Cont: Deeper than primes - Continuation 2

[jsfisher]

Thank you.

Doron, it appears that you are the one using symbolic-only reasoning. You seem to claim that since you can never write down all the a, b, c, d etc. symbols because there are infinitely many of them, the horizontal arrow in your illustration can never reach the right edge of the large square. But if you use visual reasoning, you see that it in fact does.

The terms a, b, c, d.. form an infinite geometric series, the sum of which is well known. It's X in this case.

So, using both visual AND symbolic reasoning we find that X=a+b+c+d+...

Hevneren,

Welcome to the ISF.

You may quickly come to realize that Doronshadmi has his own views about what Mathematics should be like. He has made seemingly odd statements like 2 is not a member of the set {2} and 1/4 and 0.25 are different numbers.

"Disproving" Cantor is a frequent theme.

He has an ample supply of diagrams he will cycle through. They apparently are very meaningful to Doronshadmi, and he will make sweeping statements about them, but he'll always fail to explain how they support this argument, just that they do.

As for the current diagram, it isn't hard to see what is going on. Doronshadmi has stumbled across a well-known paradox involving limits and the diagonal; without understanding the paradox, he believes it dismisses established conclusions about infinite series.

Getting Doronshadmi to explain his reasoning with any substance, however, will be impossible.

 

[doronshadmi]

As for the current diagram, it isn't hard to see what is going on. Doronshadmi has stumbled across a well-known paradox involving limits and the diagonal; without understanding the paradox, he believes it dismisses established conclusions about infinite series.

As for the given diagram, there is no paradox since (by using visual AND symbolic reasoning) 2X>X√2 AND X>a+b+c+d+... are inseparable of each other.

Actually what can we say to a traditional mathematician, which defines X=0 or X<0 for the following diagram?:

It is actually very clear that the used reasoning (symbolic-only reasoning) by this traditional mathematician has nothing to do with the issue at hand.

 

[doronshadmi]

and 1/4 and 0.25 are different numbers.

Well, ok, then.

0.01₂ happens to be 0.25 in the more common base-10.
0.01000000₂ also happens to be 0.25 in base-10.

...but according to Doronshadmi, the two are not the same number.

Doronshadmi has yet again demonstrated why his version of math is distinct from real Mathematics. The number 0.25 does not equal 0.25 for some values of 0.25.

http://www.internationalskeptics.com/forums/showpost.php?p=11398413&postcount=2065
You are never tiered on repeating on your symbolic-only reasoning, isn't it jsfisher?

http://www.internationalskeptics.com/forums/showpost.php?p=11398470&postcount=2068

Symbolic-only reasoning has nothing to do with real mathematics, as briefly explained in http://www.internationalskeptics.com/forums/showpost.php?p=11704098&postcount=2360.

...established conclusions about infinite series.

jsfisher, there is no such thing like well established symbolic-only reasoning because (among other failures of it) http://www.internationalskeptics.com/forums/showpost.php?p=11680343&postcount=2341.

As about the concept of Set http://www.internationalskeptics.com/forums/showpost.php?p=11363027&postcount=1948 is an example of visual AND symbolic reasoning about it, which can't be comprehended by using symbolic-only reasoning.

 

[doronshadmi]

Furthermore, also Fuzzy logic is addressed by visual AND symbolic reasoning, as given in http://www.internationalskeptics.com/forums/showpost.php?p=11363119&postcount=1953.

 

[jsfisher]

As for the given diagram, there is no paradox since (by using visual AND symbolic reasoning) 2X>X√2 AND X>a+b+c+d+... are inseparable of each other.

(a) To what paradox are you referring?
(b) How does your diagram establish that 2X > X√2?
(c) How does your diagram establish that X > a+b+c+d+...?
(d) In what way are (b) and (c) inseparable?
(e) Is 2 a member of the set {2}?

 

[doronshadmi]

(a) To what paradox are you referring?

There is no paradox so there is nothing to be referred to (as usual, you did not read http://math.stackexchange.com/quest...nation-about-a-convergent-series-is-requested).

(b) How does your diagram establish that 2X > X√2?

It is obvious since X>0 (or by your own words, it is "trivially true").

(c) How does your diagram establish that X > a+b+c+d+...?

2X > X√2 AND X > a+b+c+d+... are inseparable of each other.

(d) In what way are (b) and (c) inseparable?

By visual AND symbolic reasoning (clearly seen by both diagrams in http://www.internationalskeptics.com/forums/showpost.php?p=11702679&postcount=2343).

By using symbolic-only reasoning one can't comprehend it.

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

jsfisher, you actually ignore the rest of http://www.internationalskeptics.com/forums/showpost.php?p=11704131&postcount=2362, which clearly demonstrates that your questions are not asked by you in order to learn something.

(e) Is 2 a member of the set {2}?

