# half of a half of a half of a philosophic problem

So I’m sitting at my computer when all of a sudden I am overcome by a horrible thirst. Luckily, my beverage of choice is conveniently contained in a can a mere foot from my right hand.

So I reach over to grab it. Before my hand gets to the can, it first must travel half the distance between the keyboard (where it was originally) and the can. Then it must cover half the remaining distance. Then half of that. And half of that, and half of that, and so on and so on and so on.

From the looks of things, I am never going to get that drink after all.

But of course I do. This is a problem I first heard of back when I took calculus in high school, and it just recently popped up when we hit Spinoza in my philosophy class. I didnt understand how I got the can then, and I don’t understand now.

Can anyone explain?

Think about quantization… you cannot expend an infinitely small amount of energy moving your hand, so you eventually will not be able to divide by two. Or you could think about it terms of limits.

This sounds like a variation of what I heard of as the bald man’s problem. This is, when a man loses a little of his hair you are not going to call him bald. He continues to lose hairs one at a time. Is there a specific point when he becomes bald? Which hair that he lost was the difference between being bald and not being bald. Judging by there not being a specific point when the man became bald, you could falsely assert that he is not bald.

These are just philosophical tricks that cause you to think about the problem in the wrong way.

I seem to recall a little joke Mark Twain wrote about the shortening of the Mississippi River… which, due to erosion, loses a certain amount of its length as it becomes straighter…

But why cant I expend smaller and smaller amounts of energy? Is the whole idea that eventually I wont be able to tell the difference anway? And even if I could expend less and less energy, I wouldnt want to - I want to just power my hand past all that space and grab my drink. I hope it’s a beer. Or a whiskey. But I cant power past the space because I always have half of what is left before I get there. And if I could think about it in terms of limits, I would have understood it the first time I took calculus : )
And TexasSpur that is a good point about the bald guy - when is it appropriate to call a guy baldy, and when is his hair just thinning? I never really thought George on Seinfeld was bald. Mr. Clean - he is bald. Telly Savalas - bald. An egg? Bald.

The sum of the series 1/2^n from n=1 to infinity (1/2+1/4+1/8 and so on, ad infinitum) is 1. It’s a geometric series.

The flaw here is that you are not covering half the distance, and then half of that, you are covering the entire distance at once.

Regarding quantization: You cannot keep dividing the expended energy by two because energy comes in discrete (quantized) pieces, like atoms of energy.

You are not covering the distance all at once. Your hand doesn’t move from here to there instantaneously. You may, in fact be moving half way there, then half of what’s left, etc. Consider:

If Sneeze’s hand starts two feet from the drink and moves at a constant velocity of 1 ft/sec. then its motion is given by the following chart:

Elapsed time delta t Distance traveled delta d
in seconds in feet
1 1 1 1
1.5 0.5 1.5 0.5
1.75 0.25 1.75 0.25
1.875 0.125 1.875 0.125
1.9375 0.0625 1.9375 0.0625
. . . .
. . . .
. . . .
2.000000 ? 2.000000 ?

(The zeroes after the 2’s are just for for visual balance.)

Although the dot, dot, dot hides infinitely many “steps”, each step takes half the time of the previous one, so the whole procedure takes only 2 seconds.

Part of the problem seems to be generated by each one of use saying, out loud or mentally, “First I go half way, then half of what’s left, then half of what’s left, etc.” Each stated step takes time. It appears that infinitely many such steps would take an unlimited time. However, if you say it like this:

F i r s t , I g o h a l f w a y ,
t h e n h a l f of w h a t’ s l e f t,
then half of what’s left
thenhalfofwhat’s left
thhfowhtslft
thofwslt
towft
.
.
.,

I think that perception goes away. Say each line faster than the previous. By the time you get to the fifth line or so, you should sound like Mickey Mouse on helium.

Ah well, Natur abhors a vacuum, but human’s adore a pardox.

Well, the formatting in my chart was completely screwed. Make your own.

The paradox that Zeno postulated is not really a paradox at all when you carefully consider the individual tenets of each side of the equation.

