I'be raised these but they are raised often here by various people. I have threads discussing 'walls' of three types, by which I mean 'boundaries' or finite limits. Two are based on time and space as 'origins' and 'endings'. The third is to what is here and now, or the present for time and a point in space.Age wrote: ↑Sat May 04, 2019 3:02 pmSome one in this forum mentioned these "zeno's paradoxs" previously, so I looked them up. From what I can remember there is NO actual 'paradox' at all there. It is just the way that they are worded that deceives, intentionally or unintentionally, people to see things "wrongly", if that is the correct word to use here.Scott Mayers wrote: ↑Sat May 04, 2019 11:59 amI was responding more to everyone on this topic assuming only a thought experiment to why I believe it rational to infer that there is always MORE of something between any two points than we can explain EVEN if we think we've exhausted all points. I believe it is a good argument if I expanded on it properly -- something that I only 'intuitively' expressed in a small post summarily without completion.
The argument I am thinking is based on an original argument that begun with Zeno's paradox of the Arrow and rooted in the similar problems about the origins of the problem of 'rational numbers' that lead to the 'irrational numbers'. It all relates to those 'incompleteness theorems' many of us have been speaking on in various threads here.
To me, a 'paradox' is just a seemingly absurd or contradictory statement or proposition which when investigated may prove to be well founded or true.
To me, the statement; ' 'We', human beings, do NOT need money to live' IS a paradox.
The three things that are quoted as being zenos are just statements/propositions that, intentionally or unintentionally, lead to WRONG conclusions.
1. Arrow "paradox": The thing is there is NO actual 'instant', other than a conceptual one, or one caught on film, photo, picture, or print. There is, however, ALWAYS 'change' instead of any ACTUAL 'instant', which can be actually 'experienced or observed', any way.
2. As for the 'achilles and some tortoise', the deceptive words here are 'has been'. In the conclusion sentence: Thus, whenever Achilles arrives somewhere the tortoise HAS BEEN, he still has some distance to go before he can even reach the tortoise. There creates a subtle deception within you in thinking about 'ARRIVING' where the tortoise 'HAS BEEN'. If, however, the statement was about 'ARRIVING' where the tortoise 'IS', then there is NO issue here at all.
3. Dichtomoy "paradox's" conclusion; Hence, the trip cannot even begin. BUT, if it HAS begun, then there is MOVEMENT, or, if there is MOVEMENT, then the trip CAN and HAS even begun.
To me, these three statements/propositions are NOT paradoxes at all. They just use language that subtlety deceives, again maybe intentionally or unintentionally.
The paradoxes raised by Zeno were about these. I don't want to argue what one thinks is or is not sufficiently resolved. Zeno knew these were not 'true' realities but true problems one needs to overcome to make sense of the how and why questions. When he spoke of the problem he demonstrates should imply that NO movement is possible, obviously he doesn't literally think this is true but that given they ARE true, it is conflicting with what we already know of other things.
For one, the resolution, that he couldn't figure out, was that space and time are 'relative', something that had to wait for Einstein. Newton's first law asserted that anything NOT moving or moving in some constant velocity remains in their state unless something else outside of it (a force) causes it to change. The problem here is that there is no such thing as something NOT moving. Something not moving has zero velocity relative to us.
But if there is no place in space nor time where anything is not moving, then space itself has no 'background' reference. This means there is no actual point in space, like an 'address' anywhere... in principle. So we still have a different kind of paradox for fixing it using relativity UNLESS space itself is NOT REAL or 'virtual'.
If it is NOT REAL, then you would have part of your answer to how space can POSSIBLY 'expand'. Obviously if say a unicorn is not real, for instance, it can actually grow a horn and STILL be unreal.
You personally cannot assert you know these paradoxes aren't real contradictions to us without demonstrating from your own understanding how you know nature resolves this.
For the 'origin' problem, something you agreed with me (for different reasons) cannot exist, since we cannot reach outside of the universe to be certain there is nothing beyond that 'wall', Zeno's paradoxes ARE precisely as he stated but can be resolved with infinitesimals rather than infinities. In the real world at present, we know we CAN move and reach a wall that we are walking towards. But the actual reason for this is because there is still something on the other side of the wall, even if we didn't actually know what it may be. This is because we end up running into the wall for experimenting upon the challenge and get stopped.
This explanation may not fit with your but you still have to have one or you just wouldn't care to be discussing this at all. So can you tell me how you know and trust these are not paradoxes?
Okay. The Intermediate Theorem is a Calculus and logic proof that is even too hard for me to prove here. BUT we can intuit this as rational. If I have any TWO points in space, if there were NOT any other possible point in between them, it would be ONE and the SAME point, NOT TWO. So is this or is this not 'paradoxical'? We both already know that there is such thing as two different points in reality. But HOW can this be the case?Age wrote:Where was it "proved"?Scott Mayers wrote: ↑Sat May 04, 2019 11:59 amALL are based on one main concept:
In relation to space, if we take ANY two different points, we can imagine that no matter which two different points there are that we choose, there will always be another one that we can find in between those two no matter how close the first two points are. In geometry and math, this can be summarized by what is called the "Intermediate Value Theorem". Intuitively it is easy to imagine this but is actually a very very hard thing to prove but has been (and why they have called it a 'theorem')
We won't presume (pre-assume) this a 'theorem' but something that we might think as just a maybe true thing.
But, hang on, you just said it HAS BEEN proved (already). So, without ANY 'evidence' I am NOT going to might think as 'just a maybe true thing', at all.
Either you HAVE proof or you do NOT. You just stated that it HAS BEEN proved so you MUST HAVE some proof of WHERE this "proof" is.
Just because it is EASY to imagine some thing, then, to me, that in NO way even comes close to implying some thing is true, let alone inferring it is true.
I can just as EASILY imagine half way between two points in space as I can getting to a 'point' where the half way between two points is NOT even worth imagining about, and I end up just SEEING it as one point (in space).
Here is the general summary from a website on them without the proofs: NOW you ask why even state this obvious reality. WHY is this even necessary?
I need you to at least agree to this as being the case formally or you could possibly later claim you have no proof of something else that is based upon this. And note that the best Calculus books that prove everything step by step, SKIP this particular proof even though it seems obvious because it is actually very hard to prove and requires a lot of formal logic to present the case. All that matters is that you INTUIT this as true from your subjective reality.
I want you to get the sense of paradoxical flavor to the following more specific case: take two points exactly "next" to (or touching) each other. Then ask yourself, are these still two distinct points or are they one? If I say they ARE two, then it means they each lie in distinct spaces and then require asserting some 'measure' of the distance of the two points. CAN you find such a distance? If you say they are exactly ONE point away from each other, what is the SIZE of this 'point' measure mean?