Zeno's Paradox

 This is the first if the typed rough drafts, I must get it down in some form to refine it and welcome any part of the discussion.


"His essential question was how can you do an infinite amount of tasks in a finite amount of time?"
How does this idea of addition/subtraction  solve this statement.
I often introduce Zeno's paradox in my math classes as a way to discuss asymptotic behavior and limits.  I am not the only one to do this for many teachers find it a nice way to consider these things.  The general arguments against Zeno involve limits but I was wondering if there was another way around it for even philosophers don't feel that such  mathematical disprove ends the philosophic argument.  In my musings and considerations, I stumbled upon both a simpler mathematical approach and perhaps a different way to answer it in terms of metaphysics.  At the center of Zeno's argument is that you can not divide something and have nothing. This is true. Even if the limit definition says the same the thing for the limit is not about the point of dividing by infinity but what that point would look like if we could do so.  Calculus added an extra definition for continuity at that point in case we care about its existence.  If one wants to argue against the paradox using the mathematics of limits one must also include that the lim f(x)= f(a) and that f(a) exists.
Zeno, like all good Greek thinkers, loved proportions.  He argued that the arrow would never make its target because it would always be 1/2 the distance closer. This simple statement is where he skipped a stepped of mathematics, logic, and philosophy.  A proportion is a form of division.  Division is the inverse of multiplication and multiplication is repeated addition.  We do perceive, measure, and understand the passage of the arrow or Achilles by addition.  (Recall that subtraction is just a different form of addition.)  While we can never divide something to be nothing or multiply nothing to be something, we can easily add something to nothing or take away to be nothing.  This element is true in the scenario but it was not expressed in the paradox.  In all logical or mathematical arguments, we need to express these things in the beginning for the argument may lose sight of simple definitions.  We turn the passage into a problem of division, which does not allow for the zero space but our original conditions do so.
The illusion of motion is turning the addition of intervals into a zero sized one.  It is viewing flight as photograph and stating that because it does not travel in zero time so it never travels.  Consider this photograph if we were in it, for that is how Zeno presented it.  If time is frozen, we will not perceive any motion or much of anything for we will always be puppets of the infinitesimal.  This moment is one of those semi-pointless exercises in philosophy for we know the arrow to travel and we know time to pass.  Theoretical physics can change the speed of time, and even pause it for light, but not for us. 
As for that original question, of which, I need to refind the source. Consider the whole path the arrow travels as one.  It goes 1/2 the distance in 1/2 the time, and therefore makes 2 intervals.  We can keep dividing the distance and increase the number of intervals.  It will go 100 intervals of 1/100th the time.  In this manner, we can find an infinite amount of tasks in a finite amount of time.  The challenge really highlights our analytic need to make things discrete and countable and the simple continuity of a line. 

Escaping in the moment.

A few phone pictures from late.  Not sure who made the snowman but it was outside of CK's after our one snow of the year.  He was about a foot tall with a leaf hat and pine needle arms.

It's another rough of a story. Some day, I will get them more better but I need to at least get them this far for me to feel like I have done anything.

