Magnus! I could be more articulate! Thanks for pointing it out!
1 Split into two parts
1 split into 4 parts
1 split into 8 parts
etc… an algorithm!
I reverse engineered limits that converge to prove that limits don’t converge to a different number.
I’ll use the number 1 as an example for all rational, irrational, transcendental and imaginary numbers!
1 divided by 2 = 0.5+0.5
That divided by 2 = 0.25+0.25+0.25+0.25
That divided by 2 = 0.125+0.125+0.125+0.125+0.125+0.125+0.125+0.125
Obviously, this is an infinite series algorithm!
What happens in this reverse engineered sequence at the limit (the convergence)?
It means that 1=0!!!
Actually, what it means is that if infinite sequences at the limit converge to different numbers, every single possible number equals exactly zero!
That’s a proof that infinite series don’t converge at the limit (equal to) a single number!
————————————
Now let’s get to the disproof of higher orders of infinity:
This is extremely simple: I’ll use variables that I call “my cheat”, placeholders are used constantly in mathematics! If I can’t use my cheat, you can’t use math!!!
1.) rational number (a placeholder / variable)
2.) irrational number (a placeholder / variable)
3.) transcendental number (a placeholder / variable)
4.) imaginary number (a placeholder / variable)
5.) different rational number (a placeholder / variable)
6.) different irrational number (a placeholder / variable)
7.) different transcendental number (a placeholder / variable)
8.) different imaginary number (a placeholder / variable)
This is an algorithm as well. It lists every possible number as variables, representative symbols.
If you’re a stickler for details, make another list of “unlistable” Chaitin numbers!
Those are my proofs through contradiction that infinite series do not converge at limit to single numbers or other infinite series!
The other proof is that all orders of infinity equal each other in size and magnitude: there are no “higher orders of infinity”.!
That, to me, suggests that you’re probably not as much of a math dummy as you’ve made out, but rather your trigonometry and calculus teachers probably failed you.
I can tell you that at the heart of trigonometry is circles.
More usefully as a starting point, a unit circle - i.e. a circle with radius “(1)” (or (0.\dot9) if you prefer )
This is because you can now draw right triangles starting from the centre of this circle, out to the edge of the circle, then straight down vertically to the point that you can horizontally return straight to the origin.
Now you have a triangle with its longest side (that good old “hypotenuse”) the same as the circle’s radius (1), and suddenly those “sine” and “cosine” graphs have meaning - the basis of trigonometry.
The sine graph maps how long the triangle’s vertical side is for each and any angle made between the horizontal side and the hypotenuse, and the cosine graph maps how long the horizontal side is (for each and any angle made between the horizontal side and the hypotenuse).
Everything trigonometric develops from there - you can scale all sides up or down for different hypotenuse lengths, “tangent” is just the “sine value divided by the cosine value”, all the way through other terms you encounter - and you even get an insight into the world of complex numbers when the horizontal side represents the real component and the vertical side represents the imaginary component. All of this becomes accessible from just the few lines I just listed, if you just work your way up from there.
As for calculus, that all just starts at differentiation and integration and you work your way up from there in just the same way. The rest is just familiarity - explore as many different examples of the same concepts that you have the patience for, and you’ll develop an intuition that’ll help you progress further.
Differentiation is just how the gradient of some curve changes as you go along it, and integration is just the area between the curve and the x axis that you plot it on (for either the whole curve or just a part of it).
Gradient is the y value divided by the x value, and you approach it by drawing a straight line between two points in the curve, working out the gradient of that (starting simple), and seeing what “limit” is approached as you make that straight line smaller and smaller.
The area under a curve is approached by drawing thin rectangles from the x axis to the curve, e.g. with the top left of the rectangles touching the curve, and the top right jutting in or out a little to the side. Keep the widths of these thin rectangles the same, and reduce that width and see what “limit” is approached as you make it smaller and smaller, and add the areas of all the rectangles together. As the rectangles get thinner the true area is approached better and better.
Like trigonometry can lead you to complex numbers, these simple processes for calculus lead you to hyperreal numbers (through limits).
In fact, understanding limits through calculus can correct your intuitions on the title of this thread.
For differentiation, if the straight line gradient “gets to the limit of zero”, does it even have a gradient anymore?
For integration, if the widths of the rectangles “get to the limit of zero”, does it have an area anymore?
Well, obviously, both methods still approach specific answers, the curves still have gradients at any one point and areas underneath them for any required range. So the answer to both of the above questions is yes! The limit gives the answer.
This thread is no different. The limit of (0.\dot9) is no other number than (1). The consistency of the math that got you to these questions in the first place is maintained.
