## Is 1 = 0.999... ? Really?

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## Is it true that 1 = 0.999...? And Exactly Why or Why Not?

Yes, 1 = 0.999...
13
42%
No, 1 ≠ 0.999...
15
48%
Other
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10%

### Re: Is 1 = 0.999... ? Really?

In the evolution of languages, the relationship between mathematical notation and their definite ( defining) representation within the structure of language , this may be reaffirmed:

"Formal quantifiers have been generalized beginning with the work of Mostowski and Lindström".

Or are we not overlooking something?
Meno_
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:"All horses are numbers" is a true premise?

Yes. That's because the word "horse" has been defined to mean the same thing as the word "number".

You said "you can use horses to represent numbers"

Yes. I said that you can use the concept of a winged horse (represented by the word "pegasus") to represent a number such as $$1,000$$.

not horses are numbers.

In the example that I gave, the word "pegasus" represents what is represented by the concept of a winged horse (it does not represent the concept of a winged horse itself) and what is represented by the concept of a winged horse is $$1,000$$. Therefore, the word "pegasus" represents a number, specifically, $$1,000$$. It does not represent a winged horse. Note that I fully understand that such a definition is unconventional. But I wasn't talking about what is conventional in that particular post, but what is logical.

As for functions:

$$f(x) = 0$$ does not represent $$0$$. It does not represent any kind of number. It represents a function.

A function is "a binary relation over two sets that associates to every element of the first set exactly one element of the second set". Taken from Wikipedia. That's what functions are. They aren't quantities. You can't say something like "I have $$f(x) = 0$$ money". That makes no sense.

However, you can use $$f(x) = 0$$ to represent a number -- indeed, any kind of number you want. You can, for example, use it to represent $$0$$. You can take every function that you can imagine and say "This function represents this number". By doing so, you'd make every function represent some number. But by doing so, you'd also add a new meaning to the word "function". $$f(x) = 0$$ didn't previously mean $$0$$. (It's how the word "pegasus" didn't previously represent $$1,000$$ but the concept of a winged horse.)

Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:I already covered the lack of truth in your other premise when I said "there's a degree of dispute over whether all numbers are shapeless (depends on the abstract or concrete representation etc.)"

There are numbers that have a shape? Can you give me an example of a number that has a shape?

That you meant it to be clear that horse in your premise was not the same usage of horse in your conclusion is indisputably NOT clear.
And now you're blaming me for not reading nor understanding your posts?

I also said "Horses qua numbers are indeed shapeless, but what is argued here is that horses qua animals are shapeless, which is not true."

Beside that, before you decide to attack someone's position, it's a good idea to make sure that you understand their position, and that sometimes means asking questions such as "Do I understand you correctly?" and "What did you mean by this?"

Do you ever do this?

I don't think so.
I think it's more than obvious that you are horribly impatient.

What I think you need to do is to calm down and learn how to cooperate. (You are highly uncooperative.)

if your intention really was "figure things out on their own at their own pace regardless of how wrong they are" you'd approach the topic as a student and not a master who assumes every thought that occurs to them is correct without adopting a shred of humility or asking any questions when coming across something that doesn't accord to your amateur assumptions, before simply declaring them to be wrong and yourself as right. Get that sorted, and regardless of how shitty I'm being to you, my work will be done.

I do ask questions. For example, I asked you to define what the word "undefined" means to you. You never answered this question.

And I think I'm far more humble than you are.

The problem here, I believe, is your utter inability to handle disagreement i.e. people who express different opinions whether right or wrong.

Ego issues, in other words.

Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:I can't tell you, that's true, but only because I don't know what you mean by $$\otimes$$. Thus, I have to ask you to define that symbol so that I can derive the meaning of the whole expression from the meaning you assign to individual symbols. So yes, I agree.

Exactly. And since addition of real numbers is only defined for FINITE sums, we have no idea what an infinite sum is till we carefully define what we mean by it.

Magnus Anderson wrote:But this is not true. Every symbol used in the expression $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$$ is already well defined i.e. you, and pretty much everyone else, already knows what every goddamn symbol used in that expression means. You know what fractions are, you know what addition is and you know what infinity is.

Addition of real numbers is only defined for finite sums. We need to define what we mean by an infinite sum.

I can't repeat this any more times than I already have.

