1=0.999... infinities and box of chocolates..Phliosophy of Math...

Discussion in 'General Philosophy' started by Quantum Quack, Nov 2, 2013.

1. TachBannedBanned

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The notion of limit is not tied to infinities. <shrug>

3. Quantum QuackLife's a tease...Valued Senior Member

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Why did they develop calculus?
so why do they use limits when dealing with infinities?

5. Quantum QuackLife's a tease...Valued Senior Member

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try:
you can apologize now or wait a little bit ...
wiki:
http://en.wikipedia.org/wiki/Calculus

The word you are looking for is there some where and it starts with an S and ends with a Y and has the letters "ORR" in the middle.

But you can't say it can you nor can you write it?
May be you think you need to say it for my benefit, well ....you are wrong if you do

7. rpennerFully WiredValued Senior Member

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• You don't understand the real number system unless you understand the concept of continuity.
There are three ways to partition the rational numbers into two non-empty sets where all the numbers of the lower set are smaller than all the numbers of the higher set.
1. The lower set has a largest rational number and the upper set has no smallest rational number
2. The lower set has no largest rational number while the upper set has a smallest rational number
3. Neither the lower set has a largest rational number nor the upper set has smallest rational number
If you replace rational numbers with the real numbers this last case cannot happen, by definition. That is what is meant by continuity -- if you cut the real number line, you are cutting at the position of a real number. While an irrational number like $\sqrt{2}$ cuts the number line at a place that has no rational number, so a sequence of rational numbers can forever get closer to $\sqrt{2}$ without ever reaching it.​
• You don't understand the concept of continuity until you understand the concept of a least upper bound and a largest lower bound of an infinite but bounded set.
The sequence 1, 1/2, 1/6, 1/24, 1/120, ..., 1/n! ... goes on forever, eventually getting smaller than any positive number you can imagine. So do the sequences defined by 1/n, 1/n², 1/2ⁿ, 1/nⁿ, etc. Because in all these cases these sequences are bounded below by any non-negative number, it follows that that their greatest lower bound is 0, even though 0 is not in the set. You can explain this by taking the intersection of all upper partitions of rational number (or real numbers) where the elements of the sequence are a (proper) subset an in that way get tightest-fitting upper partition of rational numbers and see that 0 is the largest element of the lower partition. Thus 0 is the greatest lower bound of the elements of the sequence, a fact that has nothing to do with the way we order the elements of the sequence.​

Limits are a bit trickier. $a_1 = 1= \frac{p_1}{q_1} = \frac{1}{1}, \; a_{n+1} = \frac{ a_n + 2}{a_n + 1} = \frac{p_{n+1}}{q_{n+1}} = \frac{p_n + 2 q_n}{p_n + q_n}$ defines a sequence of rational numbers:
$1, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \frac{41}{29}, \frac{99}{70}, \dots$ that basically settle down into a neighborhood. But there is just one such number that no matter how far you go in the sequence all the higher terms are still in successively smaller neighborhoods of that number. That number is the limit of this sequence.
Now to make my point extra pretty, I picked a sequence where the odd members are always below that number and the even members are always above it so we can partition the elements of the sequence into a lower partition whose least upper bound equals the greatest lower bound of the upper partition and this is the limit of the sequence.
Lower partition: $1, \frac{7}{5}, \frac{41}{29}, \frac{239}{169}, \frac{1393}{985}, \dots$
Upper partition: $\frac{3}{2}, \frac{17}{12}, \frac{99}{70}, \frac{577}{408}, \frac{3363}{2378}, \dots$
So hopefully, it can be seen that the only number that can be the limit of this sequence is the positive solution to $\frac{x}{1} = \frac{x + 2}{x + 1}$ or $x^2 + x = x + 2$ or $x^2 = 2$, because if $p_n = \sqrt{2} q_n + \delta$ then $\frac{p_{n+1}}{q_{n+1}} = \frac{( 2 + \sqrt{2}) q_n + \delta}{(1 + \sqrt{2}) q_n + \delta} = \sqrt{2} + \frac{ (1 - \sqrt{2} ) \delta }{(1 + \sqrt{2} ) q_n + \delta} \approx \sqrt{2} + \frac{1 - \sqrt{2} }{1 + \sqrt{2} } \frac{\delta}{q_n}$ where the last approximation is justified when $\delta$ is small compared to $q_n$ and $\frac{ (1 - \sqrt{2} ) \delta }{(1 + \sqrt{2} ) q_n + \delta}$ is therefore a smaller (in magnitude) number than $\delta$.

