Moderator: Flannel Jesus
James S Saint wrote:The "..." simply means, "you can't get to the limit from here".
So you see why "∞−∞" is logically and mathematically nonsense, because it could be anything - hence why mathematicians regard such a divergence as undefined.
It's infinitely unanswerable, with an infinite range of possible answers no more correct than the last: thereby all infinitesimally correct (infinitely incorrect).
By contrast, 9.˙9−0.˙9 is far more logically and mathematically viable as the convergence to a single exact limit cannot be anything else other than precisely tending to zero.
Consider that \((0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots) = 0\) if 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.
As usual, Silhouette is doing nothing but complaining that the world is not bending to his will the way he's bending to the will of the mathematicians.
I don't think that at all. But I might guess Sil is better inclined as a mathematician than a philosopher.
Magnus Anderson wrote:As usual, Silhouette is doing nothing but complaining that the world is not bending to his will the way he's bending to the will of the mathematicians.
Mowk wrote:I don't think that at all. But I might guess Sil is better inclined as a mathematician than a philosopher.
Mowk wrote:zero is only a number through mathematical convention. 1 - 1 = nothing, no quantity to count. Zero is just a symbol used to represent that mathematically. Any number divided by zero is undefined as well. How can you divide a number by a symbol that represents nothing. It is as if nothing at all has taken place. The division can't occur because there is nothing to divide by.
Mowk wrote:Additionally, I think there is a sufficiently defined difference between 1.0 and 0.9 recurring. The respective location of the decimal point, and the ontological difference between a whole and a part. Is 0.9 recurring a well defined number, cause 1 seems fairly well defined as numbers go.
Mowk wrote:I don't see "why" "∞−∞" is logically and mathematically nonsense. Begin with boundless "what ever that is" and then subtract "what ever that is" and there is nothing left, not even the zero. The zero is sort of stuck as a symbol representing nothing, smack dab in the middle of a set that includes the infinite set of negative numbers on one side and the infinite set of positive numbers on the other.
Mowk wrote:So far, infinity is just imaginary boundlessness. I don't think it's been proven it actually exists. The difference between a philosopher and a mathematician is a mathematician is willing to forget it was an assumption, treated as if it existed, for the sake of an argument. An axiom *. Philosophy does that as well, but the philosopher, at least a good one, doesn't forget it.
* https://en.wikipedia.org/wiki/Axiom
Ecmandu wrote:If you’re just using counting numbers, nothing exists between 1 and 2, that does not instantly make them equalities!
You can look at ∞−∞ in at least two ways.
i) the second infinity is so large that subtracting it from infinity could leave absolutely nothing left, and therefore result in zero, or maybe even less than zero.
ii) the first infinity is so large that subtracting anything from it couldn't possibly get you all the way back down to zero.
The crux is that both infinities are undefined, so you can't define the result of operating on them with respect to either of them, never mind both of them.
Anything properly defined is fine,
but if you mess around with what-ifs about infinity, you break that consistency.
As I was saying, mathematics has only grown in scope as a result of challenging assumptions - hence how we arrived at treating the square root of minus one as a valid entity with complex numbers etc. etc.
Perhaps you might argue that it was philosophers who advanced the mathematics, but history will show you that the two overlap all-too-often - and no coincidentally.
A mathematician who sticks doggedly a set of accepted rules only is just as bad as a philosopher who thinks speculation without rigor is sufficient, like Magnus. To be good at either is to be good at both.
It's been conclusively demonstrated here that non-mathematicians with a chip on their shoulder about this status of theirs cannot cope with all the implications of this concept of "undefined", and that everyone with a mathematical proclivity has the ability to conceive why 1=0.˙9 so the vote at the top does more to illustrate the distribution of mathematical backgrounds on this forum than anything else - as mathematicians know, there is nothing democratic about mathematics. It's an absolute dictatorship.
