Is 1 = 0.999... ? Really?

I have three more examples I forgot to mention earlier.

• Number are often interpreted as operators. If we interpret the number 5 as an operator that stretches a line segment, then the number i is an operator that rotates a line segment in the plane. That is in fact the best way to view it. So it’s common fro numbers to be identified with functions.

• We have function spaces such as the set of all continuous functions that can be added and multiplied pointwise. With these operations of plus and times, the continuous real-valued functions of a real variable become a commutative ring.

Likewise we have famous function spaces like Banach and Hilbert spaces, in which functions are points, we have inner products, and we can do linear algebra on them.

• There is no distinction between numbers and functions in set theory. I don’t know enough about type theory to know how this is handled.

I didn’t represent an uncountable object with a countable one. There are as many continuous functions (reals to reals) as there are reals. Easy proof. That was my only point.

I’m confused about where you’re coming from. What point are you making? Morally, functions and numbers are different. But in practice we use functions as numbers and numbers as functions all the time.

No.

I disagree.

The point of contention is the standard meaning of the symbol that is (0.\dot9).

Obviously, I have to repeat it at least one more time: I am NOT talking about what I mean by (0.\dot9), I am talking about what mathematical establisment means by (0.\dot9).

That said, you might want to argue that I am wrong in my belief that mathematicians define (0.\dot9) as a sum and not as a limit.

The limit of a sum is not the sum itself.

You can call the limit of a sum by the name “sum” – and people already do that, I understand – but that doesn’t erase the difference between the concept of a sum and the concept of a limit.

Let me give you an analogy. You can call numbers horses. There’s nothing wrong with that. But you can’t say there’s no difference between numbers and horses.

The argument I’m putting forward is that (0.\dot9) represents THE RESULT OF A SUM and not THE LIMIT OF A SUM. They are two related but different things.

That said, you might want to argue that mathematicians interpret (0.\dot9) as the limit of a sum (and not as the sum itself.) By doing so, however, you would be making an exception for (0.\dot9) and similar expressions because all other decimal numbers are normally interpreted as sums.

The result of a sum is a number that is attained. The limit of a sum is a number that is approached – not necessarily attained.

Your best bet is to argue that (0.\dot9) represents a limit rather than a sum. You won’t get far by trying to deny the fact that the concept of limit and the concept of sum are two different concepts.

en.wikipedia.org/wiki/Series_(mathematics

You haven’t added anything other than reiterate that you are unfamiliar with the standard mathematical definition of the limit of a convergent series.

FWIW I’ll outline the logic.

(1) First we define the limit of a sequence a1, a2, a3, … by the “arbitrarily close” standard, which is formalized by the business about the epsilons that you may have seen. (I’ll skip writing the gory details unless requested).

(2) Then we would like to define what we mean by a notation like a1 + a2 + a3 + … The axioms for the real numbers only allow us to add up finitely many real numbers. So we have to DEFINE what an infinite sum is.

We form the sequence of partial sums: a1, a1 + a2, a 1 + a2 + a3, + … Each term is well-defined because it’s a finite sum of real numbers. Now if the resulting SEQUENCE of partial sums has a limit as given by (1), then we define the infinite sum as the limit of that sequence. It’s a definition.

That’s it. That’s really it.

Now if the question is whether that’s what mathematicians say, it is. Wikipedia agrees and so do hundreds of textbooks.

If the question is whether the mathematicians maybe got it philosophically wrong, that’s a different discussion; and one that I’m not entirely unsympathetic to. But you claim to dispute that this is how mathematicians define infinite sums, and you’re just wrong about that.

JSS started this thread. It’s his topic. That’s why I’m mentioning him. You are wondering what this thread is about. Since it’s James’s thread, it’s James who decides (or rather, who decided long time ago) what this thread is about. And what this thread is about is standard math.

I am not saying that the definition of a limit is a lie. I am not sure where you got that from. What I’m saying is that mathematicians claim that (0.\dot9) qua sum is equal to (1) and that they do so by a variety of means one of them being by pretending that an infinite sum and the limit of an infinite sum are one and the same thing. Basically, what I’m saying is that they are equivocating (not merely working with different definitions.)

I am not talking about limits, I am talking about sums. Sums aren’t based on arbitrary closeness.

I don’t think you got that right.

