Just basically speaking, this means an infinity of the finite. (The concept orange is finite) (Contradiction) or as mathematicians put it, “bound infinities”.
How do you bound an infinity (boundless)? (Contradiction)
I don’t get the gist of your argument. Are you saying there aren’t enough points to fill in all the gaps (I said points, not inches, but…), or that as soon as the gaps are out of sight, they stop being replaced?
If not enough points, how do you have not enough with an infinite amount?
Are you saying that we eventually run out of points to fill the gaps, and after the last point there’s nothing but gap? ← That would imply there’s an end to the series of point, and you know how that argument goes.
So what does infinite orange mean? Are you trying to tell me that you can have infinite oranges, but not single infinite orange? Well… if you can’t have a single orange and all those infinite oranges are equal as infinite orange, how can you have infinite oranges?
I’ve never joked, let alone joked sarcastically on ILP, even though in real life I do both abundantly.
Bound infinity has EVERYTHING to do with the discussion… it means infinity is a quantity
I’m saying there aren’t enough points. There aren’t enough points to fill the gaps that are within our sight without creating gaps out of our sight.
I am not.
We always have enough points to fill the gaps that are within our sight. But each time we fill the gaps that are within our sight, we create new gaps out of our sight.
Consider that in order to fill a gap, you have to remove a point elsewhere; and that when you remove a point, you create a gap in its place.
Here’s the infinite line with odd inches taken out:
Suppose you want to fill the first gap. How do you achieve that? By choosing an existing inch and moving it from its current place to the beginning of the line. You can pick any inch you want. There’s an infinite number of them. You can pick the first inch in the line. Let us do so. We pick the first inch in the line and move it to the beginning of the line. By doing so, we fill a gap but we also create a new gap. This is what follows:
We don’t get ( \bullet \bullet \circ \bullet \circ \bullet \cdots ). That would be creating new inches out of nowhere.
The interesting part is that you don’t have to pick an inch that is within your sight. You can pick an inch that is outside of your sight. You can pick the 100th inch or the 1,000th inch or the 1,000,000th one. In each case, you’d be creating a gap in its place. But because it’s out of your sight, it’s convenient to ignore it and pretend that the line no longer has any gaps.
It’s a trick. Something a magician would do. It’s definitely not logic.
Ok, I see what you mean. When you move the first point to fill the first gap, you get a gap after that point two points long. Then to fill that gap, you need to move the next two points, which leaves a gap four points long. It seems that as you move the points down, you get an ever grow gap moving in the opposite direction. That indeed brings into question what the line ends up looking like at the end. Do you really get an identical line, point for point, or do you get a line with an infinitely long gap at the other end (somehow still an infinite number of points away)?
This problem arises when you imagine each point taking its turn to fill the gaps. If each point takes its turn, you’d need an eternity to complete the thought experiment and answer the question above. But what about each point moving at the same time? This is how we are to imagine Hilbert’s Hotel. Each guest moves to the next room simultaneously, not one after the other. Of course, in the case of the gaps in the line, each point would have to move a different amount. The first point moves one position, the second point moves two positions, the third point moves three positions, etc…
In either case, I’m still not clear on how we’re defining “longer” and “shorter”.
Come to think of it, I’m not sure two scenarios are any different: two identical lines or one line with an infinite gap an infinite distance away. To say the gap is an infinite distance away is equivalent to saying it’s at the end of the line. But then what’s at the end of the other line? More line? For all intents and purposes, if we’re talking about “the end of the line”, I’d say it’s fair to say that’s where the lines end. So the gap has effectively been push out of the line and the two are once again identical.
(I realize the silliness of talking about the “end” of an infinite line, but I’m just following the logic that would come out of granting talk about a gap at the “end of the line”.)
Existence happens between the theoretical absolutes of 1/0.
There is no ‘one’ in reality - nor a nil. It is a mental abstraction that can refer to anything the mind detaches from space/time and places within vague space/time borders.