But as my first contributions to this thread immediately spelled out, advancements in mathematics come about by playing around with what would happen if exactly specific rules of this absolute dictatorship where treated as breakable (e.g. complex numbers). Even though arguments distinguishing 1 from 0.˙9 are wrong, it's interesting to see if anything useful happens if we treat e.g. two representations of the same quantity as different. If we were all mathematicians here, we could have moved on long ago to addressing this question - and exploring the utility of hyperreals.
The fact of the matter is that no difference between 1 and 0.˙9 can be defined - and to this the non-mathematicians cling as evidence that it's there "because it seems like it ought to be", but "it's the fault of the numbers we use" - or something woolly like that.
Silhouette wrote:I gave up trying to persuade you to further your reasoning so there's no need to lash out anymore. Chill. Think of anything I'm saying as for the benefit of others who might be able to think past only half the argument.
[A]s mathematicians know, there is nothing democratic about mathematics. It's an absolute dictatorship.
Silhouette wrote:A "difference" between \(1\) and \(0.\dot9\) evading all definition doesn't merely mean "approximately equal". It means any definition between the two representations is impossible to definitely represent, and tends precisely and exactly to no other number than zero.
Magnus Anderson wrote:Silhouette wrote:A "difference" between \(1\) and \(0.\dot9\) evading all definition doesn't merely mean "approximately equal". It means any definition between the two representations is impossible to definitely represent, and tends precisely and exactly to no other number than zero.
Look what he's saying:
- It is impossible to "definitely represent" a "definition" between \(0.\dot9\) and \(1\).
- The difference between \(0.\dot9\) and \(1\) is "evading all definition".
What kind of language is this? Certainly not a mathematical one. This is why I disagree with Mowk when he says that Silhouette is more of a mathematician than a philosopher. He's neither.
Here's my take on the matter: the difference between \(0.\dot9\) and \(1\) is \(0\) if and only if the two symbols represent the same number i.e. if and only if \(0.\dot9\) represents \(1\). If \(0.\dot9\) is a contradiction in terms, or if it doesn't represent anything at all, then it isn't \(1\).
Of course, if \(0.\dot9\) does not represent anything then we cannot calculate the result of \(0.\dot9 - 1\) since \(0.\dot9\) does not represent a number. But that does not mean the result is \(0\).
The sooner silhouette figures this out, the better.
Ecmandu wrote:Put more succinctly as I said before:
1 is not an algorithm
0.999... is an algorithm
Contradiction? An algorithm is NEVER equal to a non-algorithm!
In mathematics and computer science, an algorithm is a finite sequence of well-defined, computer-implementable instructions, typically to solve a class of problems or to perform a computation.
Magnus Anderson wrote:Ecmandu wrote:Put more succinctly as I said before:
1 is not an algorithm
0.999... is an algorithm
Contradiction? An algorithm is NEVER equal to a non-algorithm!
According to Wikipedia:In mathematics and computer science, an algorithm is a finite sequence of well-defined, computer-implementable instructions, typically to solve a class of problems or to perform a computation.
An algorithm would be something like:
1) Let \(x\) be \(0\).
2) Let \(i\) be \(1\).
3) If \(i\) is less than or equal to \(\infty\), go to step 4. Otherwise, go to step 7.
4) Add \(9 \div 10 ^ {i}\) to \(x\).
5) Add \(1\) to \(i\).
6) Go to step 3.
7) The result is equal to \(x\).
Of course, this algorithm will never halt due to the fact that \(\infty\) cannot be reached by a finite number of steps (provided that you start at a position whose index is a finite number), but that isn't really important, isn't it?
I would say that, on its own, \(0.\dot9\) is not an algorithm.
But you're right that \(0.\dot9 \neq 1\).
As Mowk hinted earlier, two decimal numbers are equal if and only if they have the same exact digits. \(\dotso0001.000\dotso\) and \(\dotso000.999\dotso\) are clearly not equal.
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