"pretty much everyone knows" is not a sufficient standard for doing math. In the 19th century mathematicians realized that logical rigor was needed to define convergence of infinite sums. The "pretty much everyone knows" standard was leading to incorrect results.
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:When you say "infinite sum" you mean "limit". When I say "infinite sum" I don't mean limit. I mean what the words the term consists of suggests: a sum consisting of an infinite number of terms.

But that has no meaning till we give it one. It's not defined till we define it. Just like $$\otimes$$.
wtf

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### Re: Is 1 = 0.999... ? Really?

Ed3 wrote:My concern is with representing Pi as an indefinite sum of Rational numbers. This is clearly wrong because an indefinite sum to n of Rational numbers is Rational for all n. Pi, as we know, is a Transcendental number.

Ed, I am sorry but I do not believe you are a professional mathematician with knowledge of foundations. If you don't know that every real number is the sum of an infinite sequence of rationals, you need to go back to your real analysis textbook.

pi = 3 + 1/10 + 4/100 + 1/1000 + ...

How you can claim mathematical credentials yet not know this, I have no idea. Truly none.

But let me offer you a proof. I hope you know that the rationals are dense in the reals. That is, every real is the limit of a sequence of rationals. So pi is the limit of the sequence 3, 3.1, 3.14, etc.

Now the sum of the infinite series 3 + 1/10 + ... is defined as the limit of the sequence of partial sums; that is, the limit of the sequence 3, 3.1, 3.14, ... That limit is pi.

If you tell me you've been a professional mathematician and you don't know these things, I'd assume you're in some more computational field or that your work is so far out there that you've forgotten the basics. That's very common.

But to claim knowledge of foundations and to not know that pi is the sum of an infinite series of rationals, is not possible.

ps -- Of course a FINITE sum of rationals is rational; but an infinite sum may be irrational.

What do you make of the famous Leibniz series, pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...?

https://en.wikipedia.org/wiki/Leibniz_f ... for_%CF%80

pps -- What do you make of the famous Taylor expansion $$e^z = \sum \frac{z^n}{n!}$$, valid for all complex numbers $$z$$?

It gives e = 1 + 1 + 1/2 + 1/6 + 1/24 + ..., an infinite sum of rational numbers. If we restrict $$z$$ to be a real variable, the Taylor series and formula for e are taught in freshman calculus.

Surely you are not going to say to my face that you're a professional mathematician and that this material is unfamiliar to you. Say it ain't so.
Last edited by wtf on Sat May 23, 2020 9:09 pm, edited 5 times in total.
wtf

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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:

$$f(x) = 0$$ does not represent $$0$$. It does not represent any kind of number. It represents a function.

Yes and no. Consider the set of all continuous functions from the reals to the reals. We can define addition and multiplication of two such functions pointwise. We define:

$$(f + g)(x) \equiv f(x) + g(x)$$ and likewise for multiplication.

With these definitions, the set of all continuous functions is a commutative ring, a generalization of the integers. Its zero element is indeed the function $$f(x) = 0$$.

Zero is just the additive identity of any ring. In a ring of numbers, zero is a number. In a ring of functions, 0 is a function.

https://en.wikipedia.org/wiki/Commutative_ring

https://en.wikipedia.org/wiki/Function_space
wtf

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### Re: Is 1 = 0.999... ? Really?

Yes, but more yes than no? Yes?
Meno_
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### Re: Is 1 = 0.999... ? Really?

Meno_ wrote:Yes, but more yes than no? Yes?

Is that for me? I have no idea why people are making a big deal about the distinction between functions and numbers. We can regard 0 as a function or as a number depending on the context. 0 is just the additive identity of any group or ring or vector space. Nothing more to it. But my biggest puzzlement is why this is such a hot subtopic in this thread. It has nothing to do with anything IMO.
wtf

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### Re: Is 1 = 0.999... ? Really?

wtf wrote:
Meno_ wrote:Yes, but more yes than no? Yes?

Is that for me? I have no idea why people are making a big deal about the distinction between functions and numbers. We can regard 0 as a function or as a number depending on the context. 0 is just the additive identity of any group or ring or vector space. Nothing more to it. But my biggest puzzlement is why this is such a hot subtopic in this thread. It has nothing to do with anything IMO.

Maybe because a hidden implied confusion that needs a more definitive clarification, ? In order to escape a feedback i & o ? , in order to relate them . I hazard.
Last edited by Meno_ on Sat May 23, 2020 8:47 pm, edited 1 time in total.
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### Re: Is 1 = 0.999... ? Really?

