Your argument rests on this: infinity minus infinity is meaningless but (0.\dot9 - 0.\dot9) is equal to zero. Not sure how to say it but don’t you realize that infinity is equivalent to a sum of ones? Like so: (\infty = 1 + 1 + 1 + \cdots). And that (0.\dot9) is also equivalent to a sum of numbers? Like so: (0.\dot9 = 0.9 + 0.09 + 0.009 + \dotso). Now, you take (0.9 + 0.09 + 0.009 + \cdots) and you subtract from that (0.9 + 0.09 + 0.009 + \cdots) AND YOU HAVE NO PROBLEM WITH THAT. You go like “Hm, okay, take first numbers from both sums, subtract them, we get – hehe – zero! Now, let us take second number, what do we get? oh yeah baby, we get yet another zero! and so on and so forth!” But you take (1 + 1 + 1 + \cdots) and subtract from that (1 + 1 + 1 + \cdots) and you tell me it’s MEANINGLESS? As in meaning-fucking-less no less. Caps lock doesn’t have big enough letters to communicate the bewilderment I am currently suffering through 
What is going on in here?
Infinity ((\infty)) times infinitesimal ((\frac{1}{\infty})) equals (1). No doubt about that. But (0.\dot9) is a different beast – one that never adds up to (1).
That’s false Magnus. Infinity times infinitesimal still equals infinity.
Sure is fascinating to think about.
You need to level up.
It’s not either true or false per se but either true or false in relation to the starting premises. If you start with the premise that (\infty + \infty = \infty), then yes, (\infty \times \frac{1}{\infty} = \infty \div \infty = \infty). But if you start with the premise that (\infty + \infty = 2 \times \infty) a different conclusion follows.
The important thing is that, whatever premises you pick, the mentioned Wikipedia proof is invalid.
I want very badly to dismiss this argument, but I am forced to admit it’s stronger than I hoped. The best response I have, which is not much response at all, is that it is question-begging: if (.\dot9 = 1), then it isn’t true that there are zero 1s when there are 9s in every decimal place.
I think we need to be careful with our language. “An infinite number” is not the same as “the number ‘infinity’”. The former treats infiniteness as a property of multiple numbers, the latter treats infinity as a number.
This might seem pedantic, but I think it matters. If there’s more than one infinity, if it can be a property of a number but not a number itself, then the equations using (\infty) as a number are at best ambiguous, so our answers will be uninsightful.
It isn’t. Both (9.\dot9) and (0.\dot9) are finite numbers.
I think those answers seem intuitive, but I think they’re playing fast and loose with notation that has a formal definition that makes those equations meaningless in a rigorous sense. Using the symbols that way is a sort of expressive math, an uncertain poetry built of the symbols of rigorous expression.
I’m willing to talk about what (\infty - \infty) might mean, I just don’t think it’s well-defined (“meaningless” might be too strong). (\infty) can refer to multiple quantities that are provably distinct. I’m not sure what the subtraction symbol means here, given that I don’t think (\infty) is a number. My most natural reading is set subtraction.
So let’s play, but I don’t want to play calvinball.
But we agree that “complex numbers” is not a category, right? Are you defining a new category with (L)?
It is the diminishing value of each place that is filled. I don’t know how to notate it but it’s the .1 in an infinite number of decimal places where it comes up short to clear the decimal point and increment the digit on the “whole” side of the decimal. In it’s form, .9 recurring; it is presented as a fraction and I don’t think infinity can change that. It may simply be the largest fraction possible. And with little more then a place holding decimal point between them they could be equal. A fraction turns into a whole through infinite recursion. Fraction or integer?
And what happens when you multiply .49 x 2, would the result not be the same in an infinite number of decimal places? Try the operations reciprocal.
1 x 2 = 2 and 2 / 2 =1. If .9 recurring is equal to 1 then the math should be that simple.
When you can invoke Murphy’s law in math? You’re having fun.
And what exactly is the effective difference?
(\infty - \infty) is the same as ((1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots)) which is in the same camp as ((0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots)). If you’re going to say that the former is meaningless, what allows you to say that the latter is not? Or vice versa, if you’re going to say the latter is equal to (0), what allows you to say that the former is meaningless?
Meaningless: that which has no meaning. When speaking of symbols (such as words) it refers to symbols that have no meaning assigned to them. By whom? Well, by someone, usually the one using those symbols. In most cases, it’s a way for people to avoid bothering to understand what the other side is trying to communicate, as in “Look, I can’t bother to understand what you’re trying to say, it’s too difficult and/or time-consuming for me, so I’m just gonna conclude that you’re not saying anything, that your statements have no meaning whatsoever, that they mean nothing, that they are meanignless”.
