What?
Unlike some here, I have no problem saying when I’m ignorant about something. It’s how you learn, and therefore legitimately win in the long term, as opposed to deny, and therefore superifically win in only your own mind and only in the short term (which is stupid).
I don’t know what Convergence theory is - I even looked it up and a cursory glance only finds references to Sociology, and nothing to do with “bounding infinities to make them quantities”.
I have no idea what that means, and I strongly suspect it’s just something you came up with and named yourself, perhaps you explained it elsewhere on the forum but certainly not here or anywhere that I read or remember reading.
I’m offering you an opportunity to detail your terminology and all the steps of its application so I can have a look at it.
It’s not shooting myself in the foot to not be able to read your mind, even if I have the intelligence to work out more of what you’re talking about than others seem to.
If you’re just talking about convergent series, then that’s fine - but your explanatory power will be increased if you use the standard terminology; in the same way that people will know what you mean better if you use the standard definitions of words.
It’s true that the construction of (0.\dot9) converges to the quantity (1) (and I just explained the difference between representative constructions and quantities myself), but I dunno what the terminology of “binding infinities” has to do with this.
I already explained that (1) is also a representation of the actual quantity (1), which doesn’t necessitate the quality of endlessness (e.g. (1.0, 1.00, 1.\dot0)) etc. makes no difference to the quantity represented.
And by contrast, (0.\dot9) is also a representation of the actual quantity (1), but necessitating the quality of endlessness (e.g. (0.9, 0.99, 0.\dot9)) etc. makes all the difference to the quantity represented.
So feel free to show how this works in practice without skipping steps or using unexplained non-standard terminology, such that you can successfully communicate all the essential details to others.
I don’t know the answer yet, but you’ve asked me to have a look at it and I’m curious anyway - you have the opportunity to explain something I don’t know to me here. Would make a first for this entire thread, so the change would be thoroughly welcome, especially if you explain it very well.
Hi gib, I wasn’t actually criticising what I quoted by you in case you were wondering, I was offering my own explanation around the concept because other people were too and I had improvements to offer.
I actually thought I covered what you refer to as “inventing a representation of the representation” through “the dot in (0.\dot9)” in what I parenthesised as “(or “ends” with indefinite continuation implied)”. The dot above the 9 to denote recursion terminates the appearance of the representation with an instruction to imply indefinite continuation. Infinite sums do this as well with the infinity above the Sigma - it looks all defined with the finite terms all laid out with finite instructions about what to do with them, except that 1 term infinity that says “do the finitely defined thing an infinite number of times”.
This is the whole trick that lures the non-mathematicians into a false sense of security in treating infinites as finites - they get fooled by the superficials of the representation and think that infinites can be defined, when it’s only the finites that are defined, accompanied by an instruction to do the defined indefinitely. Indefinitely defined does not mean defined.
But in short, I’ve not read everything you’ve said because the thread’s moving fast and there’s so much wrongness being predictably and repeatedly peddled by the less qualified for me to try and handle - but what I have read of you mostly seems to make perfect sense.
Who ever said that the lines have to be added end to end?
If it’s the infinity that’s twice as long, you have to add along the dimension that is going on infinitely. That would be consistency, but obviously it’s impossible by definition and derivation.
For example, the real numbers represented along a number line go on infinitely along one dimension only - that’s the sole dimension along which infinity applies.
Here, the number line doesn’t go infinitely up or down, toward or away from the viewer - those dimensions and all others are finitely contained.
This is what I was explaining when I clarified that any mistaken identification of different sized infinities is actually only a product of the the finite constraints around the infinity - the size refers to these finites, not the infinity itself.
If you were to add another infinite number line side by side with another, you aren’t changing the dimension along which the number lines are infinites, you’re changing a finite constraint along the dimensions that the “infinite series” are finitely bound.
Here’s a simple visual representation of why (10 \times 0.\dot9 \neq 9 + 0.\dot9).
The red line indicates something interesting. It represents (0.\dot9) but is it the same quantity as the one indicated by the green line? Obviously not. You can see that the red rounded rectangle is narrower than the green rounded rectangle. In other words, the number of terms is not (\infty) but (\infty - 1).
This is really dense of you.
If you want to say that the greens are “1 longer” than the reds, why not also say “but the reds are 9 broader than the greens”?
Again, we go back to Hilbert’s Hotel and how infinities are undefined because you can easily end up with answers that are both bigger and smaller.
Do you then try to say something like the reds are therefore 9 times bigger than the green, or perhaps the “green minus 1”, or maybe even 9-1 times bigger? But then the quantities within the reds add up to the 1 green quantity that’s not in the reds so…
Basically your argument here successfully says and shows absolutely nothing.
