A ninth of that particular pie is a particular quantity. A ninth, or 1/9, is not a particular quantity. Are you capable of understanding this?

So one pie a particular quantity, but 1 is not?

One is a particular quantity, and therefore a number; 1/9 is not. This is because a fraction must be a fraction of something in order that it signify a particular quantity, 1/9 of this, or 1/9 of that. But 1/9 on its own does not signify any particular quantity.

Oh - I thought for a minute I had it, but no, I'm not following. 1/9 must be 1/9 of this or of that, but 1 need not be one of these or one of those - it is a quantity all on its own?

Right, that's why one is called a number, it's a value representing a particular quantity (as per the definition I offered), the quantity of 1. On the other hand, 1/9 does not represent any particular quantity unless it is qualified with 1/9 of 9, or 1/9 of 90, etc..

No, I'm not following. It seems that on your account we can only use the word "quantity" to apply to discreet individuals: one pie, two pies, and so on; the word cannot be applied to partial individuals: half a pie, a quarter of a pie.

Is that your contention?

Because that seems wrong. We do talk of half a pie as being a quantity of pie.

Is that your contention? — Banno

No that's not my contention. One is a quantity, two is a quantity, three is a quantity and so is four, etc.. 1/9 is not a quantity because it is a fraction, and for it to refer to a particular quantity it must be specified what it is a fraction of. 1/9 of 9 is a different quantity from 1/9 of 18, which is a different quantity from 1/9 of 27, etc.. So 1/9 on its own does not refer to any particular quantity.

We do talk of half a pie as being a quantity of pie. — Banno

I'm not denying that people talk that way, just like they say .999... is a number. I'm saying these people are wrong. Half of this pie is a different quantity from half of that pie. So half a pie is not a particular quantity at all. Even though people might talk as if it is a particular quantity of pie, we'd be fools to believe them. If you bought half a small pizza would you complain because you expected to get the same quantity as half a large?

(as per the definition I offered) — Metaphysician Undercover

From the OED, the authoritative source of all things mathematical.

I invited anyone to provide a better definition of "number", one which would provide for these "non-particular quantities" to be called numbers, but none has been provided. So I still believe that concepts such as "real numbers" operate without an acting definition of "number", providing for all sorts of tomfoolery.

One is a quantity, two is a quantity, three is a quantity and so is four — Metaphysician Undercover

Five is a quantity, six is a quantity, and Heavens to Betsy, I do believe there are more! :lol:

There was no implicit assumption that the same thing ought not have the same name, but an implicit assumption that if the same thing does have two distinct names, there is a reason for it having two distinct names. — Metaphysician Undercover

So instead of arguing "there are two names for a thing therefore there are two things", which is a red herring, you're just arguing "there are two names for a thing and there's no reason for it having a second name therefore there are two things", which is just a red herring with weasel (obviously if a thing has two names, there's a reason it has two names... it was named twice; and obviously that doesn't count... so, the weasel is in what constitutes "a reason"). Adding a weasel to a red herring is still not an argument, though I suppose the weasel would love the snack.

I don't believe that ".999..." and "1" refer to the exact same thing. — Metaphysician Undercover

Okay.

Until then, I'll believe what seems very evident to me at this time, that these two have distinctly different meanings. — Metaphysician Undercover

But that would be silly because your premise that two names must refer to two things is a red herring. Also, it's a bit fishy:

A fraction is not a number. — Metaphysician Undercover

If you cannot agree that a fraction is a number, how are you even qualified to talk about the meaning of .999... in the first place?

So I'm asking you, who apparently does believe this, why does mathematics, as a single unified discipline, have these two distinct symbols to refer to the exact same thing. — Metaphysician Undercover

I detect some language loaded to the brim with irrelevancies.

If you could answer this for me, then you might help me to believe what you believe.

