1=.999999...?

[gay_Aleks]

Link to video.

That never happened.

 

[Sonereal]

That isn't even a good song.

 

[gay_Aleks]

That isn't even a good thread.

Fits in perfectly!

 

[Terxpahseyton]

Link to video.

 

[Sonereal]

Sorry Tolni, but Terx's music game is fire.

 

[gay_Aleks]

Foiled again! I must admit defeat.

 

[uppi]

Luckily, for all intends and purposes we so far, and in the foreseeable future, and probably forever, have to consider, infinitely close is just as good as being identical. So I see no reason why we would have to take it away from what you call "real analysis" (and which I would call "useful analysis" if we have to call it in such judging manner) and which, also, with much ease explains why the assumption that 0.999... = 1 works just as well weather it is true or not.
So the problem you see seems to be without any of the consequences you imply and hence not like a problem.

As much as I employ the "eh, close enough" approach myself, this is a dangerous thing to do. If you take out central elements of real analysis (a technical term with no judgement intended by me), it is unclear what happens. A lot that is based on that might come tumbling down. If something is just infinitely close instead of exact and then you go and do this an infinite amount of times, the result is totally undefined (the what is infinity times 0 problem).

Or it might work and you end up with some sort of para-analysis that has different axioms, but ends up with the same usefulness. Who knows? If people want to try to construct something like this, I have no qualms with that. But they should come back, when they have shown that there is a consistent system instead of making vague statements based on pseudo-math.

Regarding "whether it is true or not": What kind of truth are we talking about here? Within he axioms of standard math it is undoubtedly true. Within some kind of para-math it might not be true. So what do you think is the standard here?

So I am unclear what you think the implications of "as good as science can prove such things" are, considering that my objection towards the idea of a finite nature of physics regarding its downward scale are entirely based on the limits of what science can prove in that area.

And now you seem to agree with my original statement.
But in the beginning, you told me "Yes", as in yes, I can prove to you that this is not just an impression due to measurement limitations.
edit: But I agree this is irrelevant to the actual question. Which from my POV is weather 1=0.999... violates the rules actual quantities operate on (rather than the rules of the tautology called academic mathematics)

Let me rephrase that: In science there is no absolute proof, but the wiggle room can become extremely (not infinitely, I would guess...) small. To that standard, we have proof that there are fundamental limits to our knowledge of quantities, independent of the measurement device. That fundamental limit to our knowledge could also be a fundamental limit to reality itself. Whether it is, is a matter of debate, but in my opinion, the evidence is strongly in favor of that.

There needs to be a kind of abstraction that is grounded in reality and one that is not.
For instance, I'd argue that the rules of a sport are not grounded in reality. They are probably best understood as a matter of pure will. They are not „dictated“ by an a priory existing fact of life.
Do I have a say over the pieces of an apple I see?
Are quantities more abstract than physics?

The reality of the instances of sth and what this actually means. Math is supposed to articulate that, as you yourself said. I am saying that while academic math does a fine job in usefully articulating it, it not always entirely correctly articulates it. Whereas in the case we discuss this incorrectness is virtually non-existent. But only virtually.

I agree that there are abstractions that cannot be described as grounded in reality. That does not answer what an abstraction grounded in reality is supposed to be. You do have a say over the pieces of apple you see: You could see them as a collection of very tiny apple pieces on a molecular level. You could see it as a fraction of one whole apple. Or you could see it as a fraction of an apple tree, or apple carton in a store. Even if we constrain ourselves to separate pieces of apple, what counts as separate? If two pieces slightly stick to each other, are they still two pieces?

There are abstractions that are useful to describe reality. But of course, these abstractions are never a perfect image of reality. Therefore I do not think it makes any sense to say anything about the truthfulness of abstractions. Certainly not by comparing them with reality, as this would be a test that every one of them fails. The only thing one can say, is whether abstractions are internally consistent.

 

[onejayhawk]

We have number sets where .999... =/= 1, however reals are not one of them. If you're speaking in terms of real numbers, .999... = 1 is objectively true, and unless you explicitly state that you're working outside of reals, then the assumption is that you're working with reals.

No it is not objectively true. 0.999... is an element of (0,1), while 1 is not. Two equal numbers cannot be in a set and it's compliment. This is the real numbers.

J

 

[onejayhawk]

So you are a statistician? My guess that you've been to maths courses, but the emphasis was on the applications and not the proofs was right? The proofs were presented to you, but you brushed them of as mathematicians' nitpicking?

