Cont: Deeper than primes - Continuation 2

[doronshadmi]

But i didn't use any of those symbols. I just described what I see in your diagram. I would call that visual reasoning.

Once again, it is visual AND symbolic reasoning, do you understand this?

 

[doronshadmi]

Why? The first time wasn't enough?

Your first time in http://www.internationalskeptics.com/forums/showpost.php?p=11703103&postcount=2346 was unclear especially when you replied about a quote that I took from this link.

It turns out that I can open this link and copy short parts of it to this forum (as I did, and you replied to it) without braking ISF rule 4.

Unfortunately I understand now that other persons can't open it, because (as I have noticed now) it was blocked to anyone (except the composer) by the fellows of this site, which are, not surprisingly, traditional mathematicians that use symbolic-only reasoning.

 

[doronshadmi]

Because X√2 is the hypotenuse in a right-angled triangle with catheti X. In any triangle, the sum of two of the sides is greater than the third side.

Do you understand that 2X is a constant > X√2 in the considered diagram, no matter how many staircases of 2X there are in it?

 

[Hevneren]

But i didn't use any of those symbols. I just described what I see in your diagram. I would call that visual reasoning.

Once again, it is visual AND symbolic reasoning, do you understand this?

Of course I do. The point is that you criticized me for doing symbolic-only reasoning, when I hadn't mentioned any symbols at all.

Anyway, enough quibbling. You offered to explain things to me step by step, and I have confirmed that I understand step 1. I am now ready for step 2, if you are. I'm looking forward to it.

 

[doronshadmi]

Of course I do. The point is that you criticized me for doing symbolic-only reasoning, when I hadn't mentioned any symbols at all.

Please read what I wrote about visual-only reasoning in http://www.internationalskeptics.com/forums/showpost.php?p=11705101&postcount=2373.

Anyway, enough quibbling. You offered to explain things to me step by step, and I have confirmed that I understand step 1. I am now ready for step 2, if you are. I'm looking forward to it.

Before we continue to step 2, please answer to http://www.internationalskeptics.com/forums/showpost.php?p=11705534&postcount=2383.

 

[doronshadmi]

The three dots may be shorthand for d/2+d/2. After the d segment I can see two equal horizontal line segments, each of them d/2 long. Therefore, a+b+c+d+...=a+b+c+d+d/2+d/2=a+b+c+2d. The segments in combination cover the entire side of the square. Right? Here I have used a combination of visual AND symbolic reasoning, studying the drawing AND manipulating the symbols, so my answer should be OK.

Only if you do not distinguish between adding finitely many positive values and adding infinitely many positive values.

 

[Hevneren]

Do you understand that 2X is a constant > X√2 in the considered diagram, no matter how many staircases of 2X there are in it?

Yes.

 

[Hevneren]

Only if you do not distinguish between adding finitely many positive values and adding infinitely many positive values.

Oh, but I do!
Adding a, b, c, and 2d is adding finitely many positive values.
Taking the sum of an infinite series is adding infinitely many positive values, provided that all the terms in the series are positive.

 

[doronshadmi]

Yes.

Do you understand that the sides of the diagram are inseparable of staircases that have constant value 2X>X√2 ?

 

[doronshadmi]

Oh, but I do!
Adding a, b, c, and 2d is adding finitely many positive values.

X=a+b+c+2d (or 2X=2(a+b+c+2d)) is trivial, and we are no dealing with it in the considered diagram.

Taking the sum of an infinite series is adding infinitely many positive values, provided that all the terms in the series are positive.

Taking the sum of an infinite series is an illusion of symbolic-only reasoning.

Moreover, the term all in case of an infinite series is an illusion of symbolic-only reasoning.

 

[marplots]

Do you understand that the sides of the diagram are inseparable of staircases that have constant value 2X>X√2 ?

Only in the diagram. If we transition to formulas (away from an actual drawing and into the conceptual realm) we have to consider the case of X = 0.