...why does this sound like the hook of a con to me? My "belief" isn't relevant here (except insofar as I'm part of the math community which, technically, I am, but it's just a tiny part)... the terms here are terms of art in the math community. As mentioned before, the math community defines and uses these terms. And the way they use it, .999...=1. The definitions therefore are matters of fact. If you have any issues, it's with the proofs. But you're not pointing those out... you're just rattling about nonsense of two names having to refer to two things... it's your core broken intuition, and just propping it up with loaded language isn't going to fix what's broken here.

If we can't agree that 1/9 of a pie is a particular quantity of pie, then we can't have the conversation you want. But it's irrelevant anyway.

On the off chance that someone else is curious, yes, there's a reason that the decimal system representation of numbers gives two names for the same numbers, and it's not unusual for various systems to do so. On the off chance MU replies to this with a rebuttal, it's irrelevant... your entire two names is two things argument is dubious, so there's literally nothing to argue against including this.

Half of this pie is a different quantity from half of that pie. — Metaphysician Undercover

But isn't one of this pie a different quantity from one of that pie?

So I still believe that concepts such as "real numbers" operate without an acting definition of "number", providing for all sorts of tomfoolery. — Metaphysician Undercover

We agree on seeing tomfoolery, we just disagree on where we see it.

If you bought half a small pizza would you complain because you expected to get the same quantity as half a large? — Metaphysician Undercover

But equally, if you bought 1 small pizza would you complain because you expected to get the same quantity as 1 large?

None of what you have said makes your contention clear.

You say 1/9 of 9 is a different quantity from 1/9 of 18; Is 1/9 of three yet another quantity? But surely you must say that ⅓ is not a quantity...

But this all still leaves hanging why you think 3 is a quantity but ⅓ isn't...

If you are going to use the word "quantity" in a way that is so at odds with how everyone else uses it, you might need to put some more effort into explaining why.

@Metaphysician Undercover, are you aware of how... eccentric... you view is?

So instead of arguing "there are two names for a thing therefore there are two things", which is a red herring, you're just arguing "there are two names for a thing and there's no reason for it having a second name therefore there are two things", which is just a red herring with weasel (obviously if a thing has two names, there's a reason it has two names... it was named twice; and obviously that doesn't count... so, the weasel is in what constitutes "a reason"). Adding a weasel to a red herring is still not an argument, though I suppose the weasel would love the snack. — InPitzotl

No, I believe the two symbols have different meaning, and I've given the reasons why I believe that. You claim that the symbols refer to the same thing so I want to know the reasons why you believe this.

As indicated in the op, it is not the case that the same thing is named twice. It's very clear that "1/9X9" does not say the same thing as "1". So your claim that the same thing was named twice is false.

And the way they use it, .999...=1. The definitions therefore are matters of fact. — InPitzotl

As I've explained to fishfry already, that two things are equivalent does not mean that they are the same thing. Therefore what is on the left side of the "=" (which indicates equivalent) does not provide a definition of what is on the right side. It seems you do not know what a definition is.

If we can't agree that 1/9 of a pie is a particular quantity of pie, then we can't have the conversation you want. But it's irrelevant anyway. — InPitzotl

As I explained to Banno, it's very clear that "1/9 of a pie" does not indicate a particular quantity of pie, because pies vary in size. If your inability to accept this fact rules you out of this conversation then so be it.

But isn't one of this pie a different quantity from one of that pie? — InPitzotl

No, why would you think that? One of anything is the same quantity as one of anything else. It is one, which is a quantity. We are talking about quantity in an absolute, abstract sense now. But if we are talking about a quantity of pie, then clearly one large pie is a different quantity than one small pie.

The lesson you ought to take from this is that 1/9, as a fraction does not refer to any quantity in any sense whatsoever, because it needs to be qualified. In order to have any meaning whatsoever, we need to indicate the thing which is to be thus divided. To talk about a division without any thing divided, is to talk about a useful tool, which is doing nothing. And the tool which divides quantities is not itself a quantity.

