Tag: Paradox

The Better Solution to Zeno’s Paradox of Motion

This is about Zeno’s “dichotomy” paradox of motion.

As described by Wikipedia:

“Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on. […] This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.”

Since I’m not necessarily a mathematical Platonist, I won’t maintain that there “is” an infinite number of points to a number line; they don’t exist until you construct them individually (and then they only exists as abstractions/ideas). The paradox presumes distance should be idealized as a number line in which an infinitude of infinitessimally small distance-intervals exists already. Even if mathematics were Platonic, it would still be questionable whether an ideal number line characterizes real, physical distances.

Now, the standard calculus answer does work—that as time intervals and distance intervals diminish, they both approach the same limit (being zero) simultaneously. But that answer deals with infinities, as well as relies on mathematical equations, which are not process-oriented, and I believe that infinities are unnecessary—since distance intervals don’t exist until you create them—and that, as such, the problem could best be understood as a process—particularly a thought process, given that the problem is a thought experiment. (Yes, motion does actually happen in real life, but there’s obviously no paradox in motion intrinsically, or it wouldn’t happen; thus to contemplate the paradox, per se, is to contemplate the thought experiment as a thought process.)

So here is my deconstruction of the paradox as a thought process.

Here’s what we do: we take one step in this thought process—we imagine the thing to move half-way (say, 4m). Then we take another step in the thought process—we imagine the thing to move half-way again (2m). Then we imagine it moving again (1m), again (.5m), and so on and so on. The problem here is that, while the distance traversed diminishes for every successive iteration of the thought-loop, the time it takes to think it through, doesn’t. Thus, it appears that, while time (being easily conflated with thinking-time) carries on indefinitely, the distances traveled get shorter and shorter incommensurately with the time intervals associated with them. If one could reduce one’s thought-process time by half for each successive iteration as the distance intervals diminish by half, then perhaps one would see things differently.

The proof that this mental effect is what’s responsible for the appearance of a paradox would be that we wouldn’t formulate the argument as so: the object must travel for half the time it takes to get there, then half the remaining time, then half the remaining time again, etc., and therefore it can never get there since it takes an infinite number of steps. That formulation is symmetrical to the distance-based one and carries the same implication, but somehow it just wouldn’t work—­it doesn’t have the same poignancy as the distance-based one.

One other point that comes to mind is that the idea alone that infinitessimal distance intervals don’t exist until you imagine them resolves Zeno’s paradox because there is no infinitude of steps; there is only the circular thought process, which carries on for exactly as long as one carries it on. In the same vein, contrary to popular belief, the value pi is not “infinite”…it’s a prescription for constructing a procession of digits of indefinite length. (Actually, it’s just a value. What was described is the algorithm for expressing that value in decimal form.) The actual value of pi can actually be expressed more succinctly, and precisely, as follows:

(Maybe there’s a more concise way to express that in summation notation, IDK.) Admittedly, this is actually another prescription for iterating in a loop; it just isn’t about creating digits, per se. The more iterations you step through, the closer your approximation is to the limit л. To use this as a prescription for generating accurate digits of pi, you would have to (a) convert your value to decimal (or another base, if you preferred), and (b) know how many digits are accurate at a given iteration and discard the rest. In any case, the above

formula itself is not an approximation, and taken as a whole, it is not infinite; that is, it does not contain an infinite amount of information, or it couldn’t be shown on your screen.

Can God create a rock so heavy that He can’t lift it?

Can God create a rock so heavy that He can’t lift it? The question stems from the common knowledge that God is supposed to be all-powerful. Now, exactly what is power? Is it within the functionality of power to create a rock so heavy one can’t lift it? It seems to be like the question of what happens when an irresistible force meets an immovable object: the answer is that neither an irresistible force nor an immovable object can exist (or at least only one of the two may exists); it’s only in man’s imagination that such absolutisms are construed to be consistent with nature. It’s the convenience of words: they can be used or abused, inflected and combined in any way you want them to, seemingly able to accomplish anything. 

But the question is really not about power, but ability. If god is all-able, why can’t he create the circumstances we call “having a rock so heavy He can’t lift it”? Rather than trying to chalk this up to a physics question—since any amount of weight would be immaterial to God anyway—let’s consider it in the general case of God creating, at time x, a limitation of Himself at time x+n. That is, He creates the rock at time x and then experiences the limitation of not being able to lift it afterwards at a later time x+n. That is an interesting question, for if God is infinitely free, then how could He limit Himself, or even predict Himself, at a future time? And to say nothing of the fact that God may not operate within a framework of linear time anyway, making it ambiguous whether He may have lifted the rock before or after the time at which he decreed that He couldn’t lift it. Or did He both lift it and fail to lift it both before and after the supposed time? 

This sounds like another operation in linguistics: we think that, if God has “unlimited ability,” then He must be able to fulfill the condition of any string of words whatsoever we can come up with that would seem to indicate a state of affairs. If God is infinitely free, or if God operates outside of linear time (notwithstanding the fact that an “operation,” at least as such, is a process and requires the passage of time by definition), then the idea of “creating a rock so heavy that He can’t lift it” seems to be against His nature.

