Is 1 = 0.999... ? Really?

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?

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.

This broke my parser.

Sorry. But to avoid a tautology. Such implication implies a referential collapse into it.('self)

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.

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.