Bart vs. the Space Mutants vs. the Fundamental Theorem of Calculus

Bart vs. the Space Mutants was an insanely hard game on the old 8-bit NES Nintendo system.  I think anyone that played it before knows it.  It's harder than proving the fundamental theorem of calculus. You know how I know this?  It's because I took a Real Analysis course in college where we did actually prove the fundamental theorem of calculus, and I got a B in that course.  But I NEVER could beat Bart vs. the Space Mutants.  Not as a kid or an adult.  The furthest I got was level 4.

Of course that game isn't even the hardest one on the NES.  There were a lot of games on that system I never could beat.  Never could beat Teenage Mutant Ninja Turtles on there either.  Or Beetlejuice.  Or Home Alone.  Or Addams Family.

The hardest games I actually COULD beat were Mike Tyson's Punch Out and Adventures in the Magic Kingdom.  The Magic Kingdom one wasn't quite as bad once I figured out what the stars were for and how I could use them as powerups, and how to use the candle in the pirate level.  But even then it was no picnic.  Space Mountain was still pretty tough to me.

Those old NES games sure were tough.  Especially since a lot of them didn't have any kind of save feature.  Heck, a lot of times you were lucky to even get a password.  Some of the games I could beat with a Game Genie, but of course since it's cheating, that doesn't count.  Some of them I STILL couldn't beat even with a Game Genie.

A Chekhov's Gun Critique

The philosophy behind Chekhov's gun makes sense on the surface, where you don't put something in a story unless it comes up later, but I think there's an interesting technique that I've seen in the Sherlock Holmes short stories where stuff doesn't come up later, but still kind of serves a purpose.  

Often Watson will reference other cases at the beginning of a story, some of which actually appear as short stories themselves and some of which never make it to print.  You may ask, "What's the point of even mentioning the case if it will never come up again?"  And my answer to that would be that it makes the world appear much larger than what we actually see.

Pleasantville 2

To the best of my knowledge, there has never been any kind of sequel planned to the movie Pleasantville.  But I sure love that movie.  Grew up watching Nick at Nite in the early 90's, and so it reminds me a lot of that channel.  

I imagine the sequel would be the citizens of Pleasantville slowly realizing  more and more of the problems that come with open-mindedness that encounters no resistance.  Things like struggling single parent prevalence, profane language dominating entertainment media, addiction to technology, etc.  And then they'd slowly become more open to some of the ideas behind self restraint again.  

And when some small behavior exemplifying restraint is accomplished, a small part of the world would lose its color and return to black-and-white.  Even though not everything would return to what it once was as far as culture and behavioral customs go (racial integration in particular would remain), enough things would revert so that eventually the entire world would turn black-and-white again, slowly biding its time over the years before it becomes open to introducing color and a spirit of open-mindedness once more.  

So it's kind of like a reversal of the main idea of the first film, which focused on the problems with restraint and how the world became more colorful when restraint was cast aside.

I like this idea because it shows that to everything there's a season.  There are seasons of opening up and there are seasons of restraint.  Pros and cons to both.  Open up some, step back a bit, open up some, step back a bit, and keep the pattern going.  The cycle repeats, probably indefinitely.  I believe there are historical examples of this cycle in the past, but at least for the time being, there does not seem to be a prevalence in literature of periods of voluntary self restraint dominating a culture.  Temperance movements have popped up here and there against alcohol, but that's all I'm aware of.

The Trolley Problem in Star Trek TNG

I believe a Reddit forum somewhere discussed this, but I didn't want to participate in a conversation there, so I'll just post a little bit about it here.

Counselor Troi is studying to get the credentials to potentially move up to a captain's position.  But she keeps failing a simulation section of the exam Riker gives her, and he won't tell her why.  Eventually she discovers on her own that the reason she kept failing was because she kept the well-being of each individual person in mind, even when the health of the entire ship was on the line.  

She finally realizes her first duty is to the ship, and so when she next takes the exam, she orders the simulation of LaForge to go in and fix a warp core breach, despite knowing it was going to kill him.  That time she passed the section.

At that point, she realized she didn't want to be a captain anymore.

So the solution to this problem really depends on the duty of the decision maker.  It may mean actively ordering someone to their death if it saves the ship, or perhaps actively sending three our four ensigns to their deaths if it saves the captain and first officer.  Whatever is optimal to keep the ship afloat on its mission is key.  Even if it's not pleasant to think about.  Being an effective leader requires tough and unfair decision making sometimes and it's certainly not for everyone.  I know I often avoid assuming such a position when I can.

Functioning Under Unideal Conditions

Being able to still keep going when things aren't ideal is a crucial skill, because reality is almost never ideal.  I call this ability Functioning Under Unideal Conditions.  If turning it into an acronym that looks vaguely filthy helps you remember it, I guess that's okay.  As long as you can live by the main idea behind it.

Sometimes conditions are so bad that you aren't able to function no matter how tough you are, but if conditions are just unideal, then yeah, it's something you should be able to live with.  You have to shoot for optimal conditions most of the time because ideal is most likely too elusive.

My Unexciting Religious Testimony

I talk about religion a lot in this blog, but I don't think I've ever posted all that much on how I became religious myself.

A lot of people have very dramatic conversion stories, often about how they went down the wrong road and reached rock bottom before finally abandoning shallow pursuits and grasping a much deeper meaning of life.  I am not one of those people.  I grew up with a very religious father, and he grew up in a very religious Louisiana family.  My dad and his seven siblings were very poor growing up, but they all got along really well for as long as I've known them, even now as their days are beginning to end and they're slowly beginning to pass away.

In my elementary school days I was fairly typical.  Did bad stuff every once in a while that got me in trouble.  I'd go to church with my dad here and there and he got me a kid's Bible by Mary Batchelor with great illustrations that I'd look at from time to time without actually reading much.  Most of the time I just watched TV and played Nintendo when I wasn't in school.

One of my many cousins and I were out in the woods of Louisiana one day during this time, and even though his father was a minister, he and I spent that day saying every foul word we knew of at that time, because a bunch we still didn't know.  But then we decided we were tired of that foolishness and pretty much stopped saying bad words after that.

So when I got to middle school and high school I was actually a bit more behaved.  I never had an urge to drink or smoke or anything like that.  Adult magazines and websites were a different story and once I got into those I didn't shake that habit until I was 26 or so.  Other than that I didn't have too many problems other than being very lazy in 10th grade.  But by 11th grade I woke up and stopped being quite so lazy, and life got a bit easier after that.

