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February 22nd, 2016, 09:31 AM   #11
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OK. There's some deep philosophical sides to much of this, that I certainly don't have nailed down. However, regarding irrationals and non-terminating decimals, I would say that the representation is distinct from the number itself. In particular, you are unable to specify completely your representation of $\pi$ and so it is necessarily no better than a sequence of finite partial sums. In terms of the non-terminating decimal representation of the number $\frac13$, the decimal representation is precisely the sort of series we are describing. We assign the value $\frac13$ to the sequence $0.3333\ldots = \frac3{10}+ \frac3{100}+ \frac3{1000}+\ldots$. However, neither of these statements denies the existence of either $\pi$ nor $\frac13$. Just as we represent the original series by the number $1$ and I fully accept the existence of the number 1.

Again, for the Least Upper Bound property. I accept that completely, but that doesn't mean that an infinite sum has an intrinsic value. We can calculate a value based on the concept of limits, but just as the concept of limits is independent of the value of the expression at that limit, so the "value" of the infinite sum is independent of the limit of the sequence of finite sums. The value we assign is based on where the sequence of finite sums appears to be heading. Clearly, that value exists, otherwise it wouldn't be a value.

1+1=2 under most arithmetic systems that we use regularly. That is according to the axioms of those systems. Those systems can often be applied to real world situations, but that doesn't make such systems any more true than any other. They may very well be approximations depending on how they are applied. But more pertinent to this discussion is that 1+1 is a finite sum.

I think I'm making a mathematical point in highlighting that mathematics doesn't have a true value for infinite sums, but only values that we assign based on manipulations of finite sums. Whether you actually want to then take the philosophical step of then claiming that the infinite sum both exists and has the value we assign to it is a personal choice, but the mathematics doesn't have anything to say on that point.

There's a mild finitist position to what I'm saying here, although I wouldn't consider myself to be a finitist. In essence I am saying that for any infinite sequence of operations, the best we can do is to assign a value based on finite operations. This has nothing to say about any truly infinite sum or process. But I repeat that none of this says that any numbers do not exist or have no meaning.

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February 22nd, 2016, 11:18 AM   #12
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You've to ask yourself it there is a number between the value you find for $\displaystyle n=\infty$ and 1
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February 22nd, 2016, 12:18 PM   #13
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That would make sense if there were an infinite natural number. As it is, the value you find for $n=\infty$ is exactly the limit which is the 1 you speak of.
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February 22nd, 2016, 12:46 PM   #14
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That would make sense if there were an infinite natural number. As it is, the value you find for $n=\infty$ is exactly the limit which is the 1 you speak of.
I'm honestly not following your train of thought. You do know what a limit is. Why are you conversating like one of the forum cranks? You do know better, right? At least that's the impression I've had in the past. I have no doubt that if I challenged you to state the formal definition of the limit of a sequence of real numbers, you could do so. Yet you are pretending you haven't learned any basic real analysis. Why?

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February 22nd, 2016, 01:45 PM   #15
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Well, of course, that is what you believe
No, that part is what I know.

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So, no. I haven't "realized wrong", I simply prefer an interpretation which is more consistent with modern mathematics and physics, than an archaic one that isn't.
You can deny, but you have.

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To claim that the infinite sum $\frac12 + \frac14 + \frac18 + \ldots$ actually exists and has a value is equivalent to claiming that $2 + 4 + 8 + \ldots = -2$,
which is certainly a claim that makes me feel uncomfortable.
No, that doesn't follow at all. Makes you "feel uncomfortable!?"


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Why are you conversating like one of the forum cranks? You do know better, right?
And now you've achieved forum crank status.


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February 22nd, 2016, 01:50 PM   #16
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And now you've achieved forum crank status.
Was that intended for me? How so? There's no point arguing epsilonics with v8, he already knows all that stuff. My question to him is why he's acting like he doesn't.

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February 22nd, 2016, 04:32 PM   #17
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No, that part is what I know.
Yes, my wife "knows" that God exists too.
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No, that doesn't follow at all.
Ah, well I'm backing up my statements with reasoned mathematical thought as opposed to dogmatic denial. We'll see where we get to.
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February 22nd, 2016, 04:41 PM   #18
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$\displaystyle \dfrac12\lim_{n\to\infty}\dfrac{1-\left(\dfrac12\right)^n}{1-\dfrac12}=1$, so the series is assigned a value of $1$.
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February 22nd, 2016, 05:19 PM   #19
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Ah, well I'm backing up my statements with reasoned mathematical thought as opposed to dogmatic denial. We'll see where we get to.
You know, I would agree with you that there aren't necessarily convergent infinite series in the real world. I take math on its own terms, and apply only the criteria of logical consistency and interestingness. If the physicists or grocery clerks find math useful, that's no concern of mine. I'm with Hardy who says that the best math is useless math. [Of course history's joke is on him, as his belovedly useless number theory is now the heart of modern encryption technology].

