Before we get into the “oooh” and the “aaahh,” or maybe even “huh? I don’t get it,” we took a little of time to talk about what is it mean by “countable?” A couple of our classmates argued that if infinity is boundless and has no limit, there is no way it’s countable. However, at the end we came to an agreement that countable is a special kind of infinity and that in order to be countable, it has to have one-to-one correspondence, and it can be put in order. For instance, natural numbers (N) are considered to be countable because we can say that 1 comes after 0, 2 comes after 1, 3 comes after 2, and so on, whereas P (N) or even the real numbers (r) are uncountable because they do not have one-to-one correspondence, nor can they be put in order. We don’t exactly know what comes after 1.278563; is it 1.278564 or is it 1.2785631 or is it 1.27856305?
After understanding that the idea of infinity has the first, second, third, and so on is the essence of it being countable, we went on to talk about paradox. What is paradox? It’s not just a contradiction, but it’s a statement that could be false and true at the same time, which does not suit very well with Aristotle’s “law of excluded middle.” There are different types of paradoxes, one of which is the self-referential type that can be shown in a famous Russell’s paradox, stated “R be the set of all sets that are not members of themselves.” There is no right or wrong answer here, thus in order to “resolve” this problem, Russell decided that a huge set is not a set but a class.
Regarding the paradoxes, we went ahead and looked at 4 of the Zeno’s paradoxes that Aristotle tried to compile things to give an answer to each of them. Each paradox does not work individually; hence they work as a group. The four paradoxes are:
1. Dichotomy: in order to reach a certain location, you first have to get to half of that distant, and to get to that half of that distant, you have to get to half of the half of that distant, and so on. This is the infinite half-way points, which has no end, yet we have to pass all these infinite steps to get to that certain location.
2. Achilles: By letting his tortoise starts out first, it doesn’t matter how fast Achilles runs, he will never catch his tortoise. This is because before he can overtake the tortoise, he first has to reach p1, but when he reaches p1, the tortoise is still ahead, reaching p2. Again, when Achilles reaches p2, the tortoise is still ahead, reaching p3, and so on.
When looking at these two situations, one might think that this is easy; if time and space has the smallest unit and is not infinitely divisible, of course, we will be able to move one unit of space at one unit of time to reach that certain location and catch that tortoise. However, let move on to the third and the fourth paradoxes:
3. Arrows: clearly, we can definitely see the movement if the arrow moves from point A to point B. Assume that time is indivisible like we mentioned above, we will use the word “instant” as the smallest unit of time. If the word instant cannot be divisible, there wouldn’t be a start or an end, and thus the arrow never moves at an instant. The entire of time that the arrow moves from point A to point B will just consist of a bunch of “instant” moments. This means that the arrow has never moved, yet we see movement as the arrow goes from point A to point B.
4.Stadium: The picture below assumes A is stationary, B is moving on unit to the right of A, and C is moving one unit to the left of A at the same time. If it takes one smallest unit of time to move the purple box of B to match with the red box of A, the time that it will take for the black box of C to match with the blue box of A is also one smallest unit of time. However if time can be atomic like one assumes in the first two paradoxes, then how long will it take to have a black box of C match with the purple box of B?
Why do the 1st paradox and the 3rd one contradict?Same, why do the 2nd and the 4th contradict? Here are what i understand why 1 and 2 are put together. And 3 and 4 are put together. Are they sort of the ideas of contradictions?
ReplyDeleteWhat i understand is that the dichotomy and achilles are like infinity inside finite.Because we know where the start and the end are and there are infinite breaks between them. However, eventually we can move to the location. Same with achilles. The distance that the tortoise moves are gradually smaller and smaller, he still can make it to the end first.
The arrows and the stadium is more like infinity towards impossibility. There are infinite instant amount of time between point A and point B.We can't break the instant into smaller because it's the smallest. So it's impossible. For the Stadium example: 1 unit of time is the smallest of time for the boxes to move. However, C would pass 2 boxes of B, meaning it breaks 1 unit of time down (it'd pass the purple box of B definitely some time)
Thank you for reading :D
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ReplyDeleteThanks for the great summary! That really summed things up.
ReplyDeleteI agree with you, Mai, that the first two are like infinity inside finite. We are able to divide distance infinitely into pieces. Yet in the second two, we can't divide time into infinite pieces. In the arrows one when we look at an arrow in flight it is the same motionless arrow. so how do we see motion? and how would this be any different from an arrow hanging in the air if we can only see it as one instant?
Mai, you are thinking on the right track. To answer your question, the first two paradoxes cause us to believe that time and space has a smallest unit. On the other hand, if we keep this belief to the third and fourth paradoxes, they wouldn't make any sense. Take the first and fourth for example. If you don't have the smallest unit of space, you wouldn't never get to the door. However, if space is atomic, then how long will it take the black box of C to match with the purple box of B when both of them are moving and A remains stationary (referring to the example above)? Hope this help.
ReplyDeleteMiko,I think the way we see thing is like comparing how fast your eyes can take in the information to how fast that event is happening. We see things more on a bigger scale...way bigger than that one instant. Believe it or not, our eyes don't actually see things, but our brain is the one that control everything. I guess, instant is so small that as human, our brain doesn't have the abilit to catch that nerve impulse. Thus, we see motion (on a bigger scale) but not on the small scale like an instant.
Vorleak,
ReplyDeleteI agree with you about our eyes not recording what we see exactly the way it happens. Our eyes/brains are only able to process the "general idea" of what we saw. An example would be to record the flight of an arrow and slow it down when watching it on video, it may take a different trajectory that what you actually saw. But as the saying goes: "Seeing is believing" right?
I spent a while arguing with Prof. Hamman to no avail on why the first two don't cause issues even if I believe there is no smallest unit of time or space. However, after finishing the reading, I think I can better argue it now. In walking from a given point A to point B, we may indeed pass through an infinite amount of points, however, those points are indistinguishable. There is nothing to set them apart in our mind from just being a point along the path. With that, nothing happens along those points to obstruct movement.
ReplyDeleteIt works the same way with Achilles. There may be an infinite amount of instants before he catches the tortoise, but none of them are significant, as he does not change anything he is doing at those points.
My original argument was, treating space as infinitely divisible, if movement can happen, there has to be some actual distance we can move, no matter how small it may be. However, in moving across even that small distance, we have traversed an infinite number of points, because we can divide that distance up as much as we want. Zeno's dichotomy would fail due to this, as he says you have to get to the halfway point. He then says you have to get to the next halfway point, and all of the ones after that. Well, if you can make it to the first halfway point, you have already traversed an infinite amount of points, showing that is possible.