I believe it was in 8th grade when I finally felt called to walk up to the front of the service during the church's period of invitation and publicly commit myself to following Jesus and his teachings.  I thought about it for a good long while before I finally did.  I think I was worried about not being saved at that point.  I was baptized shortly afterward.  I started going to church regularly, but I got kind of freaked out in Sunday school with the pop religious music and socializing so I pretty much stayed in regular church.  

I read the Bible (or at least most of it) when I went to college, but even by then I didn't feel the spirit impacting me very deeply.  Not that I craved the spirit very much.  Some other religious folks there met me from time to time, but I couldn't be as committed as they were.  I think it was good to meet them here and there and I did appreciate it.  I went to church only sporadically during my college days.

My mom passed away while I was in college and I ended up moving back in with my dad and going back to the church I grew up in and was baptized in.  They asked me to do some of the sound and/or projector and I did it for a few years, but it wasn't very fun for me.  I can't call it a thankless job because I was thanked quite often by many members of the congregation.  I had a lot more fun going to lunch with the medical students after church.  We had a whole Sunday school class of just married medical students at that time because the church is close to a medical school.

A few years later I got married myself ,and my wife and I went to about two or three different churches before picking one to settle with.  I think it was at this time that I finally really started to get a feel of what Biblical doctrine was truly about, and I actually began craving a lifestyle that did not forsake the Lord.  I realized the deep truths of Proverbs and how material as old as that really can apply so much even today.  It's a book I feel EVERY young man should read because all of the pitfalls it talks about are very real.  I realized I couldn't do things on my own and needed the Lord's forgiveness for my shortcomings.  And I came to feel more and more that life is not a meaningless and random series of pointless events.  There's very much an intent behind nature, and MUCH more to the world than meets the eye.

I ended up going back to church with my dad again a few years later, and I guess I've been going with him every Sunday for the past three or four years.  It's a time I cherish because I know it's becoming limited as he progresses to an advanced age.  

Me being middle aged now myself makes me realize how fragile humanity is and how it really does need a lot of help in its battle against sin.  I know I'll take what help I can get because I certainly am not immune from falling into the traps of self-deception and failing to realize my own sins.   That doesn't go away just because I'm a bit older.  I don't care if people think I use religion as a crutch.  Why?  Because I do.  And I don't care if they think I was indoctrinated either.  So what if I was?

My life, even if it seems dull to many, has been quite fulfilling to me, where I got to live many dreams and ambitions and experienced a lot of good fortune.  There were some setbacks here and there and I'm sure those will continue.  But I hope I can be as thankful for all that God has given me as I have been over the past few years of slow spiritual awakening and realizing how amazingly frequently involved God has been in just my mediocre life.  God revealed so very much to me in my secular and religious studies and in my relationships that he never had to show me at all.  But he wanted to anyway.

Even just quiet mornings where I wake up early and take it slow before work bring me so much joy.  I don't even have my heart set on eternal life all that much because I've already been so blessed with just the temporary one God has given me.  Of course I still believe that next life is still there.  But even without it, God's goodness to me here has been so evident.  I know it hasn't been that way for everyone, and those I pray he takes special care of in the next life.

A Bad Idea About Strength

It's probably a bad idea to demand that people who are stronger than you should be even stronger than they already are.  Seems like it could potentially backfire.  

We can't forget that in this world even the strong are more fragile than they appear.

Does the Omniscient Viewpoint Matter?

Here again I'm arguing against moral relativism because it's something I like to do.  Perhaps someone has used this argument before and I've just never seen it.

Suppose we have a court case, and the man is found guilty by the jury because the evidence is compelling and strong.  But also suppose the man is innocent despite the indication of the evidence (kind of like a Sherlock Holmes story).  The jury just voted him guilty because of their limited perception, but really did their best given their lack of omniscience.  You can't bash their integrity for doing what they could with the tools they had.

Now moral relativists would classify the man on trial as immoral because his moral status was determined by consensus after some due diligence.  Even more so if the general public pretty much agrees with the verdict.  But what about the omniscient viewpoint - you know, the only one that can even potentially detect his innocence?  Does that viewpoint mean anything?

We Can’t Say Something Positive is Impossible???

I’ve spent some time lately looking at Godel’s ontological proof and trying to comprehend it the best way I can.  Axiom 5 seems the most troubling to me out of the whole argument, but today I want to focus on just Theorem 1 of the proof and the two axioms leading up to it.  Here it is below, copied from Wikipedia for quick reference:

Theorem 1 says if a property is positive, then it’s possible there exists such a property.  That seems reasonable.  Positive things do exist in this world after all.  But if we look a little more deeply at this statement, this claim is a bit stronger than it looks, because we’re saying we cannot definitively claim ANY positive property is completely impossible in all potential worlds.  We can’t call any positive activity completely impossible.

And Godel seems to use a slick method to claim this, using the vacuous truth rules, which I’m always suspicious of because I think they were just made to keep the truth value system closed to two values instead of opening up a third undecided value.  

First Godel assumes Theorem 1 isn’t true in an attempt to derive a contradiction.  He says “Phi is a positive property AND it’s necessary that for all x, NOT phi of x,” which is the opposite of what Theorem 1 says.  This makes sense.  To negate an if-then statement you assert the premise and the opposite of the conclusion. 

After that he notices that the portion of Axiom 1 stating “it’s necessary that for all x, phi of x” opposes a portion of his assumption.  Plus this portion of Axiom 1 is the beginning of a conditional.  So his assumption to start the proof of the theorem creates a vacuous truth in Axiom 1 where the axiom claims phi of x.  Because of this, by the rules of vacuous truths, the if phi of x then psi of x statement in Axiom 1 is ALWAYS true, no matter what value psi of x takes.

So Godel just takes that psi of x and makes it equal negative phi of x.  So the output of Axiom 1 after this substitution says if (phi is a positive property) and (it’s necessary that for all x, if phi of x then NOT phi of x) then NOT phi is a positive property.  Kind of a nasty double conditional.  The conditional inside the conditional was a vacuous truth.  But the outer conditional has its premise consisting of two true statements, one being phi is a positive property, which we already asserted as soon as we started the proof by contradiction, and the vacuous truth.  So both parts of the conditional premise being true will indeed yield the result of NOT phi being a positive property being a true statement.

All that mess is not easy to write out.  And we did it all just to get the statement “NOT phi is a positive property.”  Now we can finally go to Axiom 2, and claim that since NOT phi is a positive property, then that implies phi is NOT a positive property.  Easy enough.

