Mathematics

Zenos Paradox



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While I’ve never been much of a fan of mathematics, there are times when it has its uses. Take for example the measuring cup or the yardstick. Without the discovery and development of a numbering system, mankind would never have come up with the weights and measures we now use for cooking, building and calculations.

One of the things math has never helped me understand, though, is how life works. For instance, in grade school and beyond, we were always having to solve those wretched word problems. You know the ones: Train A is going this fast and Train B is going that fast, and as such, which one will get to Lompoc first? 

Then there were the mixture problems where you had so many pounds of filberts and walnuts and peanuts, and you had to try and figure out how much it would all cost once it came out the other end of the funnel- not that I’d touch the stuff, since I hate walnuts with a passion.

Anyway, throughout the years, math has always been a stumbling block for me. And since numbers and integers and fractions will always be with us, I have tried to find a way to make being around them a little more palatable. The key, I think, lies in working with them in conjunction with other disciplines, like physics, philosophy, and science in general.

When I was a kid, I was fascinated with all those molecular models in science class. Chemistry is no stranger to math and formulas, but that mattered little to me, as long as I could look at pictures and plastic models of molecules, atoms, neutrons and the rest. The same held for the Periodic Table. All those acronyms- au, se, h, o, etc. I didn’t know an isotope from atomic weight, but it didn’t matter. The arrangement of the elements in the chart was what caught my eye.

And then there were the math puzzles and tricks, whose beauty lies in the fact that, like motorized vehicles, you don’t have to know how they work to operate or enjoy them. I can build a Möbius strip with the best of them, and when it is all finished, I can point with amazement to the line I have traced on one side of the paper, and when it eventually meets itself- so to speak- I can then wax scientific about how amazing it is to gaze upon a one-sided piece of paper.

Or we could discuss the merits of the fourth dimension, and ponder what it would be like to travel from Point A to Point B without using up any time.

Now, don’t get me wrong. I have tried to learn about Einstein’s Theory of Relativity. I’ve even read, or tried to read, several of those books that claim to make E=MC squared something you could discuss at any dinner party in the land- but like the brush that grows ever denser and prickly the farther you go into the woods, so do the formulas and equations start growing thicker on the ground the deeper you venture into the that land of  “simple” explanations about relativity.

But of all the concepts and theories I have struggled to understand over the years, the most troublesome are the paradoxes- the ones so vexing that once introduced, they defied explanation or refutation for centuries to come.

And it’s not just the fact that they sometimes were (or still are) nearly impossible to prove or disprove, but that they could raise questions so baffling as to bring on a healthy case of insomnia.

The first time I encountered one of these little buggers was in a college algebra class. The teacher asked a simple question: If you were placed 20 feet from a wall, and asked to walk halfway toward it, then stop, then walk half that distance and stop, then repeat this process, how long would it take you to reach the wall?

Of course, chumps like me thought “Hey, this is no sweat. I know how to divide numbers in half! Looks like that fancy high school math is going to come in handy after all!”

But we should have known, judging by the smirk on our teacher’s face, that he was up to something.

Yet on I went.

Half of 20 is 10.

Half of 10 is 5.

Half of 5 is 2.5.

Half of 2.5 is 1.25.

Half of  1.25 is .6125.

And now I was starting to see the problem: at this rate, I’d never get there.

And with most of us students now in the first stages of befuddlement, the teacher started laying it on thick.

Not only would one’s journey toward the wall in question start getting slower and slower, but eventually one would have to admit that it was impossible to finish the trip.

And so we entered into the next stage of “huh?”

The teacher then put the icing on the metaphysical cake by stating that if one followed this situation to its logical conclusion, we could assume that motion itself is an illusion.

Not only did this induce another “huh?” moment for me, but to make matters worse, I spent so much time pondering the paradox that I missed the teacher’s explanation about why this little problem that Zeno had posed those many years ago was faulty in its construction.

For, you see, our teacher had to resort to math to explain the paradox- hence, I was good and lost.

So on I went through the day, wondering about life. If motion is an illusion, and therefore impossible, doesn’t that cast doubt on all forms of existence that I detect with my senses? Can I trust what I see going on around me?

I even tried to disprove what I heard in class. I walked toward a wall and then stopped. Then started. Then stopped. Then started. Then stopped.

Looking down at my size eleven tennis shoes, I noticed that after only a minute or two my feet were barely moving, and soon only moving in my mind. This really did seem impossible.

And having few neural pathways available upstairs with which to analyze this problem, I was stuck.

I knew there had to be a way to reach the wall, for I’d done so, many times before. So I thought and I thought and I thought. But still an explanation eluded me.

I even asked the teacher later on for an explanation- but it was over my head. In the end, I had to accept on faith that the paradox was just that- a self-contradictory statement. Motion had to be just what I thought it was, for if motion was an illusion, then mankind was hosed. Nothing would get accomplished or completed.

And though it took years for me to understand that dividing a distance up into small divisions is not the same thing as creating multiple tiny tasks, I still look at Zeno’s paradox like a wild animal in the zoo. Okay, there’s no way that tiger can hurt me- but if he ever got out from behind those bars…

Eventually I got over that problem, but pondering on it in recent times led me to another thought: How does anything ever start or end?

One moment something is at rest, and the next it is moving (and never mind how it gets to its destination, if you please!)

If I roll a ball across a table, it will soon stop moving. But when does that moment of stopping arrive?

I’m positive that much of the blame for my misunderstanding lies with a science fiction book I read years ago: “The Incredible Shrinking Man.” It concerns the trials and tribulations of a man who is exposed to some mysterious phenomena, after which he begins shrinking.

What really got my attention was the “fact” that the hero kept getting smaller, the implication at tale’s end being that there was no end to how small he could become.

Thinking about this too much could give rise to some form of OCD, I’m sure. In my case, it now seems a miracle that anything ever gets accomplished, since I can never pinpoint the exact moment when an action or thought begins or ends. It just does, and therein lies the miracle.

If I wake up in the morning and lie there all warm and comfy under the sheets, unwilling to get out of bed, at what point does my thought pattern finally switch over and move my body to action?

Now, if I think about this too much, I could come up with a good reason for not tackling any or all of those tasks my wife has lined up for me during a stretch of idle time. I can see it now:

“You know dear,” I’d tell her.” I would really like to take care of the gutters today, but since Zeno’s Paradox states that motion is impossible and hence an illusion, I honestly have to tell you that it will never happen. Sorry to be the bearer of bad news and all but…”

To which my wife would reply:

“Oh, I understand, dear- but since we’re talking about scientific things, maybe you’ve heard about the “Wive’s Theorum.” It states that any failure to complete assigned tasks will make the washing of clothes and dishes, the cooking of dinner, as well as any future conjugal visits a mere illusion-. What they call a paradox, right?”

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