Zeno and Immortality
I got this idea from ViHart recreational mathematician
One of Zeno's most famous paradox has to do with Achilles never being able to catch a tortoise that's been given a head start in a race because of the impossibility of having to traverse an infinite number of points between the two.
Using the same principle on a person x born 1976 and died 2019 can we say that x is immortal given that x had to experience an infinite number of time intervals?
Are we all immortal in some weird sense?
One of Zeno's most famous paradox has to do with Achilles never being able to catch a tortoise that's been given a head start in a race because of the impossibility of having to traverse an infinite number of points between the two.
Using the same principle on a person x born 1976 and died 2019 can we say that x is immortal given that x had to experience an infinite number of time intervals?
Are we all immortal in some weird sense?
Comments (63)
When you get to a point such as that in your reasoning, it's a cue to say, "Oops! I must have f-ed up somewhere, at least in some assumption I made."
Ok. That's really weird. Anyway what is ineffable in my post?
Strangely Terrapin is a type of turtle.
Why is that strange?
I was talking about the tortoise and Achilles paradox and a cousin responded. Coincidence! Strange.
Strangely "cousin" is a relational term.
You should be happy your "relative" wins the race and puts the Greeks to shame. :smile:
Anyway do you have any idea where I f***ed up in my reasoning?
I'm not sure but I'm beginning to doubt the whole notion of infinity.
If we're concluding that it's impossible to move then obviously we're going awry somewhere. Probably there isn't an infinite amount of points to cross.
That could be it. How do you then account for the following:
Mr x (1976 to 2019). x has to first reach 1997 and before that he has to reach 1986 and before that 1981and before that 1971 each time interval can halved indefinitely. The math says so. Is the problem with math or a subset of math infinity?
The problem is that mathematics is a way that we think about relations. The world isn't required to match that.
So the world of reason is describing a different world to the one our senses display to us which is one of motion and change.
1/2 + 1/4 +1/8... Shouldn't this tend to one at infinite? Just because we add up an unlimited amount of numbers doesn't mean we'll get a finite quantity.
Quoting TheMadFool
I don't think we are immortal because it's not like we are living an unlimited amount of intervals all of the same length: it's like cutting a square shaped cake first in a half, than the remaining half in a half and so on. You'll still eat one cake.
No. The union of all the points in that interval still lasts 43 years.
If we take time to be on a number line how many points of time are there between 1976 and 2019? Infinite, unless you want to invoke Planck time?
Wrong way to think about it. If you assume the time interval is an interval of real numbers, the measure (length) of any point is 0, but the measure of the whole interval which is the uncountable union of all points in the interval is just the usual length of the interval.
If you assume time is discrete, then the interval length is an integer number of multiples of the plank time (rounded up or down to 1 plank time).
Either way it's a finite time.
Then we're just being silly?
You might as well take time to be this painting:
We could just as well say that both are a representation of time.
It just that neither would imply anything in particular about what time is like objectively.
I guess an explanation for Zeno's paradox would apply here. A convergent infinite series.
This assertion is of course nonsense since there is no evidence whatsoever that neither time nor distance cannot be divided after some point. It's just below the ability to measure after a point.
So the argument nevertheless ties into the wording of the OP: There is very much a finite time interval (well above the Planck time) below which no 'experience' can take place, therefore shorter durations are not experienced. You have a finite number of 'experiences' in your life, and when they've run out, you're done.
Quoting TheMadFool
The math also says that you can cut the cheese as many times as you like and it doesn't give you more cheese. You f***ed up when you drew a different conclusion from the mathematics.
I also don't think Mr X ever needed to reach 1971, but that's just a nit.
The amount of partitions does no matter, provided you take the inertia in to account - it still takes the same amount of time for Achilles to catch the tortoise and to reach 2019 from 1976.
The trick is in the conflation of speed and distance.
Perhaps there's a problem with the model we're using - the number line. I mean if we do the math sure there's always a number between any two numbers but when you use a physical line to represent that idea we somehow end up with paradoxes.