And of course now we have just concluded phi is NOT a positive property, even though the very first thing we did at the beginning of proof by contradiction was to assume “Phi is a positive property AND it’s necessary that for all x, NOT phi of x.”  So we have phi being a positive property and phi NOT being a positive property.  This is a contradiction, and completes our proof by contradiction.  We have shown that if phi is a positive property then it’s possible there exists some x with that property.

But of course I am very suspicious of a vacuous truth being used.  So I asked my Gemini app if it could produce an example of a similar scenario where this proof structure is followed, and the axioms seem reasonable, but the conclusion seems like nonsense.  Mostly as an attempt to discredit the use of vacuous truths in proofs in the future.  It cleverly suggested to plug in the phi variable as “impossible crime” and the P variable as “legal.”  It also suggested phi as invisible socks and P as fashionable, and phi as boiling ice cream and P as delicious.  These seem CLOSE to fitting the axioms, but I still don’t feel they fit the axioms comfortably enough to my satisfaction when I sit down and try to plug them in.

So even though I don’t like vacuous truth arguments, I can’t seem to invalidate them the way I’d like to.  Perhaps we really do have to admit that if something is positive, then it can't be impossible, at least under the two seemingly reasonable axioms Godel used.  I really have to admire his skill in coming up with this trick.  It’s ugly, and questionable, but I can’t find a way to attack it the way I’d like to, try as I may.  Maybe someone else will have better luck.  

A pretty picture involving the Gaussian Integral

The Gaussian integral is a neat and very easy representation of the bell curve with the area under the curve exactly equal to the square root of pi.  Nice simple-looking expression.  The usual bell curve formula looks a lot worse since it's a family of curves and it also sets the area under the curve equal to one instead of the square root of pi.  Or something of that nature.

Inspired by that expression, I used Desmos to plot the picture below.  The darker shape is just a semi-circle with radius equal to the fraction one over the 4th root of pi.  But I designed the plot so that the dark semi-circle area in the picture was EXACTLY equal to the light orange area between the semi-circle and the bell curve.  I think that precise relationship makes a geometric picture such as this look quite nice, even though it's difficult to intuitively tell the areas are indeed equivalent.  The picture kind of looks like a pretty sunset if you fill it in with the right colors.  Both areas should be equal to the square root of pi over two if I didn't make any errors.

And here are the expressions I typed into Desmos to generate the aesthetically pleasing plots shown above.

I think that's enough fun for one day.

Why is the number e in the bell curve equation?

When I think of bell curves, I tend to think more of Galton boards than a graphical representation of test score averages.  Galton boards are those fun things with all the marbles in them that you flip over and most of them gather in the middle while a few go off to the sides.  Big enough Galton boards can approximate a bell curve fairly accurately.  Really interesting seeing how even though each ball follows an unpredictable path, altogether they form a shape that's quite predictable, if you have enough of them.

Anyhow, Galton boards have binomial coefficients in their mathematical calculations - like many things in probability do - and of course binomial coefficients have factorials.  But what's interesting is factorials can be represented by Striling's approximation.  This estimates n factorial to be pretty close to (root (2*pi*n))*(n/e)^n.  Hopefully I typed all that out correctly, but I'm sure it's not hard to find online.  

Well anyhow, that's how we see an e appear, if we trust formulas like that and feel comfortable looking up the derivations of those formulas later.  Stirling's approximation is involved in the approximation of a factorial, and if you plug those approximations into the binomial coefficients in the formulas similar to Galton board type models, you get a formula called the De Moivre-Laplace Theorem.  

The Galton board shows how a large accumulation of randomized left-right choices from a centralized point (which I guess we could think of as a mean) forms a bell curve, and the De Moivre-Laplace theorem gives the formula showing the e in that equation.

A Look at Why Anti-Derivatives May be Tougher than Derivatives

 The definition of the derivative is the limit as x goes to zero of (F(x+dx)-F(x))/dx.  If you're given a function, it's not too bad to plug it into this thing, do some algebra, take the limit, and see what happens.

But try going backwards.  Try taking a function, then doing algebraic manipulations to get another function into this form.  Seems to be much tougher, and I guess there are a lot more potential options you could do, but then getting them to simplify in the right way seems somewhat daunting if you're like me and not all that great at algebraic manipulation.

Let's take something simple like 2x for example, and try getting its anti-derivative by algebraically manipulating it to look like something that fits the definition of the derivative.

2x

First let's add dx to it.  Since dx will just be going to zero when the limit is taken, I guess it's kind of like adding zero.

2x+dx

Then multiply by 1, but put 1 in the form of dx over dx.

(2x+dx)(dx/dx)

Multiply the top dx through.

(2xdx+dx^2)/dx

Add zero to the numerator, with zero in the form of x squared minus itself.

(2xdx+dx^2+x^2 – x^2)/dx

Shift one of the x squareds to the front.

(x^2+2xdx+dx^2 – x^2)/dx

Realize the first three terms can combine.

((x+dx)^2 – x^2)/dx

Then realize that we now have the function x^2 in the form of the limit as x goes to zero of (F(x+dx)-F(x))/dx.

This implies that the derivative of x^2 is 2x, which means the anti-derivative of 2x is x^2.

That was a LOT of algebraic manipulation I would have NEVER thought of doing if I didn't do this problem backwards to begin with.  And this is one of the easiest anti-derivatives.  Imagine if we tried something harder.

When the tool is more interesting than the thing it was designed for....

I remember learning cursive handwriting in 3rd grade and being very excited about whatever letter I was going to learn next.  It seems obvious to me that cursive was designed to be faster than print since you don't have to lift your pen, but now that computers have taken over, the advantage of speed that cursive writing once had has now become moot.  

The capital G in particular is one of my favorite cursive letters I think.  It's just pretty to look at.  And cursive in general is pretty I think.  While it was designed for speed, now it seems to be more elegant than anything else.  It kind of seems fun to think about printing cursive G's on that old gray multi-lined handwriting practice paper for a day, even though it would serve no practical purpose.  

So the tool once used for enhancing communication speed (cursive) seems more interesting to me than the ability to print a message quickly.  The yanghaiying YouTube channel, with its sporadic focus on Chinese calligraphy, kind of reminds me of the elegance of cursive writing.

Moving on, I know some mathematics was developed as a tool for engineering or some other related field, and other mathematics was developed for its own sake, but often I feel that mathematical tools are more interesting than whatever they were designed to accomplish.  I know very little about Fourier series for instance, but it's my understanding their initial function was to deal with heat equations.  The 3Blue1Brown YouTube channel instead focuses on how they can be used to draw pictures with circles.