Meno_ wrote:Maybe because a hidden implied confusion that needs a more definitive clarification, ? In order to escape a feedback i & o ? , in order to relate them . I guess.

This broke my parser.
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### Re: Is 1 = 0.999... ? Really?

wtf wrote:
Meno_ wrote:Maybe because a hidden implied confusion that needs a more definitive clarification, ? In order to escape a feedback i & o ? , in order to relate them . I guess.

This broke my parser.

Sorry. But to avoid a tautology. Such implication implies a referential collapse into it.('self)
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:There are numbers that have a shape? Can give you an example of a number having a shape?

Sigh.

Yes. Obviously.

I just told you I would be glad to (in your words) "put to rest this joke of a conversation between the two of us" - not for the first time by any means(!), and each time you continue it or try to misrepresent me because you know I'll respond to correct you. But no, as usual, you conclude incorrectly the first thing that pops into your head that it's "ego issues" - I killed that a long long time ago. I just can't stand stupidity being stupidly attributed by stupid people to people who are not stupid.
It's not just a personal distaste, if others stupidly take stupidity seriously it can have a detrimental effect on many people, only fuelling the fire that is stupid people, which I am trying my best to douse whenever I can in a desperate attempt to feel like the wider impact of collective stupidity might be lessened even to the slightest extent. I'd be quite happy for others to perceive this attempt as pathetic - as I said, my "ego" doesn't give a shit what you think of me, - but the problem is that stupid people tend to confuse "pathetic" with being as worthless as stupid. Only non-stupid people recognise that it doesn't matter who is delivering an argument, so it shouldn't matter that it's an officially tested and confirmed fact that statistically I'm almost always the smartest guy in the room, but this doesn't guarantee that stupid people are able to recognise this. In fact, the more people there are who are stupider than you maximises the number of people who are relatively too stupid to understand this fact - not least due to cognitive biases and poor education. This takes its toll over time - there's a reason why I've singled you out to be impatient with. If you had eyes, you'd notice I'm being perfectly patient with EVERYONE else. This is even if they disagree with me. This is why you're stupid for thinking I'm an impatient and uncooperative person in general, when this is highly exceptional behaviour on my part. Obviously none of this explanation will mean anything to you, because even though it's the correct explanation, you are locked into ignoring correct explanations in favour of the first prejudicial assumption that came to your mind that best feeds your cognitive biases.

To quickly sweep up the above mess of a quote, as I have already repeatedly pointed out, quantities can refer to either the general abstract notion of "quantity" or a specific concrete instance of "a quantity". "Number", as a representation of quantity, can refer to the shape of "a number" i.e. the symbol that communicates this representation (as well as a shapeless sound of a word, or the shapeless abstract notion of number).

Just to unravel your blanket equivocation that "all numbers are shapeless"...

And yes, this is all factual and true, unlike "all horses are numbers", which is contingent upon a drastic shift away from convention, and only to the extent of representation rather than direct identity.
There's just so much oversight and truncated thought processes in everything you say...

I can handle that just fine in somebody who doesn't relentlessly present themselves as someone who is indisputably correct about a subject on which they know they lack expertise, and as though someone who does know the subject is plainly wrong in face of the first thing that pops into your head. I care far less when people are simply wrong, it's HOW you (in particular) are wrong that makes all the difference here. Again - if you had eyes you'd have noticed it's only you who is a problem out of all the other people who aren't agreeing with mathematical fact.

My problem, which again you incorrectly identify so predictably in line with common cognitive biases, is that I understand your points too easily. They're all the same things I already considered when initially coming to understand this topic. To you it's all advanced, insightful stuff, but if you weren't stupid you would at least be able to comprehend in the abstract that this isn't the same for everyone. And again this isn't my ego, clearly I don't care about what you think of me or I wouldn't be such an asshole to you, and in front of everyone else. I care that you're stupidly spreading stupidity and stupidity is an epidemic problem with the world. I care that people aren't too stupid to understand correct facts and that they don't present their incorrectness as correct with such certainty and persistence. These facts that I care about aren't "me" - I care what people think about facts (whoever is sharing them) and their approach to talking about them - nothing more.

Magnus Anderson wrote:
Silhouette wrote:"All horses are numbers" is a true premise?

