Lowering Expectations: Living in a Fallen World

  

HS #128 2026.3.12

 

Lowering Expectations: Living in a Fallen World

 

It happens all the time. Someone hits a ball (volleyball, pickleball, tennis – any sport with a boundary line) and it’s not clear whether it landed in or missed. So the teams agree to replay it. The repeat clearly misses, and then someone announces, “First one must have missed.” I counter, “No, it must have landed in – after all, we live in a fallen world.” 

 

Many tacitly assume the world is inherently just and good. We are disappointed when it doesn’t work out that way. But, especially if you grew up in the Roman Catholic or Calvinistic tradition, you have been taught since a child that the world is fallen. Why then don’t we acknowledge it? 

 

When I first came to Hope College in 1988, the provost distributed a memo to the faculty about academic integrity. It began with the question: Why do some students cheat? As a naïve newcomer, I replied, “Given the theological foundation of this institution, shouldn’t we instead ask, “Why are some students honest?” As a result of this perceptive comment, I was invited to serve on a committee. Oh well, we live in a fallen world. 

 

Do you remember the scene from the first Indiana Jones movie where Jones, in a hurry to save the girl, doesn’t have time for a manly fight with the swordsman? So Jones shoots him. The surprise is humorous, because we expected Jones to behave justly and fairly. 

 

Why are (liberally-minded) Americans so exasperated with our current President? Because we assume we live in a world where the big kid on the playground uses his strength to defend the weak and promote justice. But any elementary gym teacher will tell you that the big kid is often the bully – using his size for his own selfish interest. Why should politicians and world leaders be different? Why should our President’s attempts to acquire Canada, Greenland, Venezuelan oil, and Gazan beachfront stun us? I’m not condoning; just asking, “Why the surprise?” 

 

Ironically, accepting that we are in a fallen world can lead to better living. C.S. Lewis explained it this way: Two men are confined for a month in a poor-quality hotel. One has been told he was going to a 5-star suite. The second was told he was going to prison. Which will be more satisfied?  

 

Indeed, being content comes from lowering expectations. 

 

Let’s dig deeper. Consider this (morbid) scenario: You have been kidnapped by terrorists and shown a revolver. You are told that in two weeks one of the six chambers will be loaded with a bullet, the cylinder will be spun then pointed at you and shot. Get the idea? 

 

The second scenario is similar except now five of the chambers are loaded with bullets. 

 

My question: Which scenario would be more distressing? I’m not asking which you would choose; I’m asking which would cause more distress? 

 

I’m convinced that I’d be more distressed by the first – with just one bullet. Why? Because if five bullets, I would accept my likely fate, but would FOCUS on the HOPE that I may survive. With one bullet, my focus instead would be on the unlikely event that I am shot, causing distress. Perhaps this is why atheists fear death less than the moderately religious - because they have accepted and come to terms with the reality of no afterlife. They live their lives with lowered expectations.

 

One way of lowering expectations is by realizing that we are not special. Some teachers tell their students, “If only one of you is helped, it will all be worth it.” Each student feels affirmed because they think they are that one exception. (Likely the same reason folks spend money on lotteries. They think they are uniquely destined to win.)  Wanting students to have sober judgement that they are not special, I instead explain that, “I have higher standards - if all but one of you are helped, it will be worth it.”  It gets a laugh. 

 

Shall we end with the wisdom of Jesus? He commended the deceitful servant who, fired by his master, reduced the debts of his master’s debtors so that he would have obliging friends once he was out of work, making the best of a bad situation. 

 

Indeed, Jesus exhorted his disciples to have realistic expectations of human behavior and thus to be as wise as serpents - while as innocent as doves.  How better to live successfully in a fallen world? 

What would Lincoln do?

 HS #127 2026.2.12

 

What would Lincoln do?  

 

Getting to know historical figures, one learns of their foibles and weaknesses as well as their strengths. George Washington, great man to be sure, was stiff and aloof. Ben Franklin had strained personal relationships and was estranged from his son. Adams was described by Franklin as "always an honest man, often a wise one, but in some things absolutely out of his senses."  Jefferson was sometimes disingenuous, attacking his friends behind their backs, but unwilling to confront them face to face. 

