Getting Philosophical

 Getting Philosophical

By Bobby Neal Winters

I’ve been getting more philosophical in my old age.  People mean a variety of things when they say that.  Some mean they are getting more stoical, i.e. more willing to take what comes as it is and to endure it.  That may be true of me as well--I’ll think about that--but it’s not what I mean when I say philosophical.

What I mean is that I tend to think more about the true nature of reality.  This, and because I am a mathematician, means that I find myself spending more time thinking about what numbers are and the nature of their existence.

One can ask, do numbers exist?

The quick answer is of course they do.  You could ask a random person on the street this.  They would say sure.  If you asked them to show you three and they had been out buying fruit, they could show you three apples.

The response to this would be, “You haven’t shown me three; you’ve shown me some apples.” You could then take three of their oranges and say, “You’ve shown me three apples.  Here are three oranges.  They are manifestations of “three,” but they are not “three.”  “Three” is an idea.”

If the person you’d stopped on the street were a construction worker, you might find yourself with six pieces of fruit rammed up your backside. (Not the number six itself, which is also an idea, but the actual, physical fruit.)

The fact that numbers are ideas only gets more interesting when we push past whole numbers.

Because of the way we learn math, we’ve been conditioned to think of numbers as decimals.  Numbers are not decimals.  As has been said, numbers are ideas; numbers are words; but numbers are not decimals.  Decimals are ways in which we attempt to write numbers.

The decimal system of writing numbers is a means of doing so which is very convenient when we need to do arithmetic.  We can use them in hand-written work, sure, but they are very handy to just tap into a calculator.  I think that this has contributed to the belief that numbers are decimals.

Here’s the thing: Not every number can be represented exactly as a decimal.  The original example of this was the square root of two.  This was discovered by Pythagoras. He didn’t put it that way because decimals had not been invented yet.  More accurately, we would say that he discovered that the square root of two couldn’t be expressed as a fraction.  This isn’t exactly true either, but I am too far off in the weeds already.

It is not difficult to show this is true, but that isn’t my point today.  Let’s push on.

We refer to numbers like the square root of two as irrational numbers.

There are a lot of other irrational numbers, but the one that most people are the most familiar with is Pi.  Pi is the ratio of the circumference of a circle to its diameter.  Its decimal representation begins as 3.14159... and just goes on forever without going into a cyclic representation like 1/3 as 0.333... or 1/9 as 0.111111... .

Pi is also an example of a transcendental number, but let’s put a pin in that.  We might come back to it if I get people asking for it, but it being irrational is enough for today.

While I could sit down with you at a table at Signet Coffee Roasters with a napkin and a pen and--for the price of a chocolate chip cookie--show you that the square root of two is irrational, to show you pi is irrational is more work.

It can be done, but you have to bring a little more along with you.

Over the past few weeks, I’ve been working with a student on irrational numbers and transcendental numbers.  As a part of this, I’ve been doing some research on them and have found a nice proof that pi is irrational.

This proof is due to a man called Ivan Niven.  Niven was a Canadian-American number theorist.  His proof that pi is irrational is delightfully simple in the fact that it uses no mathematics beyond Calculus II, which is a Freshman course for math majors.

It is a beautiful proof. In its structure and because it uses such simple tools.  In its original form, it fits easily on one page of a journal with some white space to spare. The version I worked with was a little longer than that, but not much.

The notes I’ve made from it to use to explain it to my student go six pages from a yellow pad.  I don’t know what that will type up to, but it’s still not too bad.

To understand this by analogy, it’s like someone has made a chest of drawers with a hand saw, a plane, and a chisel.  All of the joins are perfect.  You can use a hand saw; you can use a plane; you can use a chisel.  Making that perfect chest of drawers is a different thing, though.

As a part of the proof, Niven uses a particular polynomial in a particular way.  At the same time I was thinking of the question on my own, the author reproducing the proof asks, “Where did Niven get that polynomial from?” I was thinking the same thing myself.

It made me feel good to know I wasn’t the only one with the question. But it’s also good to know there are minds sharper than mine.

It makes me philosophical in another way.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. Search for him by name on YouTube.

And then there were...

And then there were...

By Bobby Neal Winters

My birthday is coming up.  It’s not a special one.

By that I mean it is not a special number like 50, 60, 65, or 70;  not a round number; not a prime number; not a number associated with anything legal.  I will tell you this: It is evenly divisible by 9 and by 7, but it’s less than 100.  

