Cont: Deeper than primes - Continuation 2

[jsfisher]

You are missing my argument....

Not in the least.

Distilled to its basics, you are comparing (1/2 + 1/4 + 1/8 + ... + 2^-n) to (1/n + 1/n + ... + 1/n) [n times] for ever increasing values of n. No one (including you) is surprised the latter is 1 for all n. It is also 1 in the limit. Everyone (except you) understands how the former gets increasingly close to 1 for large n and is 1 in the limit.

Putting it all into a diagram with a diagonal doesn't suddenly make sqrt(2) relevant in any way...unless, of course, someone is mistaken about what your little staircase converges to. Then it might seem relevant, but still would not be.

 

[doronshadmi]

Not in the least.

Let's see:

Distilled to its basics, you are comparing (1/2 + 1/4 + 1/8 + ... + 2^-n) to (1/n + 1/n + ... + 1/n) [n times] for ever increasing values of n. No one (including you) is surprised the latter is 1 for all n. It is also 1 in the limit. Everyone (except you) understands how the former gets increasingly close to 1 for large n and is 1 in the limit.

Since we are dealing with a collection of endlessly added smaller values (where each value > 0 (the smallest value 0 is not one of the endlessly added smaller values)) this collection does not actually reach value 1.

Putting it all into a diagram with a diagonal doesn't suddenly make sqrt(2) relevant in any way...unless, of course, someone is mistaken about what your little staircase converges to. Then it might seem relevant, but still would not be.

Once again, the infinitely many staircases do not converge, and this fact is written as 2>√2 that is inseparable of the fact that 2>2(a+b+c+d+...), where (a+b+c+d+...) is definitely converges.

This inseparability is known only by actually using visual_spatial AND verbal_symbolic reasoning.

Moreover, by actually using visual_spatial AND verbal_symbolic reasoning, one knows that actual infinity is non-composed (can't be defined in terms collection, as seen in www.internationalskeptics.com/forums/showpost.php?p=12391433&postcount=3029).

"Putting it all into a diagram with a diagonal doesn't suddenly make sqrt(2) relevant in any way..." is exactly what one gets by using only verbal_symbolic reasoning, as clearly seen in your reply.

 

[jsfisher]

Not in the least.

Let's see:

Distilled to its basics, you are comparing (1/2 + 1/4 + 1/8 + ... + 2^-n) to (1/n + 1/n + ... + 1/n) [n times] for ever increasing values of n. No one (including you) is surprised the latter is 1 for all n. It is also 1 in the limit. Everyone (except you) understands how the former gets increasingly close to 1 for large n and is 1 in the limit.

Since we are dealing with a collection of endlessly added smaller values (where each value > 0 (the smallest value 0 is not one of the endlessly added smaller values)) this collection does not actually reach value 1.

See, now there you have several problems right out of the gate. Zooterkin is correct about your irregular usage of the term, collection, but I'm willing to step beyond that. However, I will note that neither (1/2 + 1/4 + 1/8 + ... + 2^-n) nor (1/n + 1/n + ... + 1/n) [n times] involve "endlessly added smaller values [sic]", as a collection or otherwise.

If you want to discuss limits, on the other hand, then you'll need to deal with actual, you know, limits. That means paying attention to how the term is defined and what constitutes a limit. Your "does not actually reach value 1" isn't that.

Putting it all into a diagram with a diagonal doesn't suddenly make sqrt(2) relevant in any way...unless, of course, someone is mistaken about what your little staircase converges to. Then it might seem relevant, but still would not be.

Once again, the infinitely many staircases do not converge

Of course it does (the total length of the staircase, that is). Are you unfamiliar with the meaning of converge in this mathematical sense? As has been your history, you seem to be arguing against something without understanding what that something actually is.

...and this fact is written as 2>√2

You alleged "fact" is that (1/n + 1/n + ... + 1/n) [n times] does not converge (as n approaches infinity). You inequality doesn't express that at all so to claim "this fact is written" that way is seriously deficient. Whereas it is true that 2 > √2 for all values of 2, it is equally true and equally relevant that 2 > 1.2312278.

..that is inseparable of the fact that 2>2(a+b+c+d+...), where (a+b+c+d+...) is definitely converges.

(A) You have not established any inseparability, nor any other relationship for that matter.
(B) It is not a fact that 2>2(a+b+c+d+...).
(C) "is definitely converges" does not parse in English.

This inseparability is known only by actually using visual_spatial AND verbal_symbolic reasoning.

What really is known that your cherished special reasoning techniques practiced only by you often lead to incorrect conclusions. Wrong remains wrong despite you looking at things differently.