 

On the other hand, Abraham Lincoln, whose birthday is today, was a giant – literally as well as figuratively. He deeply understood the U.S. experiment: government of the people, by the people, for the people, but also understood that to make the government work, we the people needed to be guided by our better angels. He had many close friends who loved him and recognized his greatness, and his moral sense ran deep. He was committed to finding truth, studying Euclid’s geometry to learn how to prove the veracity of an argument.  

 

But it is Lincoln’s sense of humor (often self-deprecating) counterbalanced by his melancholy that endears him to me. 

 

When accused of being two faced, he responded, “If I had another face, would I wear this one?”

Debating the short-in-stature Douglass, Lincoln was asked how long a man’s legs should be. “Long enough to reach the ground.” When one of his cabinet members died, an ambitious office-seeker asked Lincoln if he could take the dead man’s place. Lincoln replied that certainly the coffin-laid man would presumably also like to trade places. 

 

Lincoln entertained his friends with humorous true stories. His favorite concerned the Revolutionary War hero, Ethan Allen, who visited England several years after the war. Asking to use a toilet, Allen was ushered to an outhouse where he discovered the British, still smarting from their defeat, had hung a picture of George Washington on the inside wall. Exiting the biffy, Allen announced, “Good thinking, gentlemen. Nothing makes an Englishman shit faster than a picture of George Washington.” 

 

As he bore the burden of leading a fractured country, Lincoln’s humor acquired an edge. Exasperated with the lentitude of Union General McClellan to fight, Lincoln asked, “If you’re not going to use your army, do you mind if I borrow it for a while.” Lincoln admitted, “With the fearful strain that is on me night and day, if I did not laugh, I should die.” 

 

Indeed, melancholy was equally part of Lincoln’s personality. Distraught about love, his friends took his razors for fear of him killing himself. As a young man, Lincoln admitted to a friend, “If what I feel were equally distributed to the whole human family, there would not be one cheerful face on earth.” 

 

Lincoln’s humor and melancholy both emanated from his honest acknowledgment of and willingness to face the truth about the world, while earnestly seeking solutions.  Therefore, I wonder, “What would Lincoln do in our present situation?” 

 

In his article, “We Are Witnessing the Self-Immolation of a Superpower”, Garrett Graff lists six significant ways that the U.S. is presently emasculating itself by ripping away these historic strengths: i) Easy access of immigrants and foreign students to our world-class universities, ii) Generous government support of higher education and medical/scientific research, iii) Broad and easy trade access to and from U.S. markets, iv) Strict adherence to the Rule of Law, 

v) Firm network of international alliances, vi) Independent monetary policy. 

 

Taken together these have given the U.S. unrivaled influence and leadership in the world. But in the past year all have been irrevocably damaged and minimized. Would Lincoln have ideas to gain back the international friendships and respect we have lost and are losing? 

 

What about at home? The violence at the U.S. Capitol on January 6, 2021 was not as significant as the violence done to Truth. The Capitol Police are now blamed for it and those who smashed, sprayed and beat are heralded as loving patriots. More recent lies about killings by ICE agents further undermine trust in our current government. “Alternative facts” abound. 

 

These are serious concerns. How would “honest Abe” restore a nation whose current administration has forsaken truth and morality? The U.S. is in urgent need of moral, truthful, Lincoln-styled leadership which challenges us to our better angels. Lincoln declared, “I have faith in the people . . Let them know the truth, and the country is safe.” Join me in living truthfully, seeking Lincoln’s ideal.   

 

 

New Year's Resolution: Being Kind

 HS #126 2026.1.8

 

New Year’s Resolution: Being Kind

 

Those who know me well would likely agree that I have a quick temper. So, many years ago while in Hope College’s Mathematics Department, I told my colleagues and office staff that my New Year’s resolution was to avoid saying anything out of line, and that, to motivate me, I carried a $10 bill in my wallet to give to the first person I snapped at. 