    Three people stood in line behind her at the corner store.  They were all agitated at everyone taking so long since their lives were so important.  Either that or if they may get fired if they were late to work.  Regardless, their jobs were the most important ones in the world.  They had places to be and just couldn't wait on her getting something she obviously did not need at 8 A.M.
    Malinda Simpson was long past the point of caring, or at least not acting on it.  She knew they were looking at her and squinting their noses at her dingy coat overflowing with receipts.
    "Two 4'2, three 5's, and a 9."
    The clerk pulled the commercial rainbow of tickets off the stack of rolls.  She passed no judgement.  Her time selling people their daily needs taught her that all are hooked on something.  What they bought said nothing about them.  How they conversed through their bodies said much more.  She could see the condescending eyerolls of racist pigs and caught the sideways glances of the beasts needing to dominate.  There were the kind strangers helping someone buy food and even those who bought beer for the wino.
    "Everyone is a little bit of everyone and no one is like anyone," she told the young man who stopped by every morning to get coffee, cigarettes, and two honey buns.  Those words led him finish his EMT classes so he could help people, no matter who they were.
     Malinda left the store and sat in her car parked directly in front of the door.  She needed to keep it running since it might not start again.  People in the store could see it was full of clothes, food wrappers, cleaning supplies, and scratch off tickets.  Her car was her home, if home is where one sleeps.  It could just be how she got around and where she kept her few things. Malinda stayed in the parking lot of Gene's Quik Fuel.  They let her use the showers and it was generally a safe place to be.  There was a Clean Coin Mat next door.  The parking lot was were she landed after the turbulence of her actions and reactions to being ignored by her parents and then preyed upon by series of beastly men and women pushed, pulled, and spun her around in tales of our need to carry on despite the misery existence.  After one more trip to the free clinic, the nurse told her to go to Annie's Shelter for Women.  It was clean and warm and they didn't use her.  After some time, they gave her some odd jobs and she carved out a way for herself.  Few people cared but she knew that the women coming through the door felt better in a clean place.
     Malinda pulled out one of the tickets. It was red and green, Santa's Suprise, a jackpot was all but guaranteed.  She was nervous and excited.  A big smile at the dream of winning.  She started scratching of the metal powder, the action was freedom.  One of the all important agitated men behind her came out of the store and could see her eyes light hope and a real hope while she searched for winning numbers.  He wasn't sure if he felt joy or pity but he knew that his wait wasn't worthless for her. 

Time stops in the dark.

Some bad pictures of some work..

I am seeing if I can figure out how to write again. Not sure why, got a lot going on but sometimes the more you do, the more you can.  I'll probably burn out an go insane at some point but let's see if I can find some balance before then. This is first draftish...

At some point, they found themselves doing donuts in the park they grew up in.  It was not the best of ideas, but it seemed like a good one at the time.  No pain, no misery, no sorrow, just a distant sense of the need to be in the moment as the motor roared an rear tires skated across wet grass.  The tracks would disappear by spring, the headache would be gone sometime in a day or so.  A bottle of Colt 45 made them teenagers once again and it was not such a bad thing.
      Murray Strand buried his father a few weeks before and  Josh Breese poorly carried the weight of disappointment.
     The two grew up blocks apart, made fun of the same teachers, ran away to each others' houses, got drunk for the first time, loved the same women, got tattoos together, and all things boys share as they grow into life-stained men.  At times they fought, one or the other would dissappear to a new side of the country, only to be sucked back into the languid embrace of home.
     "Come by." Murray said in the message.
    "Why not," Josh figured.  It's late and everyone's in bed.  He was leaving work, a second shift at Steel America.  The man had been calling non-stop. He couldn't figure out how to be a good father, much less a decent husband.  About all he was good at was drinking and working.  Of course, the two things didn't like each other but they supported each other, a couple of codependents in a constant fight.  He figured he would just turn into an old drunk or a dead young one at some point.  He still had his pride and the silly belief he could  be a force of change but he generally killed his dreams after leaving the bar.
      "What up man?
     "Shit, same old."
     Murray and Josh embraced though it had only been a few days since they had seen each other.
     The stood on Murray's porch, watching a light rain drift down, falling like snow in the street lights.
    "Work's been kicking my ass."
    "I hear you, operator."
    Soon, they were to sharing a pack of cigarettes and reaching for the last beer.
    "Damn, I'm hungry."
    "I got an idea."
    They walked into ** Diner. It was late enough to be quiet before drunk punks stumbled in. Murray gave the man at the end of the counter $5 to play music.
    "That's how you do it there."
    It as another song about love. A man pining for his woman.  It may have been about lust but who can tell the difference.
    "Eggs are better with bacon."
    "What's Matts up to?"
    "Let's check."
    They drove over to Matts' house.  He was almost asleep but led them on a game of fifty states,
    "I bet you can't write down all 50 states."
    Josh did.  Iowa was the last one, it seemed hard to remember being wrapped up in all those other states he had driven across. He had to bum a menthol from Matts, maybe it was his girlfriend.  It went well with cheap beer.
    "Man, you should go to bed. We'll check you later."
    "Cool, bro. Good seeing y'all."
    They left and found a corner store.  Time did not exist for it had turned to the present.  They talk about relativity in physics, time and space are axises on some strange graph.  The whole problem is time is a creation of the self-aware.  Place is a fact of perceived existence.
    "That was my house."
    "Yes it was."
    They were in the playground in the park. The same place they once played baseball in.  Back before they found out about drinking and stupidity.
    "Malt liquor ain't as bad as I remember."
    "It probably is."
    Murray commented as he walked across the top of the tyke's swingset.  He always had good balance.
    "Feels good to be here."
    "That it does."
    Back in the truck, Josh decided to drive across the park.  He pressed the gas and watched the needle spike; the ass end of the truck went hither and tither until it broke free into a real donut.
   Murray held onto the handle and yelled at something.
   The moment was real and would soon be gone. Not the right thing but something needed to be done.