You can figure it out from that.

One of the things that happens to you as you grow older is that people around you start disappearing. It’s like being in one of those old murder mysteries. 

As I’ve written a number of times before, I grew up in a church that taught “the Rapture” as doctrine.  The trump would sound and people would disappear. You’d be left alone to live through the Great Tribulation. 

I found the notion to be absolutely terrifying.  Anytime I found myself alone, I wondered if everyone else had been raptured.

The Rapture in that form has not occurred--at least yet--but people are disappearing from around me, and the Tribulation is here.

Back to my point, what causes this thing I am noticing? We typically hang around with people who are close to our own age.  I remember when I was younger when I looked at people older than myself, it was like I was looking up from the base a pyramid or a skyscraper:  There were a lot of people above me.

Yes, there were a lot of people above me, but I didn’t pay them much attention because we lacked the common shared experience which is the basis for meaningful communication.  

Because of this after a certain age, some of your best friends are your former enemies, but I am digressing again.

Young people--as I was then--tend to be focused on the new.  Older people, as I found out when I became one, have been through all that.  We’ve done the new; then the new became old. We took up the new again. 

The circle goes round and round.

Now I look below me and there is a crowd.  I look above me and see that the pyramid is a lot thinner and I am a lot closer to the apex.

That’s the way it has seemed lately. It has been a year of loss.

I’ve lost schoolmates in Oklahoma.  People my age and younger. I’ve not looked at Oklahoma’s demographics.  I don’t know that the life expectancy there is less than here.  However, I am losing old schoolmates at an appalling rate.  People put on mileage faster in the oil-patch, I suppose.

But death is not a respecter of persons.  About a year ago, I lost a friend of my age who was a consummate professional in every sense of the word. The Fates simply spun his thread, measured it, and cut it.  

He was gone. It was a shock even though he’d been ill.

But most of the friends we lose aren’t from the inner circle because the inner circle by its very nature is small.  There is a phenomenon that happens; let’s call it a melting of our world.  

We go to church; we go to our service clubs.  There are people who--at first--will be missing from time to time.  Then they will only show up every once in a while.  Then they will be gone more often than not.

Then they are just gone.

You will hear someone ask, “Did you hear what happened to...?”

“No what?”

“Well...”

“Oh no!”

One less friend; one less acquaintance. 

In another sense, it’s like we are all standing on a great plain, but it is bounded by a cliff.  The ground keeps giving way, falling into the abyss below and taking people with it all the time, but we don’t notice what is happening until it gets close to us.  We can see the people between us and the edge.  Some are trying to move toward safer ground.  Some are facing it stoically. 

Some are clueless, taken by surprise.  Maybe they are the lucky ones.

Maybe not.  I don’t know.

I lost a man I knew from church this past week.  His wife had passed not long ago.  Suddenly. He’d run an errand.  When he left, she was fine. When he returned she was gone. He was never the same.  Now he has joined her.

I am close enough now to the top of that pyramid to know the stories, to cherish the stories.

The earth under my feet is still solid as far as I know, but the birthdays keep piling up.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. Search for him by name on YouTube.

The Mask and the Heart

 The Mask and the Heart

By Bobby Neal Winters

In the wake of the assassination of Charlie Kirk, there have been a lot of emotionally laden situations.  In particular, I’ve been impressed with how in control of her emotions his widow, Erika, has been.  She is a very composed, very impressive individual. I am not the first to notice this, but it can’t be said often enough. 

In the wake of this, I’ve been thinking about King David a lot. And I do mean a lot.  Not only a lot for most people but a lot for me.  We’ve been surrounded by violence and death.  But because of the stories the Bible gives us, we know our situation is not new.

David was also surrounded by death, indeed, by a lot more death than we are. He fought battles, but more personally, he lost children--which would have to be the worst of all--on several occasions.  I want to talk about two of these.

Before we get into the specifics, let me say that David was a man who--at least at an unconscious level--understood how to create a particular sort of persona, how to project a certain image: David the Poet Warrior.  And he was a Poet Warrior, don’t get me wrong, but we need to know there is a difference between being a Poet Warrior and projecting that image. It is possible to do one without doing the other. He did both.

In addition, he was also a human being.

The lost children I want to talk about are the child that was born of adultery between David and the wife of Uriah the Hittite and of David’s son Absalom. 