 

Unintended consequences. After a week or so of successfully showing my good intentions, the $10 became a booby prize because it demonstrated that some unlucky soul had done something so egregious that they had thwarted my best laid plans. Gladly, I have forgotten who won the prize. 

 

However, being kind to others may not be a bad resolution for all of us to attempt. I see admonitions to “Be Kind” with increasing frequency. A friend has as his email signature line, “If you be anything in life, be kind.” Schools and churches put it on their outdoor signs. 

 

I remember a quote from a Reader’s Digest, “Today I may have sacrificed truth for kindness, but that doesn’t greatly concern me, because I’m more sure of what is kind than of what is true.”

 

 NPR recently carried a story explaining that being kind to others leads to a better, longer, and healthier life. 

 

The reason is easy to understand. Kindness prevents unpleasant situations and promotes harmonious living. Who gets hurt when we are kind? Last fall, during the Grand Rapids Art Prize, an exhibit in DeVos Hall stated, “We all do better when we all do better.”  Don’t we all do better when we are all kind to each other? 

 

In fact, admonitions to be kind reach back millennia. Being kind to others is certainly a corollary of Jesus’s “Golden Rule” to treat others as we want to be treated. And even the ancient Stoics included kindness as one of their principles to live by. 

 

Yet, if we’re going to get philosophical about it, let’s include Aristotle’s Golden Mean. Aristotle wisely cautioned us against taking anything to the extreme. 

 

But, hey, what balance is needed in being kind? Well, let’s go back to the above quote from the Readers’ Digest. Yes, perhaps it IS easier to determine what is kind than what is true, but isn’t truth sometimes so important that it’s worth the risk even though compromising kindness? 

 

In similar spirit, my hero Teddy Roosevelt once said, “If given the choice between Righteousness and Peace, I choose Righteousness.” 

 

 Years ago, the Hope College student newspaper, The Anchor, carried a letter from a student disgusted with the destructive antics of his fellow students. I had a few thoughts to contribute, but figured that any lecture from a professor would be dismissed immediately by the offenders. So I wrote a tongue-in-cheek letter defending the juvenile behavior in the hope that, by the end, they would realize I was poking fun at them. 

 

Unfortunately, my subtle approach succeeded so well that I got several letters from faculty and administrators chastising me for my remarks. Telling my fellow-faculty racquetball partner about it later that day, he replied with words I’ve never forgotten, “You hate you always err on the side of caution.” 

 

Indeed! Yes – that is my life motto – he hit it!  It’s easy to “play it safe” in life. To keep out of trouble by not taking chances with people. And being kind to others MAY be motivated by avoiding risk. In fact, there is a descriptive phrase heard here-abouts, “West Michigan Nice” describing the avoidance of conflict by eschewing substantive discussion. 

 

Instead, the worthy challenge is to be kind in a substantive, even sacrificial, way that cares for the other person, while treating them with the respect which may include honesty and risk. 

 

Are you familiar with the Antarctic explorer, Earnest Shackleton, whose ill-fated voyage caused him and his men to abandon ship and trek back home over the ice-covered continent? His trip is detailed in the riveting book, The Endurance.  In Munising MI, I met the daughter of the only American on that venture. He told her of “shenanigans” by Shackleton and the officers when they all drew straws to determine who would get the warm fur-lined sleeping bags. It “just so happened” that ALL of the good bags were drawn by the sailors, leaving the poorer bags for Shackleton and the officers. 

 

THAT’S kindness - that’s kindness that is sacrificial and that matters. 

Living with Ambiguity

 HS #125 2025.12.11

 

Living with Ambiguity

 

Any pickleball player will tell you that it is essentially a social game. One goes for the exercise, but just as much for the social interaction. The inventers have certainly added immensely to the quality of life of retired folk in particular. 

 

And that social interaction, as always, allows one to see how folk are different from each other. 

 

Case in point: Some balls are clearly in and some are clearly out. But others bounce so close to the line that (given the speed and the bifocals) one can at best make a good guess. Good sportsmanship necessitates that if it is truly unknown, you give the point to your opponent. Yet, even then, there are some bounces that I think are most likely in my favor, but I can’t be sure. So I will sometimes hesitate and ask my partner to verify, or check to see if someone on the other team had a clear view. (In fact, depending on the angle of sight, the same bounce can appear clearly in or clearly out.) 

