to the actual fundamental developments of Maths that are deeply rooted in researches of the human brain.

Now days mathematics is an aggregation of frameworks that even contradict each other, exactly because of the lack of Unity consciousness among human beings.

Mathematics, together with more scientifically developed methods, is going to play a main factor in developing Unity consciousness among human beings, where drafts in this direction are seen in http://www.internationalskeptics.com...postcount=2342 and http://www.internationalskeptics.com...postcount=2438.
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Old 17th February 2017, 06:02 AM   #2505
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Oh no, I'm obsolete. Still not looking forward to the day when all humans have Doron's consciousness
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Old 17th February 2017, 09:24 AM   #2506
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Originally Posted by doronshadmi View Post
jsfisher, MetalPig and Hevneren,

http://www.internationalskeptics.com...postcount=2500 is exactly the right result about the considered diagram.
And yet, it isn't. Perhaps if you stopped pretending your diagrams showed what you want to be true and stopped fighting basic definitions in Mathematics, you might actually figure out your diagrams prove nothing while suggesting the opposite of your baseless beliefs.
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Old 17th February 2017, 10:43 AM   #2507
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Originally Posted by doronshadmi View Post
Please show me an infinite geometric series that is not a collection of infinitely many things.
Still stuck with your insistence on enumerating every member of the sequence in order to arrive at the sum?
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Old 17th February 2017, 11:33 AM   #2508
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Originally Posted by zooterkin View Post
Still stuck with your insistence on enumerating every member of the sequence in order to arrive at the sum?
Traditional Mathematics can't deal with the simple fact that a series of infinitely many ever smaller positive values (as given in the considered diagram) does not have a fixed sum (where in the considered diagram this fact is notated as 2X>2(a+b+d+...)).
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Old 17th February 2017, 11:48 AM   #2509
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Originally Posted by jsfisher View Post
Perhaps if you stopped pretending your diagrams showed what you want to be true and stopped fighting basic definitions in Mathematics,
Perhaps if you stopped pretending your symbolic-only reasoning showed what you want to be true and stopped forcing it as if it is really basic for actual definitions in Mathematics, you might actually figure out my diagrams prove that 2X>X√2 AND 2X>2(a+b+c+d+...) are inseparable in the considered diagram, despite of your baseless symbolic-only beliefs.

The near future of real mathematics is inseparable of human brain research (as briefly given in http://www.internationalskeptics.com...postcount=2504) and the tradition of symbolic-only reasoning is going to vanish.
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Old 17th February 2017, 11:59 AM   #2510
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Originally Posted by doronshadmi View Post
Traditional Mathematics I can't deal with the simple fact that a series of infinitely many ever smaller positive values (as given in the considered diagram) does not in some cases has a fixed sum (where in the considered diagram this fact is notated as 2X>=2(a+b+d+...)).
FTFY, you're welcome
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Old 17th February 2017, 12:01 PM   #2511
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Originally Posted by doronshadmi View Post
The near future of real mathematics is inseparable of human brain research (as briefly given in http://www.internationalskeptics.com...postcount=2504) and the tradition of symbolic-only reasoning is going to vanish.
So this is the dawning of the age of doronetics? We're certainly living in interesting times.
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Old 17th February 2017, 01:08 PM   #2512
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Originally Posted by Hevneren View Post
So this is the dawning of the age of doronetics?
No, this is the dawn of the age of the science of human consciousness development into Unity consciousness, and mathematics is going to play a main role in this development.
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Old 17th February 2017, 01:31 PM   #2513
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Originally Posted by doronshadmi View Post
...your symbolic-only reasoning...
Is this supposed to be meaningful in some way?

Logical reasoning follows a set of rules to draw its conclusions. Got anything like that?
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Old 17th February 2017, 03:50 PM   #2514
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Originally Posted by jsfisher View Post
Logical reasoning follows a set of rules to draw its conclusions. Got anything like that?
Logical reasoning follows a set of rules based on both visual AND symbolic brain skills in order to draw its conclusions. Got anything like that (for example http://www.internationalskeptics.com...stcount=2504)?
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Old 17th February 2017, 04:01 PM   #2515
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Originally Posted by Hevneren View Post
FTFY, you're welcome
Typical result of dichotomous visual | symbolic brain skills that is going to vanish simply because real mathematics is done by naturally using both visual AND symbolic brain skills.