Yes. That's because the word "horse" has been defined to mean the same thing as the word "number".

Yes, as I said, that's what a "given" is. It's not what a truth is. You can posit falsity as a given for the sake of valid argument. It just won't be sound argument.
It's not a fact that a horse is a number even if you posit it as a given. Stop equivocating.

It should be quite obvious that for $$f(x)=0$$, $$f(x)$$ represents the number $$0$$. It also represents a function - more specifically the "way" (a doing) to get to the number (a being) that it already represents. Depending on the number in the set with which there is this binary relation - as in your copy/paste from Wikipedia - the function might represent a different number, but it represents a number regardless.

I really don't see any point in continuing this, you're not going to get it, nor ask questions if you don't agree, which you won't.

Silhouette
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:
Magnus Anderson wrote:There are numbers that have a shape? Can give you an example of a number having a shape?

Sigh.

Yes. Obviously.

I just told you I would be glad to (in your words) "put to rest this joke of a conversation between the two of us" - not for the first time by any means(!), and each time you continue it or try to misrepresent me because you know I'll respond to correct you. But no, as usual, you conclude incorrectly the first thing that pops into your head that it's "ego issues" - I killed that a long long time ago. I just can't stand stupidity being stupidly attributed by stupid people to people who are not stupid.
It's not just a personal distaste, if others stupidly take stupidity seriously it can have a detrimental effect on many people, only fuelling the fire that is stupid people, which I am trying my best to douse whenever I can in a desperate attempt to feel like the wider impact of collective stupidity might be lessened even to the slightest extent. I'd be quite happy for others to perceive this attempt as pathetic - as I said, my "ego" doesn't give a shit what you think of me, - but the problem is that stupid people tend to confuse "pathetic" with being as worthless as stupid. Only non-stupid people recognise that it doesn't matter who is delivering an argument, so it shouldn't matter that it's an officially tested and confirmed fact that statistically I'm almost always the smartest guy in the room, but this doesn't guarantee that stupid people are able to recognise this. In fact, the more people there are who are stupider than you maximises the number of people who are relatively too stupid to understand this fact - not least due to cognitive biases and poor education. This takes its toll over time - there's a reason why I've singled you out to be impatient with. If you had eyes, you'd notice I'm being perfectly patient with EVERYONE else. This is even if they disagree with me. This is why you're stupid for thinking I'm an impatient and uncooperative person in general, when this is highly exceptional behaviour on my part. Obviously none of this explanation will mean anything to you, because even though it's the correct explanation, you are locked into ignoring correct explanations in favour of the first prejudicial assumption that came to your mind that best feeds your cognitive biases.

To quickly sweep up the above mess of a quote, as I have already repeatedly pointed out, quantities can refer to either the general abstract notion of "quantity" or a specific concrete instance of "a quantity". "Number", as a representation of quantity, can refer to the shape of "a number" i.e. the symbol that communicates this representation (as well as a shapeless sound of a word, or the shapeless abstract notion of number).

Just to unravel your blanket equivocation that "all numbers are shapeless"...

And yes, this is all factual and true, unlike "all horses are numbers", which is contingent upon a drastic shift away from convention, and only to the extent of representation rather than direct identity.
There's just so much oversight and truncated thought processes in everything you say...

I can handle that just fine in somebody who doesn't relentlessly present themselves as someone who is indisputably correct about a subject on which they know they lack expertise, and as though someone who does know the subject is plainly wrong in face of the first thing that pops into your head. I care far less when people are simply wrong, it's HOW you (in particular) are wrong that makes all the difference here. Again - if you had eyes you'd have noticed it's only you who is a problem out of all the other people who aren't agreeing with mathematical fact.

My problem, which again you incorrectly identify so predictably in line with common cognitive biases, is that I understand your points too easily. They're all the same things I already considered when initially coming to understand this topic. To you it's all advanced, insightful stuff, but if you weren't stupid you would at least be able to comprehend in the abstract that this isn't the same for everyone. And again this isn't my ego, clearly I don't care about what you think of me or I wouldn't be such an asshole to you, and in front of everyone else. I care that you're stupidly spreading stupidity and stupidity is an epidemic problem with the world. I care that people aren't too stupid to understand correct facts and that they don't present their incorrectness as correct with such certainty and persistence. These facts that I care about aren't "me" - I care what people think about facts (whoever is sharing them) and their approach to talking about them - nothing more.