As such, given “math in the first place” such that we can arrive at this topic’s question at all, following it through we get a clear, exact and definite, correct and indisputable answer. Like I said - math allows this by virtue of being strictly precise to the core, such that we can arrive at such a specifically defined question at all.
The conceptual difficulty of a zero length gradient, or the area of a rectangle with zero width is just that - a conceptual difficulty. Mathematics resolves this difficulty by yielding exactly correct answers that make absolutely perfect and consistent sense once you get to them, regardless of any oddness you find in logically getting to an answer that turns out to be true.
I mean, I just resolved the entire thread yet again in another single post. And it’s no reflection on me, I’m just passing on mathematical knowledge that already exists.
The correct answer was there before this thread even began - all that’s elusive is the humility and honesty in approach by the non-mathematicians who think they casually stumbled upon some insight into mathematics that mathematicians got wrong all this time, and relentlessly insist that their inexperienced thoughts have clout (in spite of all valid explanation by actual mathematicians to the contrary), pretending they’re open to learning/understanding the actual truth, and are here only for rational debate despite all evidence to the contrary.
It’s as much of a poison here as it is in politics and anywhere on the internet, where a bunch of amateurs want a shortcut to tasting what real creative innovation and large scale usefulness is like by pretending to themselves and others that they can be treated like an authority who knows better than countless experts on the subject, because they think they have “a special something” that others don’t - or at least they want to think of themselves as such. That’s why this thread is not much better than an exercise in identifying psychological biases and logical fallacies - the same as almost every thread on this board in fact. With the notable absence of obvious academic philosophical education here, I only really come here myself to test out layman reactions to certain ideas I have, to test if there’s anything obvious that I’ve missed, or if I actually know a solution to something that others are discussing I can offer them help if they want it and I enjoy trying to teach those who are willing to be taught. I just despise those who do not want to be taught, because they bring everyone else down with them simply out of their own shortcomings and weaknesses - and there are plenty of these types, who collectively make everything worse for everyone.
It’s all so human, of course - purpose seems to be a deep need. This desire can inspire one to really study and become a genuine expert in a subject, or it can frustrate the weak who are scared that they might never be able achieve genuine expertise in a subject before they’ve really even tried, so their wishful thinking and narcissistic eagerness to indulge fantasies of their own greatness get the better of them and they act them out like their very identities depend on it - because they do depend on it. It’s a deeply entrenched sickness that I’m hardly going to cure, so I’ll simply call it when I see it, and honestly recommend the humility they need, that they will pretend they have, but will probably never truly attain as it’s the polar opposite of what’s making them sick in the first place. It’s everywhere on this forum and ones like it, and if there were any actual philosophical experts here they probably long departed because of all these pretenders who tend to hang around to feed their habit instead of going elsewhere to gain actual expertise.
Yep, that’s (e), you know the one. Amazing number, pops up everywhere and for very good reason.
Think about this argument of yours for a minute.
If every number of any type actually equals zero, then by “transitivity” every number equals every other number. So there is only one quantity that is “no quantity”, with infinite (false?) representations.
If that were the case, then the math would not exist to get you to the start of your reasoning in the first place, that essentially uses math to conclude that math is undone.
So, “given math, there is no math”. A contradiction, no?
Either we then conclude that math is therefore bunk, useless, meaningless, doesn’t work, is false.
Or we conclude that there’s an issue in the process you used to arrive at this conclusion.
You’re obviously quite taken with the drama of the first of these potential conclusions. Now I’m suggesting you complete the thought process by investigating the second of these potential conclusions.
Is there an issue with how you arrived at the first potential conclusion?
As in the post of mine that I linked to you, perhaps the plurality of apparent solutions to these equations of yours is a symptom of a flawed methodology. Perhaps doing “more work” could lead you to a different, singular and unique solution that maintains the consistency of the math that you’re using to arrive at your conclusion, and which led you to begin your argument at all? There’s nothing wrong with noting that an answer is “undefined”. Maybe that’s the answer, maybe you just need to try a new methodology to give the previous one some better context?
Either way, I don’t think your line of thinking has come to an immutable end that everything is zero, which basically just means math is broken - just because of some back-of-a-napkin calculation.
Whose credibility? (I believe the answer is yours.)
And why should I care? (I believe that I shouldn’t.)
You are missing the point. There are sequences that aren’t algorithms. (\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dotso) is an example of a sequence that isn’t an algorithm.
Algorithms are (some algorithms, that is) specific type of sequences, namely, sequences of instructions on how to perform certain task. Most algorithms, however, are closer to being trees than sequences.