So if you are saying that in real life the series 1/2 + 1/4 + ... does not actually converge, then I am in full agreement with you. If that's what you mean, then you're right.

But I don't understand your reasoning about the purely mathematical aspect. You agree that we define the sum of the above series as the limit of the sequence of partial sums 1/2, 3/4, 7/8, ...

Now the limit of a sequence is as you know defined in terms of two things: (1) A rigorous definition of the real numbers from the axioms of set theory; and (2) The epsilon-N definition of the limit of a sequence, which I'm pretty sure you're already familiar with so I won't go into the details.

But once you have the real numbers and the epsilon-N definition of the limit of a sequence, the definition of the limit of a series is built on that.

So which part of this purely mathematical set of ideas do you disagree with? In other words, tell me where you first become unhappy:

* The rules of standard predicate logic;

* The ZF axioms of set theory;

* The construction of the real numbers;

* The definition of the limit of a sequence of real numbers;

* The definition of the sum of an infinite series as the limit of the sequence of partial sums of the series.

Is that a fair question? I'm trying to determine exactly where you are rejecting standard math and claiming that a limit isn't a limit. After all a definition can not be right or wrong, it's just a definition.

If I tell you a foozle is a frisbee and I show you a foozle, then you know it's a frisbee, whether these words have meaning or not. Likewise if you believe in the axioms of set theory you are pretty much on a freight train to the end of the line, because it all follows logically.

On the other hand if you are arguing some flavor of constructivism, that is perfectly legitimate, but I do just wish you'd consider the mathematical universe you are thereby committed to. For example the constructive real line has more holes than points. That's why I don't like constructivism. It doesn't satisfy my intuition of a continuum.

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February 22nd, 2016, 06:56 PM   #20
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I'm honestly not following your train of thought... Yet you are pretending you haven't learned any basic real analysis. Why?
No pretending here. I'll try to explain.

I wrote an awfully long post that doesn't really say what I want it to. So I'll try with a shorter one.

The argument boils down to the semantic distinction between assigning a value to an infinite object and claiming that adding up each of the terms in that object produces that value.

Finite addition and infinite addition are different. They obey different rules. Finite addition is commutative and associative. Infinite addition is neither. It's not commutative because conditionally convergent series can change their value when you re-order the terms. And it's not associative because non-convergent series can be made to have different (and finite) values by grouping the terms in different ways ($1-1+1-1+\ldots \ne (1 - 1) + (1 - 1) + \ldots \ne 1 - (1 - 1) - (1 - 1) - \ldots$). Since they are different, we therefore need to: prove that all of the terms of an infinite sequence sum to a value; and prove that the value is what we say it is.

There are two basic ways of getting a value for an infinite sequence.

The first says that given a value $\epsilon \gt 0$ we can find a number $N$ such that any finite partial sums with more than $N$ terms will be within $\epsilon$ of the value we want the infinite sum to have. But all of these sums remain finite. The method simply assumes that the finite sums are a good guide to what an infinite sum is equal to. It also assumes that they are equal to something in the first place. We can easily create functions where $\lim \limits_{x \to a} f(x) \ne f(a)$ and even where at least one of the limit and the function are not defined. Why should these limits be any more reliable?

The second approach is to assign a value to the infinite series an then deduce what that value must be. Again, there is no proof that the infinite series equals anything.

For divergent series, the first approach is useless because it can't deliver a number. The second approach delivers highly counter-intuitive results such as $2+4+8+16+\ldots=-2$. Of course, the main reason such results are counter-intuitive is that our intuition is based on finite summation. There's no reason at all why that should be a good guide. But in order to have a consistent theory of infinite summation, we must assume that the series has a value.

But that value doesn't necessarily have to be what you get by adding all of the terms together. It may not make sense to decompose an infinite sum in that fashion anyway. For example the values of $\zeta(-2n-1)$ are equal to $\int \limits_{-1}^0 \sum \limits_{k=1}^n k^{2n+1} \, \mathrm d n$.
Yeah, you can put a finitist spin on this, but that's not where I'm coming from at all.

My point of view is, I believe, not possible to disprove unless you can prove that the value of an infinite sum is equal to the sum of each of the infinitely many terms - and the key to that is how you prove it for the divergent series. I don't think such a proof exists at the present time.
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