Kinship of Mathematics and Science

 HS #129 2026.4.9

 

Kinship of Mathematics and Science 

 

In May 2024, we discussed whether mathematics is invented or discovered.  (Remember?) Now, we explore another long-battled question about mathematical knowledge: Does one learn mathematical truths from observing the natural world, or are we born with mathematical knowledge inside of us? 

 

Kant believed that unlike scientific knowledge of the natural world gained via experimentation, mathematical knowledge “rests upon no empirical grounds.”  He claimed we know mathematics a priori (prior to experience); we could discover it in a dark closet without ever seeing the world. 

 

I respectfully disagree. 

 

It seems to me that mathematics is initially learned via experience. It took Bertrand Russell and Alfred Whitehead ten years and 379 pages of logic to prove that 1+1 = 2, but the rest of us know it by counting our fingers. Given a circle, the length of the circumference divided by the length of the diameter always gives the same ratio, which we call pi. How do we know? Experience. Indeed, I send my math classes to athletic fields with ropes of different lengths where they pace off the circumference and the diameter. Always get pi.  Another example:  Why do we know that the order of adding numbers doesn’t matter (7+9 = 9+7)? Because we’ve tried it many times and it works. It's an axiom of the real numbers;  we are convinced of its truth via experience. Why do I know that five is a prime number? Because no matter how I try to divide this many dollar signs $ $ $ $ $ into equal piles of size greater than one, I can’t. 

 

Why do we know that two points on a plane determine exactly one straight line? Having seen straight lines, we can visualize it. Conversely, why can’t we picture a four-dimensional object? (Salvador Dali came the closest – see his Crucifixion (Corpus Hypercubus).) Four-dimensional objects are just as math-legit as three-dimensional objects, but we don’t have the benefit of seeing them, so we find them difficult/impossible to visualize. Similarly, it’s impossible to imagine the next color in the rainbow – the ones we’d enjoy if our eyes could see ultraviolet or infrared light. We humans have an embarrassing lack of imagination! 

 

So, as with experimental science, it seems we learn foundational mathematical truths from experience. However, once those initial truths are determined, then mathematics proceeds quite differently from science. 

 

A coin illustrates the difference. Having observed that both sides of a coin are HEADS, we DEDUCE that when flipped, it will always show HEADS.  Similarly, having all the information they need (their axioms), mathematicians proceed deductively, using logic to jump from one conclusion to another. They are confident that “proved true” means it IS true. 

 

On the other hand, if without looking at a coin, you flipped it 100 times – always getting HEADS, then you might – via induction – become convinced that it is a two-headed coin, even though you will never know for sure.  Similarly, scientists never fully assume the truth of their foundational assumptions. Instead, they acquire knowledge inductively by observing regularities and patterns in nature and then draw hopefully-correct conclusions – a process called the scientific method.  Karl Popper realized that, like the tossed coin, scientific experiments can never prove anything true.  One becomes increasingly convinced, but never certain. 

 

Indeed, consider these historical conclusions of physics: i) energy is conserved, ii) mass is conserved, iii) an object’s length and mass doesn’t change, iv) velocities in the same direction can be added. The truth of all of these were challenged by the brilliant mind of Einstein, and now have now been shown to be wrong via experiments.  

 

Mathematics, historically, also has had incorrect conclusions: i) every line has another which is parallel, ii) every number can be written as the ratio of two integers, iii) if something is true, it is possible to prove it true, iv) if you take some away, you have less. How simple and obvious – but all shown to be wrong! No experiments were needed, just pencil and paper through mathematical argument. 

 

Surprisingly, mathematics and science are also similar in challenging a fundamental truth. Even though they proceed by disparate methods, physics (via Schrodinger’s Cat) and mathematics (via sizes of the infinite) agree that, paradoxically, there exist things which are neither true nor false. That is, the world can’t be neatly divided into true things and false things.  Reeks of  postmodernism! 

 

How neat! Mathematics and science, each in its own way, reveal that our world is rich with wonder and surprise. 

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.