Posts Tagged ‘mathematics’
One of the more interesting paradoxes comes to us from Zeno via Aristotle: the dichotomy paradox. The paradox is quite simple, but perhaps best stated by Aristotle in his Physics, 239b11:
The first [paradox] asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal.
Or, said in plain English, in order to traverse a certain distance, you must first reach its half-way point. As the paradox infers, you must then traverse that distance’s half-way point again. And again. And again, ad infinitum. You can mathematically represent this by (via Wikipedia)
So you can see immediately that should you have to first reach the distance’s half-way point before reaching the goal, you’ll never actually reach the goal.
There are some proposed solutions, only one of which I find particularly convincing. But let’s go through some of the losers first.
Aristotle himself argues that you may be able to find the solution by also having to divide time as you do distance. Therefore, you have 1/2 the time to traverse the 1/2 of the distance, 1/4 of the time to traverse 1/4 of the distance, and so on. I don’t find this solution even somewhat convincing (although he believed Zeno would be satisfied), although I do find it incredibly interesting that, perhaps for the first time in the history of humanity, we see time and distance — or space, if you will — as one. That wouldn’t be mathematically proposed until Hermann Minkowski and Albert Einstein proved it later in his theory of special relativity. So, while unsatisfactory, it’s interesting nonetheless.
Diogenes the Cynic (via Simplicius) states that because we see that people can indeed traverse distances, therefore they must somehow be able to traverse the infinite divisions of the distance. But, as Zeno points out, perceptions can be deceiving, and not all that we see is necessarily true. But, in my opinion, he’s on to something.
The solution I propose is the same as another argument from Aristotle. Perhaps Stanford University‘s Metaphysics Research Lab puts it best:
In his response Aristotle drew a sharp distinction between what he termed a ‘continuous’ line and a line divided into parts. Consider a simple division of a line into two: on the one hand there is the undivided line, and on the other the line with a mid-point selected as the boundary of the two halves. Aristotle claims that these are two distinct things: and that the later is only ‘potentially’ derivable from the former. Next, Aristotle takes the common-sense view that time is like a geometric line, and considers the time it takes to complete the run. We can again distinguish the two cases: on the one hand there is the continuous run from start to finish, and on the other there is the run divided into Zeno’s infinity of half-runs. The former is ‘potentially infinite’ in the sense that it could be divided into latter ‘actual infinity’. Here’s the crucial step: Aristotle thinks that since these times are geometrically distinct they must be physically distinct. But how could that be? He claims that the runner must do something at the end of each half-run to make it distinct from the next: she must stop.
What Aristotle is proposing here is that traveling is continuous, while Zeno’s dichotomy paradox relies on there being some ceasing of motion by the person traveling the distance. Said simply, if you’re traveling to a goal, you needn’t stop at each half-way point, but rather you traverse right over it and every subsequent half-way point. The dichotomy paradox is only applicable should you have to discontinue your motion at each point.
So really this paradox, like many others, can find its problem — and not really a solution — in a misplaced, misperceived, or just a completely wrong premise. This paradox isn’t any different, as its premise infers that you must, like distance, divide the motion of the traveler. But, as we and Diogenes the Cynic see every single day of our lives, people do reach their destinations, and they never once must stop at each half-way point of their intended target.
Ask anyone who’s received a ticket, and you’ll hear one persistent excuse: “They just do it to make money.” Being familiar with law enforcement, in my experience this excuse couldn’t be further from the truth. In the vast majority of cases, officers only pull over offenders who either grossly endangering themselves or others, or they believe the offender may have other infractions with which they can also charge him or her. But that doesn’t make the excuses go away. In fact, some other departments and organizations have supported them.
The Governors Highway Safety Association says,
Addressing other misperceptions, Hunter said, “We don’t have quotas. That’s always been a big debate. No, we don’t. It doesn’t happen that way for us. (But) we expect troopers to work.”
State troopers go where the problems are, she said. “People think we’re making money, but what we’re doing is responding to the calls of the citizens.”
Meanwhile, however, the same article states, “A speeding survey by the Governors Highway Safety Association found that 40 state police departments or highway patrols, including Washington’s, issued more than 8.1 million citations for speeding in 2003, generating as much as $2.3 billion in revenue.”
And that’s a lot of money.
Government representatives aren’t helping the misconception, either. As this article by MSN Money states,
“It is primarily a tool in many communities to raise revenue,” Louisiana state Rep. Hollis Downs, who represents two parishes in north-central Louisiana, says of the town’s aggressive traffic enforcement — what others might call speed traps.
But considering the retributivism rate of “minor” traffic offenses, it appears that this type of penalty has lost its effect. I’d like to propose another procedure.
Countering the public’s oft-used excuse, I’d like to institute a penalty of time and education as opposed to monetary-based consequences on first-time minor traffic offenses. What I propose includes the following: simply, the penalty of a first-time minor traffic offense would be an hour-long video educating the offenders of the further consequences of continuing such behavior.