 

[doronshadmi]

Only in the diagram.

We are talking exactly about the considered diagram by using visual AND symbolic reasoning, no less, no more.

If we transition to formulas (away from an actual drawing and into the conceptual realm) we have to consider the case of X = 0.

If you are using symbolic-only reasoning, which has nothing to do with generalization about the considered diagram.

 

[Hevneren]

X=a+b+c+2d (or 2X=2(a+b+c+2d)) is trivial, and we are no dealing with it in the considered diagram.

That's OK with me.

Taking the sum of an infinite series is an illusion of symbolic-only reasoning.

Here we disagree. Of course you can't add together the terms one by one because there are too many of them, but nevertheless the sum can exist. 0.9999... is a well-known example.

Moreover, the term all in case of an infinite series is an illusion of symbolic-only reasoning.

We can avoid the word "all". Let me rephrase what I said: Taking the sum of an infinite series is adding infinitely many positive values, provided that the series doesn't have any term that isn't greater than 0.

 

[Hevneren]

Do you understand that the sides of the diagram are inseparable of staircases that have constant value 2X>X√2 ?

If you mean that all the horizontal components of the steps will add up to X, and all the vertical components of the steps will add up to X, no matter how small you make the steps, then yes. X + X = 2X, and 2X > X√2. That never changes.

 

[doronshadmi]

Here we disagree. Of course you can't add together the terms one by one because there are too many of them, but nevertheless the sum can exist. 0.9999... is a well-known example.

This is another example of an illusion based on symbolic-only reasoning, 0.9999...₁₀ = 0.9₁₀ + 0.09₁₀ + 0.009₁₀ + ... does not have a fixed sum.

We can avoid the word "all". Let me rephrase what I said: Taking the sum of an infinite series is adding infinitely many positive values, provided that the series doesn't have any term that isn't greater than 0.

Adding infinitely many positive values does not have a fixed sum, this is an illusion of symbolic-only or visual-only reasoning.

 

[jsfisher]

Only in the diagram. If we transition to formulas (away from an actual drawing and into the conceptual realm) we have to consider the case of X = 0.

There is nothing in his diagram to preclude X = 0. Moreover, since he first asserted 2X > X√2 and then followed with the diagram as proof of his assertion, he also cannot claim that X cannot be negative, either.

Be that as it may, Doronshadmi still has made no progress showing the relationship between (and the inseparability of) his two inequality assertions.

 

[doronshadmi]

Hevneren, please answer to http://www.internationalskeptics.com/forums/showpost.php?p=11705733&postcount=2389.

 

[jsfisher]

Adding infinitely many positive values does not have a fixed sum

Doronshadmi, try as you may, you cannot disprove Mathematics by rejecting its definitions. Many infinite series have fixed sums, including the one implied by your oft-repeated diagram.

 

[doronshadmi]

Doronshadmi, try as you may, you cannot disprove Mathematics by rejecting its definitions. Many infinite series have fixed sums, including the one implied by your oft-repeated diagram.

jsfisher, try as you may, your symbolic-only reasoning is full of illusions, also in the case of the considered diagram.

 

[doronshadmi]

If you mean that all the horizontal components of the steps will add up to X, and all the vertical components of the steps will add up to X, no matter how small you make the steps, then yes. X + X = 2X, and 2X > X√2. That never changes.

This is exactly where your visual-only symbolic-only reasoning fails.

"all the horizontal components of the steps will add up to X, and all the vertical components of the steps will add up to X, no matter how small you make the steps" is no more than your illusion.

By using visual AND symbolic reasoning about the considered diagram, one immediately understands that 2X > X√2 exactly because there is no fixed amount of infinitely many staircases, and since a+b+c+d+... is inseparable of 2X>X√2, then X>a+b+c+d+... (or 2X>2(a+b+c+d+...)).

Another diagram that is based of visual AND symbolic reasoning, is given in http://www.internationalskeptics.com/forums/showpost.php?p=11702679&postcount=2343.