Problem solving, thinking, and some effort...

Some posts are written down on paper first, this ain't one of them.  It's really just disguised laziness.  I like the challenge of problem-solving.  I like teaching math and I liked the part about math where you work real hard to figure something out and then figure it out.  It is a special accomplishment and why people always view math as hard and a measure of intelligence.  It is often a challenge but it is not any more of a measure than any other study, we just measure it on a different part of the scale than we do other things.
I work to try and help people build their schema of math understanding and develop that part of reasoning and symbolic logic.  Unfortunately, I am not the most organized individual so I haven't refined how to best reach my goal.  My goal also involves changing others' viewpoints and that just don't always work so well.  People are a stubborn species and teenagers are a stubborn subset of people.  Getting them to buy into a different view of the world, particularly one based in the unexciting system of algebra, feels like a pointless fight at times.  My only advantage is that the students do believe that something about the subject is good for them, even though they have to ask "When will I use this?" every class.
That supremely frustrating question is why I am shifting to a class more focused on problem-solving rather than mechanical-computational skills.  I am also moving in that direction because modern students have astoundingly little space or need to recall facts and skills.  They are absorbing information at peak levels.  They know fifty times more bands that I did at the same age.  So much of the working memory I need access to for traditional learning is used up on things far more interesting to teenagers.  So....how do I teach a subject that requires a significant amount of this cognitive process? I have faith, perhaps too much, in my ability to present the information to people in way that makes sense but the same people often don't store it.  I am trying to circumvent this issue by using a problem-solving approach or a method in which I guide students to construct their understanding of mathematical systems.  Both methods require thinking and both require effort from the students.  Effort is a challenge in the subject.  Student often ask for help as soon as they read a problem. He or she may work on it for a minute or two but rarely long enough to reason through it.
I have worked on a problem-solving approach for a few weeks in one of my classes and I am seeing signs of change, though.  I hear fewer more questions and students seem to be following my guidelines to find some manner of solution.  Next up, setting up the problems so that individuals develop an understanding of the rules of higher math.  Until now, we were working mostly with old computational skills.  I am adopting my system from Exeter's Harkness math but I am confined to less time and with a broader range of students.  I do know they are more engaged and that students who could care less for a lecture on solving quadratics enjoy they can find different ways to solve a problem.  Watching people use different methods has taught me a great deal about math over the past few years.  Ehh, I just got tired and stuff so I will continue this later...

Some pictures, a story start, and some rambling.

Still going.

   Let's see if I can get to the words on here.  I should be helping at his school or cleaning the house.  Instead, I am going to see which of the mundane sentences I jotted down last night make any sense here.  Writing on the computer is not an organic process for me and it really takes away much of the power of expression.  I relied on the machine form during school and in boring work, that reliance pushed me from the habit of writing as a release and as a place to make the inner discussion real.

