Is 1 = 0.999... ? Really?

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 #-o

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 :smiley: You really have no ability to change your mind do you :smiley: 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 :smiley: 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.

Hi wtf,

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.

To make the leap to a Transcendental number, a limit must be taken.

There is no formula to find the nth digit in a decimal expansion of Pi so writing the nth digit becomes problematic. It could take hundreds or thousands of years to find, even with the best supercomputers. This makes us resort to what should be the dreaded “…”.

These ellipses are relatively infamous for causing confusion. I have heard of someone earning her PhD in philosophy simply by studying ellipses.

If we were to think of Chomsky, we might expect that mathematicians should better understand ellipses, as they are used in their field, because they should have a common background. But it’s becoming clear to me that at least occasionally we don’t.

Case 1:
If the “ …” means, explicitly, that we are formally taking the limit as n goes to infinity for S(n) and we do the rigorous work to prove it ( or at least give a passing reference to a respected source), then I agree that your representation is correct. However I believe that we should explicitly disclose that we are referring to the limits and not simply to the sequences. I would favor the abolishment of the “…” s term in favor of the limit term in this case.

Case 2:
However, in most cases, “…” is left vague and simply means that we should intuitively follow some perceived pattern. In this particular case, it would lead us to incorrectly conclude that Pi is a Rational number. Additionally, since there is no formula representing the nth digit of Pi, it could lead some to shear madness.

Thanks Ed

Gonna stop you right there at your very first sentence.

“All horses are numbers” is a true premise?
You said “you can use horses to represent numbers”, not horses are numbers.

Horses are numbers is not true.
That humans use specific words (sounds/symbols) and not horses to represent numbers as a matter of convention is true.
Given that centaurs mean 100 is a given, which happens to be false both literally and as a matter of convention. That you COULD use horses “as numbers” has nothing to do with truth, never mind “being a true premise”.

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.)”

Gonna stop you right there at your SECOND sentence too.

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.
All you said was that the conclusion about horses doesn’t follow from the premise about horses.
And now you’re blaming me for not reading nor understanding your posts?
As I’ve just demonstrated, I read you VERY clearly, and now you’re trying to claim that I didn’t read your posts???

My god, reading what you have to say is like wading through thick shit - and so far I only covered the very first two sentences that you managed to shit out.

But whatever - this is why I ignore most of what you say - it’s worthless drivel.

I’ll close this “joke of a conversation between the two of us” by telling you I’m so relieved you are aware that a syllogism isn’t valid if a conclusion treats two words across two different premises as meaning the same thing - when in fact they mean different things - in order to come to a conclusion that would only be valid if they did mean the same thing.
See, aren’t I nice? See how I recognise the rare thing you say that isn’t completely false?

So now that I’ve validated a single thing that you’ve said, you can go back to continuing your presentation of yourself as an expert on mathematics, having declared yourself as no expert on mathematics, and complain that it’s others who are equivocating.
FYI, I’m not trying to program you into being right, I’m trying to get you to sort out your attitude - 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.

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?

Yes. That’s because the word “horse” has been defined to mean the same thing as the word “number”.

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).

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.)

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

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.)

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.

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.

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.

But that has no meaning till we give it one. It’s not defined till we define it. Just like (\otimes).

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 + …?

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.

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.

en.wikipedia.org/wiki/Commutative_ring

en.wikipedia.org/wiki/Function_space

Yes, but more yes than no? Yes?