Magnus Anderson wrote:
Silhouette wrote:"All horses are numbers" is a true premise?

Yes. That's because the word "horse" has been defined to mean the same thing as the word "number".

Yes, as I said, that's what a "given" is. It's not what a truth is. You can posit falsity as a given for the sake of valid argument. It just won't be sound argument.
It's not a fact that a horse is a number even if you posit it as a given. Stop equivocating.

It should be quite obvious that for $$f(x)=0$$, $$f(x)$$ represents the number $$0$$. It also represents a function - more specifically the "way" (a doing) to get to the number (a being) that it already represents. Depending on the number in the set with which there is this binary relation - as in your copy/paste from Wikipedia - the function might represent a different number, but it represents a number regardless.

I really don't see any point in continuing this, you're not going to get it, nor ask questions if you don't agree, which you won't.

Wow.

Even in the realm of mathematics, this sort of truculence goes on.

Some might suppose that one of them is right and the other is wrong. Or that one of them comes closer to whatever the necessarily correct answer finally is.

All this and nary a trace of dasein, conflicting goods and political economy. Or none that I can discern.
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles

Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382

iambiguous
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### Re: Is 1 = 0.999... ? Really?

wtf wrote:And since addition of real numbers is only defined for FINITE sums, we have no idea what an infinite sum is till we carefully define what we mean by it.

Addition of real numbers is only defined for finite sums. We need to define what we mean by an infinite sum.

That's one of the points of our disagreement. I have no idea how to resolve it.

What I'm going to do now is merely restate my position:

$$0.9 + 0.09 + 0.009 + \cdots$$ is an expression that is meaningful and an expression that was meaningful long before the concept of limit was introduced.

Here's something that supports that claim:

We all know that $$0.9 + 0.09 + 0.009 + \cdots$$ is greater than $$0.9$$. Moreover, we can know this without knowing anything about limits. I hope you agree with this. How can that be the case if the expression is truly meaningless? How can you say that $$x$$ is greater than $$0.9$$ if $$x$$ is truly meaningless? You can't.

Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

To wtf:

Here's a hypothesis as to what you're trying to do.

What you're trying to do here is you're trying to force the expression to be something other than it is.

You are asking "What terminating decimal is equivalent to this non-terminating decimal?" The answer to this question is "None". But you can't accept this because you find it unsatisfying. You want it to be something else because you have a need for something else.

Consider a different example.

You want to know the result of $$5 \div 2$$ but you want it to be an integer. The result is not an integer but you nonetheless want it to be an integer. You know that $$5 \div 2$$ is $$2 + 5 \div 10$$ but since you want the result to be an integer you need to do something about it.

The RIGHT way to approach this problem is to substitute $$2 + 5 \div 10$$ with an integer that is closest to it. That's either $$2$$ or $$3$$ and the one to choose is to be decided by your needs.

The WRONG way to approach this problem is to fail to recognize that the result of $$5 \div 2$$ is NOT an integer and to instead insist that the result is undefined and thus in need of being defined e.g. by changing the definition of division to that of integer division.

In the same way that $$5 \div 2$$ is not undefined merely because it is not equal to an integer, $$0.\dot9$$ isn't undefined merely because it isn't equal to a terminating decimal.

Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

iambiguous wrote:Even in the realm of mathematics, this sort of truculence goes on.

Ooo nice word, I actually didn't know that one and had to look it up

Eh, it's been building up for a long time now. I began civil as always, but this one guy has been "I'm not a mathematician, but I know for certain that the mathematical consensus is wrong, and everyone who doesn't agree with me is definitely flatly wrong and their arguments either never existed or are irrelevant" for way too long by now, without learning anything...

Would you say that dasein affects how correct mathematical equations are?

Silhouette
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:I understand your points too easily

No, you don't. And we shall see that shortly.

Magnus wrote:There are numbers that have a shape? Can give you an example of a number having a shape?

Silhouette wrote:Sigh.

Yes. Obviously.

"Number", as a representation of quantity, can refer to the shape of "a number" i.e. the symbol that communicates this representation (as well as a shapeless sound of a word, or the shapeless abstract notion of number).

What makes you think that when I said "All numbers are shapeless" I actually said "All written symbols that we use to represent quantities are shapeless"?