Betaine HCL Testimony

Even though I prefer religious testimonies to testimonies about non-religious experiences, I think my particular experience here is worth mentioning - particularly to anyone struggling with chronic severe indigestion.  

I had testicular cancer way back in 2007 and radiation was part of the recommended preventative treatment so it wouldn't spread to other organs.  That radiation was no joke and I had to take STRONG anti-nausea medication in order to function during that time.

Many years later, in 2019, I began to have major digestive issues.  Constant, constant belching and bloating.  Massive indigestion.  Significant weight loss as well.  An endoscopy revealed no definite cause.  Slight gastritis, but nothing else.  At least it seemed to rule out anything big and noticeable like cancer again.  Whether what I was having was caused by or related to the radiation treatment many years before, I have no idea.

Eventually through strong persuasion by good friends that left me no excuses, I tried an unconventional approach with a chiropractor who had a simple test to measure stomach acid.  This was in 2024.  He determined that my stomach acid was very low and prescribed Betaine HCL pills.  These were supplements that provided additional stomach acid when the stomach acid I made naturally fell short.  

I started out taking as few of these pills as possible because I was scared to death of them.  I thought they'd eventually eat a hole through my stomach and kill me.  I suppose that's still a possibility, but now I usually take 15 of them a day (5 per meal), which is the maximum amount the chiropractor recommended, and think nothing of it.  And they certainly make me feel a lot better.  I still bloat up here and there when I eat a little too much, but it's not CONSTANT like it was before I took the medicine.  

So even though we see a seemingly endless amount of antacid commercials for heartburn on TV, I guess some people have the opposite problem and need MORE stomach acid.  I must be one of the few that fall into that category.  

Atlanta Nights

Atlanta Nights is one of the worst books ever.  A bunch of science fiction writers got together to write the worst book they could and trick a self-publishing company into accepting it, and their plan actually worked.  

I may be the only person in the history of the world to read this book from beginning to end TWICE.

Even the people that WROTE the book probably didn't fully read it through once.  They wrote it with each writer taking a separate chapter I believe.

I reviewed it on Amazon many years ago.  But I don't think my review stressed how filthy the book was.  I mentioned it, but didn't stress it enough.  I remember a scene that was quite disturbing involving body parts stored in a room.  Private body parts too.  The book really went out of its way to be tasteless.  

But even with gross stuff like that mixed in here and there it was still quite hilarious for the most part, even during some tasteless scenes.  In one scene, a lady shoots an intruder in her house and then genuinely tries seducing him during his massive blood loss, recognizing him as a former lover after firing her gun.  Pitiful.  

The typos were by far my favorite part.  One guy in one chapter kept rubbing his beard and pushing back his hair.  But "back" was spelled "backs."  So it was a typo.  Only this was written like six times in the chapter.  With the SAME typo every single time.  Also there was one part in that same chapter where someone was told to back off.  But the same typo was present there as well, so it read, "Backs off, you!"  Intentionally hilarious.

Why 26 is a Special Number

I found out a while back that the number 26 is the ONLY number out of all the natural numbers that is immediately after a perfect square and immediately before a perfect cube.  Indeed, 25 equals 5 squared, and 27 equals 3 cubed.  26 is sandwiched perfectly in-between these.  But how do we know there are no other instances of this anywhere?  If we list out ALL perfect squares and ALL perfect cubes, how do we know that no other number will be exactly 1 before and 1 after anything in the two lists?  That's a whole heck of a lot of numbers to test.  Especially since neither list ever ends.

Our first strategy is to write the equation x^2 + 1 = y^3 - 1 and simplify it to x^2 + 2 = y^3.  Okay, that part seems simple.  But where the heck do we go from here?  I was stuck.  So I went to the math stack exchange website, because it had the question along with several answers.  A link is right here:

https://math.stackexchange.com/questions/874226/y3-x2-2-rightarrow-y-3-27-is-the-only-cube-exceeding-a-square-by

But I couldn't really understand the answers I saw for a long time.  Then after pondering over it for a long while and asking AI questions and stuff, I finally got the main idea.  I tried to post that main idea I had as an answer on the math stack exchange site to supplement the other answers, but it was not well received and was promptly deleted.  Probably because the audience for the site is not comfortable working in imprecise layman terms.  So I'm reposting what was deleted here:

"This is a very interesting question to me, and the path to its answer is even more interesting. Since my formal graduate level education is non-existent in this subject and I'm not comfortable at all with a lot of mathematical jargon, I had to think long and hard to understand the solution, even AFTER looking at the other answers here.

But the simplest way I can communicate the main idea behind the answer to the question is this: while it would be WONDERFUL to get this answer without leaving the domain of the integers, it's pretty darn tough to do. So what you do is you examine a domain that contains all the integers, but is actually LARGER than the integers and you get the solutions there. If you only have one solution in that larger domain, then you're only going to have one or less solutions in the integer domain. That's the main idea.

 

So what you do is you take this equation and look at its solutions in the domain that contains all integers AND all integers with scalar multiples of the square root of negative two added to them.

 

Numbers in this larger domain are of the form a+b(root -2) so rewrite the equation x squared plus 1 equals y cubed minus 1 where both x and y take on this form and then work with the algebra and see what happens. This is pretty much what Adam Hughes already did in his answer - better than I could ever do - but I know I had a hard time even comprehending his answer without understanding the main idea about the larger domains first.

 

For the equation you pose in your question in this a+b(root -2) domain, there's only one answer, and it is an integer, and so there's only one answer in the smaller domain of the integers.

 

This post may seem redundant to those comfortable with the material, but for novices I think it's a vital point to slow down on and emphasize in order to fully comprehend. And to put in plain English with little jargon if possible."

So with that main idea in place we factor the x^2 + 2 in the equation as (x+root(-2))*(x - root(-2)), and we can work in the larger domain.  So x^2 + 2 = y^3  becomes (x+root(-2))*(x - root(-2)) = y^3.

Then we can write the y in the a+b(root(-2)) form and take the cube of that form to see what happens. (a+b(root(-2)))^3 is straight-forward to expand but a lot of effort.  Thankfully the Adam Hughes answer in the site already does that part.  He has a good answer but for people who don't know what norms or integer rings are, they're going to be lost on how to even begin.  I certainly was.

Uncertainty Principles and the Failed Quest for the Worst Movie Ever

When I first started watching Mystery Science Theater 3000 on the Sci-Fi Channel back in 1997, after several black and white movies and a few color ones, all of a sudden episode featuring the film called "The Incredibly Strange Creatures Who Stopped Living and Became Mixed-Up Zombies" sprang up and caught me by surprise.  Sure, the movies before this one were quite laughable, but this one clearly stood out.  I instantly thought it was the worst film I had ever seen and that I ever would see.  But then a few months later "The Horror of Party Beach" shows up.  I still thought Incredibly Strange Creatures was worse, but Party Beach was a major contender.