‘One’ is an idea, representing an arbitrary moment/place.
Like all ideas it can be defined by the mind - synthesized, manipulated, redefined and redefined, combined, in ways that go beyond the real.
It is a linguistic representation of a mental abstraction, created by the translation of sensual stimuli.
0.9999 is not one…it is a movement towards an absolute that does not exist, and therefore can never be attained.
It represents the fluidity of existence, in relation to the mind’s abstraction.
Why does everyone think that 0.999… has to “build up”? Like it’s a process that needs time to complete?
If I say, “suppose you had a queue of 50 people,” do you say, “wait… okay, the 50th person has just been added. Now we can talk about it.” What if I say, “suppose you had a queue of 100 people”? Do we have to wait twice as long before continuing with the conversation?
0.999… is simply notation. It just stands for the idea of an infinite number of 9s. You’re supposed to imagine you already have an infinite number of 9s.
(And really, it doesn’t even represent that; it represents a quantity; the debate in this thread is: what is that quantity? Is it 1 or the next number before 1?)
Thanks for the masterclass on carrying and borrowing - you could have stuck to your kid’s website though - it’s that basic. No wonder you’re more qualified than professional mathematicians to determine these issues.
Are these processes how you’re distinguishing the standard algorithm from addition “in general”?
You realise that carrying and borrowing are only superficial accomodations for numerical systems that use positional notation? Without positional notation, let’s say we’re using a conceptual numerical system that had an infinite number of digits, you wouldn’t need to carry or borrow, because they don’t actually do anything to the quantity itself that is being denoted. The standard algorithm is addition with superficial decoration - your argument that they’re different is like saying a house is different to a house with graffiti on it.
This is probably indicative of a major difference between mathematicians and non-mathematicians: that mathematicians understand quantity at a deeper level since they traverse across multiple different representations depending on the job at hand and the best way to go about it. They don’t get hung up on the extraneous and incidental, they don’t confuse appearance for essence. This will be why non-mathematicians get so bewildered by (1=0.\dot9) because their intuitions lack adaptability to different representations of the same thing.
It’s not wrong and as mathematicians will tell you - explaining why is always necessary.
Simply saying it’s wrong backed by a vague objection and saying it’s unnecessary to prove it more is a far cry from mathematical proofs that can require obnoxious amounts of backing just to be complete.
Your main objection is that (\frac{9.\dot9}{10}\neq0.\dot9) right?
As above, you’re getting “carried” away by the superficial - thinking that (\frac{9.\dot9}{10}) literally shifts all the digits 1 space to the right (or the decimal point 1 space to the left) instead of that just being appearance, meaning there’s an extra “unmatched 9” at the “end” of the endless sequence of 9s compared to the “end” of the endless sequence of 9s in (0.\dot9)
The whole reason I explicitly used this notion for (\frac{s}{10}=\frac{9/{10^0}+9/{10^1}+9/{10^2}+…}{10}=\frac{9/1+9/{10}+9/{100}+…}{10}=9/{10^1}+9/{10^2}+9/{10^3}+…=\sum_{x=1}^\infty \frac9{10^x}) was to emphasise the one-to-one correspondence for “each decimal place” in both (\frac{s}{10}) and (s) but by representing each as a fraction instead of a decimal place - to try and help you from getting confused about superficial appearance of decimals.
This matches the (9/{10^1}) in (s) with the (\frac{9/{10^0}}{10}) in (\frac{s}{10}) (i.e. each 9 in the tenths column together for both s and s/10)
It does NOT match the (9/{10^1}) in (s) with the (\frac{9/{10^1}}{10}) in (\frac{s}{10}) (i.e. not the 0.9 in s with the 0.9/10=0.09 in s/10)
No doubt this still went over your head and you still got confused by the appearance of (9/{10^1}) looking the same in either, even though one was divided by 10 in (\frac{s}{10}) (obviously) and the other in (s) wasn’t (obviously).