You can do arithmetic with (0.\dot9) but you can’t do arithmetic with (1 + 1 + 1 + \cdots)? Really? You can do arithmetic with convergent series but you can’t do arithmetic with divergent series? W-why? Is it perhaps because you’re REFUSING to do so? As in, you don’t wanna do it because it does not support your present conclusions?
Look closer at what you’re saying.
You’re saying that ((0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots) = 0) makes PERFECT SENSE but that ((1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots) = 0) makes NO SENSE.
Are they infinite sums of non-zero terms or finite sums of non-zero terms? They can’t be both. Their RESULT can be a finite number, sure, just like how the result of (1 \times 1 \times 1 \times \cdots) is a finite number, but they are nonetheless expressions involving an infinite number of terms. The entire point is that WE’RE DOING ARITHMETIC WITH AN INFINITE NUMBER OF NON-ZERO TERMS. And if we can do it in some cases (such as with (0.9 + 0.09 + 0.009 + \cdots)) why can’t we do in other cases (such as with (1 + 1 + 1 + \cdots))?
You think that ((1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots)) is not well-defined and at the same think that ((0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots)) is well-defined?
How are complex numbers not a category?
It is an edge case, and so when it’s put this way it seems very ad hoc. But this is kind of an ad hoc way of putting things.
I have a related question: is (0.\dot9) rational or irrational? If it’s rational, what’s the ratio?
I can think of a few. The biggest is that “an infinite number” is different from “the infinite number ‘(\infty)’”: the former implies multiple infinities.
Another is that numbers have properties and operations, and I don’t think any standard operations have an obvious meaning on infinite numbers, and I’m not sure what properties apply or what we could use to infer those properties.
Why? Why not the same as ((2+2+2+…) - (3+3+3+…))? Especially considering that you probably don’t think ((1+1+1+⋯) = (2+2+2+…)), why should we pick one or the other as a stand-in for (\infty)?
So I guess my answer to your question is that I don’t think (\infty - \infty) means the same thing as ((1+1+1+⋯)-(1+1+1+⋯)).
Though I later acknowledge that “meaningless” is perhaps too strong, here I mean it as saying that just because you can string some symbols together doesn’t mean that they express a coherent concept. A “square circle” is meaningless in the sense that, even though the words that compose the phrase are perfectly meaningful, the phrase doesn’t point to a coherent concept.
2 points:
- (.9) can be expressed as a convergent series, but it isn’t a convergent series.
- A divergent series is undefined in the limit. So we can do arithmetic with the limit of a convergent series and not with the limit of a divergent series, because we can do arithmetic with things that are defined and not with things that are undefined.
They are the finite limits of infinite sums. There is no tension at all there. And again, they can be expressed infinite sums, but they are numbers. I’m about 95% sure that every number can be expressed as multiple infinite sums.
Sorry, typo, should have said “‘complex numbers’ is a category”. I was replying to your rejection of identifying what kind of number (L) is, and you offered the argument that (i) does not fit into any pre-complex-numbers categories. But we agree that complex numbers are a category of numbers. So (L) too should fit into some category. So you should be able to say that it’s in some category, or it’s some new number like (i) was and it needs a brand new category of numbers, and you have some rigorous definitions and properties of numbers like (L) (or even some vague, hand-wavey, there-should-be-something-(L)-shaped-in-roughly-this-vicinity type answer).
You can pick any.
So you don’t think that (1 + 1 + 1 + \cdots = \infty)?
Obviously, you refuse to understand the meaning that your interlocutors assign to the symbol (\infty) so maybe I should just stop using it, instead substituting it with an infinite series.
What does ((1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots)) equal to? Is it (0) or is it (1 + 1 + 1 + \cdots)?
If you say it’s (0) then the second part of this post is irrelevant to you and so there is no need for you to respond to it.
“Square circle” is a logical contradiction. Are you saying that (\infty - \infty = \infty) is a logical contradiction? I don’t see why. Remember that I am not the one claiming that (\infty - \infty = \infty). On this forum, that would be Phyllo. Outside of the forum, that would be mathematicians.
That’s like saying (2 + 2) can be expressed as (4) but it’s not (4). In a way, it’s true. They are two different expressions. They aren’t the same expression. But also, they are expressions representing the same thing. They are two different expressions representing the same idea. In that sense, (0.\dot9) and (0.9 + 0.09 + 0.009 + \cdots) are two different expressions representing the same idea.
The Wikipedia proof I constantly quote does not do arithmetic with the limit of convergent series. It does arithmetic with convergent series.
They start with:
(x = 0.999\dotso)
Then they multiply both sides by (10).
This gives them:
(10x = 10 \times 0.999\dotso)
Then they substitute (0.999\dotso) with (0.9 + 0.09 + 0.009 + \dotso) telling us that the number of terms in the sum is (\infty).