Over the years I was exposed to more MST3K movies, particularly from earlier seasons, and man there were some bad ones.  I kept looking for one that was clearly the worst.  And titles like "The Wild, Wild World of Batwoman" were pretty much on par with Incredibly Strange Creatures.  Then years later Rifftrax comes along and boy, they really found out how to scrape the very bottom of the barrel.  Santa and the Ice Cream Bunny might have actually been worse than Incredibly Strange Creatures.  And of course later we start seeing titles like Rollergator and Birdemic, and more recently stuff like Suburban Sasquatch and The Amazing Bulk.  This stuff is utter trash.  So much of it could really claim to be tied for first place as the worst movie ever.  But the one that made me truly give up my quest for the worst movie ever in earnest was Fun in Balloonland.

Does that mean Fun in Balloonland is the worst movie ever?  Not really.  My exposure to it simply establishes that finding a movie that's definitively the worst ever is too vague of a question.  IS Fun in Balloonland even a movie?  It's slightly shorter than a full length feature, the credits seem to be limited or non-existent, and half of the movie is clearly just parade footage with no story whatsoever.  Fun in Balloonland is borderline a home video.  So can a home video even qualify as the worst movie ever?  I mean, it's not a real movie in that case.  The worst movie ever is not a movie at all.  So definitions start to break down.  It's like the uncertainty principle in physics.  You try to know momentum and you can't know position, and you try to know position and you can't know momentum.  At least that's how my untrained mind understands the principle.  

And with Balloonland, it's like you try to measure its quality and you sacrifice measuring how close it is to an actual movie, and you try to measure how close it is to an actual movie and you sacrifice how you can measure its quality.  Because you'd judge the quality of home movies differently than you'd judge the quality of regular movies.  At least I'd think you would.

For years I tried to put this uncertainty principle for bad movies into more mathematical terms, proving that attempting one type of measurement always disrupts the other, but have failed in my attempts to do so.  So I guess I've given up proving that there can be no worst movie ever, just like I gave up actively looking for the worst movie ever.  This is what Fun in Balloonland did to me.

Strange Bean Novels From 20 Years Ago

When I first read Alice's Adventures in Wonderland I absolutely loved it.  It was wonderfully goofy and weird, and I thought it would be fun to try to write something like that myself.  So I did.  I wrote the weirdest, goofiest thing I could think of.  Long, long ago.  But I didn't actually post it online until fairly recently.  I guess I delayed it because I thought I'd eventually try to clean it up more and make it better and try to market it again.  But I got too busy with other things and decided I probably wasn't going to work on it anymore and to just leave it as it is.  Especially since the likelihood of publication was slim to none, leaning far more towards none.  

But I had a lot of fun writing it and a lot of fun re-reading it even though it has its flaws.  Hopefully I uploaded it with no sections missing and all the sections in order.

I called this unpublished book Boomo the Bean Visits Confusion Country, and a link to it is here:

https://confusioncountry.blogspot.com/

But after I wrote that one I also wrote a crazy sequel, Boomo the Bean Sees Through the Hourglass, and the link to it is right here:

https://boomohourglass.blogspot.com/

Also, if you click on my blog profile, you'll find the links there as well.  I put illustrations in the books too and that was a big part of the fun.

Causes, Effects, and Natural Selection

Science is about causes happening and effects stemming from those causes.  Goal setting is about changing causes to reach desired effects.  In goal setting you are not observing effects as in science.  You are dictating them.  

Natural selection appears to be a goal to sustain life.  Strict cause and effect should have no goals.  I am not arguing that natural selection is false here.  But I certainly don't mind asserting that nature has intended goals.  There's a term already in existence for this assertion - teleology.  And there's also a term for its counterargument - teleonomy.  

When I asked Copilot AI about teleology, it suggested an argument based in teleonomy that variations in species occur randomly, and environments filter out the variations so that only the successful variations survive.  Sure seems to be more of a deterministic argument than a free will argument.  

But I believe that the striving we experience in life goes far beyond unintentional programming.  Can't prove it of course.  But I can assert it vehemently.    

Obligation vs. Preference

It should seem clear that obligation is not determined by preference.  Certainly not at the individual level because it can potentially erase accountability.  But then we have to think, is obligation determined by collective preference?  I would imagine not.  Otherwise books like Extraordinary Popular Delusions and the Madness of Crowds never would have been written, and peer pressure wouldn't have a negative connotation.  

Why Does EVERYONE Say Their Parents Were Bad?

I don't know if I visit the most representative locations on the internet as far as statistical samples go by frequenting Yahoo news articles and political Youtube videos and such, but I swear, in almost every forum or comment section, for every one person I see complimenting their parents and calling them a blessing, I see about 20 or 30 talking about how toxic their parents were.  And we're not talking about teenagers that are still in the middle of maturing and dealing with discipline issues.  These are people in their 30's and 40's who've had ample time to look back.  

My question is, why the dissatisfaction?  Are there really THAT many bad parents out there abusing and neglecting their kids?  Is there just too much expectation regarding parental roles where they have to be absolutely perfect to even be considered decent?  I mean, WHY do so many people have problems with their parents?  I had awesome parents, so it boggles my mind that so many people apparently did not.  

I can't stress enough that I have NO problems with my parents and what they did in my upbringing whatsoever.  They were good people that tried their very best and did WAY more than they ever should have for me.  Were they flawed human beings?  Of course.  Who isn't.  But once I grew up, and possibly even before then, I could certainly tell that they clearly had my best interests at heart and put in SO much effort and sacrifice for my own wellbeing - particularly my mom.  

This realization is not dampered when I look back and think how scared I was when my mom yelled at me or how mad I was at my dad for using the belt when I got really out of line.  They did these things because I needed correction.  Not because they were on some power trip or thought it was fun.  Because they darn sure didn't seem to have fun correcting me.  Not to mention they bought me more toys than I could even count over those 17 or so years.  And they didn't cheat on each other or spend all night at the bar.  

Are parents like mine really so rare?  If so, I guess I can see why the divorce rate is so high.  But it still boggles my mind that so many parents would be problematic enough to not even be considered decent, much less good.  

Even if I classified mine as phenomenal, that could still potentially be an understatement.  The fact is they deserve more love and respect than I could ever give them.  Thank you mom and dad.  Regardless of how your peers behaved, you were shining examples of a traditional family structure.

