Posted: Sep 05, 2017 11:40 am
Unfortunately, I only have a limited amount of time (heh) to write this, so since Thomas Eshuis and Greyman (among others) have already demolished quite a lot of your claims, I'll just pick a few of the funnier ones...
Yes, "abstract math" can have such "woo" as claiming that 1 + 1 can equal 10, which would be clearly absurd, because only counting on our fingers can apply to reality. For your own safety, I'd recommend that you stay far away from anything whose basic operations relies on such trickery! (As for why your "inverse" construction of a continuum is flawed, see below.)
Now, I know that apologists often are quite immune to any knowledge that isn't at least a thousand years old, so this might be a bit of a stretch: have you heard of limits? It's only been around since Newton's time, and it provides a quite handy and useful way of dealing with such questions.
Well, his concept of "atomic" is also quite outdated. "Atom" literally means indivisible, and if you paid any attention to the Korean peninsula in the recent weeks, you should bloody well know that "atoms" are anything but. Now, space (and space is a far more interesting subject here than particles) may well be quantizable at Planck length intervals, and if you really want to base your argument on modern physics, be my guest, but what an ancient Greek said on the subject is hardly likely to provide any relevant information...
Your main problem in the quotes above is that you try to disprove a mathematical concept by appealing to reality. Mathematics doesn't work that way. It is an axiomatic system, so "evidence" isn't an issue, only proofs are. If you want show that infinities are logically incoherent, you actually need to prove it, which in turn would require you understand the underlying mathematics. If you want to continue to blindly dismiss areas of mathematics just because they don't agree with your preferred conclusions, you are throwing the baby out with the bathwater; if you want to claim that only Wortfish-infinities are applicable to reality, please justify how you think concepts like calculus has no real world application...
And this was just too fucking hillarious not to comment on:
First of all, you do realize that there are irrational numbers <1 (including negative ones)? This makes your claim that "the inverse of all irrational numbers [...] lie within 0 and 1" blatantly false. Secondly, you have definitely NOT "always stated" that; here you said "I have shown that any real and rational number between 0 and 1 can be expressed as the inverse of a positive integer multiplied by 10^n and, for >0.5, summed with the inverse of 2" (not to mention your claims about continuums earlier), which directly contradicts your statement above, since the square root of 2 isn't an integer.
But the funniest part is when you try to dismiss irrational numbers by comparing them to "rational approximations". A "rational approximation" is not an irrational number, and there are two subtle clues to this that you may have missed: the term "rational", which indicates that it is not irrational, and the term "approximation", which means "close to", not "equals". That we can construct a rational approximation to an irrational number doesn't really tell us anything; it certainly doesn't tell us that your claim that "any real [...] number between 0 and [0.5] can be expressed as the inverse of a positive integer"* is credible in any way. The only way this could lead us to any interesting conclusions is if you can show that for each possible rational approximation, there is at most one irrational number that is best approximated with that particular approximation. Since there are 2ℵ0 irrationals in the interval between any two rationals, I won't hold my breath waiting for your proof...
* That you don't even realize that the "multiplied by 10^n" is a thoroughly unnecessary addition should tell us something...
Wortfish wrote:But I quite frankly fail to see your point here. You are simply admitting that there are (at least) as many rationals between 0 and 1 as there are positive integers, which (by a sane mathematical definition) there are an infinity of...
My point is that we are engaged in numerical trickery by pretending there is some infinite continuum between 0 and 1 when all we are doing is taking inverses of unbounded positive integers. Unfortunately, abstract maths does entail a lot of this woo!
Yes, "abstract math" can have such "woo" as claiming that 1 + 1 can equal 10, which would be clearly absurd, because only counting on our fingers can apply to reality. For your own safety, I'd recommend that you stay far away from anything whose basic operations relies on such trickery! (As for why your "inverse" construction of a continuum is flawed, see below.)