The first of these deaths happened when the child, a boy whose name is not given, was a baby.  According to scripture, the child was stricken because David had stolen Uriah’s wife (whose name isn’t given either, but we know it was Bathsheba) and killed Uriah.  

While the child was ill, David openly fasted and wept for the child.  His actions were extreme enough that when the child died, David’s servants were reluctant to tell him because they were fearful of what he would do.

As David was very tuned-in to his household, by their very quietness, he was able to tell that the child had died.  When he elicited that information from them, ”[he] arose from the earth, and washed, and anointed himself, and changed his apparel, and came into the house of the Lord, and worshipped: then he came to his own house; and when he required, they set bread before him, and he did eat.”

When asked why he fasted and wept when the child was ill, but not when the child died, he replied, “I fasted and wept: for I said, Who can tell whether God will be gracious to me, that the child may live? But now he is dead, wherefore should I fast? Can I bring him back again?”

It can be roughly summarized that he was putting on a show of grief, in hopes of a different outcome. He’d affected the persona of grief.

The second death I want to talk about happened to his son Absalam. 

Absalam was not a baby. Far from it.

Absalam was in the process of leading a revolt against King David, he was riding along by himself on a mule and got his long hair caught in the limbs of a tree.  While he was stuck there, some of David’s soldiers, acting against David’s orders, killed Absalam with spears.

Upon learning of this, David’s response was much different than before. He cried out: “O my son Absalom, my son, my son Absalom! would God I had died for thee, O Absalom, my son, my son!”

Such was the strength of his emotion that Joab, his main general, had to caution him for fear of demoralizing the army. (For the sake of brevity, I am leaving a lot of backstory and nuance out of this. Read the book.)

I pick these two incidents to see the change that occurred to David over the course of the passage of time.

In the first of these, David put forth the face of mourning with an attempt to sway God Himself, who can see into every human heart, even that of a king.

In the second, David’s love for his son was so strong that in spite of the fact his son was in open, violent revolt against him, his grief upon his death overcame any ability to put forth the image required of a king.

Our leaders, the ones we like as well as the ones we don’t like, bear a burden: Very rarely can they be themselves. More precisely, they have to be very careful in the parts of themselves they reveal to the public. They wear a public mask.

But God sees the heart and will heal the pain.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. Search for him by name on YouTube.

Fall, the Smell of Grass, and Shame

 Fall, the Smell of Grass, and Shame

By Bobby Neal Winters

We are coming into the Fall of the Year.  The nights are getting longer, and so the days are getting shorter. It’s getting a bit cooler, and there are more fallen leaves on the ground.  

And the grass is different. It knows something is going on.

This last is important because we are moving toward the last mowing of the year. We know not the hour that it will come, but all the signs that it is coming are in place.

I am being deliberately ambiguous here.  I don’t know how many more times I will mow.  If the weather conditions are right, I may have just mowed for the last time this year and simply don’t know it yet.

It’s hard to tell, because, as I said, the grass is getting different.  It’s fall grass.  What I mean to say is that the grass is a little different at every station of the mowing season, and the fall grass is here.

For one thing, the smell is different.  

Yes, the grass smells different--the mown grass that is--at every part of the mowing season. I state this as a plain fact without supporting argument.  If you know, you know, and if you don’t know words will not be able to convey anything to you.

I’d like to believe that if I were put into a coma and then woke up at an unknown time later with a blindfold on, I could tell what time of year it was just by the smell of the grass.

But if I couldn’t tell, there are people who could because the smell is different, and that’s a fact.

Not only is the smell different, the grass itself is not as heavy as it is at other times of the mowing season.  It is so heavy in the spring, I have to stop and change batteries in my mower.  As the season progresses and the relative amount of moisture in the grass decreases, I can get it on one charge.

Because of the current lightness of the grass, right now would be an ideal time for me to mow with my battery-powered mower except that it--much to my disappointment and shame--is broken.

Let’s talk about that.

Back in 2020, during the lockdown, I bought my first battery-powered lawnmower.  I liked it: I didn’t need to buy gas for it; I didn’t need to keep it oiled; it was quieter than my gas mower; it was easier to start.

I wrote about it.  I had people say they had read my article and followed suit by buying their own battery-powered mowers.

I felt good, like I was helping to save the planet from global-warming one lawn-mowing at a time.  They would put up statues to me; they would name a Monday Holiday after me.

