I am gravely disappointed that so many people have trouble have difficulty resolving Zeno's paradoxes. I have resolved the one's I've looked at, and I will start with the most famous example: Achilles and the Tortoise.
"[I]t is impossible for [Achilles] to overtake the tortoise when pursuing it. For in fact it is necessary that what is to overtake [something], before overtaking [it], first reach the limit from which what is fleeing set forth. In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if it is less than that which what is pursuing advanced … .And in the time again in which what is pursuing will traverse this [interval] which what is fleeing advanced, in this time again what is fleeing will traverse some amount … . And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount." http://plato.stanford.edu/entries/paradox-zeno/#ParPla
I'm curious to see the Mises community tackle it. I'm optimistic in that I don't think it will take a few of you very long to do so. After some discourse, I will gladly offer my solution.
Oh, and aside from the unique Road Warrior expression that I started with, it's not disappointing really, but that's such a cool phrase I had to use it.
Thoughts?
I guess a restatement of the paradox would be:
1. The limit of an infinite series is never actually reached by any finite partial sum. [Laws of Math]
2. Any measurable action we perform on this Earth can be subdivided [conceptually] into an infinity of partial actions, each also measurable, whose sum [in the sense of the sum of an infinite series] is the measure of the original action. [For it is valid to create a model of the universe that makes sense to us, and this model is perfectly sensible].
3. But in the real world, only half our model is applicable. For "subdivision" in the sense of 2 is real, but "addition" in the sense of 2 isn't.
This is because subdivision in our minds is possible, being merely a concept, and this subdivision accurately reflects reality.
But summation happens in the real world, where it is not possible to perform an infinite summation, neither by humans nor by nature. Nature is not a cosmic glue that melds all the infinite partial actions into one. Therefore no summation is possible. Therefore the tortoise never catches the hare.
So what's the flaw? The flaw is in assuming his model makes sense. Slicing something up into a finite number of pieces makes sense to us, and we justified to feel we know the laws that apply to finite slicing, because we see countless examples of it every day of our lives.
But we have never seen anything divided into an infinity of pieces. Therefore we have no basis for assuming we know the laws that apply to infinite slicing in the real world [even granting that such a thing is possible]. So we have no basis for the assertion that nature is not a glue that melds infinitely many objects together. On the contrary, we see that it is.
EDIT: Taking it a step further, when we slice up something like a foot race into infinite pieces, implicit in the very act of slicing is the assumption that the sum of the slices is the original object. Otherwise we would not be justified in slicing it in the first place.
For example, when we physically slice a salami, we can argue along the lines of Zeno's paradox thusly:
The whole is equal to the sum of its parts. Therefore all these slices of salami are equal to the original salami. But you can make thirty sanwhiches, say, out of those slices, but you cannot make even one sandwhich out of the original salami, because it is too big to fit into a sandwhich.
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It's easy to refute an argument if you first misrepresent it. William Keizer
Smiling Dave: This is because subdivision in our minds is possible, being merely a concept, and this subdivision accurately reflects reality.
I agree with you on the first part, but I disagree on the latter. How is mental subdivision an accurate representation of reality?
Smiling Dave: So what's the flaw? The flaw is in assuming his model makes sense. Slicing something up into a finite number of pieces makes sense to us, and we justified to feel we know the laws that apply to finite slicing, because we see countless examples of it every day of our lives. But we have never seen anything divided into an infinity of pieces. Therefore we have no basis for assuming we know the laws that apply to infinite slicing in the real world [even granting that such a thing is possible]. So we have no basis for the assertion that nature is not a glue that melds infinitely many objects together. On the contrary, we see that it is.
His model does make sense in the realm of mathematics, though, and this is where I see Zeno's flaw. Mathematics doesn't exist in reality. It is inherently a mental construction, just as you pointed out in a different way in your text above. Laws of physics, biology, properties of matter . . . none of these actually exist in reality.
We can take a system of any kind, superimpose it onto reality, and while it might line up as a perfect stencil (for instance, I'm not going to walk off a cliff anytime soon even though a system tells me I will fall to my death and I'm saying the system is fake), the system itself is purely subjective. It is the same with infinity, as you pointed out. There is no way to even conceptualize infinity much less see it in nature. Thus there is no way for us to manipulate it and use it or view it outside of a subjective construct that doesn't actually exist.
Zeno's example of Achilles does not work in reality. Achilles would blow by the turtle in all accounts, and we know this from experience. Can we mathematically explain it? No, and that's why his paradox frustrates so many people. But the truth of the matter lies in the distinction between the mental system of mathematics or physics and what is real. Zeno maintained that his paradox showed that movement and change is an illusion, but it really doesn't. Mathematics and all subjectively constructed systems are the illusion.
Max Planck worked out the smallest possible unit of distance or time that can conceivably be measured. No one has actually measured anything nearly that small, of course.