This leads to:
(10x = 10 \times (0.9 + 0.09 + 0.009 + \dotso))
How do they calculate the result of the right side of the expression?
Certainly not by asking “What’s the limit of (0.9 + 0.09 + 0.009 + \dotso)?”
What they do is they multiply every term by (10), like so:
(10x = 10 \times 0.9 + 10 \times 0.09 + 10 \times 0.009 + \dotso)
This leads to:
(10x = 9 + 0.9 + 0.09 + 0.009 + \dotso)
Note that the number of terms in the resulting sum is the same as before: (\infty).
Then we “split off” the integer part:
(10x = 9 + 0.999\dotso)
The number of terms in the resulting (0.999\dots) is (\infty - 1) since the resulting sum is produced by removing one term from the previous sum. (\infty - 1), they tell us, equals to (\infty), which makes the number of terms the same as before. However, saying that (\infty - 1 = \infty) means that (\infty - \infty \neq 0) making (\infty) a non-specific number.
Thus, we can substitute the resulting (0.999\dotso) with (x), like so:
(10x = 9 + x)
Then we subtract (x) from both sides:
(10x - x = 9 + x - x)
And this is where the problem lies. We don’t get what they tell us we get.
We don’t get (9x = 9) because (x - x) is not (0) since (x) represents a non-specific number.
AN INFINITE NUMBER OF NON-ZERO TERMS MINUS AN INFINITE NUMBER OF NON-ZERO TERMS IS NOT ZERO NON-ZERO TERMS IF WE ALREADY ACCEPTED THAT INFINITY MINUS INFINITY IS NOT ZERO.
THAT’S THE ENTIRE POINT.
… So (1+1+1+\cdots = \infty = 2+2+2+\cdots = 2\times\infty)?
But (\infty -1 \neq \infty)?
It is charitable to assume that I’m capable of understanding, but uncharitable to assume my interlocutors have made their meaning clear.
I don’t know. I agree it seems intuitive, but I think that’s deceptive. By associativity, (1+1+1+\cdots = (1+1)+(1+1)+\cdots = 2+2+2+\cdots )
(= 1+(1+1)+(1+1+1)+\cdots= 1+2+3+4+5+\cdots) And I have it on good authority that that last expression is equal to (-\frac{1}{12}), but I don’t know why.
And ((1+1+1+\cdots)-(1+1+1+\cdots)) could mean taking each term in the one on the right and subtracting it from the one on the left. But normal algebraic manipulation also permits a pattern of (1+1-1+1+1-1+1+1-1+\cdots).
I also take your point that this has implications for (9.\dot9-9), but it is noteworthy that this doesn’t have similar problems. There’s no way to change the outcome by mixing different parts of the series, because the parts don’t overlap: (9 + .9 + .09 -.9 + .009 + .0009 - .09 + \cdots) That difference seems significant, what do you make of it? I have a suspicion that this is related to the divergent/convergent distinction (e.g. that the formal way of saying that is true for convergent series and not for divergent series; that is a weakly held belief, like 65% confidence).
I’m saying that it’s meaningless in the sense of not pointing to a coherent concept. Another example that doesn’t use a logical contradiction would be “the number of dreams it takes to power a Saturn V rocket”. All the words are meaningful, but they don’t point to a coherent concept.
Right, but the idea and the expression are distinct. That you can use an infinite series to express the idea does not make the idea infinite. Or, if we want to insist that it does, then it’s true of any number that can be expressed as an infinite series, which I’m pretty sure is any number. And so in that case there is no distinction being made.
This is not my understanding of that proof.
To give an analogy for the argument I understand you to be making: the number 2 can be expressed as the product of two complex numbers (-2i\times i). So you point out that (0.\dot9) can be expressed as an infinite series, and so you treat any math using (0.\dot9) as “do[ing] arithmetic with covergent series”, but I wouldn’t call the expression (2+2), “do[ing] arithmetic with [complex numbers]”.
Am I to understand that you don’t think any convergent series equals an integer? Otherwise this is completely breaking of like all of algebra.
It is no mere hyperbole to assert that the seemingly moat simple things are the most complicated?
There must be a transformation at the end of the line, such as was described by meno, in his paradox , such as later treats partial differentiations in a reassamblad Fourier transformations; where the transformation encompasses matter/energy mutations?
The question is what’s the procedure by which we find the result of subtracting one infinite sum from another. There must be such a procedure, right? Otherwise, how can we say that (0.\dot9 - 0.\dot9) is (0)?