This was exactly why it was a rephrasing of something you said to me, that “You have no clue what you’re talking about, so maybe you should just stop it and instead stick to the topic, what do you think?”
The whole point of me saying it was to demonstrate to you that what you just said was “one of the most pointless things you can say in a forum discussion.”
God, you miss literally everything…
You wonder why I’m getting so frustrated?
And you think I’m proud of it?!
I couldn’t be further from proud of the fact that even the most clear and obvious explanations mean nothing to you.
I love how you’ve just brought up Hilbert’s Hotel too, because it’s just another one of those thought experiments that demonstrates the indeterminate sizelessness of infinity (the hotel with infinite rooms in it) and therefore disproves all your nonsense about sizes of infinity. There’s still only 1 infinity (the hotel) whatever happens to it, and regardless of any finite operations/adjustments that you make to its rooms, it’s counterintuitively still full whilst also able to have spaces. Operating on infinity does nothing definite to the fact that it’s still infinite.
This is what indeterminate/undefinable/infinite/endless amounts to.
You can emphasise that, no matter how many 9s in (0.\dot9) fill the gap between it and (1), there’s always an infinite number of smaller infinitesimal gaps that could be filled, and I can emphasise that infinite recurring 9s will always fill these gaps and both/neither of us would be fully right, because infinity is undefinable, which is why things like (\frac1\infty) are invalid because it’s the same as saying (\frac1{undefinable}).
And since there’s nothing definite about how many 9s there are in (0.\dot9) (including any “spare 9s” compared to (9.\dot9)), this is the quality of endlessness that changes all your finite understandings of progressions like (0.9+0.1=1) and (0.09+0.01=1) and so on such that (1=0.\dot9) - there’s no gap because the indeterminacy of endlessness/infinity means there’s no way to determine any gap.
Yet another set of explanations that really really should put this topic to rest.
No doubt it won’t. Disappoint me further.
You are correct.
I was hasty in my expressions.
There is no movement towards the absolute, but away form it. Linear time is a movement towards increasing chaos, not increasing roder - that lies in the indeterminate past, as a hypothetical point in space/time - Big Bang.
It is a nihilistic inversion corrupting human language.
The more accurate image would be a movement towards absolute nil - away from the theoretical one.
The arrow of time points away from a singularity - the near absolute point we use to measure time - and represented by Yin/Yang.
Even in the metaphor of the line, the individuals in line are deteriorating - fragmenting, as they wait.
One is reference to order, since the mind cannot conceive chaos.
All language, including mathematics, measures order, so it tends to be positive, when, in fact, order is declining over space/time, not increasing.
Representations are constructions that you have to build up before the construction “equals” a quantity (that was conceptually “already there” and didn’t need building).
Building the representation of (0.\dot9) needs endlessness, so writing out the representative construction never ends (or “ends” with indefinite continuation implied) with the quantity that it does equal, which is 1.
Any “gap” is at most hinted at by the representation, not the quantity that it’s building up to, but even then it is only hinted at to the uninitiated who impatiently second guess what the representation would look like if it were to be completed at any given finite point(s) along the way - and they thus conclude that it’ll never get there and there will always be a gap.
You don’t go (\frac{circumference}{diameter}\neq3.141592…\neq\frac4{1}-\frac4{3}+\frac4{5}-\frac4{7}+\frac4{9}-\frac4{11}+…\neq4\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}) because there’s always got to be some “gap” at any conceptual “end” point to the representation implied by that contradiction in terms, that prevents it from ever equalling (\pi). No, they all equal the quantity “pi” because of the fact the construction contains some “incomplete” indetermination to it, not in spite of it.
A fraction with finite terms (C\div{d}) isn’t resolved, the explicit decimal places format isn’t finished, the infinite expansion of explicit fractions doesn’t terminate, the infinite series never gets to that “infinity” - the representation of (\pi) just pretends the irrational and transcendental quantity of pi is finite to make it easier and more compact to accept that all these incomplete constructions really do equal (\pi) in exactly the same way as (0.\dot9) really does equal (1).