Exploring Oppression's Limits... On the Weak End

I wonder, how oppressed would Cinderella consider herself to be if her stepmother and stepsisters still held household authority over her, but forced her to do little or no chores, and she spent most of her time playing on a computer all day.  I imagine if they continued to verbally torment her, that would still be oppressive (or at least abusive).  But I'd think it would take more to claim she was genuinely oppressed besides her not be the one in charge making the decisions.

I think to be a true oppressor, one has to exercise enough authority that they transfer most of the shared workload to the one they're oppressing, and not carry it themselves.  But I'm not sure how much distinction we make between terms like jerk, bully, oppressor, and abuser on a daily basis, and that may factor into this thought exercise here.  Those terms probably each emphasize slightly different negative characteristics.

Mr. Friend

My son's favorite stuffed animal growing up was a frog he called Mr. Friend.  It was hard not to like Mr. Friend.

Jet Jaguar Can Deadlift

 

AI Caught Making an Error - Oops!

When I ask AI questions usually, either it doesn't make any error, or it makes errors I don't notice.  But this time it definitely made a noticeable error.  I asked it to verify the number 1984 in binary, and I believe 11111000000 is correct, but that translates only to 1024+512+256+128+64.  There should be NO 32 in the sum as AI stated here.  

The 11111000000 stands for 1*2^10 + 1*2^9 + 1*2^8 + 1*2^7 + 1*2^6 + 0*2^5 + 0*2^4 + 0*2^3 + 0*2^2 + 0*2^1 + 0*2^0

Hope I typed all that out right and didn't make an error myself.

Have it Now is the New Way to See

Years ago I wrote a satirical variant of Billy Joel's "It's Still Rock and Roll to Me."  Not an improvement, and probably hasn't aged well.  But I don't think I published it anywhere, so might as well put it up now:

“What’s the matter with the song I’m playing

Will you tell me that it’s not the time?

Maybe I should sing of fame and fortune

And pretend that everything is fine

Nowadays everybody wants something sunny

Guarantees of fancy cars, and piles and piles of money

Everybody’s pushing all this new style on me

‘Have it now’ is the new way to see

 

What’s the matter with the way they’re thinking?

It just seems like a twist of time

Back in old days things weren’t as easy

But there wasn’t nearly so much whine

Nowadays people go dancing all around town

They have tons of fun while their love of working goes down

Good days, good pay, good times, all the way

‘Have it now’ is the new way to see

  

Oh, we get up every morning and we see in the paper

More bad news that we’ve already seen

Scientists working oh so hard everyday

To find the reason for every bad thing

Try to keep it all running

 

How about signing up with big business,

Where you use all of your great wit?

We can say for sure that you’ll love it

And you’ll never want to call it quits

You know if you say ‘no’ that you’ll get weaker

But saying ‘yes’ will make you a great and rich speaker

Oh man, what a plan, get it now, while you can

‘Have it now’ is the new way to see

 

Best of all it’s not considered cheating

And there can never, ever be too much

Once you get your foot in the door and make money

 You’ll never, ever lose your touch

Good days, good pay, good times, all the way

‘Have it now’ is the new way to see

 

Everybody’s pushing all this new style on me

‘Have it now’ is the new way to see”

 

My Best Man

One of my favorite people ever was the guy who was good enough to be best man at my wedding.  He was a handsome, fun, and talented fellow.  Studying mechanical engineering and could bench press over 400 pounds.  This boosted his ego to the point that some people found him somewhat insufferable despite the high amount of charisma that he also possessed.  But I loved it because he reminded me of all of the heels I watched on professional wrestling.  So I did my best to actually feed his ego even MORE just to see what happened.  It was a lot of fun, and he seemed to eat it up.  It sure was an easy way to become his friend.

But at the end of the day, he was smart and sensible enough to realize his limitations (even though he'd never admit them out loud) and never let his ego get TOO out of check as to be unhealthy in a way he couldn't handle.  His life never became unstable, and he ended up working at Lockheed Martin for many years.  I hope he's still doing well.  Because even though it seems like I just fed his ego as some kind of joke, I really did like the guy.  He was my best man for a reason.

The Desire For a Vision

Some people really seem to want to have a spiritual vision at some point in their lives.  Either to confirm their faith, or just because it sounds like it would be something really interesting to experience.  But I'd rather not experience such a vision myself.  In Biblical literature, God never has a vision for somebody unless he also has a special mission for that person.  Usually a difficult one.  That alone kind of puts a damper on my desire for any kind of spiritual vision.  I think I'm fine without one.  Especially since I quite enjoy that blurry line between fantasy and reality.  I wouldn't be a pro-wrestling fan if I didn't, or a fan of imaginary numbers and the interesting properties of holomorphic functions.

Perhaps an unconfirmed faith is sometimes best.  It's not like we'll be able to confirm everything we wish for in our lifetime anyway.  I'm in my mid-forties and I know for sure there are a lot of things I will never know before I pass away.

Maybe I had mysterious but meaningless visions before without ever knowing they were visions.  I believe I briefly mentioned in a prior post before something about an older woman in one of my community college art classes back in 2001, and her husband was a holocaust survivor that challenged me at arm wrestling one time.  As far as I know this was reality.  But can I verify it or can anyone else verify it either?  Probably not.  So someone can claim it was a vision and how can I prove them wrong?

Graphing Congruences in Modular Arithmetic And Turning a Cartesian Graph Into a Polar One

Even though this isn't anywhere close to Fields Medal level stuff, and it's not very deep, I think it's still pretty cool.  I haven't seen it before, but I'm pretty sure someone somewhere already did something like this.

What if we want to transform a graph in cartesian coordinates into one that looks like polar coordinates, where the x-values circle around some center point, and the y-values measure a distance from that center?  X would be comparable to Theta and Y would be comparable to R.  So we'd do a transformation as follows:

Theta = X times (( 2 times pi ) over a modulus)

R = Y

This modulus would cut the distance we have to travel around the center point into equal pieces.  We'll use the modulus 7 for this demonstration.

So if I wanted to take the arbitrary function Y=X^7 (this is an unrelated 7 - sorry I'm using the same number twice) and graph it in this system, I could do so, but first I'd have to do some easy algebra and express X in terms of Theta instead of Theta in terms of X.  Just divide both sides by (( 2 times pi ) over the modulus 7).

We end up getting R = ((7 times Theta) over (2 times pi)) raised to the power of 7.  That is something we can graph in Desmos.