Given ((a_1 + a_2 + a_3 + \cdots) - (a_1 + a_2 + a_3 + \cdots)), how do we calculate the result? And you seem to be arguing my case (: Do we pair the terms like this ((a_1 - a_1) + (a_2 - a_2) + (a_3 - a_3) + \cdots) resulting in (0) or do we pair them like this ((a_1 + a_2 - a_1) + (a_3 + a_4 - a_2) + (a_5 + a_6 - a_3) + \cdots) which may be non-zero say if every (a_n) is equal to (1). And to make it worse, the number of ways we can pair the terms seems to be infinite . . . so which way is the right way?
How can you say that (0.\dot9 - 0.\dot9\ = 0) without having an answer to that question? Consider that ((0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots) = 0) only and only if we pair the terms like so: ((0.9 - 0.9) + (0.09 - 0.09) + (0.009 - 0.009) + \cdots). But as you say, we can also pair the terms like so ((0.9 + 0.09 - 0.9) + (0.009 + 0.0009 - 0.09) + (0.00009 + 0.000009 - 0.009) + \cdots) in which case the result isn’t zero but approaching zero. And this is a serious problem. It would mean we’re trying to prove that (0.9 + 0.09 + 0.009 + \cdots = 1) by taking for granted that (0.09 - 0.0801 - 0.008901 - \cdots = 0). That would make it . . . not much of a proof. And certainly, noone ever mentioned this anywhere in the proof.
?
But there is no end of the line.
Because (0.\dot9) isn’t an infinite sum, any more than (4) or (\pi) is an infinite sum. (0.\dot9) is a real number, and 0 is the additive identity, so (0.\dot9-0.\dot9=0)
Yes. Also not much of a disproof.
But we aren’t doing something repeatedly, and nothing is changing. I think this is unintuitive but crucial: (0.\dot9) is not changing, it isn’t moving, it isn’t approaching; it is static.
Also, still wondering about your answer to the rational/irrational question.
Opps, was hoping to get back to edit before someone responded.
So the argument was in evaluation of the question 1 = 0.999… ? Really? And I am continuing with decimal notation.
The quoted part is the result of the rephrasing.
So sorry, I didn’t mean to imply any thing other then our minds were changing.
In other words change the question from a math question to a true false question.
Is 0.9 equal to 1. False
Is 0.99 equal to .1 False
Is 0.999 equal to .01 False
This is a pattern that is represented infinitely by the expression 0.9 recurring = 1, and the answer is… it is infinitely false that 1 = 0.9 recurring. From the first decimal position to the infinite, all at once.
What happens when logic and math disagree?
And I respect Magnus for smelling something fishy in the mathematical proofs. This idea of the infinite and how it is applied, is mathematically slippery.
“But there is no end of the line.”
The end of the line rests on the argument that has expanded from meno through the geometers through the conical hyperbole of calculation, which lead to the quantic/uncertainty generated partially reconstructed Fourier series which defined the MC2 transformations.
In fact, it could not have been surmised inductively, anyway.
Meno was right, and Leibnitz through Kant showed metaphysics to have been shut reductively
So how the connect, the touted synthesis?
Maybe Russel’s sense data is right on point, after all, as right as the well repeated proposition:
‘if God does not exist, he had to be created’
The point about transformation is significant because in a sense, something does come from an appearent Nothingness, and that is the point to the underlying functional math analysis , where partial differentiations are subsumed by the calculus of indiscernibles.
So the 'end of time’s is inherent in the declaration that -(.99=1.00) ; because of the same partially derived reconstruction by Fourier transform, validates such a pro-position.
The end is transformed from a logical level, (Russel-Wittgenstein) to one that is inductively not reduceable, is subsumed by the differential levels of quantifiability.
Put it in linear language, upon whose architecture the modality of it has to accorded to: simply: the end(s) justify the means of the reconstruction- that, the original transformation can be signified,
The transformed end can not be defined within but without recourse to a reconstructed generation of progression: from meno through Leibnitz/ Kant, all the way to the missing key of Principia Mathematica:
The ‘sense data’.
This hotly contested concept, overcomes the epoche within which liberal arts still vanquishes, unable to mimic its more quantitatively able cousin , the human brain.
The end of any road is the preintegrated calculus of missing pieces minus the ones left unfilled, but such a presumption fails, unless a transformative filler can replace it.
So, the end of the road could not be in a one dimensional map, but does exist in the calculable certainty in the existence of the perfect monad.
Without that, atomism would/ could not have come about signaling the need: to overcome it.
I shared this with St.James on occasion, and I think there was some concurrent albeit partially limited agreement, there.
The transformed end can not be defined within but without recourse to a reconstructed generation of progression: from meno through Leibnitz/ Kant, > and all the curious minds asking the questions here at ILP, and everywhere < all the way to the missing key of Principia Mathematica: >and beyond <
Come on everyone… in your best Buzz Lightyear impersonation.
Amen.
I have just changed my vote from other to no; pseudo confidently.
Curious if the argument has come up else where in the discussion.