Then after a couple of years, that mower died; se morto’.  Something in the safety system--the stuff that is put in to keep the consumer from killing or maiming him- or herself--broke down. 

I tried to fix it, but at that time there was precious little help online for a DIYer to fix a battery-powered mower.  I decided that maybe I’d just been unlucky, so I bought another.  Same make, slightly newer model.

After the same amount of time, the same thing has happened.  

This time I didn’t even think about buying a new one.  Fool me once, shame on you; fool me twice, shame on me.  Well, they fooled me twice.

In the intervening couple of years, there has been an increase in the battery-powered mower ecosystem. There is help online about how to fix it, and that is good news. The bad news is that most of that help boils down to getting around the mower's safety system.

Don’t get me wrong.  I don’t want you to think that I’m putting on airs.  I am from Oklahoma for goodness sake and safety isn’t a big concern to my people.  A siphoning hose for gasoline is referred to as an Oklahoma credit card after all.  The only thing wrong with that phrase is I don’t know of anyone ever giving any gasoline back.

The thing is that I am going to have to think about it a while. If I’m going to have to learn something.

Anyhow, I put the mower away in the potting shed, brought out the old gas mower, filled it with gasoline, and started it.  It only took about three pulls.  I went a little overboard and put some oil in it. It hadn’t had oil since at least 2007. You gotta love Briggs and Stratton.

I wear hearing protection while I use it, so at least it won’t hurt my own hearing, and I’m having to buy gasoline.

I also have a gasoline-powered chainsaw now, which is not only loud and produces carbon dioxide, but it cuts down trees.

I don’t think there are going to be any statues for me.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. Search for him by name on YouTube.

The Method of Partial Fractions

 The Method of Partial Fractions

By Bobby Neal Winters

I am teaching Calculus II again after 22 years.  None of my students had been born yet the last time I taught the course.  They are all young, fresh, and full of energy. Their brains are so sharp. They keep me honest on my “arithmetic.”

I put the scare quotes around “arithmetic” because most people would call it algebra, but to a mathematician algebra is something different.  If you are not a mathematician, my telling you what we mean by algebra wouldn’t help; if you are, you already know.

I try to make it easy to tell me when I’ve made a mistake and how to do that kindly because that is a skill that will be useful to them even if they don’t use a single thing they’ve learned in Calculus II.

That does bring me to a question that does come up from time to time: When will I ever use this?

It is a legitimate question.  They--and the Great State of Kansas, bless it from border to border--are paying good money for them to learn this material.  When will they use it?

An easy answer is that I don’t know.  There are a lot of students, and there is a lot of material.  They have different backgrounds, different talents, different ambitions, and different plans.  I don’t know what those plans are, what all of them entail, nor how they might change.

What I do have is a collection of mathematical material that was invented by Isaac Newton and Gottfried Leibniz about 300 years ago, has been found useful by scientists and engineers, and has been refined over the course of a couple of centuries.

It is like a huge toolbox that is full of expensive, well-used tools. I am teaching them that the tools exist and the best way that I know to use them.

To continue with this metaphor, I’ve been having to clean up and sharpen the tools because I’ve had the opportunity to use them for the last two decades.  

Well, that is not entirely true.  While I was in administration, I didn’t continue to teach calculus, but I did continue to teach a course called Introduction to Analysis.  This is a course in which the theoretical foundations of Calculus are taught. To really explain what this means in a way that would satisfy a fellow mathematician would require a lengthy article that not many would read.  Not “many might” be a great exaggeration of the number. For the current readership, let me just say that Newton and Leibniz were scary smart and did things that folks like you and me have difficulty understanding.  The theoretical framework that has been set up makes it accessible to a few more.

Recently, I caught myself with a topic I had not seen at all in the last two decades: the Method of Partial Fractions.  Again, this is one of those things I am not going into detail for the current readership, but folks who’ve had a course in high school algebra (fairly recently) would be able to understand.

Metaphorically, this is a chisel in the toolbox that is Calculus II, and I needed to sharpen mine.  Because of this, I came into the office before church on Sunday morning and spent a couple of hours with the equivalent of a whetstone and a leather strop. It was a yellow pad and a pencil, but a whetstone and a leather strop sounds much cooler.

It brought back a bunch of memories, not all of them comfortable.  Believe it or not, I was something of a know-it-all in my college days. (There will be a pause here to let those who knew me during those days spew whatever they are sipping as they read this through their noses.) I remember when my Calculus II teacher was teaching me this method.  I thought I knew a better way to do it than he did.

