Cont: Deeper than primes - Continuation 2

[doronshadmi]

You continue to argue to disprove definitions. Definitions are not subject to proof. They are, after all, definitions.

So, complain all you like. Mathematics doesn't care you don't understand.

Definitions can be wrong, insufficient etc., since they are defined by human beings at the level of thoughts, and so is the case about any aspect of Mathematics.

Once Mathematics becomes some kind of religion, quotes like yours above, are written.

And it is well known that religious' dogmas (which is actually the attempt to fix (freeze) thoughts) can't be criticized by their believers.

 

[jsfisher]

Definitions can be wrong, insufficient etc., since they are defined by human beings, and so is the case about any aspect of Mathematics.

Insufficient for some purpose, sure. Wrong, no. Moreover, if you wish to displace them with doron-sanctioned versions, your definitions would apply to only the doron branch of Mathematics. The already established stuff won't change on your disapproval alone.

 

[doronshadmi]

Wrong, no.

The definition of convergence does not prove that, for example, 0.999...₁₀ is identical to 1.

The best it can do in case of the sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) is to define 1 as a limit of it (which is trivial, since no term of this sequence > 1).

So you are wrong each time you are claiming that by this definition one enables to deduce that 0.999...₁₀ is identical to 1.

Since "0.999...₁₀ is identical to 1" became your dogma, you simply can't criticize it.

For example, you do not understand Katz & Katz quote in http://www.internationalskeptics.com/forums/showpost.php?p=10667254&postcount=614 exactly because you do not understand that you can't construct the real number system without transfinite cardinality |R|.

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

Slings and Arrows, thank you for providing the links to Katz works, it helps to clarify the danger of Dogmatism about mathematical development.

 

[jsfisher]

The definition of convergence does not prove that, for example, 0.999...₁₀ is identical to 1.

Alone? No, of course not. The meaning of limit is needed to make the final connection.

It's really 'limit' that's giving you all this trouble, doron. Mathematics offers a very concise, precise, and appealing meaning for limits, even limits like the following:

L = limit(x -> infinity) F(x)​

I won't repeat the meaning, here. It is easily found on the web. What would you, doronshadmi, have us use for such a limit?

In the past, you have been rather unsuccessful defining anything. You've told us what something will be like, what it's related to, but never actually what it is. Perhaps you can do better this time.

What's your precise definition for this type of limit where F() has the reals for its domain and co-domain/range?

 

[jsfisher]

...The best it can do in case of the sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) is to define 1 as a limit of it...

Neither I nor any other mathematically capable person has ever defined 1 as the limit of that sequence. Please, if you have a point to make, be so kind as to omit the wild fabrications.

 

[doronshadmi]

Neither I nor any other mathematically capable person has ever defined 1 as the limit of that sequence.

What is the limit of the sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) according to the definition of convergence, and if it is 0.999...₁₀, than how exactly 0.999...₁₀ is identical to 1?

http://en.wikipedia.org/wiki/Limit_of_a_sequence#Examples may help you (where the example there is given by (0.3₁₀, 0.33₁₀, 0.333₁₀, ...), 0.333...₁₀ and 1/3).

 

[doronshadmi]

Mathematics offers a very concise, precise, and appealing meaning for limits, even limits like the following:

L = limit(x -> infinity) F(x)​

I won't repeat the meaning, here. It is easily found on the web.

Please provide some link on the web that appealing meaning for limits, and then please explain it in details.

As long as you do not explain it in deitals, all you have is a string of notations without notions.

(You may use also http://en.wikipedia.org/wiki/Karl_Weierstrass#Soundness_of_calculus).

So, the stage is yours, I am waiting.

 

[jsfisher]

Please provide some link....

No. Your constant attempts to shift the burden away from yourself is tiring. You are the one objecting to how 'limit' is defined. You should be the one well-versed in its meaning if your objections are to be taken at all seriously.

If you don't know how 'limit' is defined in Mathematics, just say so.

Meanwhile, since you claim to have a better idea for 'limit', it falls to you to present what that better idea is. So, again, how would doronshadmi define limit as expressed in

L = limit(x -> infinity) F(x)​

where F() is a function with domain and co-domain/range in the real numbers?

 

[jsfisher]

Neither I nor any other mathematically capable person has ever defined 1 as the limit of that sequence.

What is the limit of the sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) according to the definition of convergence, and if it is 0.999...₁₀, than how exactly 0.999...₁₀ is identical to 1?