A concrete example about the fundamental concept of set, which is based on the natural activity of visual AND symbolic brain skills, is already given in http://www.internationalskeptics.com...postcount=2438.
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Old 17th February 2017, 08:01 PM   #2516
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Originally Posted by doronshadmi View Post
Logical reasoning follows a set of rules
This part is correct.

Quote:
based on both visual AND symbolic brain skills in order to draw its conclusions.
This part is incorrect. It does nothing but get us back to the direct perception nonsense you tried to peddle once before.


Be that as it may, if you do have a set of rules for reasoning (based on whatever you imagine it to be based on), please, tell us all about these rules. I'm sure they will be fascinating.
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Old 18th February 2017, 02:10 AM   #2517
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Originally Posted by jsfisher View Post
This part is incorrect. It does nothing but get us back to the direct perception nonsense you tried to peddle once before.


Be that as it may, if you do have a set of rules for reasoning (based on whatever you imagine it to be based on), please, tell us all about these rules. I'm sure they will be fascinating.
Since you simply unable to realize that your responses are based on your visual AND symbolic brain skills, lets make it more easy for you.

The axiom of mathematics:

Given X, no sub-X is X.

In order to really understand this string of symbols both your visual AND symbolic brain skills are simultaneously used, whether you aware of it, or not.

So the useful state of mind is to be aware of it during your mathematical work, and this awareness is the mathematical basis for the development of Unity consciousness among human beings.

What is already given in http://www.internationalskeptics.com...stcount=2504)? is done by being aware of it during mathematical work, which in turn becomes a useful tool for the development of Unity consciousness among human beings.
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Old 18th February 2017, 07:46 AM   #2518
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Originally Posted by doronshadmi View Post
The axiom of mathematics:

Given X, no sub-X is X.
This is not an axiom of Mathematics.

Quote:
In order to really understand this string of symbols both your visual AND symbolic brain skills are simultaneously used, whether you aware of it, or not.
This is a bare assertion. You do a lot of assertion, but never go beyond that.


Be that as it may, what, I ask again, are your rules for your version of reasoning? They do not correspond to those of mathematical logic, so it would be helpful those playing along at home trying to follow your line of reasoning if they knew what your rules were.
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Old 18th February 2017, 09:46 AM   #2519
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Originally Posted by jsfisher View Post
This is not an axiom of Mathematics.
I agree with you, because this is The axiom of Mathematics:

Given X, no sub-X is X.

No assertion of any kind, and it is extremely useful by naturally using both our visual AND symbolic brain skills, as already given in http://www.internationalskeptics.com...stcount=2504)?

From now on I will reply to you by using The axiom of Mathematics, any attempt from you to ignore it will be a clear sign that you do not really wish to discuss with me about Mathematics and its further development.

Now about this axiom, it simply says that that no expressions that may emerge from X, are X, simply because X is non-composed by definition that is known both visually AND symbolically.
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Old 18th February 2017, 10:38 AM   #2520
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Originally Posted by doronshadmi View Post
I agree with you, because this is The axiom of Mathematics:

Given X, no sub-X is X.
You are re-asserting the same falsehood. Got anything to show otherwise?

Quote:
No assertion of any kind
It certainly was an assertion. That is a simple truth by the very nature of your statement. It question was was it a bare assertion. I'll leave it to you to divine the difference.

Quote:
From now on I will reply to you by using The axiom of Mathematics, any attempt from you to ignore it will be a clear sign that you do not really wish to discuss with me about Mathematics and its further development.
You may leap to whatever conclusions you like; it is only when you attempt to pass them off as fact that you run into trouble.

Be that as it may, you seem more concerned with the past developments in Mathematics, and that misinterpreting some drawing disproves well-established things.
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