Neither. The problem is it could just simply be that there is a "smallest time". As in the smallest amount of time that could exist. Similar to how the smallest amount of an element that could exist is called an "atom". I'm not very knowledgeable on this but from what I understand there is such a time. It's called planck's time. Look it up.
I don't see a problem with the model, moreso with its presentation.
A lens can magnify or reduce the size of an object, so you can manipulate its relative size.
If you zoom in, the length seems long - too long; but if you zoom out for instance, it'll look curiously short.
Numbers are infinitely divisible.
A line is infinitely divisible??? Zeno's paradox
Quoting Gregory
That illustrates what I mean by the model being inadequate for the purpose. Physical lines have limits. Mathematical lines don't
Where's the issue?
:smile:
A line is NOT infinitely divisible. Numbers are.
And you passed this infinitely divisible line, just as you pass from one infinitely divisible digit to another.
Is there an issue?
At which point we should try to figure out what the ontological facts about time are supposed to have to do with the concept of numbers.
Quantification (numbers) is the problem and also the solution.
It's the solution because once we have the numbers we can understand.
[quote=Galileo]Mathematics is the language of the universe[/quote]
It's the problem because what can be done with math can't be done with reality. Zeno's paradoxes.
I think it can be best explained as:
All things in the universe are things that are mathematical
BUT
Some mathematical objects are not things in the universe
It's just the problem, because there's no reason to believe that time (or space for that matter) works just like our concepts re numbers.
:up:
I am guessing that when moving, we are not traversing infinite points because of how impossible that is. I am guessing that, just like movement that we see in a computer screen, we disappear and appear in a different point.
Assume everything, to the smallest particles would be still without anything moving anything else. The fundamental forces wouldn't effect anything. What would it matter if a microsecond or a milennium would pass by? If after two milennia things started to move again, we wouldn't notice the two milennia that past just by.
The mistake in the OP, going all the way back to Zeno, is thinking that discrete dimensionless positions in space and discrete durationless instants in time are real. Instead, it is continuous motion through continuous spacetime that is real, while positions and instants are useful fictions that we create for the sake of description and measurement.
Quoting Terrapin Station
Indeed, mathematics is the science of drawing necessary conclusions about hypothetical states of things, which may or may not match up with any real states of things.
Quoting TheMadFool
Again, a continuous line or interval of time does not consist of discrete points or instants at all, but we can mark any multitude of points or instants along it to suit our purposes. In other words, contrary to Cantor, there is a fundamental difference between a continuum and a collection.
Quoting Filipe
What Cantor got right is that there is likewise a fundamental difference between an infinite collection and a finite collection, such that we cannot reason about them in the same way. The multitude of real numbers between any two arbitrary values is the same, because they can be put into one-to-one correspondence with each other.
No, the collection of all combinations of the subjects of a collection--even an infinite collection--is always of greater multitude than that collection itself. The integers and the rational numbers can be put into one-to-one correspondence with each other, but not with the real numbers, because those are of the next greater multitude. There is another multitude greater than that, and another greater than that, and so on endlessly--which is why an infinite collection of any multitude can never be "large" enough to qualify as a continuum.
Quoting Gregory
That depends on what you mean by "parts." The portions of a continuum are indefinite, unless and until they are deliberately marked off by limits of lower dimensionality to create actual parts. For a one-dimensional continuum like a line or time, those limits are discrete and indivisible points or instants that serve as immediate connections between portions, but the portions themselves remain continuous--which is why they can always be divided further by inserting additional limits of any multitude, or even exceeding all multitude.
Again, all the different combinations of subjects of any collection--including any infinite collection--is of greater multitude than the collection itself; i.e., there are not "enough" subjects to be put into one-to-one correspondence with their combinations. The real numbers correspond to all the different combinations of rational numbers, so the real numbers are of greater multitude than the rational numbers; i.e., there are not "enough" rational numbers to be put into one-to-one correspondence with the real numbers. Put another way, the real numbers are a "larger" infinity than the rational numbers.