You completely disregard what I wrote. I denied your claim that I have ever defined the limit of the sequence {0.9, 0.99, 0.999, ...) to be 1. I have not.

Instead of fessing up to your error, you attempt to sidestep it with this non sequitur.

 

[doronshadmi]

No.

In that case you are unable to support your claims, and all you have is a string of notations without notions.

Your evasion about this subject has been noted.

 

[doronshadmi]

I denied your claim that I have ever defined the limit of the sequence {0.9, 0.99, 0.999, ...) to be 1. I have not.

{0.9, 0.99, 0.999, ...) is gibberish, because you used have used "{" and no base is used.

 

[jsfisher]

{0.9, 0.99, 0.999, ...) is gibberish, because you used have used "{" and no base is used.

That's it? Your argument devolves to ridicule of typographic errors?

 

[doronshadmi]

If you don't know how 'limit' is defined in Mathematics, just say so.

If you don't now how limits a defined in http://www.internationalskeptics.com/forums/showpost.php?p=10666379&postcount=612 (including all of its links), just say so.

Meanwhile, since you claim to have a better idea for 'limit', it falls to you to present what that better idea is.

Done in http://www.internationalskeptics.com/forums/showpost.php?p=10666379&postcount=612 (including all of its links).

Your system is some particular case (with a loss of generality) of my system.

 

[jsfisher]

In that case you are unable to support your claims.

My claim was that the definition for limit was readily available on the web. It is not like that sort of claim needs much support, nor would I have expected it to have fallen into dispute. But for you, I will retract the claim and replace it by, "the definition for limit in Mathematics is readily available on the web to all except doronshadmi."

Nevertheless, it was and always has been your claim that was at issue, here. But that's ok. No one here actually believed you could provide an actual definition for limit.

 

[jsfisher]

If you don't now how limits a defined in http://www.internationalskeptics.com/forums/showpost.php?p=10666379&postcount=612 (including all of its links), just say so.

No where in that post, nor any of its links, nor any of the links found in those links, and so on, do you define limit. Given any value, L, and any function, F(), your definition should let us conclude whether L is the limit of F(x) as x goes to infinity.

Got anything like that?

 

[doronshadmi]

My claim was that the definition for limit was readily available on the web. It is not like that sort of claim needs much support, nor would I have expected it to have fallen into dispute. But for you, I will retract the claim and replace it by, "the definition for limit in Mathematics is readily available on the web to all except doronshadmi."

You are the one objecting to how 'limit' is defined. You should be the one well-versed in its meaning if your objections are to be taken at all seriously.

Done in http://www.internationalskeptics.com/forums/showpost.php?p=10666379&postcount=612 including all of its links.

Once again, your evasion about this subject has been noted.

Nevertheless, it was and always has been your claim that was at issue, here. But that's ok. No one here actually believed you could provide an actual definition for limit.

Your religious dogmatic approach about Mathematics airs its view once again by using phrases like "No one here actually believed ...".

Belief has nothing to do with mathematical development.

 

[doronshadmi]

No where in that post, nor any of its links, nor any of the links found in those links, and so on, do you define limit. Given any value, L, and any function, F(), your definition should let us conclude whether L is the limit of F(x) as x goes to infinity.

Got anything like that?

Done in http://www.internationalskeptics.com/forums/showpost.php?p=10666379&postcount=612 (including all of its links).

Once again, your system is some particular case (with a loss of generality) of my system.

 

[zooterkin]

Your religious dogmatic approach about Mathematics airs its view once again by using phrases like "No one here actually believed ...".

Belief has nothing to do with mathematical development.

You might have a point if that's what the sentence you quoted was talking about. Actually, it was assessing your capabilities, and the fact that you appear to have missed even that tends to support the assessment.

 

[zooterkin]

Done in http://www.internationalskeptics.com/forums/showpost.php?p=10666379&postcount=612 (including all of its links).

You'll be able to quote the specific part where you gave that definition, then?

 

[jsfisher]

Done in http://www.internationalskeptics.com/forums/showpost.php?p=10666379&postcount=612 including all of its links.

Ok, I don't see it, but let's tentatively assume it is there. Please show us how you'd apply that definition to the limit of:

F(x) = 1 + 1/x², when x is not an integer, and
F(x) = 1 + 1/x, otherwise.​

as x goes to infinty.

Your religious dogmatic approach about Mathematics airs its view once again by using phrases like "No one here actually believed ...".

Belief has nothing to do with mathematical development.

You might want to re-read the text you quoted. Your response bears no resemblance to the point the text was making.