There is difference between "2" (or "There is 2" and "2 (or "There is 2 which") is a member of {2}".

 

[Hevneren]

Getting Doronshadmi to explain his reasoning with any substance, however, will be impossible.

Thank you, jsfisher. But maybe it will be fun.

 

[jsfisher]

It is obvious since X>0 (or by your own words, it is "trivially true").

I didn't ask if it was trivially true; I asked how your diagram showed this trivial fact.

2X > X√2 and X > a+b+c+d+... are inseparable of each other.

Whether they are or are not, how does your diagram show that X > a+b+c+d+...?

By visual AND symbolic reasoning. By using symbolic-only reasoning one can't comprehend it.

So far, you have merely asserted it, but not shown they are inseparable.

There is difference between "2" and "2 is a member of {2}".

You dodged the question. Is 2 a member of the set {2}?

 

[phiwum]

Thank you, jsfisher. But maybe it will be fun.

At first.

 

[doronshadmi]

I didn't ask if it was trivially true; I asked how your diagram showed this trivial fact.



Whether they are or are not, how does your diagram show that X > a+b+c+d+...?



So far, you have merely asserted it, but not shown they are inseparable.

All is needed is already given in http://math.stackexchange.com/quest...nation-about-a-convergent-series-is-requested.

1) Once more don't read http://math.stackexchange.com/quest...nation-about-a-convergent-series-is-requested (jsfisher, if this site can't be viewed, then please just say it).

2) Since you are using symbolic-only reasoning, it is (probably) obvious that you can't comprehend that "the staircase line apparently converging to the diagonal without its length converging to the length of the diagonal" (that's it, 2X > X√2), and series a+b+c+d+... are inseparable of each other (that's it, X > a+b+c+d+... and no partial sums (a, a+b, a+b+c, ...) are involved here).

You dodged the question. Is 2 a member of the set {2}?

Only if one defines it as "There is 2, such that it is a member of {2} (or if you will, it is a member of S)".

You still do not comprehend that without such definition, nothing forces 2 to be a member of any set.

More generally, your symbolic-only reasoning can't comprehend the difference of being a set, and being a member of a given set.

In this particular case your symbolic-only reasoning can't comprehend the difference between {{},{{}}} (or {{{}}}) and {{{},{{}}}} (or {{{{}}}}).

 

[doronshadmi]

Getting Doronshadmi to explain his reasoning with any substance, however, will be impossible.

Thank you, jsfisher. But maybe it will be fun.

Well, the real fun begins if one enables to comprehend mathematics by visual AND symbolic reasoning, which is something that U 2 do not (yet) do.

 

[Hevneren]

Hi Hevneren, welcome.

Like jsfisher, you are using symbolic-only reasoning.

In my figure a+b+c+d+...clearly does not add up to X, exactly because of the fact that in this figure (by using visual AND symbolic reasoning) 2X>X√2 AND X>a+b+c+d+... are inseparable.

But I find it easy to separate them. They're just inequalities. I remember seeing the first one in school, but not the second one. Is your point that both of them are true in the context of your diagrams? I may have jumped to the conclusion in my previous post, when I said that a+b+c+d+... is the sum of an infinite geometric series. 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.

More details are given in (can't post links yet).

What details? That is the post I orginally responded to.

Also please look at (sorry, still can't post links yet).

The second link takes me to a list of academic papers, all of them apparently containing the words "visual and symbolic reasoning". Clicking any of the links on that page takes me to an abstract. I can't find any relevant information here. Maybe you were trying to convince me that "visual AND symbolic reasoning" is a valid concept? I'm sure it is.

 

[doronshadmi]

The segments in combination cover the entire side of the square. Right?

Only if (by this diagram) 2X=X√2, which is defiantly not the case.

Here I have used a combination of visual AND symbolic reasoning,

No, you did not used a combination of visual AND symbolic reasoning exactly because you separated the visual AND symbolic fact of 2X>X√2 from the visual AND symbolic fact of X>a+b+c+d+..., exactly as both of them visually AND symbolically are related to the considered diagram:

What details? That is the post I orginally responded to.

The details that are already found in http://math.stackexchange.com/quest...nation-about-a-convergent-series-is-requested.

If you can't open this site, please just say it.

I said that a+b+c+d+... is the sum of an infinite geometric series.

What you have said is based on symbolic-only reasoning, where partial sums like (a, a+b, a+b+c, ...) are involved.

By using visual AND symbolic reasoning as done by the considered diagram, no partial sums are involved.

The second link takes me to a list of academic papers, all of them apparently containing the words "visual and symbolic reasoning". Clicking any of the links on that page takes me to an abstract. I can't find any relevant information here. Maybe you were trying to convince me that "visual AND symbolic reasoning" is a valid concept? I'm sure it is.