 

[doronshadmi]

See, now there you have several problems right out of the gate. Zooterkin is correct about your irregular usage of the term, collection, but I'm willing to step beyond that.

www.internationalskeptics.com/forums/showpost.php?p=12405089&postcount=3039 clearly shows that Zooterkin does not have any useful thing to say about the considered subject.

Here is a quote taken from Wikipedia:

"A series is convergent if the sequence of its partial sums {S₁, S₂, S₃, …} tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number ℓ such that for any arbitrarily small positive number ε, there is a (sufficiently large) integer N such that for all n ≥ N, |S_n- ℓ| ≤ ε."

jsfisher, "tends to ...", "become closer and closer ...", "for any arbitrarily small positive number ε, there is a (sufficiently large) integer N such that for all n ≥ N, |Sn- ℓ| ≤ ε" are all equivalent to "a collection of endlessly added smaller values (where each value > 0 (the smallest value 0 is not one of the endlessly added smaller values)) this collection does not actually reach a given accurate value, called limit.

Of course it does (the total length of the staircase, that is). Are you unfamiliar with the meaning of converge in this mathematical sense? As has been your history, you seem to be arguing against something without understanding what that something actually is.

No, the staircases' length is permanent (it does not "tends to ...", "become closer and closer ...") to the diagonal length.

You alleged "fact" is that (1/n + 1/n + ... + 1/n) [n times] does not converge (as n approaches infinity).

It is finitely converges, so it is not relevant to the considered subject.

You inequality doesn't express that at all so to claim "this fact is written" that way is seriously deficient. Whereas it is true that 2 > √2 for all values of 2, it is equally true and equally relevant that 2 > 1.2312278.

You are still missing that 2>√2 and 2>2(a+b+c+d+...) are inseparable, if they are defined by using visual_spatial AND verbal_symbolic reasoning skills.

What really is known that your cherished special reasoning techniques practiced only by you often lead to incorrect conclusions. Wrong remains wrong despite you looking at things differently.

You can't claim that as long as you try to value my framework by using only your verbal_symbolic reasoning.

 

[jsfisher]

Here is a quote taken from Wikipedia:

"A series is convergent if the sequence of its partial sums {S₁, S₂, S₃, …} tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number ℓ such that for any arbitrarily small positive number ε, there is a (sufficiently large) integer N such that for all n ≥ N, |S_n- ℓ| ≤ ε."

jsfisher, "tends to ...", "become closer and closer ...", "for any arbitrarily small positive number ε, there is a (sufficiently large) integer N such that for all n ≥ N, |Sn- ℓ| ≤ ε"

Yes, in a rather choppy form, that's kinda sorta how limits are defined. And from that one can deduce that the summation of 2^-n where n varies from 1 to infinity does in fact equal 1.

...are all equivalent to "a collection of endlessly added smaller values (where each value > 0 (the smallest value 0 is not one of the endlessly added smaller values)) this collection does not actually reach a given accurate value, called limit.

That the sequence of partial sums (is that what you are referring to by 'collection'?) does or does not "actually reach" something isn't relevant. The infinite sum does "actually reach" 1.

Of course it does (the total length of the staircase, that is). Are you unfamiliar with the meaning of converge in this mathematical sense? As has been your history, you seem to be arguing against something without understanding what that something actually is.

No, the staircases' length is permanent (it does not "tends to ...", "become closer and closer ...") to the diagonal length.

To the diagonal length? Of course not. Why did you once again bring in the irrelevant length of the diagonal? The issue was whether you understood what mathematical convergence meant in this particular context. You have confirmed you do not.

You alleged "fact" is that (1/n + 1/n + ... + 1/n) [n times] does not converge (as n approaches infinity).

It is finitely converges, so it is not relevant to the considered subject.

"Finitely converges"? You are inventing more terminology again? Be that as it may, (1/n + 1/n + ... + 1/n) [n times] does converge as n approaches infinity, and you were wrong to state otherwise.

You inequality doesn't express that at all so to claim "this fact is written" that way is seriously deficient. Whereas it is true that 2 > √2 for all values of 2, it is equally true and equally relevant that 2 > 1.2312278.

You are still missing that 2>√2 and 2>2(a+b+c+d+...) are inseparable, if they are defined by using visual_spatial AND verbal_symbolic reasoning skills.

Your fictitious super powers of reasoning are just that. Wrong is a constant for you.

What really is known that your cherished special reasoning techniques practiced only by you often lead to incorrect conclusions. Wrong remains wrong despite you looking at things differently.

You can't claim that as long as you try to value my framework by using only your verbal_symbolic reasoning.

Another fiction on your part.

 

[doronshadmi]

Yes, in a rather choppy form, that's kinda sorta how limits are defined. And from that one can deduce that the summation of 2^-n where n varies from 1 to infinity does in fact equal 1.