Now, if I cut the distance around the origin into 7 parts, it would just be the angle Theta = (( 2 times pi ) over 7).  For some reason Demos really doesn't like me graphing that and won't let me do it, so I use the graph Y = tan(( 2 times pi ) over 7) times x, since that's the same thing.

So this line shows one-seventh of the distance around the origin.  But only the upper right part of the line stemming from the origin.  The lower left part is just a continuation of it, and does not represent another portion of 7 parts around the line as far as I know.

Modular arithmetic would tell us that once our Y=X^7 equation crosses this line, that equation is congruent to 1 (mod 7) at the R value where it crosses.  When the equation crosses the right side of the x-axis, that's when it's congruent to 0 (mod 7).  We could do this with other values as well if we were willing to draw more lines (and I'm not).

Anyway, we can test it out.  Here in this picture, we see that our X to the 7 function, the red curve, crosses the 1 (mod 7) line, the green line, right where that blue circle is.  The blue circle is just R = 8^7.  A pretty big number.  Over 2 million.  Is it congruent to 1 (mod 7)?  Indeed it is.  It's equal to 299,593 times 7 plus 1.  

Let's try another one.  This time we see the graph crosses the line at another purple circle, much larger than the last blue circle.  In fact, the blue circle at R = 8^7 looks almost like a small dot here.  This purple circle is R = 15^7.  Is this also congruent to 1 (mod 7)?  Sure is.  It's equal to 24,408,482 times 7 plus 1.  Nice to see computers confirming what the math shows.  And it's fun to graph this thing and zoom in and out on it.  Reminds me of the old Spirograph toy from a long time ago.

And that's all I've got to say about modular arithmetic here.

Now I want to do a conversion of a cartesian graph into a polar one using this same transformation, just because I think it looks interesting.

Let's do a nice and easy graph of a system of linear equations and note their intersection point.  Here the diagonal green and blue lines intersect at the red x=1 line and the orange y=7 line.

Applying our transformation to this system, we see the red x=1 line is now diagonal at the origin, the orange y=7 line is now a circle, and the two diagonal lines that cross each other there have become some kind of crazy spirals.  Very complicated looking.

So I just took something nice and easy and made it way more complicated for no reason.  I just think it's something kind of worth looking at, at least for me.

Proverbs 21:30

Proverbs 21:30 states "There is no wisdom, no insight, no plan that can succeed against the Lord." 

This verse is really about the futility of going against divine authority, but for some reason it reminds me of a common implication that if there is no final authority, then anything is permitted.  This is a common argument in apologetics I believe.

And vacuous truths, which I mentioned not too long ago, look somewhat similar.  If the premise is false, then any conclusion is permitted.

So if there isn't a God, apologists would say anything goes using their usual arguments.

But also if there is one, and we declare there to not be, again we'd reach the conclusion that anything goes, using a vacuous truth argument.  (Or if Proverbs 21:30 turned out to be true but we believed we had insight that was against the Lord.  Then any conclusion would go there too.)  

So in declaring God as a falsehood, whether accurate in actuality or not, apologists are going to reach the idea that anything goes.  Either using a standard argument or a vacuous truth one.

Certainly not a brilliant observation or anything, but a questionable and minor observation easy to put in a personal journal for a day. 

The Black Sun

I was fortunate enough to witness the April 8, 2024 solar eclipse in person.  I had to travel a bit to reach the path of totality, but it wasn't terribly far.

The experience was a bit different than I pictured it would be.  Instead of it feeling like nighttime in the middle of the day, it was more like a sunset in the middle of the day, where it didn't get near as dark as night, but still much darker than it usually is during the day.  It was enough to hear the sound of roosters crowing in the distance.  

But it was odd to look up at the sun and see it completely black in color.  I took a picture with my phone, but I guess since I didn't have any special photography equipment, it didn't capture the moment correctly at all, and the sun looks pretty much as bright as normal in the picture.  

Still, as short as the experience was, and as disappointing as my documentation of it was, the memory of the sun completely black in the sky with a glowing ring around it sticks with me.  Definitely a once in a lifetime experience.  I'll never see it again.

Why There's No Quintic Formula - The Easiest Explanation I Have So Far...

Here's currently the best explanation I have of why there is no general formula in radicals for the roots of a quintic polynomial. It's more for my quick reference than anything else. It may not be totally accurate but maybe it's close.

Vieta's formulas put the roots of a general polynomial equation with unspecified variable coefficients in a position to be swappable, but potentially roots nested inside other roots can pop up, and they can't be easily swapped if they're not conjugates. If a general quintic formula in radicals existed, it would violate this conjugate swapping structure.

I think the biggest pitfall I've gotten caught in is not realizing that the only requirement for SPECIFIC examples where we already know the coefficients is that they respect the nested conjugate rule and don't violate it. Their roots don't have to be all swappable with each other. But when we don't KNOW the coefficients and are dealing with a general variable-coefficient case, we have to allow all roots to be swappable because a specific polynomial may potentially have that property.

These ideas build on a video I uploaded a while back here: https://www.youtube.com/watch?v=qOHkF26EKfg

A HUGE Difference Between Worldly Thought and The Teachings of Jesus

It seems common in our culture today to get as close as you can to the line without crossing it.  Get as drunk as possible at parties, as long as you don't become an alcoholic.  Date as young of a lady as possible as long as she's over 18.  Cuss and get in someone's face as much as possible, as long as you don't pull out a gun.  Stuff like that.  

This is very much the opposite of the teachings of Jesus.  His teaching advocates taking extreme precautions against even the potential to sin.  That's probably why he implied that it was worth sacrificing even a limb like an eye or a hand if it meant keeping you far away from the kingdom where permanent sin dwells.

Even from a secular standpoint, Jesus's teachings make a lot of sense.  Just for precautionary reasons if nothing else.  Why put yourself unnecessarily in a situation where you have so little room for error?  Can someone call themselves anything other than foolish if they accidentally step over a line they spent so much time treading?  

Weak Argument Against Vacuous Truths

Vacuous truths never felt intuitive to me, ever since I first learned about them.

Here's one reason why.  It's a weak reason and not a strong reason.  But it's strong enough in my potentially deranged mind for me to feel I should mention.  Forgive me if there are any typos, because they are pretty easy to make here.

Look at the statement NOT(if A then B) and the statement A and NOT B.  They both have the EXACT same truth table, if you draw the truth table out or ask a competent AI program to do it for you.

But I would argue A and NOT B is actually a stronger statement than NOT(if A then B), making the statements not equivalent.  A and NOT B is clearly not a conditional statement.  A happens no matter what.  Does A also happen no matter what in the statement NOT(if A then B)?