You say 1/9 of 9 is a different quantity from 1/9 of 18; Is 1/9 of three yet another quantity? But surely you must say that ⅓ is not a quantity... — Banno

Now you've struck the heart of the problem. Some quantities cannot be divided in certain ways. It is impossible. Three cannot be divided by nine, it is impossible. Nevertheless, mathemagicians are an odd sort, very crafty, wily like the fox, devising new illusions all the time. They like to demonstrate that they can do the impossible. Some people even believe that they actually do what is impossible. That is a problem.

But this all still leaves hanging why you think 3 is a quantity but ⅓ isn't... — Banno

Let's start with this definition. A "quantity" is something which can be measured. The simple act of counting, 1,2,3,4,5,etc., when there are no objects being counted, is an act of measuring imaginary things. These imaginary things are called "numbers". So a numeral represents an imaginary quantity, which is called a number. A quantity is something which can be measured and in this case the measurement is counting. Now look at "1/3". It represents a ratio, which is a specific relationship between two distinct quantities, or numbers. A relationship between two numbers is not the same thing as a number, therefore we ought not try to represent it as a number.

The relationship between two numbers (indicating determinate measurable quantities), is not necessarily a measurable quantity itself. When it is not, there's a word for this "incommensurable". Why create the illusion that incommensurable things are actually not incommensurable, and insist that this illusion is truth.

are you aware of how... eccentric... you view is? — Banno

It's just that i think about extremely trivial things, which is not a common trait. But the important things are already over thought so why not?

I take my definition of "number" from OED: "an arithmetical value representing a particular quantity and used in counting and making calculations". Notice specifically the criteria "particular quantity". This rules out the possibility that .999... is a number. — Metaphysician Undercover

As I've explained to fishfry already, that two things are equivalent does not mean that they are the same thing. — Metaphysician Undercover

I don't believe that ".999..." and "1" refer to the exact same thing. — Metaphysician Undercover

You're all over the place here. You have a definition of number that refers to a value (read the newer version of OED; cf to definition 1b of your revision). 1 and .999... being equivalent means they refer to the same value. And don't think I didn't catch that suddenly "refer to" changed to "are"; nevertheless, it's common language to use forms of "to be" to represent equivalence under equality. If .999... represents the same "particular quantity" that 1 does, they refer to the same value, which is what it means to say that they are the same thing.

Therefore what is on the left side of the "=" (which indicates equivalent) does not provide a definition of what is on the right side. It seems you do not know what a definition is. — Metaphysician Undercover

Your "therefore" is thwarted by the definition of a number. Equivalence under equality means that the left hand side has the same value as the right hand side. Your OED definition of number is that of a value. Therefore, equivalence in this context means referring to the same number, since it's the same value. And you're complaining about tomfoolery?

But isn't one of this pie a different quantity from one of that pie? — InPitzotl
No, why would you think that? — Metaphysician Undercover

...

As I explained to Banno, it's very clear that "1/9 of a pie" does not indicate a particular quantity of pie, because pies vary in size — Metaphysician Undercover

Because pies vary in size?

One of anything is the same quantity as one of anything else. — Metaphysician Undercover

Apparently not. One pie is the same as one pie even if they are different sizes, but one ninth of a pie is not the same as one ninth of a pie because they are different sizes. I know special pleading when I see it. Again, you're all over the place.

If your inability to accept this fact rules you out of this conversation then so be it. — Metaphysician Undercover

Uhm... but...:

Some quantities cannot be divided in certain ways. It is impossible. Three cannot be divided by nine, it is impossible. Nevertheless, mathemagicians are an odd sort, very crafty, wily like the fox, devising new illusions all the time. They like to demonstrate that they can do the impossible. Some people even believe that they actually do what is impossible. That is a problem. — Metaphysician Undercover

...yet:

So I'm asking you, who apparently does believe this, why does mathematics, as a single unified discipline, have these two distinct symbols to refer to the exact same thing. — Metaphysician Undercover

...and:

Until then, I'll believe what seems very evident to me at this time, that these two have distinctly different meanings. — Metaphysician Undercover

What conversation pray tell are you even talking about? How can .999... have a second meaning if .9 means 9/10 and 9/10 is allegedly a problem? And how come you can't be honest about what you're inviting me to do? The problem isn't that you're missing that conversation about why there are numbers that have two representations in the decimal system... the problem is that you don't believe decimals are possible because you have a quixotic quest against fractions, and yet you present to claim that you believe .999... has a meaning at all. I'm not the problem here, MU; I can easily have that conversation with someone who isn't so wrapped up in your fictional world of fraction-denial. I just can't have this conversation with you because you can't face the fact that there's a thing to discuss.