That’s not an algorithm. That’s a sequence of mathematical expressions (written poorly.)
1 = 1
1 = 2 x 1/2
1 = 4 x 1/4
1 = 8 x 1/8
1 = 16 x 1/16
1 = 128 x 1/128
etc
What do you mean when you say that the other side (the right side, I guess) converges to zero? The right side is at all times equal to 1.
That’s a sequence of instructions (an algorithm), whereas (1, 2, 3, \dotso) is a sequence of numbers (not an algorithm.)
I have no idea what you’re talking about. Unless you make an effort to clarify your position, this is the end of the discussion between the two of us. You have no right to complain at this point.
(\frac{1}{8}) is NOT the right side of that equation, it’s merely one small part of it. The right side is (\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}) and it’s equal to (1).
And that’s why you lost credibility in this thread. Like I stated before. You have your fancy symbols… but plain English is something you fail to grasp.
I’m not speaking this in complex ways.
I wish I could dumb it down for you, but I’m not sure that I can.
I’ll try again…
1=1/1
1=0.5+0.5
1=0.25+0.25+0.25+0.25
Etc…
On the right side, the numbers get lower and lower and lower.
For example: the next step is:
1=1/8+1/8+1/8+1/8+1/8+1/8+1/8+1/8
…
Eventually, in this series, if limits converge, every number to the right will equal zero.
I honestly cannot make it simpler than that. I’m sorry, it’s impossible to me to make it simpler!
I complain because we live in a zero sum world. Where there’s a winner, there’s a loser.
I complain about my losses and I complain about my victories. I hate this world. If you had any sense, could put just 2 neurons together in that brain of yours, you’d hate it too.
It makes perfect sense and I understand the argument, and you’ve explained it fine for anyone with a brain.
The limit of each number on the right is indeed 0, as all the denominators tend towards infinity. Add all those limits together to get 0, when the left hand side remains 1 throughout.
The problem is that you have an infinite number of fractions on the right hand side.
This undefined element is what you’re not respecting, because you can’t only pay attention to “any number of zeroes added together is zero”. You also have to pay attention to “an infinite number of positive numbers added together is infinite” and also “any number of that same number’s reciprocal added together is one” etc.
1=“0” isn’t the only answer you can get, but it’s the only answer you’re paying attention to.
I’m not being biased when I say “look at all the possible solutions, not just one”. That’s literally the opposite of being biased.
I’m sorry, but it’s just not a valid proof that there’s something wrong with math. Nice try, but no cigar.
What’s interesting to me here silhouette, is that if this series doesn’t converge at zero (as you state) then 0.9… cannot equal 1. You’re arguing against yourself here!
I’m creating a very specific box with this proof:
If my proof is false, YOU’RE the one destroying math, not me!
Math goes on just fine if convergence is false! If convergence is true … that’s the end of all math!
if the fractions do actually reach their limit of 0, then the right hand side properly equals 0 when the left hand side still equals 1, and the difference between (0.\dot9) and (1) can also genuinely reach its limit of 0, but math is therefore broken? And,
if the fractions on the right hand side of your equations don’t reach their limit of 0, leaving something to work with to maintain equality with the left hand side of 1, then there is the same kind of “something” between (0.\dot9) and (1), and mathematical consistency is maintained?
You let me know if I understand the message.
What I’m trying to say is that those aren’t the only two options.
The limit of the equations you’re presenting leads to an undefined number of fractions, such that whether each fraction reaches their limit of 0 or not, the right hand side of the equation isn’t neatly “0” (or anything else) until you do more work and fully narrow down a singular defined answer only.
By contrast, there’s no extra work to do to get the difference between (0.\dot9) and (1). There is no other number that it can be than the singular defined answer of zero. You could arbitrarily phrase the difference as (\lim_{n\to\infty}(n\times\frac1n)=0) just like in your argument to try and force the same complication, because the the difference may as well be (\sum_{n=0}^\infty\frac0{10^n}=0), but we already know the singular defined answer never gets to a point where it’s larger than 0 no matter how far you go down the decimal expansion.
It’s simply not the same situation, so the complication you stop at with your argument, instead of working it through to get beyond the undefined element and finding a way to fully get to a singular defined answer, doesn’t apply. It doesn’t prove anything by itself, never mind anything about (1=0.\dot9), because you stopped short and concluded a singular defined answer without doing more work to resolve the undefined element that could just as easily yield a different answer for you to stop short at instead, if you wanted.
Let me know if you, in turn, understand my message.
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