 A Story in Progress
     "You jerk."  The lady shouted as she pulled into the elementary school to pick up her kids.  It was hot out, her husband was pretty much useless and the line of cars to get into the school lot was not moving.  She needed to turn left and some asshole and some asshole in a big truck wouldn't stop to let her forward.  To make it worse, he pretended he didn't even notice but he must have for they were just yards apart, sitting in their traveling recliners.  He looked like one of the young men who had come into her office and made work less bearable.  He knew nothing and pretended like he knew everything, that she should be on her knees before him.  At least the old boss wanted her bent over the table so she could get some pleasure from it.
     "You jerk.!"  He heard the woman yell as she passed.
     "Damn lady, I didn't mean to do nothing." He wondered as he could feel the heat sweat out his back and his legs stuck to the seat.  "Does she freak out like that with her kids?"
    He was worried, bothered by the moment.  He was a jerk, that he could admit but not to some stranger and not for being stuck in school traffic.  He wasn't a jerk merely for existing.
    Maybe if these lazy folks wouldn't drive right up to the school. Damn, I hope I get an email from the people soon.  Every time he saw the yellow light next to the E it reminded him that he was broke.  Try as he might, work was not easy to find.
     The messed up thing about it all was work was not the chore.  All these people around searching for a way to make rent and scrambling to pay some old debt or to buy some milk,wasting a few dollars on beer flavored rotten barley to forget, finding someone to blame but really just wanting to work, to feel like they had meaning and to have some cash, they were everywhere.  Begging outside of stores and serving up fries, pushing papers and making traffic stops.  They attacked each other because it was the only thing they could think to do.  Others just watched T.V. on and endless reel.  

     Not that it's a new idea but it's one worth explaining.  What does another person live through?  I was on a selfish version of this thought game, "They just don't know my responsibilities or my concerns, strengths, limitations, motivations, trials, etc.  This self trudgery and fairly useless line of reasoning occurred on yet another walk to the store to get another beer to mollify the worry of the day and give me freedom to dream of something more than the moment.  I was going along the familiar stretch of sidewalk as my son ran about  figuring out how to throw a frisbee.  We found it on the strip of grass that catches all the trash from the evils of the road.  A red VW bug kept passing up and down the street, a couple about my age enjoying driving the loud little car.  Various other people drove by dressed in machine finery or wrapped in torn and patched metal getups.
    All these people going around and meeting demands in their lives were worn down by bearing the challenges of the day or exalted by the events great or small, new love, finding $20, a healthy baby, a kind word.  Here we all were going through time-space in some controlled idea driven by wants and needs.  Consuming on past consumptions.  Actions mandated by habits, choice at moments when the future seemed different or lacked any reality.  Another drink, a new car, a late night fling, doing what one wanted and then the obscure shackles of obligations were locked.  The choices may have  been the best, to not call the woman at the bar back, to not join the Army, but there was and is no way of knowing.
     Scientists are at work proving that time is an illusion.  Perhaps they are right in some sense that it is part of a time-space construct but we know it to be a real law that we live by.  Our mortality is one of the absolute truths, if not the only one.  It is fun to think of time dilating around planets and we may get some useful stuff from it but it ain't the world we live in.  Rain fall, counting a beat, a measure, a rythem as it builds into the puddle, falling over its breaks into a swirl, slipping into a gurgle, falling into the creek and beyond.  All along, we see the present, recall our past, and imagine a future.
     Kindness, sympathy, caring, love: the forces that push for  better moment are how we know that we do understand each others little worlds.  Greed and violence are the actions we take when we only act for ourselves.  I find myself in another conflict and perhaps with a greater understanding of greed.  Seeking, begging, lying, and stealing in order to provide support for my family.  Is it not an honest goal?  But the end provides poor support for the means for it makes me a weaker man  and further scars my being.  Festering sores that I pick at in worry and shame.  Afraid to admit to those I love and further leaving a part of me rotten.

     Seems that I either lost direction or forgot where I was going.  It's the problem with no plan.  Bugs are bit annoying and my stomach is burbling from too much coffee.  Can't seem to shake this filthy nicotine habit.  A car horn goes off in the distance, probably a forgotten alarm.  Maybe a kid found the panic button and now it has stopped.  The bright yellow corvette is going on its laps.  Still....no plan, just gonna revive a habit that never stuck like it should.  Why do the good habits require maintenance of effort while the bad ones just come in to control.

Flowers, children, and shapes.

Sitting in a class with some tunes and folks busy at work making kites.  On my best days, I would think my class was awesome, not so sure about my bad days. It's always a challenge figuring out what the process of useful learning is.  What is worthwhile and what is interesting, what helps build a students understanding and what applies to a real development of knowledge.  I have finally learned that it is okay just to follow the guides and textbooks when
I ain't so sure where to go.  I wish I had swallowed that bid of pride in earlier times but I am not really good at doing that until I keep getting hit in the head with the same board of stubborn imbecility.