The point of contention is how mathematicians define (0.\dot9). (Not how they define the term “infinite sum”.)

Your claim is that (0.\dot9) represents the limit of the sum (which you also call “the infinite sum” or “the sum”.)

My claim is that (0.\dot9) represents the infinite sum itself and not the limit of the sum.

Does that help?

James’s idea of infA is not only not standard, it’s logically incoherent. As were his other ideas.

I gave you the Wiki link and the outline of how the definition works. It’s logically solid.

But first you say that you are merely disputing that mathematicians define infinite sums the way I say they do.

But now you say they are equivocating, meaning that you disagree with the official definition. So you ARE challenging the official standard definition.

I just gave you the Wiki link. Sequences are based on arbitrary closeness; and infinite sums are defined as limits of SEQUENCES of partial sums.

It’s described with perfect clarity on the Wiki page.

“A series ∑an is said to converge or to be convergent when the sequence (sk) of partial sums has a finite limit. If the limit of sk is infinite or does not exist, the series is said to diverge.”

en.wikipedia.org/wiki/Series_(mathematics#Convergent_series

The infinite sum IS the limit of the sequence of partial sums. Read the Wiki page if nothing else.

How can you read that page and claim it says something other than what it says?

The fact that James’s (infA) is not a standard mathematical representation does not mean this thread has nothing to do with standard math.

I already told you I am not talking about limits. What’s the problem? Note that when you say “infinite sum” you mean “limit”.

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. That’s precisely how (0.\dot9) is interpreted by mathematicians: as a sum consisting of an infinite number of terms. That’s what I claim. You get distracted by the fact the word “infinite sum” has been given additional meaning – that of a limit.

It has no meaning except by virtue of how it’s defined. How do YOU define “a sum consisting of an infinite number of terms?” How do you define it? Lay out your formal definition and we can kick it around. I’m pretty openminded. But you keep saying there’s some secret definition that you won’t share with me.

Bear in mind that the axioms of the real numbers only provide for the addition of finitely many terms. To define an infinite sum we must do exactly that: define it.

How do you define it?

I have no idea what you’re looking for and I have no idea why.

The term “infinite sum” simply means “a sum consisting of an infinite number of terms”.

If this isn’t enough for you to understand what is meant by the term, you’ll have to help me understand what’s unclear about it.

On the other hand, how can you know what the limit of an infinite sum is without first knowing what an infinite sum is?

How can you know what the love of a woman is without first knowing what a woman is?

How do we know there is any such thing; let alone how to compute it or evaluate it?

The basic axioms of real numbers say that any FINITE sum of real numbers is defined and exists.

If you then give me an expression such as 1/2 + 1/4 + 1/8 + 1/16 + … then what on earth could it possibly mean? The basic rules of the real numbers do not say!

In math we try to define everything from the ground up. There is a formal definition of +, there’s a formal definition of real numbers, and so forth.

So if you see an expression – which has no defined meaning in math – like 1/2 + 1/4 + 1/8 + 1/16 + … ; then how do you know what it means? You have to define it, just like you have to define integrals and derivatives and topological spaces and quantum field theories in physics.

Look at it this way. Suppose I ask you, what is (3 \otimes 47) You can’t tell me. You would have to first ask: How is (\otimes ) defined. Right? I hope you can agree.

It’s the same for 1/2 + 1/4 + 1/8 + 1/16 + … The sum of any FINITE number of terms is defined; but not the sum of infinitely many. We have to define that first. Otherwise we have no idea what it means; or if the notation can even be made logically consistent with the rest of math.

It turns out that it can. By the definition given on the Wiki page. By defining the sum of an infinite series as the limit of the sequence of partial sums, each one of which is finite hence defined, we are defining a NEW notation in terms of things we already know. That’s science! That’s logic.

ps – I wanted to add that I think we’ve arrived at a good place. I see the core issue.

You don’t realize that everything we write down in math must be formally defined and shown to be sensible. We can define the number 3. We can define plus and times. We define everything. Even the things we’ve taken for granted since we were children, must be formally defined once we formally study the subject.

So if we want to talk about infinite sums, we have to define them.

In biology, we can’t just say, “A flying elephant is an elephant that files,” and then open research labs to study them. Since they don’t exist, it’s pointless. Likewise with infinite sums. They don’t have any a priori mathematical existence. We have to define them and show that our definition makes sense in the context of the rest of math.