That's all I need to show you that you're wrong: tell you that what I said is not what you think I said.

But since you love conventions, I'll add that if you go to Wikipedia you can see that they make a very clear distinction between numbers and numerals.

https://en.wikipedia.org/wiki/Number
Wikipedia wrote:[I]ndividual numbers need to be represented by symbols, called numerals; for example, "5" is a numeral that represents the number five.

Is it so strange that the word "number" most commonly refers to actual numerical quantities and not to symbols (written or spoken) that refer to numerical quantities?

And yes, this is all factual and true, unlike "all horses are numbers", which is contingent upon a drastic shift away from convention

When I said "All horses are numbers" I did not mean "All horses (as conventionally defined) are numbers". What I meant is "All horses (as I defined the term here in this post) are numbers".

I did not count how many times you misinterpreted the simplest of my sentences but I believe it's a lot of times. This goes against your claim that you understand my points too easily. You don't.

The point of that post of mine (where I spoke of how horses can be used to represent numbers) was to show how changes in definitions can lead to logically invalid arguments. I wasn't discussing conventions.

I don't care about what you think of me or I wouldn't be such an asshole to you, and in front of everyone else

Being an asshole is a common sign of having an over-sensitive ego. You are that kind of guy. The overly sensitive guy who has all kinds of silly rationalizations for his irrational behavior. But once again, that's not the subject of this thread, so let me keep this nice and short.

Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:
Would you say that dasein affects how correct mathematical equations are?

Nope. Not even close. Dasein is an existential contraption relevant only in regard to moral and political value judgments in the is/ought world. Again, if I do say so myself.

Mathematics [I presume] is anchored wholly in the either/or world.

And, that being case, I presume further that there is in fact one correct answer. But the problem here for mathematically challenged dummies like me, is that I can't connect the dots between the arguments and the world of actual material/phenomenal interactions.

Human or otherwise.
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles

Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382

iambiguous
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### Re: Is 1 = 0.999... ? Really?

Some interesting stats. $$16\%$$ of Silhouette's response to the question "Can you give me an example of a number that has a shape?" deals with the actual question and $$84\%$$ of it is about who he is. There are $$4,601$$ characters in total.

Guess which subject is more interesting to him. The subject of who he is.

Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

iambiguous wrote:But the problem here for mathematically challenged dummies like me, is that I can't connect the dots between the arguments and the world of actual material/phenomenal interactions.

That's because there is no connection!

Mathematics, especially modern math, is highly abstract and not bound by any physical considerations. Nobody is saying -- well at least I'm not saying -- that .999... refers to anything in the real world. On the contrary, I maintain that it does not. Nor does 1! The fact that .999... = 1 is a formal exercise in pure math; and has no point of contact with the real world as described by contemporary physics.

Physics $$\neq$$ Math.

Now of course you will say, well isn't math used in physics and biology and economics and everyday life at the grocery store and so forth.

Yes. And this is a philosophical mystery. Why is it that math, which is ethereal, way out there, and free of any "ontological burden" to be about the real world; nevertheless turns out to be supremely useful, in fact essential.

This question is discussed in the famous paper of Eugene Wigner titled, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

The title says it all. Math is so abstract that it is unreasonable to think it applies the world; but it does.

So the best thing to do is remember that math has nothing to do with anything except itself. If others find it useful. the mathematicians are happy but that's not their reason for doing math. Of course I'm exaggerating a position but I hope I'm making my point.

It's the job of philosophy to explain why math, a purely conceptual enterprise, is useful at all. Much has been written.

Historically, people used to believe that math described the real world, and that math = physics.

The split between math and physics was caused by the advent of non-Euclidean geometry in the 1840's. That's when everyone realized that math could not provide certainty about the world; and that if math couldn't, then NOTHING could.

That was the birth of postmodernism and the cultural relativism of contemporary society. If rationality can't tell us what's true, then maybe rationality's just a tool of oppression and not a path to truth at all. This all started with non-Euclidean geometry.
wtf

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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
wtf wrote:Addition of real numbers is only defined for finite sums. We need to define what we mean by an infinite sum.

That's one of the points of our disagreement. I have no idea how to resolve it.

Look up the field axioms for the real numbers and it's resolved. Addition is a binary operation defined on a set. It inputs two elements of the set and outputs a third. We only define x + y for real numbers x and y. By induction we can extend the definition to arbitrary finite sums. But there is NO definition or meaning to an infinite sum.