This "fact" is based on verbal_symbolic-only reasoning, which is not sufficient in order to comprehend that 2>√2 and 2>2(a+b+c+d+...) are inseparable of each other, as defined in
http://www.internationalskeptics.com/forums/showpost.php?p=12404383&postcount=3030, by using visual_spatial AND verbal_symbolic reasoning.

That the sequence of partial sums (is that what you are referring to by 'collection'?) does or does not "actually reach" something isn't relevant. The infinite sum does "actually reach" 1.

It is mostly relevant that a collection of endlessly added smaller values (where each value > 0 (the smallest value 0 is not one of the endlessly added smaller values)) can't actually reach a given accurate value, called limit, exactly because collections can't be defined in terms of actual infinity, as addressed in www.internationalskeptics.com/forums/showpost.php?p=12391433&postcount=3029 by using visual_spatial AND verbal_symbolic reasoning.

Your fictitious super powers of reasoning are just that. Wrong is a constant for you.

No fictitious super powers of reasoning are involved here. One simply has to choose to use visual_spatial AND verbal_symbolic reasoning.

Anybody can choose to do that, including you, jsfisher.

 

[jsfisher]

No fictitious super powers of reasoning are involved here. One simply has to choose to use visual_spatial AND verbal_symbolic reasoning.

It is something you made up about yourself. It is fictitious. You use it as a smoke screen to avoid defining anything or clarifying what you actually mean or proving anything.

If you cannot express yourself in ways others can understand, it is your failing, not theirs.

 

[doronshadmi]

It is something you made up about yourself. It is fictitious. You use it as a smoke screen to avoid defining anything or clarifying what you actually mean or proving anything.

If you cannot express yourself in ways others can understand, it is your failing, not theirs.

I define anything in my framework by using visual_spatial AND verbal_symbolic reasoning.

This reasoning is naturally available to anyone, and all one needs is to use it.

No fictions, smoke screen etc. are involved.

By using visual_spatial AND verbal_symbolic reasoning, one defines that, non-composed objects (for example, non-composed circle, non-composed sides of a given square etc.) can't be defined in terms of collection of objects on them.

By this notion, given a non-composed object, it is defined as an actual-infinity with respect to any amount of infinitely many objects on it (the collection if infinitely many objects on it is defined in terms of potential infinity with respect to it).

This notion can't be known by using only visual_spatial reasoning or only verbal_symbolic reasoning.

 

[jsfisher]

I define anything in my framework by using visual_spatial AND verbal_symbolic reasoning.

No one believes you.

 

[doronshadmi]

No one believes you.

Using visual_spatial AND verbal_symbolic reasoning does not involve belief, as very simply addressed in http://www.internationalskeptics.com/forums/showpost.php?p=12391433&postcount=3029 and www.internationalskeptics.com/forums/showpost.php?p=12404383&postcount=3030.

More details can be seen in https://philosophy.stackexchange.co...here-both-potential-and-actual-infinity-are-u.

 

[jsfisher]

Using visual_spatial AND verbal_symbolic reasoning does not involve belief...

No, only gullibility.

 

[doronshadmi]

No, only gullibility.

Well, this is the best that one can get by using only visual_special or only verbal_symbolic reasoning in order to air his/her view about visual_special AND verbal_symbolic reasoning.

 

[jsfisher]

Well, this is the best that one can get by using only visual_special or only verbal_symbolic reasoning in order to air his/her view about visual_special AND verbal_symbolic reasoning.

As has been said many times before, if you want to comment on Mathematics, you need to understand the Mathematics you are commenting on, lest you appear to be an idiot. Your insistence that there is this special reasoning ability only you possess accomplishes nothing other than enhancing the appearance.

 

[doronshadmi]

Your insistence that there is this special reasoning ability only you possess ...

Using visual_spatial AND verbal_symbolic reasoning is definitely not my special ability that only I possess.

Anyone can use it in order to do Math.

As has been said many times before, if you want to comment on Mathematics, you need to understand the Mathematics you are commenting on, lest you appear to be an idiot.

It has been said many times before that verbal_symbolic only reasoning is not useful in order to deal with Mathematics that is based on visual_spatial AND verbal_symbolic reasoning.

Simple as that.

Again, here is a quote taken from Wikipedia:

"A series is convergent if the sequence of its partial sums {S₁, S₂, S₃, …} tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number ℓ such that for any arbitrarily small positive number ε, there is a (sufficiently large) integer N such that for all n ≥ N, |S_n- ℓ| ≤ ε."


In my example (S₁=1/2, S₂=1/2+1/4, S₃=1/2+1/4+1/8, …) where each partial sum is an accurate value.

