A limit is by definition something that will not be exceeded. We can be absolutely sure that 1/1 + 1/2 + 1/4 + 1/8 ... will never add up to infinity, because the limit of the partial sums is 2, which means it will never ever ever add up to more than 2, and only "at infinity" will even add all the way up to 2. — Pfhorrest

No. Calculus is formulated in terms of finite sums and limits. You can't jump to infinity and expect the rules of finite arithmetic to apply. Jumping from the finite to the infinite is an infinite distance and we don't know what happens there.

It is not. For any decimal fraction, the limit of both the corresponding sequence and the corresponding series is always finite, which is obvious enough: it is always between 0 and 1, there can be no infinity here. — Andrey Stolyarov

You need to read back a few pages to see what I'm saying. It is like this-
It is being asserted that 1/10 + 1/100 + ...taken to an infinity of terms is 1.
If we take a finite number of terms they converge to a limit. But a limit is not a sum. It is what a finite sum converges to. We can't know what an infinite sum is. It may well be 1 but it may also be infinity. We don't know because you can't apply finite arithmetic to infinity. You can't jump to an infinite sum and assume it is 1.

For this particular case, you can either take the limit of the sequence 0.9, 0.99, 0.999, ..., or you also can work in terms of so called series (see https://en.wikipedia.org/wiki/Series_%28mathematics%29) and consider the series of 9/(10^n), that is, 9/10 + 9/100 + 9/1000 +..., the result will be exactly the same. — Andrey Stolyarov

Earlier on someone wrote a very convincing 'proof':
x = 0.999...
10x = 9.999...
10x = 9 + 0.999...
10x = 9 + x
9x = 9
x = 1

All well and good. But what does x = 0.999... mean? In terms of infinite sums let the sum be S.
The last two lines give:
9S = 9
S = 1.

But what if S is infinite? That is, what if an infinite sum of terms is infinite?
Then we have $9\infty = 9$
$\infty = 1$

That's the thing, we don't really know what S is because you can't apply finite arithmetic to infinite sums.

No, you're not getting the point. There's no need for any "infinite sum" — Andrey Stolyarov

But this is what is being asserted: 1/10 + 1/100 + ...taken to an infinite sum of terms. Let S be this literal infinite sum. What is S? Is it 1 or $\infty$? That is what the problem is, we don't know what a literal infinity is because finite algebra does not apply to literal infinities.

The diagonal of a square, for example, measured in the units that the sides are measured in, is how long? Is that length not a number? Or did something magic happen? — tim wood

In geometry the length of the line - in this example $\sqrt{2}$ - is exact. But the decimal expansion representing it is not, unless we go to an infinite number of places.

that 0.999... is taken as another representation for 1 — Andrey Stolyarov

It may well be that the infinite sum is 1 but mathematicians were suspicious about such a concept because infinity is not a number. This is why calculus is formulated in terms of limits, not infinite sums.

I think the system of real numbers allows that "number" remain undefined, indefinite, and this is why "the real numbers" is not a fixed system. Rigorous defining of "number" has been withdrawn for the sake of producing the real numbers. — Metaphysician Undercover

Yes exactly, calculus works in practice. You can sum $10^{1000000000000000000000000}$ terms and that's fine because it is a finite sum.

But what is $10^{\infty}$?

Show me a definition of "number" which allows that .999... is a number. — Metaphysician Undercover

Kummer did not believe in real numbers; "God made the integers and all the rest is the work of man". This is a bit extreme as real numbers - whatever they are - are cool. Without them we would not have calculus.

It does though. It defines the sum of an infinite series as the limit that the partial sums approach. — Pfhorrest

I understand what you are saying but a literal infinite sum is not considered in calculus. If partial sums are added they approach the limit. If more terms are added it gets closer to the limit and so on. This is how people like Cauchy formulated calculus. They don't consider literal infinite sums. And this is the question I am raising: is the concept of an infinite sum coherent?
When infinitesimals were invented/discovered it was asserted that, they are so small, no matter how many of them are added together, you end up with another infinitesimal. This is why Berkeley balked at these 'Ghosts of departed quantities.' They were so small you couldn't do anything with them.

As I see it, the conclusion that was being reached was that an infinite sum of zeros add to 1. And this did not make sense. So infinitesimals were invented. This is why I have reservations about literal infinite sums.

$0.999...$ is the limit of the sequence of partial sums $s(n)$ — fdrake

Yes, 'partial' sums. That means the sums are finite. Calculus does not speak about literal infinite sums. It speaks about finite sums approaching a limit. As more terms are added indefinitely, the limit is approached more closely. That is what calculus is saying.

What I am saying is that a literal infinite sum is probably an incoherent concept.

0.999... is the limit of {0.9,0.99,0.999,...} — fdrake

https://en.wikipedia.org/wiki/Limit_(mathematics)#:~:text=In%20mathematics%2C%20a%20limit%20is,)%20%22approaches%22%20some%20value.&text=The%20concept%20of%20a%20limit,direct%20limit%20in%20category%20theory.