Currently the second link of academic papers (https://scholar.google.co.il/schola...symbolic+reasoning"&hl=en&as_sdt=0,5&as_vis=1) unfortunately demonstrates how visual AND symbolic reasoning is not used (yet) as main mathematical research among the majority of mathematicians, and the "best" they currently do is researches about mathematical education, in order to understand how to translate visual AND symbolic reasoning into symbolic-only reasoning, which is obviously doomed to fail simply because visual AND symbolic reasoning can't be restricted into symbolic-only or visual-only reasoning.

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

Symbolic-only reasoning provides "la-la-land" mathematics, which most of it has nothing to do with real life.

Visual-only reasoning provides "la-la-land" mathematics that is doomed to fail by optical illusions, which most of it has nothing to do with real life.

Real mathematics is not less than visual AND symbolic reasoning, which actually enables to deal with real life.

I'll be glad to know about real mathematical works (found, for example, in https://scholar.google.co.il/schola...symbolic+reasoning"&hl=en&as_sdt=0,5&as_vis=1) that are essentially based on not less than visual AND symbolic reasoning as the foundation of mathematics.

 

[Hevneren]

... the visual AND symbolic fact of X>a+b+c+d+...

In what way is that a visual fact? Looking at the figure, it still looks like the segments cover the entire distance X. Where is the gap? Or should I look somewhere else in the diagram?

The details that are already found in http colon slash slash math dot stackexchange dot com/questions/1890108/some-explanation-about-a-convergent-series-is-requested.

If you can't open this site, please just say it.

Ok, I'll say it now. The link you just provided gives a "Page not found" error. But your previous link was to the post I originally responded to, and I couldn't find any details there.

What you have said is based on symbolic-only reasoning, where partial sums like (a, a+b, a+b+c, ...) are involved.

By using visual AND symbolic reasoning as done by the considered diagram, no partial sums are involved.

No, there are no partial sums involved. What I can see in your diagram is a straight line, that has been "cut up" by a number of divider lines. Nothing has been taken away from it. Still you seems to think there's a gap somewhere. And I fail too see how you can call this line of reasoning symbolic-only - there are no symbols involved. If i were to use symbols, I would end up with the sum of an infinite geometric series. The conclusion would be the same either way.

 

[doronshadmi]

In what way is that a visual fact?

You are still missing it, it is not less than visual AND symbolic fact.

Ok, I'll say it now. The link you just provided gives a "Page not found" error.

Next time please say it right at the beginning in order to avoid confusion.

But your previous link was to the post I originally responded to, and I couldn't find any details there.

Now you are missing what a wrote about link https://scholar.google.co.il/schola...symbolic+reasoning"&hl=en&as_sdt=0,5&as_vis=1 in my previous post.

No, there are no partial sums involved.

In that case you are not using the traditional mathematics of limits, do you aware about your approach here?

What I can see in your diagram is a straight line, that has been "cut up" by a number of divider lines. Nothing has been taken away from it. Still you seems to think there's a gap somewhere.

By using visual AND symbolic reasoning in the diagram below, since 2X > X√2, a+b+c+d+... can't be but < X (they are inseparable of each other).

And I fail too see how you can call this line of reasoning symbolic-only - there are no symbols involved.

X,2,√,>,+,a,b,c,d,... are definitely symbols.

If i were to use symbols, I would end up with the sum of an infinite geometric series. The conclusion would be the same either way.

This is again the result of using symbolic-only reasoning, about the following diagram, which is not less than visual AND symbolic reasoning:

You are still obviously unsuccessfully comprehend it by using symbolic-only reasoning.

 

[Hevneren]

You are still obviously unsuccessfully comprehend it by using symbolic-only reasoning.

Yes, I don't comprehend what you are trying to claim here. Would you care to explain? What do you actually mean by "visual AND symbolic reasoning", and how should I apply it to your diagrams? I have made a couple of attempts, but you claim that my conclusions are wrong. Why?

BTW, please try to keep your internet links straight. You have several times referred me to some URL. When I comment on what I find there, you reply with talking about some other link that I've never heard of. It's a bit hard to follow you if you change the subject and the links without warning.

 

[doronshadmi]

Yes, I don't comprehend what you are trying to claim here. Would you care to explain?

Let's do it step by step.

1) Please explain why 2X>X√2 in the following diagram (no matter how many staircases of 2X there are):

 

[Hevneren]

Let's do it step by step.

1) Please explain why 2X>X√2 in the following diagram (no matter how many staircases of 2X there are):

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.

 

[jsfisher]

jsfisher, if this site can't be viewed, then please just say it

Why? The first time wasn't enough?

 

[Hevneren]

What I can see in your diagram is a straight line, that has been "cut up" by a number of divider lines. Nothing has been taken away from it.

X,2,√,>,+,a,b,c,d,... are definitely symbols.

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