 

Now what’s interesting is that some with whom I play appreciate (or at least tolerate) such indecision, while others will snap back, “Just call it - it was either in or out!” 

 

Little do they know that such a response ties up my thinking for the next half hour pondering the difference between ontology and epistemology. Briefly, ontology concerns “what is” while epistemology concerns “what we know.” I’m in hearty agreement that in fact either the ball was in or it was out. Yet, as anyone who watches D1/professional sports knows, even looking at a play in slow motion from several angles can yield different opinions. 

 

Indeed, there’s a story of two baseball home-plate umpires arguing about how to make the ball-strike calls. One says, “I call them AS THEY ARE!” The second responds, “They ain’t nothing til I’ve called them!” They are having a philosophical phight!

 

Which umpire is correct? Is one clearly correct, or do you appreciate that each may have a valid point? Can YOU live with ambiguity?  

 

Perhaps you heard of the judge who upon hearing the prosecutor, announced, “You’re right!”

Then hearing the defense, announced, “You’re right!” The bailiff whispered to him, “Your Honor – they can’t BOTH be right.” The judge replied, “You’re right too!” 

 

My pastor father once told me that the reason for huge attendances at mega-churches with ministers who “simply tell it like it is” (think Falwell) is because many cannot tolerate ambiguity, instead needing sure, simple answers. I agree. 

 

In fact, there is a psychological experiment where the subject is given a picture book with many pages. The first page clearly shows a drawing of a cat, the last page has a drawing of a dog. But the pages in between show drawings that very slowly, subtly evolve from one to the other, each picture just minutely different from the previous. 

 

So when asked to identify the pictures, some will respond, “cat, cat, cat, not clear, not clear, not clear, dog, dog.” Others, however, refuse to admit ambiguity. They can’t reasonably switch from cat to dog on any two adjacent pages because of the minute change, so they begin with “cat, cat . . “ and continue saying cat even though the picture is clearly that of a dog. 

 

Those who read my columns regularly might anticipate that we’ll now turn to religion. 

 

I taught a senior seminar class at Hope College for twenty years in which on the first day of class I asked students to introduce themselves including: Give your number on a scale from 0 to 10, 0 indicating “I am sure there is no personal God” to 10 indicating “I am certain that there is a personal God.”  Some were safely at the extremes – some were safely in the middle. 

 

C.S. Lewis admonished his readers to make a decision in faith and stick with it – that living in any room is better than staying undecided in the hallway. I disagree. I realize it’s likely a matter of temperament, but on many significant issues in which I find no compelling evidence, I prefer living undecided. For me, it seems that that is the best way to live with integrity, rather than artificially making a call when I’m not confident. 

 

I liken it to being bounced around in the waves – completely at the mercy of swells of truth coming from all directions. Unsettling perhaps, but a wild ride. 

 

 

Thankful for Holland

 HS #124 2025.11.13

 

 Thankful for Holland

 

Several weeks ago, I saw a movie with testimony of a former Holland resident who said that as a teenager, she found Holland boring and somewhat ethnically snooty (Ain’t Dutch, ain’t much.)

Given that teenage years are often ones of discontent, and given that I’m presently 67, the age statistically found to be the happiest (perhaps the best balance of health and freedom), I still found the comment almost laughable. 

 

Granted, Holland is proud of its history. Our annual Tulip Time and the Van Raalte statue in Centennial Park remind Hollanders of our proud heritage. On the other hand, I recently sent Hope College President Matthew Scogin an email suggesting that Hope consider changing its mascot from the Flying Dutch(men). The Flying Dutchman was a legendary ghost ship – kind of cool, but perhaps it has outlived its usefulness to inspire. My own choice would be HOpe College HOney Badgers. Essentially a Wolverine’s little cousin, honey badgers are not only super smart and ferocious fighters, but are also rather handsome. Check out: https://www.youtube.com/watch?v=c36UNSoJenI    and 

 

https://www.youtube.com/watch?v=MHGNsZVE5Ik. You’ll be impressed!