You just accepted that 0.999... is the limit of {0.9,0.99,0.999,...}, and equal to 1 — fdrake

Yes of course. I have not said it is not the limit. I said 0.999... is not equal to 1 if we are talking about a literal sum. If we are talking about a limit, yes, the limit is 1. I keep saying a sum and a limit are not the same thing.

Great. That means you accept
0.999... = 1 — fdrake

Only according to a strict interpretation of the ' = ' sign: 1 is not the sum. It is the limit of the sum. So 0.999... = 1 does not mean it is literally 1. It means 1 is the limit.

Just a comment about posting math material, symbols, equations, etc. I doubt if anyone here uses it, but MathType is very easy to use and is WYSIWYG rather than coding for each symbol. — jgill

The first post on this math forum explains how to us the math tags.

https://thephilosophyforum.com/discussion/5224/mathjax-tutorial-typeset-logic-neatly-so-that-people-read-your-posts

There isn't. What you're looking for is the infinity symbol above the Sigma. — Kenosha Kid

I just left it out for brevity. I'm sure you know what i mean.

That is exactly how it is meant. That is what 0.999... means. — fdrake

Ok, I'll accept that. But what we are talking about here is subtle and the " = " sign in calculus can be misleading:

$\sum 1/x$ is a literal sum.

lim $\sum 1/x$ is not a sum. It is the limit towards which the sum (over the range) converges.

That is the difference.

fdrake — EnPassant

What is the difference between
$\sum$ = x and
lim $\sum$ = x ?

What is the limit of the series {0.9,0.99,0.999,...}? Call this x.
What does the symbol "0.999..." represent? Call this y.

Is x=y ? — fdrake

It depends on how you read these expressions. I'll grant you that 0.999... can be identical to the limit of the series if that's how you interpret it. But if you do you interpret it as a limit not as equals.
You can say 0.999...= 1 if by that you mean the limit of 0.999...

fdrake — fdrake

Yes, but that is the limit which is different from equals. When you say 1 = you are saying 1 = the limit not simply 1 =

It should be written
lim $\sum$
not simply $\sum$

x = 0.999...
10x = 9.999... — Michael

Nice. But it still begs the question: what does it mean to say x = 0.999...?
It means an infinity of 9s but what can that mean when infinity is not a number?
You have to say x = lim 9/10 + 9/100 + 9/1000 + ...
And we are back to square 1. (Uh, I mean square 0.999...)

How do you know it's infinity and not, say, an octillion? — InPitzotl

Because I know it is not any nameable number.

The sequence elements tend towards the limit. The limit is not a sequence element. 0.999... is the limit. It is equal to 1. — fdrake

Note that in the article cited in the op they don't write

$\sum_{i=1}^{\infty}$ etc = x

They write

lim $\sum_{i=1}^{\infty}$ etc = x

These are two different concepts.

If you don't understand these issues, you should read through jorndoe's document. If you have any questions regarding its content, ask in thread and I will try and address them for you. — fdrake

I have read through it. These are mathematical expressions and as such they are symbols. They represent infinity. But mathematicians were aware of these issues when formulating the calculus and they cautioned against saying 'equals'. They said we should say 'Tends towards the limit'

What is 1/3 in decimal? — Michael

Infinitesimals have never really been understood rigorously. Have you heard of Berkeley's "Ghosts of departed quantities"? Below 'Fluxions' means infinitesimals.

Ghosts of departed quantities
Towards the end of The Analyst, Berkeley addresses possible justifications for the foundations of calculus that mathematicians may put forward. In response to the idea fluxions could be defined using ultimate ratios of vanishing quantities (Boyer 1991), Berkeley wrote:

It must, indeed, be acknowledged, that [Newton] used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?[6]

Edwards describes this as the most memorable point of the book (Edwards 1994). Katz and Sherry argue that the expression was intended to address both infinitesimals and Newton's theory of fluxions. (Katz & Sherry 2012)

Today the phrase "ghosts of departed quantities" is also used when discussing Berkeley's attacks on other possible foundations of Calculus. In particular it is used when discussing infinitesimals (Arkeryd 2005), but it is also used when discussing differentials (Leader 1986), and adequality (Kleiner & Movshovitz-Hadar 1994).

Which is exactly why you write 0.999... — fdrake

Yes, I understand what you are saying. But if infinity is not a number how can you have an infinity "of"?

0.999... IS the limit of the sequence {0.9,0.99,0.999,...}, which IS 1. — fdrake

.999... could not be the limit. To write the limit you'd have to have .999999999999999999999999999999999999999999999999999 - an infinity of 9s. And we can't write that, whatever it means.

.999... is a symbol for 'an infinity of' 9s. But what does that mean?

What is 1/3 in decimal? — Michael

I have no idea!!! I suspect there is 'an infinity of 3s' but what does that mean? That's the crux of the biscuit.

So, how many numbers are there? — InPitzotl

I don't know because I don't know if 'how many' applies to infinity. At the beginning of the theory of limits mathematicians were careful to say that we should say a series 'tends towards' a limit. It is a conservative statement.