The original paradox states (approximately) that if a tortoise wishes to travel from point A to point B it must first traverse a given distance of (let’s say) halfway. In front of the tortoise will now be a shorter distance that remains to be traversed. If the tortoise then traverses half of that distance, there remains yet another shorter half-length of the distance to travel. In theory, the distance between points A and B contains an infinite number of these half distances. Ergo, to traverse them would require an infinite amount of time regardless of your velocity. This is the pivotal error in this paradox.

Yes, there are an infinite number of Euclidean points in between points A and B. For that matter there are an infinite number of mathematical points between any two points you can locate. This is an artifact of Euclidean geometry. The nonmaterial nature of Euclidean points allows them to mathematically manifest in such a fashion. However, in the material world there is no such theoretical luxury as the ability to identify an infinite number of points. To identify all of them would take an infinite amount of time (is there some resonance here?), and is therefore, unachievable.

We live in a delimited material system that has a finite structure. This finite aspect is what overturns Zeno’s paradox. There is a fundamental graininess to our material world. It is one of the great quests of particle physics to identify the basic building blocks of our material universe. Some say that quarks are those components. Others are now leaning towards string and superstring theory, to define the basic structure of reality. Whatever may be the case, a given distance is physically delimited and cannot contain an infinite amount of any dimensional quantity. It is for this reason that we are able to walk across the room or travel anywhere else for that matter. We live in a world of finite distances and finite quantities of matter.

As a scientist, the realization of the flaw in Zeno’s paradox was a personal epiphany. It demonstrated to me how scientific analysis must be set aside in the accomplishment of everyday tasks. You may spend an infinite amount of time theorizing, but the job just will not get done. This is where I first located the correct and central function of faith in my own character. I must have faith that I can traverse what is otherwise an infinite number of scientific points in space, even if I only wish to make it out the door in the morning.

So when I brake my car in order to come to a complete stop, the car gets progressively slower, but at what point does it actually stop?

There must be a point in time at which it has only just stopped, indicating that at the previous point in time, the car was moving at some infinitesimal speed. Therefore, has the car decelerated at an infinite rate? Surely that’d be a job for the airbags!

It kinda suggests to me that time moves in frames like a motion picture. I’m no expert on the space-time continuum, but I think the “movie frames” idea doesn’t quite cut the mustard. So, what gives?

Take a square, say a yard on each side. Area, 1 square yard, right?

Devide it in two. Now the areas are half plus a half, equals one.

Devide one of the halves in half. Now the areas are a half plus a quarter plus a quarter, still equals one square yard.

Devide one of the quarters in half. Now you have a half plus a quarter plus an eighth plus an eighth, still equals one square yard.

Obviously, you can keep doing this for ever. The conclusion you are forced to is that the sum of a half plus a quarter plus an eighth plus a sixteenth plus a 32nd plus a 64th plus a 128th etc. etc. deviding an infinite number of times is ONE.

So although you have a infinite number of elements which you are adding together, they sum to a finite result. Fun, huh? This is a convergent series, whereby adding an infinite number of progressively smaller elements converges to a finite result.

In same way, you have expressed the distance elements and time elements required for you to reach your can of beer as a geometric series.

The paradox arises from assuming that summing the infinite number of time elements results in an infinite amount of time. In fact, the series is convergent so it results in a finite figure. There is no need to worry about quantisation or the graininess of the real world.

I take exception to this. Are you seriously suggesting that a scientific analysis of Zeno’s paradox would come to the conclusion that it would take an infinite amount of time to get from here to there?