But again, it's irrelevant, because your two-names-means-two-things premise is still as dubious as it ever was.

I wonder if MU believes in negative numbers either, or just the naturals. Does zero count to him?

A ninth of that particular pie is a particular quantity. A ninth, or 1/9, is not a particular quantity. Are you capable of understanding this? — Metaphysician Undercover

1/9 is a proportional relationship. In geometry if the side of the square is $1$ the diagonal is $\sqrt{2}$ and the proportion is $1:\sqrt{2}$

Apparently not... see the underlined as evidence for your continued confusion of the same point. The sum is by definition the same as the limit. — InPitzotl

I missed this post. Yes, the definition of the sum is the same as the limit. But I am talking about an actual sum. An explicit infinite sum that you can write down. This of course is not possible because it requires an infinity of calculations. That's the point I'm making. I understand sums and limits but that is not really how the question can be answered. It can only be answered by an explicit infinite sum. What makes me suspicious is the paradoxes that exist at infinity.

It's just that i think about extremely trivial things, which is not a common trait. But the important things are already over thought so why not? — Metaphysician Undercover

I think your problem lies with the distinction between pure and applied maths rather than a distinction between 1 and 1/9. 1, 1/9, -1/9 , 0.9, 0.999i are all numbers in the realm of pure maths.

Then when you start applying them to pies you enter the realm of applied maths which is a different realm. 1 pie and 1/9 of pie take on meaning but not so much -1/9 pie or 0,99i pies.

But I am talking about an actual sum. — EnPassant

I realize that, but there's no "actual sum" to speak of outside of this definition. In principle I could give an intuitive argument for why .999...=1 using the idea that each digit in the decimal is dialing in on the "address" of the number it refers to. In such an argument I could say that if you have an infinite number of 9's after the string .999, then the resulting string dials in on the address of 1 itself. But in using this argument and applying it to a repeated decimal, I would in effect be using the limit definition.

What makes me suspicious is the paradoxes that exist at infinity. — EnPassant

I understand that as well... but analogously, divergent infinite sums can't use the definition above; only convergent ones. But that's precisely why we would apply this definition to infinite sums for convergent infinite sums (and in certain cases we can apply a definition to divergent sums, but with different definitions).

But I think in the bigger picture this thing boils away, because in the state of affairs that we're in, the "actual sum" for a convergent infinite sum has been defined, by this definition. In essence, the infinite sized string .999... is like a word, and we have assigned a formulaic definition for all words matching this pattern, including that one... and by that definition, .999...=1. We could say then that the address .999... has been assigned in such a way that the number assigned to it is 1.

Or phrased another way, I'm not sure you can actually say what it is you're disagreeing with meaningfully. To do so you would have to reify .999...'s definition without using the provided one, and claim that whatever reified thing you came up with "has a problem". But what problem? If the problem is it hasn't been defined, then it's a bit vacuous. If the problem is we don't know what it is, then you're presuming it has a meaning... in what sense does it have a meaning? What meaning did you assign to it? That's the problem I'm raising here... you cannot talk about this reified "actual sum" unless you can talk about it, and I'm not sure you've convinced me there's a thing to talk about.

There's no logical or conceptual problem with infinite sets, like, say, $\mathbb{N}_+ = \{1, 2, 3, \ldots\}$.
Would be kind of tedious for physicists and cosmologists to have to check whether their results had exceeded "the largest number". :D
Just have to remember that ∞ isn't a real number, can't be shuffled into arithmetic calculations (+-×/) just like that.