Then you don't know what the symbols mean and should read the OP's article! — fdrake

But does anybody know? Intuitively yes, we can see that the limit is 1. But limit is not the same as equals. The argument is subtle. What is being said is 'After an infinity of 9s'. That is what I am suspicious about. I'm not sure what 'an infinity of' means. Or if it is a coherent statement.

That ... MEANS the thing on the left IS the limit. 0.999... IS the limit of the sequence {0.9,0.99,0.999,...} — fdrake

I don't think so. .999... describes the series. The limit of it is infinitely far away.

Another argument, more or less following similar thinking, is whether a number could be found between 0.999... and 1.000... (like the mean).
If no such number can be found, then we might reasonably say they're one and the same. — jorndoe

But the problem is what does .999... mean? How many 9s are we talking about? An infinity of them, of course. But what is an infinity 'of' something?

Actually it does, that's why they use the equals sign. It's the entire essence of calculus. — Pantagruel

True. But they don't mean .999... = 1. They mean the Limit of the infinite sum = 1. There's a difference.
The sum being 9/10 + 9/100 + 9/1000 + ...

because that’s just what decimal notation means. And since the limit of the series of partial sums of that infinite series is 1, that means the total sum of that infinite series represented by 0.999... is also 1, so 0.999... = 1. — Pfhorrest

But it has to do with the way language is being used. 'The total sum' is not the same thing as 'equals'.

The total sum cannot be explicitly demonstrated. It is really a concept that cannot be substituted with 'equals'.

the ... means the limit is taken. IE, .999... = 1 — fdrake

Yes but 'limit' is not the same as 'equals'.

It comes down to how the language is used .9999... equals 1. This is not true. In calculus one is only allowed to say that a series converges to a limit. In this case the series is .999... and the limit is 1. That is, incorrectly speaking, .999...becomes one after an infinity of 9s. But you can't have 'an infinity of' 9s because infinity is not a number. So no, .999... is not 1. It only converges to 1. That is all the link you posted is showing. They are replacing 'tends towards the limit 1' with 'equals'.
You can say The limit = 1 but not .999...= 1

Thought is being and being is thought. In our modern ossified world we tend to believe thinking is just abstract machinations and 'logic' and such things. It is much broader than that.

Science has shown remarkable capability of verification, prediction and use.
How is is this possible if it is only the appearance of external reality (phenomena), not the external reality itself (noumena)? — Arthur Rupel

Look at it this way. If we surmise that what we perceive is analogous to what is really out there we may be able to make progress. For example, if we see something moving in a circular manner we can surmise that there is something out there that is changing in a cyclical manner - not necessarily a circular manner. So our experience is telling us something real.

Also, if we perceive something, assuming the analogy as above, we can then focus on a smaller detail of the perceived object. Then on a smaller detail and so on. We can then determine how these details are related to each other and assume that these perceived relationships between parts are also analogous to the actual relationships between the parts of the perceived object.

This process of endless division into smaller parts should be coherent and analogous to the coherence between the parts of the object itself. If there is coherence in our phenomenological perception, down to the finest detail, this coherence can be understood to reflect the actual coherence of the object itself because the object is the source of our perception.

In other words, if our perception is, in any way, analogous to what is really there, it should be analogous down to smaller and smaller details. We can then make a detailed map of what is really there.

The only other possibility is that nothing in our perception is analogous to what is actually there, which is philosophy gone mad.

So it doesn't mean anything? — tim wood

Isn't child abuse perverse? I don't see why one's subjective point of view needs to be meaningless. The sadist is acting out evil. The masochist wants freedom from self - without giving up self.

What is? — tim wood

That depends on one's subjective point of view.

if I adduce enough arguments to show that time is unreal, time might stop. In other words, there is a recognition that since one can speak however one pleases, that one can in some sense 'make true' whatever one pleases, just by talking about it. — Snakes Alive

Isn't this exactly what Lazerowitz is doing? Talking himself into his own truth? Maybe all philosophers do that, lol.

Sexuality is spirituality in bodily terms. Sexual distortion is spiritual distortion. Think about it...

What I've referred to as the (mono)theistic 'command to love' seems akin to masochistic rape-fantasy or self-abnegation: — 180 Proof

Masochism is a distortion of spirituality, not the other way round.

You have a point there I suppose. What means you by wise obedience? — TheMadFool

It means thinking about why religion tells us how to live. This is what Buddhism means by 'wise living'. But, of course, sometimes religion is corrupt.

To entertain the idea of false/misguided obedience is to sow the seeds of disobedience that ultimately leads to the rejection of god. — TheMadFool

The hope is that it would lead to wise obedience. Slavish obedience is not a good thing. Obedience should be understood rather than rejected at face value. There is a worthy goal in wise obedience.

one of the most effective methods to make normal people do immoral things is to convince him/her that s/he is doing god's will. — TheMadFool

That would be false or misguided obedience.