I wanted to add that there is a reason WHY we like to formalize things very strictly.

In the 19th century mathematicians started studying things like trigonometric series and Fourier series and such, and there were a lot of questions about what exactly it did mean for a series to converge, and what was the difference between regular and uniform convergence.

Historically, mistakes were made. Someone would think they proved convergence but they were wrong. It gradually became clear that they needed to have a precise formal definition of convergence. So this didn’t happen in a vacuum. There was a practical need to have clear definitions because things were getting messy.

So 19th century math started with the same mindset that you have. We “know” what the sum of an infinite series is and we can work with them intuitively. It only became clear gradually, over time, that a formal theory was needed. That’s why the 20th century was all about formalization. Not for the sake of being formal for its own sake; but to avoid errors caused by intuitive and imprecise thinking.

As usual, either determined to misrepresent or unable to understand a simple point.

I’m not saying function and number mean the same thing.

I’m saying they’re the “doing” and “being” versions of the same thing: quantity. You “do” a function to arrive at a quantity, and a number is a representation of what a quantity “is”.
Quantity means the same thing as quantity, but “doing” does not mean the same thing as “being”.
This couldn’t be less equivocated.

A valid but unsound syllogism, which has absolutely nothing to do with your closing line about (0.\dot9)
Non sequitur. Irrational nonsense. Pointless.

Just listen to wtf.
Unlike you, I’m able to identify when somebody knows more about a subject than I do, and should be listened to and asked questions, rather than told with certainty that he’s wrong and you’re right. You refuse the opportunity with me, don’t throw away an even better opportunity with him.

I assume this implies that the sex that your parents had that led to your conception is a “doing” version of you?

You are the sex that they had? There’s no difference humans and sex? Humans are sex?

Not a valid syllogism.

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.

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. Therefore, you should be able to logically derive the meaning of the expression as a whole (instead of claiming it’s undefined and thus in need of a fresh new meaning.)

Let’s take (0.9 + 0.09 + 0.009 + \cdots) as an example.

Are you going to tell me you don’t know that whatever this expression represents is greater than (0.9)?
Note that If you know that what it represents is greater than (0.9), then you also know that what it represents is not equal to (0.9).

In the same exact way, you can (and really should) know that whatever this expressions represents is less than (1).
That’s pretty much what James’s proofs (posted on the very first pages of this thread) amount to.

Note that the subject of this thread isn’t “What’s the result of (0.9 + 0.09 + 0.009 + \cdots)?”
The subject of this thread is “Is (0.9 + 0.09 + 0.009 + \cdots) equal to (1)?”

We can know that the result of a sum is less than (1) without knowing the exact result of that sum.

Oh my GODDDDddddDDDDddddDDDD

Will you stop with the fucking “This thing that I unthinkingly believe is a fact” bullshit?!

Look it the fuck up.

If the conclusion of a syllogism is logically necessitated, then it’s valid, even if the premises are completely false.
Soundness requires that the premises are true.

Now man the fuck up and admit you are wrong, sit the fuck down, and listen to people who know about topics you don’t and ask questions when you have a thought rather than flat out proclaiming it to be true simply because it seemed like it probably was to you. God forbid you actually learn something.
Jesus fucking christ, you’re infuriatingly dense.

See, I told you you were either determined to misrepresent or unable to understand a simple point.

A function represents a quantity in terms of its making, it’s not “its history before the quantity was ever represented as a number” like sex is before the birth of a child.
A number can subsequently be formulated into a function, as well as vice versa, but a baby can’t subsequently become the sex that created it as well as vice versa.
Your analogy is misleading - anything to avoid learning a correct solution to something that’s already been solved, right? You add in this extraneous temporal element - presumably because the act of working through a function takes time before you work out the answer, even though the answer was already there before you even started working.

You go on about others equivocating when they aren’t, and it’s all you fucking do. It comes up time and time again - all you want to do is stop thinking at the point where things seem like they support your premature presumptions rather than pushing the thinking all the way through to the end, having sufficiently countered it from all possible angles - like mathematics already has done long before you poked your nose through the doorway and presented yourself like you knew it all already. If you actually were a mathematician you’d sufficiently know about these possible counters to challenge your naive ponderings, and give them the appropriate context to justify the certainty of your assertations. You probably think I’m just flexing when I keep reinforcing this point - but it’s got absolutely nothing to do with me. It’s a simple fact of objective logical thinking, which you need to at least begin to learn before you can start even thinking about presenting your points like you are doing.