 

Be that as it may, Holland as a whole seems to embrace good living for all of its residents. My list below is available to all – and many are free or inexpensive. 

 

Our nearby beaches are obvious, but it’s easy to forget to take advantage of hometown riches. If you haven’t done so recently, take a trip to Oval Beach (Saugatuck), Saugatuck State Park (includes beautiful fall color hike and the Felt Mansion), Laketown Beach (dune to climb), Holland State Park, and Grand Haven State Park. I still get “thankyous” from former Hope students who I introduced to those parks during their college years. For something even closer, walk down the lakeside sidewalk of Kollen Park as the sun sets. I am often there and am surprised more don’t take advantage of it. 

 

Kayaking is available at Pigeon River a few miles north of Holland, or New Richmond Park on the Kalamazoo River just a few miles south of town. I’ve seen osprey and blue heron lately. The Black River is a quick paddle with sounds of Holland all around. The Nature Discovery Center (with its Bird of Prey exhibition and best-I’ve-ever-seen taxidermy) provides a fun afternoon’s outing. 

 

For indoor recreation, the Holland Aquatic Center is one of the best in the country – including weights, cardio, indoor track, and exhilarating water slide. Pickleball courts (both indoor and outdoor) abound. Outdoor are all free and enough indoor that competition keeps the prices low. I just now returned from Holland Charter Township Community Center – one more first-class facility available at a decent price – and free for Silver Sneakers. For those over 55, Evergreen Commons provides all sorts of recreation and activities. 

 

And we have more coming – a multi-season ice rink and mountain bike course. 

 

What about the arts? In the last twenty years, Hope College has built an “arts corridor” along Columbia Avenue including music, theatre, dance, and painting/sculpture. Many of the events are free admission, and the Miller Auditorium provides a space second-to-none. Local churches (Free@3 at First Reformed Holland, Second Series at Second Reformed of Zeeland)  offer monthly programs by area artists in a beautiful space. 

 

And the food! It’s easy to forget, when walking along 8^th Street, that Holland’s population is only 35 thousand. Many such Midwest towns of this size are excited just to have an Apple Bees. A dozen years ago, offered a faculty position at Carleton College in Northfield, Minnesota (also home of St. Olaf College), I was surprised on my visit to find relatively few good options for dining. I decided to stay in Holland. Even compared to other such college towns, Holland has it good. And we have a new Indian restaurant! 

 

Then there’s our location. We are a short hop from Saugatuck and Grand Haven, two interesting places. Grand Rapids offers the only professional ballet in the state (fantastic!), the Grand Rapids Men and Boys Choir (rarely found out of England), DeVos Hall with many first-class shows, and Meijer Gardens for art of all kinds. Chicago is a three-hour train ride away. 

 

AND! – have you thought of this! Holland is positioned at the far western edge of the Eastern time zone. So we get long summer evenings in which to enjoy all of these opportunities. 

 

So, raise your glass in thankfulness to Holland. If it’s empty, head down to the Farmers Market. At this writing, the Honey Crisp apples are still available.  

 

 

 

The Risk of Belief (Published in Ottawa News Network)

 ONN #2 2025.10.30

 

The Risk of Belief

 

Dear West Michigan friends – 

 

Before my column proper, a quick introduction. Ten years ago, Sarah Leach invited me to write columns for the Holland Sentinel. I have enjoyed it greatly. For the previous dozen years, I had been known as “the mathematician with Elvis – the calculus corgi.” That changed, to “I read your column.” As you all likely know, a couple years ago, Sarah was removed from the HS and subsequently began the Ottawa News Network. Wanting to continue my readership, I submitted my columns to both. Recently, the HS explained that they can only take columns exclusive to them.  So, I’ll continue my monthly columns with the HS, and will write occasional (perhaps more personal) columns for ONN, including this one and a recent one, “Why Must the Beautiful Die.”  You can find ALL my columns at timothypennings.blogspot.com. 