That the limit (when it exists) is unique is something that's explained usually at the first course of undergraduate maths too: If a sequence (a_n) had two limits, say a and b. Then it would be:
For all e>0 there is a n_ea such that |a_n -a| < e whenever n>n_e
and
For all e>0 there is a n_eb such that |a_n -b| < e whenever n>n_e.
Now choose e=|a-b|/3 and m = 1+max {n_ae, n_be}. Then:
|a-b| <= |a-a_m | + |a_m - b| < |a-b|/3 + |a-b|/3 = 2/3 * |a-b|.
This is a contradiction, since obviously 2/3* |a-b| < |a-b|.
Thus, a sequence can have only one limit.

As another point, R is not by itself a measure space. It becomes a measure space once you define a measure there. There are multiple choices how R can be a measure space.

Can you also elaborate how R (with a suitable measure) being a measure space has anything to do with the thing at hand?

Also, if you know what a measure space is, then it's odd that you should say that a real number is not measurable (with the Lebesgue measure for example), since that's not how the term is applied.

And lastly, you still haven't explained how some difference "is not defined", since the field axioms for R require that for each x and y in R there is x-y in R.

Now you are starting to feel around the edges of the difficulty.

Limits cannot be unique in the real numbers, but they can be unique in a dense subset, like the rational or algebraic numbers. In such a subset, differences (or measures) are always defined. However, the definition of limits allows an uncountable infinity of points, not just one. However, the relationships of these points are largely undefined.

I like to picture it as a flatspace of pixels. The number of pixels is countable and the set is discrete. Around each pixel is a halo, representing the associated continuous elements. We have no definition for two points that are continuous, yet we have sets that are continuous. In the pixel paradigm, all the "halo" points continuous with the pixel point, for some hypothetical definition of continuous points. The measure would appear to be zero. Yet this uncountable set is composed of distinct points. The unique part is the element of the discrete set.

Limits, and several other definitions and methods, are a way to reduce a continuous set to a much more tractable discrete set. This whole thread is a reminder that it is a simplification. I like your term "suitable measure". The point is that there is none, yet there are uncountable points.

J

 

[brennan]

I suppose brennan ultimately referred to how consistency always is self-contained, so you can wrap up whatever freaking nonsense you like as long as you do the work to have it somehow account for what is established knowledge and to have it be internally consistent.

Yup.

There are no a'priori truths in rigorous philosophy. Any internally consistent worldview is equally valid. Nobody has ever come up with a method of proving a damn thing beyond doubt.

 

[Sonereal]

Any internally consistent worldview is equally valid.

No it isn't.

 

[hoplitejoe]

Yup.

There are no a'priori truths in rigorous philosophy. Any internally consistent worldview is equally valid. Nobody has ever come up with a method of proving a damn thing beyond doubt.

turns out, real philosophy was just accepting everyone views without question.

socrates would be proud.

 

[Terxpahseyton]

@uppi
I was preprepared and already working on giving you a comprehensive response, when I realized that we were just diverging from the main point of mine (which you have avoided to address) into different meanderings of philosophical topics, and had the impression that until we together rethought the whole of philosophy, we won't get anywhere, and I think I speak for both us that this not a path we want to or will walk.

So please do me a favor and forget philosophy and math and whatever education is occupying your head for a minute and let's focus on simple primitive causality, the kind even my dog comprehends.

If I repeat something as much as I want, 10 times, 1000 times, 1000000000000000000000000000000000000000000000000000000000000000000000000000 times... and it doesn't achieve what I want to achieve - and if moreover I know for a fact that no matter how much countable times I try it won't work - how does it make sense to propose that doing so infinitely achieves it?
Can you answer this question? Infintely saying "I shall be a god" will not make me a god, because saying "I shall be a god" is well established to not make me a god. Similarly, if I add 0.09 to 0.9, it wont be one, and if I add 0.009 to 0.99 it wont me one and I can continue this until I grow old and die and it still won't be one and I know that my ancestors can do the same thing forever and it never will be one but simply stating that it goes on for forever makes it one because... reasons. Because it is consistent within mathematical rules. But the question how this illogical feat is accomplished never is even worth mentioning. Because math says it is true. Because a tautology says so.
Tautologies are by definition meaningless beyond their own system of axioms and conclusions... So IMO only a fool would even consider answering such an inherent contradiction with math rules. You may just as well say because Jesus told me.

 

[hoplitejoe]

Can the statement 1=0.9999... even exist in a "primitive causality" as naive as yours? Your insistence on removing mathematics from a mathematical question is enough to make the question unintelligible.