That’s correct.
((9 + \underline{0.\dot9}) \div 10 \neq \underline{0.\dot9})
The two underlined numbers don’t have the same number of 9’s.
In other words, the two infinite sums don’t have the same infinite number of non-zero terms.
Therefore, they aren’t the same number.
That’s my claim. It’s up to other people (which includes you) to point out flaws within my argument if they wish to do so.
That’s precisely the point of our disagreement. The (0.9) in (s) is paired with (0.09) in (s \div 10).
Urgh.
I have been pointing out the flaws in your argument because I wish to help you learn.
Yet even the non-mathematician expressing mathematical expertise over professional mathematicians doesn’t want to learn, because what? You’re an adult? You want to feel competent, or at least not incompetent? You see how I responded to Ecmandu at the start of this post? We’re all students, and the less you’re ruled by your insecurities, the better you’ll learn if you simply admit you’re NOT an expert and also not ACT like you’re an expert nonetheless.
The (0.9) in (s) is NOT paired with (0.09) in (s \div 10). Let me explain in yet another way for you to not listen to…
The first term in (s) is (\frac9{10^0}). This looks like “9”.
The first term in (\frac{s}{10}) is (\frac9{10^1}). This looks like “0.9”
But this doesn’t mean we match the unit column with the tenths column - that would be to be fooled by superficial appearance.
In both cases the first term is dealing with the quantity in the unit column.
It’s only positional notation that puts the resultant decimal representation in the tenths column, but appearance doesn’t transfer the first quantity to be matched with the second term.
To think it does is - as I explained - getting fooled by how things look rather than the essence of the quantity to which they’re referring: which is the first term, which operates on the quantity in the units column.
Is this all coming together for you yet, or are you still distracted by all the pretty lights?
The second term in (s) also looks like “9/10” in fractional notation.
The second term in (\frac{s}{10}) also looks like “9/100” in fractional notation.
This is slightly less misleading, but if you still can’t resist the urge to translate it back into decimal notation to make it look like the one-to-one correspondence is “misaligned” such that there’s a “spare” unmatched digit at some fictitious “end” to infinity, then you’re going to continue to miss the point.
This isn’t to say that I’m not sure about my claims regarding Wikipedia proofs. I’m pretty confident about them.
Have you considered that, given your position as a non-mathematician, maybe you shouldn’t be pretty confident about them?
You’re hedging here, with your lip-service “I might, in fact, be wrong about why Wikipedia proofs are flawed”, and your confidence in your simplistic “proofs”.
You actually do need mathematical expertise to deeply understand “2 + 2 = 4” - have you not seen the size of the proof that 1+1=2?
It’s for this reason that “a bunch of mathematicians” cannot get away with “2 + 2 = 5”, like some kind of political mob taking power.
But this does not prevent non-mathematicians from simply accepting and memorising the equation on a superficial level, like even children do from a very young age.
If you think the mathematical world works like a direct democracy rather than an elitist meritocracy where proving yourself wrong actually ascends you up the ranks, then this is just another example of your superificial non-mathematical intuitions and assumptions.
Not everything is complicated in life, indeed. Mathematics is, even if you have an uncomplicated understanding of it.
I actually have an uncommonly high tolerance threshold in person, which I am proud of and I consider it a personal failure when it fails. I am willing to publically talk about such weaknesses as well as my strengths - the public element does not affect my authenticity whatsoever, so it certainly doesn’t make me proud of my frustration. The fact that you consider the public element to be a factor is no doubt why you’re having so much trouble accepting the degree to which your mathematical abilities can currently enable you to deal with topics like this one. If you actually listened to people who know what they’re talking about, and your admission of being a non-mathematician ought rationally to be followed by this line of behaviour, you might actually stand a chance to learn and grow to become a mathematician - so you’ll progressively empower yourself to competently and legitimately deal with topics like this one! If that prospect doesn’t tempt you, then I don’t know what to say - enjoy the limbo of self-denial, intellectual stasis and long debates with people getting angry at you “because that’s their flaw”.
A hotel that is both full and not full. Not a logical contradiction at all.
Again! The whole point completely passes you by. WHOOSH.
The whole point of the hotel is to show the contradictions presented by the intrinsically indefinite nature of infinity, and therefore to not treat them as finites that can have sizes. It’s the finite constraints upon which infinity can operate that give the series any “size”, it’s not the infinity that therefore has size. The fault is not with the thought experiment, but with the subject of the thought experiment - the thought experiment is just a messenger. Again - you get fooled by the superficials!
Are you really sorry for disappointing me though?
I don’t think you’re hiding the fact that you’re saying this rhetorically, but maybe you really should seriously consider the possibility that being sorry might actually help us all out?