Ah, alonicist, you’ve run into one of the age old paradoxes of vB code. I’m guessing you were trying to space things out to look nice, but vB ruined your hard work by taking out the extra spaces. Many have run into this problem, and after thorough scientific analysis, Zeno (or was it Euclid?) discovered that by putting {code} {/code} (with substituted for {}) you can force a monospaced font. For example:

``````
This
is
inside
of
{code}

``````

People are actually taking credit for his stuff by renaming it? Anyway from: http://www.mathpages.com/rr/s3-07/3-07.htm

"Of the 40 arguments attributed to Zeno by later writers, the four most famous are on the
subject of motion:

``````  The Dichotomy: There is no motion, because that which is moved must arrive at
the middle before it arrives at the end, and so on ad infinitum.

The Achilles: The slower will never be overtaken by the quicker, for that which is
pursuing must first reach the point from which that which is fleeing started, so that
the slower must always be some distance ahead.

The Arrow: If everything is either at rest or moving when it occupies a space
equal to itself, while the object moved is always in the instant, a moving arrow is
unmoved.

The Stadium: Consider two rows of bodies, each composed of an equal number
of bodies of equal size. They pass each other as they travel with equal velocity in
opposite directions. Thus, half a time is equal to the whole time."``````

Just recently, I proposed a question of a similar nature to a friend of mine: If we have a quanta of energy, why not a quanta of space or quanta of time? This sparked a small debate between us, but unfortunately it led us to no satisfying conclusions. We are only high school students and thus have a limited understanding of the physical workings of our universe. I’m afraid that all we were able to accomplish were a few theoretical/philisophical discussions lacking any application of mathematical formulae. We are not well versed in quantum mechanics or the special theory of relativity so our discussions were quite limited.

However, during this time, we wrote one another a few interesting e-mails and there are some passages written by my friend that I’d like to share:

It sounds logical that, since energy and matter are related, a quanta of mass could be possible. Mass, being something tangible, is an extension of space. If it were to be found that there was indeed a quanta of mass, could it not also be assumed that space (distance) was quantisized as well? And, from this same inductive reasoning, we can go even further by saying that if space is quantisized, then time might be as well (space and time being the same thing according my buddy Al).

However, I had problems imagining that there could be a quanta of time. I mentioned this to my friend and he responded thus:

If there were to be a quanta of time, that must mean there would be a ‘smallest interval of time’. But, no matter how small of an interval you choose, time must be able to flow from the begining of that interval to the end of it. And the whole idea of a ‘flow’ of time would negate the existance of a ‘smallest interval’. So, one might go on and ask whether time may even be even be illusory in nature.

But you ask, if time may not exist then why do we ‘percieve’ it? Well, maybe our whole concept of time is wrong. We like to picture our universe as 3-dimensional space that is propelled by time, and so we think that objects exhibit motion because time allows for it. But perhaps it is the other way around… perhaps it is not time that allows motion, but motion IS time. Objects shifting from one quantisized position in space to the next gives the illusion of a flow of time, when in fact everything happens instantaneously. This is somewhat like what ‘The Loaded Dog’ was suggesting: “time moves in frames like a motion picture.”

So, I don’t really know if I helped out the OP at all, but his original question brought all this to mind again, and I thought I’d share it. Also, before a load of people tell me how wrong I might be in the above interpretations, let me remind you once again that I’m aware of my ignorance when it comes to understanding the workings of this universe. This post is basically a summary of an interesting conversation I had with a freind of mine. It does not necessarily have to make perfect sense, but I think it’s interesting nonetheless.

About the guy losing hair, when is he bald?

Just substitute “age” for hair, and do the same math.

I know, Myself

Arthur, you guys are engaging in some great reflective thinking. I don’t pretend to know much in this field, but I believe that mass is quantisized. All mass is composed of quarks and electrons. Now why are those not quanta? BTW, energy and mass are not related. They are identical. E=MC squared. Not E is related to MC squared.

It also appears to me that time and motion are the same. I think that what Einstein meant by time being the 4th dimension. You cannot locate any object unless you pinpoint the position, which is pinpointing the time. Because time is quantisized does not mean it does not flow. Energy and matter are quantisized. That doesn’t mean they don’t move or that more complex energies and matters do not exist. I don’t see the logic there. Perhaps I’m missing something.

So, Arthur, what am I missing?

Didn’t we first quantize time so that we could rationalize one moment from the other, tell two different moments apart, since the first clocks, we made clocks so we could perceive time better, as the saying goes ‘time flies when you’re having fun’

Perhaps Sneeze as we knew him is still reaching for the drink, and the Sneeze from the Dark Side of the Moibus strip is just now wondering if he is indeed thirsty at all?