The so-called "fallacy" comes from the failure to recognize this difference between sets and their members (or between transfinite numbers and integers, for that matter), and we do actually have mathematical models that can explain how such things could work. Your entire counterargument still rests on an insufficient and out-dated model, and simply arguing from the (likewise out-dated) authority of Aristotle doesn't help you.
So you believe Achilles can never catches up with the tortoise and Zeno's arrow never reaches its target?
Now, I know that apologists often are quite immune to any knowledge that isn't at least a thousand years old, so this might be a bit of a stretch: have you heard of limits? It's only been around since Newton's time, and it provides a quite handy and useful way of dealing with such questions.
Aristotle was the one to bring the terms "actual" and "potential" into it, which was what I was talking about. But Democritus may well have been first to the particular conclusion you mention, but he had the same lack of modern mathematical tools, and thus his results are equally out-dated. Without set theory or analytical limits, there is only so much you can do... To put it simply, you need to show how these speculations apply to modern mathematics and/or physics, instead of just making an argument from authority.
Well, in the case of Democritus, his hypothesis became the basis for atomic theory. There is no evidence that we can divide particles indefinitely into infinitely indivisble components.
Well, his concept of "atomic" is also quite outdated. "Atom" literally means indivisible, and if you paid any attention to the Korean peninsula in the recent weeks, you should bloody well know that "atoms" are anything but. Now, space (and space is a far more interesting subject here than particles) may well be quantizable at Planck length intervals, and if you really want to base your argument on modern physics, be my guest, but what an ancient Greek said on the subject is hardly likely to provide any relevant information...
Your main problem in the quotes above is that you try to disprove a mathematical concept by appealing to reality. Mathematics doesn't work that way. It is an axiomatic system, so "evidence" isn't an issue, only proofs are. If you want show that infinities are logically incoherent, you actually need to prove it, which in turn would require you understand the underlying mathematics. If you want to continue to blindly dismiss areas of mathematics just because they don't agree with your preferred conclusions, you are throwing the baby out with the bathwater; if you want to claim that only Wortfish-infinities are applicable to reality, please justify how you think concepts like calculus has no real world application...
And this was just too fucking hillarious not to comment on:
Wortfish wrote:Calm down. I have already stated that the inverse of all irrational numbers like 1/sqrt(2) lie within 0 and 1 and cannot be definitively expressed in terms of integers. This applies also to sqrt(2) -1 and to PI -3. However, we can always arrive at a rational approximation to any irrational number. Now, the inverse of 2414213565 multiplied by 10^9 is pretty good. I can get ever closer to sqrt(2) -1 by adding more digits. Indeed, the bigger the number, the more accurate the approximation.
First of all, you do realize that there are irrational numbers <1 (including negative ones)? This makes your claim that "the inverse of all irrational numbers [...] lie within 0 and 1" blatantly false. Secondly, you have definitely NOT "always stated" that; here you said "I have shown that any real and rational number between 0 and 1 can be expressed as the inverse of a positive integer multiplied by 10^n and, for >0.5, summed with the inverse of 2" (not to mention your claims about continuums earlier), which directly contradicts your statement above, since the square root of 2 isn't an integer.
But the funniest part is when you try to dismiss irrational numbers by comparing them to "rational approximations". A "rational approximation" is not an irrational number, and there are two subtle clues to this that you may have missed: the term "rational", which indicates that it is not irrational, and the term "approximation", which means "close to", not "equals". That we can construct a rational approximation to an irrational number doesn't really tell us anything; it certainly doesn't tell us that your claim that "any real [...] number between 0 and [0.5] can be expressed as the inverse of a positive integer"* is credible in any way. The only way this could lead us to any interesting conclusions is if you can show that for each possible rational approximation, there is at most one irrational number that is best approximated with that particular approximation. Since there are 2ℵ0 irrationals in the interval between any two rationals, I won't hold my breath waiting for your proof...
* That you don't even realize that the "multiplied by 10^n" is a thoroughly unnecessary addition should tell us something...