Suppose A doesn't happen and B does.  Is this an instance of NOT(if A then B)?  What if A doesn't happen and B also doesn't happen?  Also an instance of NOT(if A then B) perhaps?  I would say anything whatsoever that is anything other than if A then B fits the criteria of NOT(if A then B).  

Let's try an example to flesh out this line of thought.

Think of the actions of A and B being the following:

A = the action that you honked your car horn yesterday

B = the action that your car windshield broke.

There are four possibilities involving your horn and windshield yesterday:

1) your horn honked and your windshield broke

2) your horn honked and it didn't break

3) your horn did not honk and your windshield broke

4) your horn did not honk and your windshield didn't break.

Which of these 4 possibilities fit the 2 statements NOT(if A then B) and A and NOT B?  

A and NOT B would say that you definitely honked your horn yesterday and your windshield did not break.  Only possibility #2 fits that.

NOT(if A then B) would say it's not true that if you honked your horn yesterday, then your windshield broke.

If you did honk your horn, then only possibility #2 would fit that.  If you didn't honk your horn, then we don't know what would have happened for sure.  But mostly likely honking your horn wouldn't actively have caused your windshield to break.  This is NOT A and NOT B. Possibility #4.

Is possibility #4 a subset of the statement "it's not true that if you honked your horn yesterday, then your windshield broke"?  I would say yes, based on probabilistic assumptions alone. The only way this would have been true is if some mischievous individual installed some device in your car on purpose that would smash your windshield every time you honked your horn.  At least if you interpret the conditional if/then as definite cause and effect and not just random occurrence of each.  

We could alter our truth table for this specific example, but if we wanted to make a general table, we'd probably have to set up a U for unknown on vacuous truths.  Or maybe M for meaningless.

Still, there are pros and cons to using U for vacuous truths. They're more conservative, but they leave the logical system open instead of closed.  Plus they introduce a ternary or three-valued logic system (which some logicians already have invented) and they're a bit more complicated than logics with truth values only limited to T or F.

So even though I don't think vacuous truths always adhere to reality despite being pushed as such, they do allow closure in a fairly simple logical system, and there are definite benefits to that.  Not worth changing, but probably worth questioning more than they're currently questioned in introductory logic classes.

The American Dream

One of the most popular wrestlers ever was Dusty Rhodes, who symbolized the American Dream.  The American Dream was his nickname as well.  

Most people equate the American Dream with owning a home, but it can pretty much be anything.  My dad was quite impoverished growing up, and when his mom said she didn't have any money to buy him a Coca-Cola Classic, he said when he grew up he was going to get a refrigerator stocked with plenty of Coca-Colas so he could have one whenever he wanted.  With this determined mindset, he went to college in Louisiana where he struggled to pass calculus, taking it three or four times before he finally got the credit he needed, and ultimately became a civil engineer.  Today he has a refrigerator and I've never seen that refrigerator without a Coca-Cola inside.  The man lived his dream.

When my wife was growing up she always wanted a super-fluffy cat.  But this proved a greater challenge than she thought initially.  She wasn't able to get one right away, and then had a sweet cat she dearly loved that lived to be at least 20 years old.  This cat wasn't fluffy, but he was her best friend and kept her very happy.  When he passed away, it broke her heart.  Then she finally decided it was time.  She ended up getting the super-fluffy cat she always dreamed of, even though it took a few decades.  She still misses the cat that was her best friend, but she loves her super-fluffy cat too.  

And when I was in high school, I loved wrestling even more than I do now.  I watched old stuff from the video store, and watched either WCW or WWF every week for a good four years or so.  Always looked forward to the Monday night Nitro or RAW that followed the monthly Sunday pay per view in particular to find out what happened at the big show.  I did purchase a few pay per views, but they were at least $60 a show, so I probably bought fewer than I could count on one hand.  I bought a few VHS tapes as well, but they were $30 each, and that's a lot of money to a kid with no job.  Couldn't buy very many of them.

I said one day I'd watch all those pay per views I couldn't watch growing up.  Then I found out about the WWE network, I guess not too long after it came out.  Every pay per view I ever wanted to watch was on that network, for a reasonable monthly fee.  But even though I had a degree and a job at that time, I was married and still didn't have quite enough money to buy the subscription.  Or the time to watch it for that matter.  I said when I finish grad school I'll get the WWE network.

Well, eventually the network was acquired by Peacock.  So I said I'd get Peacock when I finished grad school, and I didn't have but a few classes left at that point.  But then my wife tells me she's getting Xfinity for our TV service, just because it came with our internet.  And Peacock was included for free.  So just a few months before finishing grad school, I finally got to watch all the pay per views I wanted to watch.  In particular I wanted to see the Shawn Michaels and Undertaker cell match from Bad Blood 1997.  And I finally got to see it after 23 years of waiting.  I got to live my dream.

Now a lot of those matches are on the WWE Vault and WCW YouTube channels for free, and Netflix has the WWE pay per view library also.  But the main thing is I got to live my dream.  Just like my dad got to live his, and my wife got to live hers.  Even though it took each of us a while to get what we had our hearts set on.  We lived the American Dream.

Hulk Hogan Sign from WrestleManias 6 and 18

This is kind of amazing to me.  In April of 1990, WrestleMania 6 took place in the Skydome in Toronto, Canada with Hulk Hogan vs the Ultimate Warrior in the main event.  And you could see a big banner of Hogan clearly in the audience as the Ultimate Warrior made his entrance.  

https://www.youtube.com/watch?v=IRtOTIykZNE

A full 12 years later, WrestleMania took place in the Skydome again, with Hulk Hogan going against the Rock in a highly anticipated match at WrestleMania 18.  And either the same exact banner or a REMARKABLY similar one could be seen in the audience as the Rock and Hogan stared each other down.  The only difference is it's in a different seat this time.  But pretty much the same sign, in the same building, with the same wrestler in the match.  Twelve years later.

https://www.youtube.com/watch?v=eJ9zibElS5w

Sparing the Rod and Spoiling the Child

This old phrase is often attributed to the book of Proverbs.  It has fallen out of fashion in recent years due to the increasingly negative view of corporal punishment in the western world.  But if you generalize it beyond the meaning of corporal punishment, there's something interesting to note.

It really implies that the wise party should never yield control to the unwise.  And not even because of the adverse effects on the wise, but because such yielding does NO favors for the unwise either.  Power in the hands of the foolish does not aid the foolish in any way that matters.