1/2+1/4+1/8+… on the contrary does not have an accurate value exactly because a collection of endlessly added smaller values (where each value > 0 (the smallest value 0 is not one of the endlessly added smaller values)) can't actually reach a given accurate value, called limit, because collections are no more than potential infinity, as addressed in www.internationalskeptics.com/forums/showpost.php?p=12407181&postcount=3050.

https://en.wikipedia.org/wiki/Limit_of_a_sequence#Formal_definition is a definition that is based only on verbal_symbolic reasoning, which can't be used in order to distinguish between potential infinity and actual infinity (where again, actual infinity goes beyond the notion of collection (a collection is an aggregation of objects)).

 

[jsfisher]

Using visual_spatial AND verbal_symbolic reasoning is definitely not my special ability that only I possess.

True. It is an imaginary skill you claim, but, in fact, do not have.

You use it for proof-by-assertion. You imagine it, so it must be true.

However, as a proof technique, it does not work, and Mathematics does not redefine itself to match you fantasies.

 

[jsfisher]

here is a quote taken from Wikipedia:

"A series is convergent if the sequence of its partial sums {S₁, S₂, S₃, …} tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number ℓ such that for any arbitrarily small positive number ε, there is a (sufficiently large) integer N such that for all n ≥ N, |S_n- ℓ| ≤ ε."

Everything up to "More precisely" is figurative, meant to convey understanding without the exact details. Yet, you prefer to take it as the literal truth, misinterpret it, and ignore the important part:

a series converges, if there exists a number ℓ such that for any arbitrarily small positive number ε, there is a (sufficiently large) integer N such that for all n ≥ N, |S_n- ℓ| ≤ ε.​

The infinite summation of 2^-n for n = 1 to infinity satisfies the "more precisely" part just fine. It therefore converges.

https://en.wikipedia.org/wiki/Limit_of_a_sequence#Formal_definition is a definition that is based only on verbal_symbolic reasoning, which can't be used in order to distinguish between potential infinity and actual infinity (where again, actual infinity goes beyond the notion of collection (a collection is an aggregation of objects)).

No. You don't get to redefine parts of Mathematics you don't understand so you can reject them.

 

[doronshadmi]

True. It is an imaginary skill you claim, but, in fact, do not have.

You use it for proof-by-assertion. You imagine it, so it must be true.

However, as a proof technique, it does not work, and Mathematics does not redefine itself to match you fantasies.

It is a proof technique which relies on visual_spatial AND verbal_symbolic reasoning.

 

[doronshadmi]

... and ignore the important part:

a series converges, if there exists a number ℓ such that for any arbitrarily small positive number ε, there is a (sufficiently large) integer N such that for all n ≥ N, |S_n- ℓ| ≤ ε.​

The infinite summation of 2^-n for n = 1 to infinity satisfies the "more precisely" part just fine. It therefore converges.

I do not ignore this part, as clearly seen in

In my example (S₁=1/2, S₂=1/2+1/4, S₃=1/2+1/4+1/8, …) where each partial sum is an accurate value.

1/2+1/4+1/8+… on the contrary does not have an accurate value exactly because a collection of endlessly added smaller values (where each value > 0 (the smallest value 0 is not one of the endlessly added smaller values)) can't actually reach a given accurate value, called limit, because collections are no more than potential infinity, as addressed in www.internationalskeptics.com/forums/showpost.php?p=12407181&postcount=3050.

which is ignored by you, exactly because it does not fit your verbal_symbolic-only reasoning.

The infinite summation of 2^-n for n = 1 to infinity satisfies the "more precisely" part just fine. It therefore converges.

Since we are talking about summation of infinitely many things, it is no more than potential infinity that can't reach (it is inaccessible to) the non-composed, which is actual infinity, and potential infinity does not have an accurate summation.


Given endlessly added smaller values (where each value > 0 (the smallest value 0 is not one of the endlessly added smaller values)), they don't have an accurate value.

 

[doronshadmi]

No. You don't get to redefine parts of Mathematics you don't understand so you can reject them.

jsfisher, verbal_symbolic-only reasoning can't be used in order to criticize notions that relies on visual_spatial AND verbal_symbolic reasoning, about the considered subject.

On the other hand, visual_spatial AND verbal_symbolic reasoning enables to criticize notions that relies on verbal_symbolic-only reasoning, about the considered subject.

https://en.wikipedia.org/wiki/Limit_of_a_sequence#Formal_definition is a definition that is based only on verbal_symbolic reasoning, which can't be used in order to distinguish between potential infinity and actual infinity (where again, actual infinity goes beyond the notion of collection (a collection is an aggregation of objects)), as addressed in www.internationalskeptics.com/forums/showpost.php?p=12407181&postcount=3050.

 

[jsfisher]

It is a proof technique which relies on visual_spatial AND verbal_symbolic reasoning.

Only according to you.