Mathematicians are well aware of it, and it is not a problem at all. The real numbers are of greater multitude than the rational numbers, but the even and odd numbers are of the same multitude. We would never "run out" of even numbers to pair with the odd numbers in a one-to-one correspondence. We would never even "run out" of even numbers to pair with the integers, despite the fact that there are only half as many of them on any finite interval. Again, we cannot reason about an infinite collection in the same way as a finite collection, and we also cannot reason about a true continuum in the same way as an infinite collection.
Because the real numbers correspond to all the possible combinations of rational numbers, and therefore are necessarily of greater multitude than the rational numbers themselves--which are of the same multitude as the natural numbers, along with the even numbers, the odd numbers, the whole numbers, the integers, etc.
Quoting Gregory
"Countable" is defined as being of the same multitude as the natural numbers, and thus applies to the rational numbers, the even numbers, the odd numbers, the whole numbers, the integers, etc. "Uncountable" is defined as being of a multitude greater than that of the natural numbers, and thus applies to the real numbers.
By the way, I am using Peirce's terminology by referring to the "multitude" of a "collection," rather than the standard terminology that refers to the "cardinality" of a "set."
Density is irrelevant to multitude, and in any case the whole numbers are of the same multitude as the odd numbers.
For any collection A that has n subjects, the collection B of all the possible combinations of A's subjects has 2^n subjects, and 2^n > n for any value of n (whether finite or infinite). Therefore, B is always of greater multitude than A; it is commonly called the "power set" of A. The real numbers correspond to all the possible combinations of the rational numbers, so the collection of real numbers is of greater multitude than the collection of rational numbers.
Suit yourself, but I will go with the mathematicians on this. Cheers.
Cantor wrongly thought that the real numbers constitute a continuum, but as I noted previously, they can only constitute an infinite collection--one whose multitude is greater than that of the rational numbers. His own theorem proves that there is another collection of even greater multitude, and another greater than that, and so on endlessly. Consequently, a true continuum cannot consist of discrete subjects (like numbers or points) at all.
Quoting Gregory
Right, but that paradox stems from the same mistake of treating discrete points as if they were somehow continuous. It reflects a limitation of such standard models of continuity, which are adequate for most mathematical and practical purposes. Banach-Tarski does not arise in a better model of true continuity, such as synthetic differential geometry (also called smooth infinitesimal analysis).
To get to where TP is at one point, takes less and less time for Achilles, until the distance between the two and the time to reach the point both reach zero.
You can only think about this in terms of Calculus.
The closer Achilles gets to TP, the less time it takes PROPORTIONALLY. Zeno assumed that it takes a non-proportional time. So by the time Achilles reaches TP, and the distance between them is zero, then the time elapsed to travel that distance is also zero.
IN other words, when the time elapsed between a prevous point in time to the time Achilles reaches TP is zero, the distance between the two of them is also zero.
Consider a sum of fractions: 1+1/2+1/4+1/8+1/16... which is the sum of all 1/2**n, for n= positive integers zero to infinity. There is a proof that proves that the sum is equal to 2. Don't ask me for the proof, because this was taught to us in grade 12, an immense amount of time before I forgot the proof. But I do live with the conviction that the proof exists!
Anyhow, if the distance halves between Achilles and TP and the time halves for achilles to reach the point where TP is, then somehow magically you can marry the two: (the sum of 2(**-n) and (Zenos paradox)), and there you go, bob is your uncle.
I am old-fashioned, and old-brained. If i remembered every proof and every thing I ever learned in school, then I would be crazier than I am not now.
The introduction of an infinite series is not a solution, and would actually support his impossibility argument. While Achilles moves a distance x, the tortoise moves a distance y.
The underlined word is key, implying simultaneity. It was another poster (yrs ago) who called attention to the fact that the rate of motion was constant. Thus the time to overtake the tortoise was separation/(difference in speed). Zeno was demonstrating the nonsensical results from assuming space as a continuum.