As I recall Wiki doesn't have a very good article about this but I found another writeup. Note that addition is a function that inputs two variables and outputs a third. Under this system infinite sums don't exist or have meaning. We have to define their meaning.

http://web.math.ucsb.edu/~moore/2axiomsforreals.pdf

Magnus Anderson wrote:What I'm going to do now is merely restate my position:

Its truth value would not change with repetition.

Think about what it means for addition to be defined as a binary relation or a function that inputs two variables and outputs a third.
wtf

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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:$$0.9 + 0.09 + 0.009 + \cdots$$ is an expression that is meaningful and an expression that was meaningful long before the concept of limit was introduced.

You are historically correct. Infinite sums were used by Newton in the 17th century 200 years before the modern theory of convergence was developed.

But even then, when "everyone knew" what infinite sums were, .999... was still 1. There was never a historical time when it was anything else. All that changed was that we figured out how to prove it logically from first principles.
wtf

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### Re: Is 1 = 0.999... ? Really?

Wtf wrote:Note that addition is a function that inputs two variables and outputs a third. Under this system infinite sums don't exist or have meaning. We have to define their meaning.

I understand very well that addition is a function that inputs two numbers and outputs a third number.

However, I don't see how such a definition (together with other relevant definitions) isn't enough for us to logically derive the meaning of the expression that is $$0.9 + 0.09 + 0.009 + \cdots$$.

Your conclusion (that this isn't impossible) looks like a non-sequitur.

I believe you will have to make your argument a bit more explicit because as it is it is very hard, if not impossible, to evaluate its validity.

How does the conclusion (that addition is not defined for infinite sums) follow from the premise (that addition is a function that takes two numbers and returns a single number)? Where's the link?

Hopefully, you are NOT saying that just because addition is a function that takes TWO inputs and not AN INFINITE NUMBER of inputs it follows that addition is not defined for infinite sums. Using that kind of logic, one can also conclude that addition is not defined for any sum that involves any number of terms other than $$2$$ which you agree is not the case.

Its truth value would not change with repetition.

The point was to make my position clear and visible so that other people can address it and not address a strawman.

Magnus Anderson
Philosopher

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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:I just can't stand stupidity being stupidly attributed by stupid people to people who are not stupid.

Did I explicitly state that you or anyone else is stupid? I don't think so. Indeed, I don't know what would be the point of doing that. To disagree with someone is not to say that they are stupid.

This takes its toll over time - there's a reason why I've singled you out to be impatient with. If you had eyes, you'd notice I'm being perfectly patient with EVERYONE else. This is even if they disagree with me. This is why you're stupid for thinking I'm an impatient and uncooperative person in general, when this is highly exceptional behaviour on my part.

Yes, I do have eyes, and yes, there is a reason you singled me out to be impatient with. However, what you think is the reason you singled me out is something that can be disputed.

By the way, note that I did not say that you are impatient ALL OF THE TIME and with EVERY person who disagrees with you.

I can handle that just fine in somebody who doesn't relentlessly present themselves as someone who is indisputably correct about a subject on which they know they lack expertise, and as though someone who does know the subject is plainly wrong in face of the first thing that pops into your head. I care far less when people are simply wrong, it's HOW you (in particular) are wrong that makes all the difference here.

The way I see it, you have no problem with people who disagree with you so as long they do not threaten your ego or otherwise frustrate you. Is that right?

I don't think I present myself as "indisputably correct". What I did here in this thread is 1) I presented my opinions and 2) I did not change them in response to what you and other people had to say. This does not mean I am correct, of course, it can simply mean that I fail to see the point that other people are making despite their efforts. But also, it can mean that none of you have any arguments whatsoever.

They're all the same things I already considered when initially coming to understand this topic. To you it's all advanced, insightful stuff, but if you weren't stupid you would at least be able to comprehend in the abstract that this isn't the same for everyone.

I don't think the things we're discussing here are advanced and insightful. And whether you or anyone else thinks the subject of this thread is advanced and insightful is, to be perfectly honest with you, completely irrelevant.

"ego issues" - I killed that a long long time ago

it's an officially tested and confirmed fact that statistically I'm almost always the smartest guy in the room

Note that these kinds of things are typically said by people who are insane.

Magnus Anderson
Philosopher

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