 

This column, a bit different, is correspondence with my friend, author/Hope Professor David Myers when he alerted me to the recent essay, “Fully Maga-fied Christianity” by Peter Wehner in The Atlantic. 

 

Dave wrote to me: “Tim, you might enjoy this - actually, it’s the last sentence that I liked best.” 

 

The last few sentences were: 

 

“This world is broken but beautiful, a gift from God, and the good in this world is worth fighting for. One lifeon this Earth is all we get, and, in the words of the pastor and theologian Frederick Buechner, “at the very least we are fools if we do not live it as fully and bravely and beautifully as we can.” God is far moreresplendent than the theologies and doctrines about God that we humans construct. And, as the writer Rachel Held Evans put it, “faith is always a risk. No matter what we believe, there’s always the chance we might be wrong. But the story of Jesus is just the story I’m willing to risk being wrong about.”

 

 

I replied: 

 

Dave – 

 

This evening in choir (GR Choir of Men and Boys), we had a guest - a former teacher and presently a minister with the Orthodox Presbyterian Church. After choir, I approached him with a direct question which led to a 75-minute conversation on the street of GR (a policeman actually pulled up - perhaps wondering what was going on). Thankfully, he was self-assured enough (with good reason - he was obviously well educated and thoughtful) to take my question in stride: How old is the earth?

 

His answer: Somewhere less than 10,000 years. 

 

My reply: A Presbyterian - under the umbrella of Calvin - acknowledges all truth as being God's domain and, so I would think, you would find it difficult to accept an understanding of the Bible that is in direct opposition with all the various disciplines of science which point to the age of the earth and universe in the billions of years. I explained that the ratio of 4.5 billion years vs 10,000 years is like walking in a forest of 200-year-old trees and thinking they had been planted that morning. 

 

However, he thought that in time, scientists will come to recognize that our present scientific understanding is entirely wrong. That, in fact, the world is best/only explained as being one in which there were numerous acts by God unexplained by science except as being a miracle.  He REALLY BELIEVED IT - - that is what struck me. He reminded me of Johnny Depp's character, Ichabod Crane, in “Sleepy Hollow” when he first encounters the headless horseman, and the next scene shows him in a fetal position in a corner of a room with all the people who had told him that the headless horseman existed. He exclaimed, "You don't understand! There really IS a headless horseman!” Crane assumed the people’s belief was insincere, whereas he really KNEW it to be true.  

 

Similarly, this young pastor wasn't making his claim out of obligation - he really believed it, all because of his devotion to read and understand the Bible as (he claims) it has historically been understood by the church.  For my part (I was very candid), I was aghast that such a thoughtful, educated person would live his life without appreciating the true age (and hence grandeur) of the universe in which we live. That he would go into museums and read articles and books apparently shaking his head that, "Nope - all of this is totally wrong. All of these tens of thousands of scientists - geologists, physicists, cosmologists who have researched for decades - have all come to the wrong conclusion about the world we live in.”  To me, that's a real tragedy. It does violence to truth itself. Hence my passion about it. 

 

Case two: Just now I heard a BBC article about hundreds of glaciers disappearing each year in Switzerland. On TV this evening a story about how African farmers are finding ways to change their practices in light of global climate change – either change or starve. Another recent story: The coral reef off Florida is dying before our eyes because of warming sea water.  A constant stream of such evidence and yet some are convinced that it's not happening. 

 

How can (why do) people allow themselves to go through life believing a lie? 

 

So how do these examples of living in deliberate ignorance compare with willingly risking being wrong about the story of Jesus?

 

Most importantly, beliefs of a purely spiritual nature don't affect one's view of physical reality and so don't hinder the advancement of human knowledge. 

 

Secondly, as you have documented and eloquently expressed, religious beliefs can lead to a “spirituality which helps makes sense of the universe, gives meaning to life, opens us to the transcendent, connects us in supportive communities, provides a mandate for morality and selflessness, offers hope in the face of adversity and death, and nurtures purpose, love, joy, and hope.”   