To his credit, he said nothing and let me do it my way.  I got the right answers; the math was correct; but it was a lot more work. 

It took me 44 years to figure that out.

I understand why he let me do it that way; I believe it was the right thing to do; I would do the same thing myself in the same circumstances.

But it does make me smile a little right now.

But that is neither here nor there.

The alert reader will notice a couple of things.  The first of these is that I’ve not used this particular technique in more than 20 years and before that I only used it whenever I was teaching it.

These alert readers will also notice that we learn other things while wrestling with hard material:  We learn to wrestle with hard material.

Things will be easy for these students that other people think are hard.

I believe that it has helped me, at least.

So, if you’ve had algebra within the last couple of years, you can bring your yellow pad with you up to me at the coffee shop, and I will tell you more about the Method of Partial Fractions than you ever wanted to know.

Maybe I already have.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. Search for him by name on YouTube.

The Value of a Picture

 The Value of a Picture 

By Bobby Neal Winters

On a Saturday in early September, in the entrance to Timmons Chapel on the campus of Pittsburg State University, I had a vision.  It was a flash of movie dialog going through my head:

“You come to me on the day of my daughter’s wedding, and you ask of me a favor.’

It was my wife asking me to put together a tripod that was to hold a poster that contained pictures of the bride and groom as babies with baby food smeared across their faces.  

On one hand, it was a worthy thing, well within my skillset, all the parts were there, no tools needed, no glue necessary.

On the other hand, it was less than an hour to the wedding; I was dressed in a coat and tie; and I didn’t have a work bench.  All I had were the “directions.”

(In my wife’s defense, she hadn’t known that the tripod required actual assembly or we would’ve assembled it ahead of time.)

Let me say this: The directions weren’t useless; they were worse than useless. They were written on a tiny piece of thin paper, and consisted of tiny pictures with tiny labels. Connect part G to part H, when you can’t tell what parts G and H are from the instructions.  This sort of thing continued until the last, and perhaps, only useful line of the directions:  Look at the picture on the box and put it together.

So, after 15 minutes of fumbling, I looked at the picture on the box and put it together, with my wife and two of my daughters (not the bride who was otherwise engaged) helping.

A picture is worth a thousand words just as they say.

Description of physical objects is difficult.  When I was doing mathematical research in the topology of 3-manifolds, I sweated blood in describing the pictures I saw in my head.  Was I right in my descriptions?  The first time anyone reads one of my papers, we might know.

But I am not the first person to have had this difficulty, far from it.  This is a problem that goes back literally thousands of years. To know this is true, all you need to do is read Chapter 26 of the Book of Exodus where the construction of the Tabernacle (the tent the Children of Israel used to worship) is described.  If you have a Bible near to hand, I would encourage you to take it in hand, open to that chapter, and read a while.

It reminds me of an exchange from “Paint Your Wagon”:

“You should read the Bible, Mr. Rumson.”

“I have read the Bible, Mrs. Fenty.”

“Didn’t it discourage you about drinking?”

“No, but it sure killed my appetite for readin’!”

My point isn’t to discourage you from reading the Bible but to emphasize how difficult it is to describe reality without pictures.  Even using a lot of words and taking admirable care in description, describing the Tabernacle was hard.  

Can  you make one from reading it?

I am sure you can, maybe several of them that look quite different from each other.  Drawing a picture of the Tabernacle would’ve been helpful with a list of parts and tools needed, but that idea hadn’t come yet apparently.

But things like tents aren’t the only challenge.  A lawyer came to Jesus asking him how he might inherit eternal life.  A discussion ensued in which it was suggested the lawyer should love his neighbor as himself.

This sounds like a simple rule, but the lawyer asked for clarification.

Jesus responded with the story of the Good Samaritan.

A story isn’t literally a picture, but it serves a similar purpose.  It builds a picture in our minds, a picture we can relate the words, the concepts to.

The poster I referred to at the beginning--the one on the tripod--had pictures of the bride and groom with baby food on their faces.  It reminded us that these two responsible, grown-up people were once babies in our arms. To some degree, they are still babies in our hearts, but as parents we need to now let that go.  While that connection can never completely be overcome, we have to know that they must be more important to each other than we, the parents, are to them.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. Search for him by name on YouTube.