For example, I’d like to include a section of the video entitled, “Is it worth it?” I came up with a rather basic equation that tells you exactly how much time you save, or lack thereof, by speeding. By using the equation:
where d is the distance traveled in miles, v is the velocity of the vehicle in miles-per-hour, and t is the time in hours it takes to traverse the distance at that particular velocity. In order to find the time in minutes, which is more appropriate in the case of everyday travel, you simply take the variable t and multiply it by 60.
By simply showing and explaining this equation with some everyday situations, the viewers will see that excessively speeding — or even speeding at all — truly has little effect on the time it takes to traverse typical distances. For example, according to this poll by ABC News some years ago,
Life for commuters can be heaven or hell. They report an average one-way commute time of 26 minutes (over an average distance of 16 miles).
So let’s use an example of 16 miles as the typical everyday drive, and we’ll use a velocity of 55 miles per hour:
Which is 17 minutes, 30 seconds.
Compare that to speeding 15 miles-per-hour over the speed limit, which may qualify for more than a simple speeding infraction:
Which is 13 minutes, 42 seconds.
So by putting your own life and others’ lives in danger, you only accomplish decreasing your travel time by a bit less than 4 minutes.
The main form of punishment in this case may very well not even be the education, but rather that it takes time to complete, even if it’s just an hour. Paying a speeding ticket or simply paying a lawyer to “fix” the ticket isn’t really that much of an inconvenience, with exception to the pocketbook. But to take time away from the individual might encourage him or her not to do it again, and subsequent punishments would include explicit monetary damages, as well.
I truly believe that creating this educational procedure would at least somewhat encourage the offenders to reconsider their actions while simultaneously giving the public a chance to understand that tickets aren’t given out as a form of revenue but rather as a punishment and deterrent.
Some years ago I came across an interesting list of paradoxes, and one caught my eye and enthralled me. The paradox argued that, in mathematics, 1 and 0.999… are equal. It argued this on the basis of the explanation below:
Simply multiplying each side by 3, we will see the paradox:
thus,
It’s a pretty easy paradox to understand, and yet its conclusion is mind-boggling. Can it truly be correct that 
? Well, not so fast.
After some awe-inspired contemplation, I found the solution. When considering paradoxes, the solution is often found in a mistake in a premise. That was the case in this mathematical problem, as well.
The premise of is incorrect. Instead, I deduced that there must be a missing infinitesimal (PDF) in the equation. That is, an incredibly small number — the smallest number possible — was missing from this equation: 
.
When correctly stated, we can now see that what we have been taught ever since we were young — that — is incorrect; rather, the correct equation is 
. I can prove this by simply stating the original problem with the mathematically correct solution:
Thus,
I think what interests me so much about this equation is that, like much of philosophy, the solution went against what I had been taught. Oversimplifying the problem led to an incorrect understanding and thus an incorrect solution. But with a little ingenuity, the answer was right there, hidden in the smallest number possible.
The Monty Hall paradox is an often misunderstood statistical problem whereby the solution is counterintuitive. The problem is stated as:
A thoroughly honest game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. “First you point toward a door,” he says. “Then I’ll open one of the other doors to reveal a goat. After I’ve shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door.” You begin by pointing to door number 1. The host shows you that door number 3 has a goat.
Do your chances of getting the car increase by switching to door 2?
Conventional wisdom says that you have an equal probability of getting the car regardless of whether you switch doors. This is because the host showed you a goat, thereby reducing the problem into a choice of two. However, that solution isn’t correct. The correct solution is that switching doors increases your chances of getting the car. In fact, the probability of winning by switching is an astounding 2/3. How is this so?
The easiest way to show the solution is by example. Behind door 1 is a car, door 2 has a goat, and door 3 has a goat. If you choose door 1, the host will show you either goat. Switching the your initial door choice leads to you losing. However, if you choose door 2 or 3, the host will show you the other goat. Switching inevitably leads to the car. Therefore 2 out of 3 times switching will result in you winning the car.
But how is this related to quantum physics? Wave-particle duality, of course!
Wave-particle duality is, simply, how objects seem to exist in two states: one as a particle, the other as a wave. In the famous double-slit experiment, electrons are shot at an object that has two slits in it. You would expect, as with particles, that you would merely see two lines where the electrons went through. Instead, you see a wave pattern. So small objects do seem to exist in both states.
So how is the Monty Hall paradox and wave-particle duality related? Well, I’m not sure yet. It’s something to think about further.
My thought thus far is that, within the framework of quantum physics, small objects exist as either particles, waves, and/or not at all. By observing the object we are, in essence, choosing a door: always the one that has the car. Observing it means it exists, and so the other two doors encompass the other 2/3 probability; that is, you can open either of the other two doors, and you will see the object exist in that state. So now we have opened two doors: the object exists and exists in a state we are observing.
So instead of applying complex mathematics to principles like non-locality (PDF) and n-dimensions, we can reduce it to simple statistics.