Is that your contention? — Banno
We do talk of half a pie as being a quantity of pie. — Banno
(as per the definition I offered) — Metaphysician Undercover
One is a quantity, two is a quantity, three is a quantity and so is four — 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.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
Okay.I don't believe that ".999..." and "1" refer to the exact same thing. — 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:Until then, I'll believe what seems very evident to me at this time, that these two have distinctly different meanings. — 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?A fraction is not a number. — Metaphysician Undercover
I detect some language loaded to the brim with irrelevancies.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
...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 you could answer this for me, then you might help me to believe what you believe.
But isn't one of this pie a different quantity from one of that pie?Half of this pie is a different quantity from half of that pie. — Metaphysician Undercover
We agree on seeing tomfoolery, we just disagree on where we see it.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
If you bought half a small pizza would you complain because you expected to get the same quantity as half a large? — 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. — InPitzotl
And the way they use it, .999...=1. The definitions therefore are matters of fact. — InPitzotl
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
But isn't one of this pie a different quantity from one of that pie? — InPitzotl
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
But this all still leaves hanging why you think 3 is a quantity but ⅓ isn't... — Banno
are you aware of how... eccentric... you view is? — Banno
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
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.I don't believe that ".999..." and "1" refer to the exact same thing. — 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?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
...But isn't one of this pie a different quantity from one of that pie? — InPitzotl
No, why would you think that? — Metaphysician Undercover
Because pies vary in size?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
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.One of anything is the same quantity as one of anything else. — Metaphysician Undercover
Uhm... but...:If your inability to accept this fact rules you out of this conversation then so be it. — Metaphysician Undercover
...yet: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
...and: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
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.Until then, I'll believe what seems very evident to me at this time, that these two have distinctly different meanings. — Metaphysician Undercover
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
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
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 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.But I am talking about an actual sum. — 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).What makes me suspicious is the paradoxes that exist at infinity. — EnPassant
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
From this we conclude that any positive quantity added infinitely sums to infinity — EnPassant
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.
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
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