It’s NOT a logically valid argument.

Oh wow You really have no ability to change your mind do you Literally at the most basic logical level when the simplest concept is presented to you in the simplest possible way, you can’t/won’t admit you had it wrong You’re handing over your incredibility to everyone on a silver platter - YOU still can’t/won’t see it, I’m sure, but at least now everyone else can and will at the most basic level.

Did you even read the link?

The syllogism you presented:

1. All numbers are shapeless.
2. All horses are numbers.
3. Therefore, all horses are shapeless.

The link presents this argument:

1. All cups are green.
2. Socrates is a cup.
3. Therefore, Socrates is green.

In both cases at least 1 of the premises are false - perhaps there’s a degree of dispute over whether all numbers are shapeless (depends on the abstract or concrete representation etc.) - so there’s even grounds to say both premises are false in both cases.
And yet!
The link CLEARLY prefaces its argument as follows: “The following argument is of the same logical form but with false premises and a false conclusion, and it is equally valid” - and it’s comparing said validity with "the following well-known syllogism:

1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal."

EQUALLY VALID.
This is what validity means.
“Therefore, Socrates is mortal”, “Therefore, Socrates is green” and “Therefore, all horses are shapeless” - ALL EQUALLY LOGICALLY VALID, given their preceding premises.

Only the first of these three is a true conclusion, the latter two are false conclusions. It’s the form that makes them each valid. Not the truth or falsity of the conclusion - that’s an issue of soundness.
Scroll down just a tad to the “Soundness” section, and it will very clearly tell you “Validity of deduction is not affected by the truth of the premise or the truth of the conclusion.” and “In order for a deductive argument to be sound, the argument must be valid and all the premises must be true.”
As the link clearly demonstrates with yet another example, “the initial premises cannot logically result in the conclusion and is therefore categorized as an invalid argument.”:

1. All P are not Q.
2. S is a P.
3. Therefore, S is a Q.

That’s what “NOT a logically valid argument” means.

So now.
As I said - sit down, man-child, and man the fuck up and admit you were plainly and clearly wrong.
LISTEN to people who know what you’re talking about and ask questions.
Stop fucking acting like every possible thought that occurs to you is undoubtably unequivocally true and everything else is completely false.
Grow up and learn something for the first time in your life.
Maybe then we can put to rest this joke of a thread - we’re all waiting on the slowest bulb in the box, and that’s you, Magnus.

The first syllogism (the one I presented) has true premises. “All numbers are shapeless” and “All horses are numbers” are both true. However, the conclusion does not logically follow and this is because the word “horse” means one thing in the second premise and another in the conclusion. In the second premise, it means “number” and in the conclusion it refers to an animal. And that’s precisely why the conclusion does not follow. It’s an instance of equivocation. It looks like it logically follows but it doesn’t really.

You need to READ and UNDERSTAND my posts before you declare a victory.

You need to listen to your own advice.

You need to sit down, man-child, and man the fuck up and admit you were plainly and clearly wrong. You need to LISTEN to people and ask questions if you suspect you don’t understand what is it that they are saying (instead of presuming you know what they are talking about.) If you don’t understand what they are saying, but you nonetheless proceed to attack their position, you will end up attacking a strawman i.e. a position they do not hold. Stop fucking acting like every possible thought that occurs to you is undoubtedly unequivocally true and everything else is completely false. Grow up and learn something for the first time in your life. Maybe then we can put to rest this joke of a conversation between the two of us – we’re all waiting on the slowest bulb in the box, and that’s you, Silhouette.

Your number one problem is that you’re a control-freak. You are utterly incapable of holding a civil conversation with people who disagree with you and you are always looking for a way to program them into being right (instead of merely addressing their arguments and letting them figure things out on their own at their own pace regardless of how wrong they are.)

Anyways:

Equivocation isn’t logically valid.

This is not a logically valid argument:

1. Only man is rational.
2. No woman is a man.
3. Therefore, no woman is rational.

That’s because the word “man” means one thing in the first premise (it means “human”) and another thing in the second premise (it means “male”).

It’s not that the premises are false. The premises are true. It’s that the conclusion does not follow from the premises.