Yet somehow, non-mathematicians will generally all accept any of these notations of (\pi) but not the notation of (1) as (0.\dot9). Forgive the pun, but this is entirely irrational.
I’m only able to make sense of some of what you say because the intelligence tests on which I score highly are mostly testing how well someone can extract patterns and sense from increasingly obscure sets of information. When Magnus says to me “It seems like you’re one of those rare people who can understand Ecmandu”, he seems to think that even though my ability to understand even extends as far as it does with you, I don’t understand his extremely simple explanations - the poor guy’s understanding is so consistently backwards.
I don’t have “man up” to address any argument, whether it’s one I support or not - I’ll address it fairly and rationally without partiality either way. I’d be quite happy to analyse any new arguments in this same way, but unfortunately I can’t quite get to the bottom of what you’re building here. Sometimes I feel like I’ve grabbed hold of something, but it goes away when I look into it a bit further.
In the seeming absence of some of your workings, and apparent jumps in your “notes”, I think maybe it boils down to some non-standard terminology that you’re using, like “convergence theory” (I don’t understand the exact process of getting the progressively different results that you’re presenting) or terminology that you’re using in a non-standard way such as the exact distinction “carrying left to right” and “carrying right to left”. Maybe an understanding of them will help me piece together these seemingly bizarre equations of yours.
What they lack is ability (or at the very least, incentive) to abandon logic when dealing with this subject.
The subject isn’t so complicated that it requires mathematical expertize. You don’t need mathematical expertize to understand that “2 + 2 = 4”. A bunch of mathematicians can’t declare that “2 + 2 = 5” and claim that the dissenters are wrong because they are merely amateurs.
The claim is that the two symbols ((0.\dot9) and (1)) do not represent the same quantity. A number of arguments have been put forward. You’re supposed to point out their flaws. The quoted doesn’t do that.
It’s important to note that it’s not necessary to show why Wikipedia proofs are wrong. There’s a much simpler proof that shows that they must be wrong – whatever their real flaws are.
I’m examining Wikipedia proofs for fun, not out of necessity. I might, in fact, be wrong about why Wikipedia proofs are flawed, but such a scenario wouldn’t render the main proof (which is a very simple proof) invalid.
This isn’t to say that I’m not sure about my claims regarding Wikipedia proofs. I’m pretty confident about them.
Thankfully, not everything is complicated in life, so not everything requires extensive proofs.
Again, the above isn’t pointing out flaws within an argument.
I don’t wonder, I know. You suffer from low tolerance threshold. Not something to be proud of.
Considering that you talk about it publicly, yes.
If you can’t accept that Hilbert’s Hotel is nonsense, there is, I am afraid, no hope for you.
Notice that it’s not merely counter-intuitively full and not full. It does not merely go against intuition. It goes against logic.
Intuition is neither right nor wrong on its own. It’s that thing that is sometimes wrong and sometimes right. You can never know. So if a claim is counter-intuitive, it does not necessarily mean that it’s right or that it’s wrong. But if it goes against logic, then we have a problem, Sir.
A hotel that is both full and not full. Not a logical contradiction at all.
Now you’re conflating different concepts. Infinity/endless is not the same as indeterminate/undefinable.
That’s not true.
If you can’t determine whether there is a difference between two numbers, it does not follow that there is no difference between them.
Sorry to disappoint you.
Here’s a couple of super simple arguments for you to address. And when I mean “address”, I mean “point out their flaws”.
Who ever said that the lines have to be added end to end? That was the point of having apples, oranges, and pears, each an infinite set.
And Ecmandu, there is a serious difference between “the concept orange” and “the concept of AN orange”.
Equally, there is a significant difference between “AN infinite line” (a finite quantity of a thing) and “the infinity of a line” (a quality of a single thing not being finite).