 

You and I both live richer lives because of the community, inspiration and opportunities for service that our church offers. However, personally, while I understand those who would “take a risk” with their religious beliefs, I think the best life is one focused unflinchingly on truth - on truth as one honestly comes to understand it.  Is there a personal God? I don’t know. But if there is, I will let God lead the dance and I will respond in kind in the hope that a personal and good God, a God worth knowing and worshipping, values that sort of life and commitment. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mathematics and other Magic

  

HS #123 2025.10.9

 

Mathematics and other Magic

 

Here’s a riddle my father told me when I was young: Three men enter a hotel and pay $10 apiece for their rooms – a total of $30.  After they go to their rooms, the manager realizes he has overcharged them, so he gives the bellhop $5 to split between them. But the dishonest bellhop pockets $2 and then gives each of them $1. 

 

So each of the three men has now paid $9.  That’s $9 x 3 = $27. Plus there is the $2 which the bellhop kept. $27 + $2 = $29.  Where did the extra dollar go? 

 

It’s simple 3^rd grade arithmetic – try to solve it before reading further. 

 

Nope – only 30% of you have tried to solved it. We don’t go on until everyone tries. 

 

OK – some of you have now succeeded. I’ll explain it to the rest of you. First, where are the 30 dollars?  The manager has $25, the bellhop has $2, and each of the three men has $1 - a total of 30. 

 

So how did I intentionally confuse you with my question?  I misdirected your attention by suggesting that you ADD $27 and $2 together, even though there was no reason to do so. Instead of adding them, you should have subtracted the $2 from $27. Why? Because subtracting them answers the question: How much money did the hotel make? The hotel made 3 x $9 = $27 MINUS the $2 which the bellhop kept which gives: $27 - $2 = $25. 

 

This sort of intentional misdirection lies behind much magic. 

 

Years ago, Barbara Walters had the psychic Uri Geller on her show. As Walters and viewers watched, Geller seemingly effortlessly bent a thick key just by lightly stroking it with his finger. As I understand, Walters called it one of the most spiritual moments of her life. Years later, the Amazing Randi reproduced the trick for Walters, showing how both he and Geller had quickly bent the key when he had diverted Barbara’s attention. Then, with the key already bent, he held it in such a way as to conceal that it was bent, and then made it look as if he were effortlessly bending it with (he claimed) psychic power. I have fun entertaining my math students with similar tricks. 

 

Now back to mathematics. Have you heard of Zeno’s Paradoxes? Zeno was a math-magician of the highest caliber who has misdirected the thoughts of his readers for millennia - usually involving the infinite. Here’s an example: A hare and a tortoise are in a race in which the tortoise has a 10-meter head start, and the hare runs ten times faster than the tortoise. Zeno argued that it’s impossible for the hare to pass the tortoise because by the time the hare has run to the starting point of the tortoise, the tortoise has run another meter. By the time the hare reaches that point, the tortoise has run 1/10^th of a meter. By the time the hare has reached it, the tortoise in turn has run 1/100^th of a meter and so on. Thus, whenever the hare has reached where the tortoise was, the tortoise has advanced. Conclusion: The hare can never pass the tortoise. 

 

Isn’t this mind-boggling? Of course, we know it’s not true - there must be a flaw in the reasoning somewhere because obviously the hare will in fact pass the tortoise. But where’s the flaw? If we didn’t know the truth from experience, we’d likely find the argument convincing.  

 

As with the other examples, your gaze has been misdirected. If you took a pie and split it in half, then again, then again, and continuing, you would get increasingly smaller pieces. But putting them all back together again you would still get just the one pie. Agreed? Cutting the pie this way illustrates that ½ + ¼ + 1/8 + 1/16 + 1/32 + . . . = 1. Conclusion: An infinite series of numbers can sum to a finite number. 

 

The hare and tortoise example is similar. The number of seconds needed to catch the tortoise (assuming the hare runs 1 meter/sec) is 1 + 1/10 + 1/100 + . . . = 1.1111 . . . which is 10/9 seconds (check with your calculator).  Zeno tried to confuse you – to misdirect your gaze – by breaking that time into an infinite number of pieces.  Now that you know his trick, you can sleep again at night.