Thus I put to you, does 1=0.9999? The answer is blob.

Edit: a more useful response because apparently there is that little to do in my life.

If I repeat something as much as I want, 10 times, 1000 times, 10000000000000000000000000000000000000000000000000 00000000000000000000000000 times... and it doesn't achieve what I want to achieve - and if moreover I know for a fact that no matter how much countable times I try it won't work - how does it make sense to propose that doing so infinitely achieves it?
Can you answer this question?

This is just really basic mathematics again, go onto wolfram alpha and plot y=1/x, what you will see going up the y bar is a line that looks like it is going to cross the x axis, but as it gets closer to the x axis it starts going more and more vertical, till it looks like it should be touching, but it isn't (if you look at the equation this is for obvious reasons).

Drawing this manually we can never reach 0, as we can never make a line infinitely long, so in a physical case, the question of does the line ever touch 0 is meaningless, as we have no such line that we could test it with, as it is a physical impossibility. With mathematics, we can extend to this hypothetical (physically impossible) limit of infinity, and say, yes at the limit the line touches 0.

This is equally true for 1=.9999..., if you want you can easily make a graph which has the same thing around the number 1. You can't seperate this question from the mathematics behind it, to do so makes the question pointless.

 

[Terxpahseyton]

Imagine your ancestors never die out. There will always be distant ancestor of yours. Moreover, this ancestor will always dutifully continue to try to write the number of 0.999.... Generation after generation they ad another digit, another "9" to the number.
And never dieing out, they will continue to do so forever.
Will they ever reach 1?
Of course not. They can't.
However, when math says that 0.999... rather than being continued forever does continue forever, it suddenly is 1.

Explain to me how that does make sense. Without axioms you pull out of your heat, even if that heat is "math".

Guess what. You can't. So as far as I am concerned, you simply ignore the contradiction. Great argument. But hey, you know math, so surely that doesn't matter...

 

[azzaman333]

No it is not objectively true. 0.999... is an element of (0,1), while 1 is not. Two equal numbers cannot be in a set and it's compliment. This is the real numbers.

J

0.999... is not an element of (0,1) when working in the set of real numbers.

 

[onejayhawk]

0.999... is not an element of (0,1) when working in the set of real numbers.

Of course it is. It's constructed to be a member. Only the limit lies outside the set.

J

 

[azzaman333]

Of course it is. It's constructed to be a member. Only the limit lies outside the set.

J

It is not constructed any more than pi is constructed. It's a number with a value, and the value does not fall within the set (0,1) in the set of real numbers.

 

[Kyriakos]

Yup.

There are no a'priori truths in rigorous philosophy. Any internally consistent worldview is equally valid. Nobody has ever come up with a method of proving a damn thing beyond doubt.

That is quite correct, yes, in the views of many ancient notables too (including Socrates).

The reason it is true is that we neither sense nor express what we sense or think in a manner which itself is not a medium: we are not 'directly' sensing anything, we use our particular human organs/mind to do so. Moreover we do not 'directly' communicate anything, we use language which itself is not discreetly tied to what we sense or think.

As a sidenote of some importance, in at least late 5th century/onwards greek philosophy this is a very fundamental issue, namely that the various syllogisms or even dialectics (which itself is not about 'truth', btw, but about constructing a rigorous system of arguing while trying to be consistent, which is why Aristotle wanted to replace it with 'syllogism' that is more tied to axioms and thus can be inherently 'true' in the confined system it rests upon) are not presenting something we can pick up as 1-to-1 referring to our own individual thinking system or anyone else's. We only sense our own thinking process. So Socrates did argue that even if one makes a statement that is 'clearly' false (eg: if you jump from the top of a high building you will fly away) then provided the person stating is is indeed of the honest view it is true we have to note that while it is false in the system commonly used, it is true for what this person had in mind (maybe he thought that 'fly away' means 'free-fall'; maybe he thought of something even weirder, but inherently consistent, etc, maybe he even was talking in full allegory consciously or not).

 

[hoplitejoe]

Guess what. You can't. So as far as I am concerned, you simply ignore the contradiction. Great argument. But hey, you know maths, so surely that doesn't matter...

If you are trying to think this using real world logic you are never going to get it. As soon as you try to imagine an end to the line of nines, the number stops being 0.999... and starts being a 0 followed by a certain number of 9's, which is not 1.

0.999... is a mathematical concept, it cannot exist in reality. The fact it can't isn't a "contradiction" because this is a mathematical question, it exists purely in abstract. To ask the question insisting that we don't use maths it pointless as the question relies on maths to make any sense.