It's simple: just because you can keep coming up with intervals by which the tortoise has not yet overtaken doesn't mean that that is true for *every* interval. Just because I can endlessly keep coming up with sentences that start with the 2 words "Just because" does not mean that all sentences that start with the word "just" must have "because" as their second word.
Why anarchy fails
There is no such thing as a paradox. As with every other paradox all you are doing is using a camoflage language and misapplying a set of rules to the mechanics of what is going on via language tricks. It's a worship of words as a "fixed idea".
"As in a kaleidoscope, the constellation of forces operating in the system as a whole is ever changing." - Ludwig Lachmann
"When A Man Dies A World Goes Out of Existence" - GLS Shackle
The solution is simple if you accept that time and space are discrete on a fundemental level and slicing into infinity (whatever that actually means) is nonsense since it implies there is nothing in reality itself.
Given that time and space are discrete it follows that the tortoise can be stationary at one time period when Achilles actually moves. So it is then possible for them to come along side each otehr and then Achilles can clearly pass the tortoise.
The atoms tell the atoms so, for I never was or will but atoms forevermore be.
Yours sincerely,
Physiocrat
vive,
Could you kindly point out where the language trick, the misapplication of the rules, actually is in this particular case?
Or are you saying that all of mathematics and theoretical physics is unreliable, since it is only a language?
When a practicisng physicist finds something in his lab that doesn't agree with established laws of physics, should he just shrug his shoulders and say, "Oh well, the laws of physics and math are just a camoflage language. I guess we all misapplied them all these years"?
Gamma rat,
If we throw in Max Planck the paradox will be that when we get to the stage that the rabbit has to move a distance so small it cannot be measured, then we cannot know who will be ahead. And yet we see the rabbit always wins.
vive la insurrection: There is no such thing as a paradox. As with every other paradox all you are doing is using a camoflage language and misapplying a set of rules to the mechanics of what is going on via language tricks. It's a worship of words as a "fixed idea".
This is very well put. I agree. There is the system of language and mathematics that we use to define the world, but the definition is not the world itself. Also, true, paradoxes are impossible . . . they are merely a concept that we can abstractly grasp. For instance, the classic "What would happen if an unstoppable force met an unmoveable object?" Even though we can comprehend the paradox's cause, we cannot grasp the effect. For instance, nobody can conceive what would actually happen if such a force and object met. One might even say we can "know" that no such forces or objects exist in the universe, though not everything in the universe has been witnessed, so there can be no scientific certainty of it.
I do have a question I'm curious about, though: what is your take on the origin of the universe? Or is there an origin to the universe? Since you maintain that paradoxes cannot exist (and I agree), I'm anticipating that you do not believe that something came from nothing.
Physiocrat.
Slicing into infinity has been clearly defined. It does not imply that there is nothing in reality itself, only that a single point in space has no dimension. Existing and having a dimension are not the same thing.
Euclid made just that assumption, that a single point has no length, and it has been universally accepted for thousands of years to this very day, and nobody had any problem with it.
Also, Newtonian physics [AKA the only thing we've got] assumes that time and space are not discrete on a fundamental level. Nor does it assume that the tortoise is stationary at some point.
So we can restate the paradox this way: Given the universally accepted assumptions of Euclidean Geometry and Newtonian Physics, how do we resolve the paradox?
vive is right: there are no paradoxes, it's always just misuse of language. And yes, geometry is full of mystical silliness, but it is used without major issue most of the time because the main applications have found ways of avoiding contradiction when using it, though unfortunately this sometimes requires messing up the language even further. Paradoxes that are themselves well-formed are therefore red flags that the system you're working in has a flaw somewhere - that is, a misuse of language in the fundamental aspects of the underlying theoretical framework. Zeno's paradox as phrased above never says the hare doesn't overtake the tortoise; it just gives a method of endlessly designating a sequence of intervals over which the hare hasn't yet overtaken.
Smiling Dave:Euclid made just that assumption, that a single point has no length, and it has been universally accepted for thousands of years to this very day, and nobody had any problem with it.
Right - a point is an abstract mathematical construct. It has zero extension in any dimension under consideration. Therefore, any extension can be said to consist of an infinite number of points. Yet the extension is (presumably) finite. How does this make sense?
I think it makes sense if one stops thinking of a line, area, volume, etc. as being composed of points at all. One can instead think of these things as separations or relations between/among points. Indeed, the very notion of location implies a relation to something else.
Geometry isn't even necessary to think about this. How many rational numbers are there between 0 and 1? Infinitely many. Yet the extension between 0 and 1 is said to be finite. Is there a paradox here? I don't see how.
The keyboard is mightier than the gun.
Non parit potestas ipsius auctoritatem.
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Autolykos,
The right Math course answers all your q's. They have been chewed over by great minds, and they came up with answers.