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9. Quantum QuackLife's a tease...Valued Senior Member

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oh but I did.... but you wont accept that I did... and that's your problem not mine.

"To accommodate the reality of the infinite, mathematics had to develop calculus and a way of placing limits so that the infinite became workable and math gained an significant increase in utility because of it.
Sir Isaac Newton, a devoutly religious person, I believe, was instrumental in evolving mathematics ability to work in a way that was compatible with the divine "Infinitely Mono-Theistic Universe". Science then had a mathematical tool [Calculus] to allow them to cope with working on that which, Sir Isaac Newton must have, due to his strong religious beliefs considered as divine.
Science has been attempting to incorporate the divine [ infinity ] ever since in it's pursuit of understanding the universe [God]. "

Perhaps you may have missed this bit in your elementary introduction to calculus? Or perhaps you were not smart enough to work it out for your self at the time?
Perhaps both are true as you keep asking me to tell you what you should already know?
So what do you know Tach?

Just think Tach, every time you use calculus you use it to attempt to calculate God into a corner with in a circle... [chuckle] and as you know a circle has an infinite number of corners..

10. TachBannedBanned

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No, you didn't, you only compounded your problems by demonstrating that you don't know the difference between "infinity" (as in $\frac{1}{\infty}=0$) and "infinitesimals" (as in what calculus deals with). Besides, you made not one but several false claims I challenged you to backup and you didn't. So, your two choices have narrowed down to one: admit that you made up all the rubbish you posted, all by yourself.

11. Beer w/StrawTranscendental Ignorance!Valued Senior Member

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Come on!

That post deserves the cesspool for this thread.

12. Beer w/StrawTranscendental Ignorance!Valued Senior Member

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:EDIT:

Slow internet, pressed post before seeing Tach's response.

13. Quantum QuackLife's a tease...Valued Senior Member

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I answered by saying that the development of limits in calculus was because those that were responsible for the evolution of mathematics recognized that 1/infinity can never equal zero, which is why they developed the process of bounded sets, limits, etc. 1/infinity may infinitely converge towards zero but never actually get there for to do so contradicts the definition of infinity.

The mere existence of calculus is my evidence.
Now show other wise or be considered as a fringe troll.

14. TachBannedBanned

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Yet, all textbooks clearly show $\frac{1}{\infty}=0$ (more generally, they show $\frac{a}{\infty}=0$), so you made up the above all by yourself.

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seconded

16. Quantum QuackLife's a tease...Valued Senior Member

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why's that! Didn't you know that Sir Isaac Newton was a devout mono-theist?
And you have been promoting and using his work all this time... gosh!

Notes: Beer w/straw and Tach who seconded the motion, want to put Sir Isaac Newton in the cesspool.... [ chuckle ]

17. Beer w/StrawTranscendental Ignorance!Valued Senior Member

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Wow! When Tach accused you of making stuff up...

English is your first language right?

18. TachBannedBanned

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You got it wrong, not Newton.

19. Quantum QuackLife's a tease...Valued Senior Member

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"Sending stuff into the "cesspool" and into "Iggy land" doesn't change the reality nor the truth"... sorry...
I suppose you want to put Godel in the cesspool as well.. for his devout religious views?
Just think... contemporary math and physics was primarily founded by "bible bashing zealots".. gosh what a wake up call!

20. Quantum QuackLife's a tease...Valued Senior Member

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Enough fun hey?!
Seriously, didn't you guys know that the founding fathers of science and mathematics were devout religious people?
And that you are using their work all the time when you attempt to condemn those who have intuitive relationships with that which inspired those great men?
Albert Einstein,
wiki: http://en.wikipedia.org/wiki/Albert_Einstein#Political_and_religious_views
I personally am in accord with Albert Einsteins view...
What view are you in accord with?

21. someguy1Registered Senior Member

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Calculus has nothing whatsoever to do with infinitesimals.

The entire point of calculus (and its more formal uncle Real Analysis) is to eliminate consideration of infinitesimals, replacing them with rigorous definitions of the real numbers and limits.

22. Quantum QuackLife's a tease...Valued Senior Member

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and therefore allow mathematics and science generally to cope with the infinite in a way that they could make sense of.

23. Beer w/StrawTranscendental Ignorance!Valued Senior Member

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An infinite universe interferes with my idea of an infinite God, sorry.