And there are any number of ways to write 1.
0.5 + 0.5 = 2 - 1 = 1 × 1 × 1 = $\frac{\sqrt{2}}{\sqrt{2}}$ = $1^5$ = $1 + \sum_{n \in \mathbb{N}_+} (1 - 1)^n$ = ...
No numerical difference.

And we can reason about $\sum_{n \in \mathbb{N}_+} 9(\frac{1}{10})^n$ and such if we're careful. (y)

Why would any of this be a problem anyways...?
(I didn't see the formalities implying a contradiction here in the thread.)

you cannot talk about this reified "actual sum" unless you can talk about it, and I'm not sure you've convinced me there's a thing to talk about. — InPitzotl

What I'm saying is very simple. Suppose you had a kind of God calculator that would print out the actual addition of 9/10 + 9/100... to an infinity of terms, what would that be, 1 or infinity? That's what I mean by the actual sum.

To show how quirky infinite sums are consider the following (this is not meant to answer anything, it is just to illustrate how strange things become at infinity)

Theorem 1
Define 1/x such that 0 < 1/x < 1. If 1/x is summed to itself infinitely often, the sum is infinity. From this we conclude that any positive quantity added infinitely sums to infinity

Now sum 1/2 + 1/4 + 1/8...in view of the above theorem. No term in this series is zero, they are all positive quantities. So we are summing an infinity of positive quantities, some of the bigger than others...

Again, I'm not trying to answer anything here but it is worth contemplating.

From this we conclude that any positive quantity added infinitely sums to infinity — EnPassant

Only in the special case you describe of adding the same thing to itself forever. Diminishing quantities act differently. Otherwise Achilles could never pass the tortoise, or even get started running.

One of those two series you gave diverges; it does not have a limit. The other converges: it has a limit. The second one never gets anywhere close to infinity no matter how long you run it. It would only ever even get up to 1 if you ran it forever, with your “God-calculator”.

This is exactly what limits are for. Only a series without a limit sums to infinity. A series with a limit sums to that limit.

rethinking...

1/2 + 1/4 + 1/8...to k terms
> 1/x + 1/x + 1/x...to k terms — EnPassant

This becomes false as soon as k = x.

1/2 + 1/4 + 1/8 = 0.875
1/3 + 1/3 + 1/3 = 1

This becomes false as soon as k > 2. — Pfhorrest

I know, I made a mistake. Let me rethink how to formulate it...

You say it in your notes:

The proof above is standard calculus. You may also come across a variety of philosophical,
semantic, arithmetic, algebraic, precedence arguments[22], some of which are interesting or
relevant in their own right. Perhaps a source of confusion is that the number 1 figures nowhere in the sequence (0.9, 0.99, 0.999, ...). Another possible source of confusion could be the Archimedean properties [23][24][25] : neither ∞ nor infinitesimals[26] are real numbers [27][28]. Either way, calculus has real applications, proven in action, just ask physicists and engineers.

Yes, we know the answer, we know how it works.

The sorry fact is, that we cannot either describe or simply cannot understand infinity as clearly as we would want. Or infinitesimal and it's relation to numbers.

One of those series diverges; it does not have a limit. The other converges: it has a limit. The second one never gets anywhere close to infinity no matter how long you run it. It would only ever even get up to 1 if you ran it forever, with your “God-calculator”. — Pfhorrest

But that begs the question: you say it don't sum to infinity because it don't sum to infinity. That is the very thing that is being questioned. I know the limit is 1. But that limit is defined by finite arithmetic. I am asking what really happens at infinity. An infinity of positive quantities are being summed and any positive quantity summed infinitely, is infinity.

The point of a limit is that the sum never exceeds it. No matter how many terms you add to that convergent series, it will never exceed 1. Why then would you think it could ever add up to infinity? If it could, that would make it a divergent series, one with no limit, by definition.