So, the question then becomes, “can God defy God’s own nature?”, as a god that cannot lift a given rock on some specific occasion is, apparently, not all-powerful. Or perhaps more accurately, since this is purely analytical, it’s “can God behave in a way that belies His own nature/definition?” So here we see that what the question hypothesizes is a logical paradox and is equivalent to the question of whether God could create a square circle—although that still leaves open the question of whether the fact that God can’t, for example, create a square circle makes Him not all-powerful. I don’t think it does because, if God is all-powerful, presumably omnipotence is a real thing. A square circle, on the other hand, is merely an absurdity concocted by linguistical combinatorics. (Yes, you could say that the thing in question is whether God’s omni-power is a real thing, but since for the sake of argument we don’t know whether it’s a real thing, and do we know that a square circle isn’t a real possibility, the square circle case therefore does not impinge on the case of omnipotence, at least if you accept my premise that the use of language can be a weaselly son of a bitch.)

That being said, in the many dimensions in which God does His godding, there may exist one very real sense in which He has created a rock He can’t lift—that is, real in the sense that we personally, physically experience it on a daily basis. Imagine that God makes a rock He can’t lift by continually deciding that He can’t lift it, until He decides that He can lift it again! And furthermore, maybe He could decide to make a rock that He can’t decide to be able to lift, until He decides that He can decide that again! Or He could decide to make a rock that He can’t lift until He decides that He can decide that He’s able to decide to lift it again. And so and and so forth. It’s easy to imagine that if He creates enough levels of this He could get lost in it and forget that He has the power! Maybe He could create an experience where He forgets His true self altogether! At least temporarily. Could we possibly call that experience…being a human being?  Indeed, if God is omni-present, then we must, for there can be no part of a human that is ever anything but God.

Cheap labor vs. jobs

Imagine you have a servant who does all the stuff you don’t particularly want to do for, say, 5 cents an hour. He vacuums your house, he builds your furniture, etc. Or even imagine that you have a person you can buy a new car from for $100. Now imagine that, instead of the subjects being you and your servant, they are your country versus another country that gives your country goods and services for really cheap prices. How could this possibly be a disadvantage?

The problem—or at least the widely-perceived problem—is, of course, that when people get goods and services for cheap from overseas, they no longer pay the people in their own country to do the work, so people lose jobs. This is why we have protectionistic measures in effect such as tariffs, quotas, etc. But the same very basic principle that makes it a benefit to an individual person to receive the benefits of labor for little cost should logically also apply to an individual group of people or country, therefore the concern that receiving the benefits of cheap labor overseas somehow harms us is fundamentally irrational.

The only difference in the case of the whole country from that of the individual is that in this case, some members will gain the benefits from this type of transaction while others won’t, i.e. those who don’t can’t buy anything because they’re unemployed. Therefore, if only the benefits of this kind of transaction are absorbed by the receiving nation wisely, or in other words, if the received goods and services are distributed evenly (or evenly enough) among the country’s populace, it should be an overall boon to the country and everyone should benefit.

The reason this isn’t the case is that, under our current capitalistic ideology, we’re way too afraid of the idea of giving anyone economic benefit who didn’t earn it through labor. If we simply accept that we can be wealthy enough as a nation for some people to live well without having to work, then the availability of cheap labor overseas should never be a problem..

The same principle above applies to the threat looming on the horizon—and to some degree happening now—of robot labor replacing our jobs.

Of course, the details of how to wisely redistribute the wealth resulting from cheap labor would probably be complex, especially due to the issue that if nobody has any incentive to work, due to being able to receive economic benefit for free, then the country as a whole won’t have anything to offer foreign countries in exchange for goods and services; but I believe the problem is solvable and should be solved.

The details of how to wisely redistribute goods are outside of my intentions for this article, but one thing I will mention is that if people are guaranteed a sufficient, but minimal, living (such as shelter, food, water, clothing and possibly basic medical care) without the need to work, then the desire to have a much better living might still act as a sufficient incentive for a sufficient number of people to join the workforce.

I suppose a legitimate concern might be “what if there aren’t even enough jobs available for those who want to work to support the whole nation because they’re all replaced by overseas jobs?” And given that jobs could be done for lower wages to create more openings, I think this is equivalent to (or isomorphic to) the concern that our own labor may become too cheap due to its naturally shrinking until it matches the price of overseas labor.

To this concern I would say that there must be something already advantageous about the natural resources, infrastructure, government, culture, education, intelligence, place within the international community, or something of the country receiving the benefits of cheap labor that affords it its superior economy, or in other words that gives its people the luxury of working for higher wages or gives the country a stronger currency; and this has to be some advantage other than simply a practice of paying people more money to work, because doing that would end up simply causing prices of all goods and services to go up and hence raising inflation and weakening the country’s currency. And therefore, whatever advantage that country has already would remain even if they adjusted distribution and the workforce so properly reap the benefits of cheap labor overseas.