New National Anthem?

If we ever get a chance to select a new national anthem, I'd vote for Hulk Hogan's Real American entrance music when he was in the WWF.  I listened to it countless times over the years watching old wrestling tapes and YouTube videos, and to this day I still feel a sense of pride in the country when I hear it.

Desire Idea

I had a thought this morning:

Perhaps the people that desire capitalism the most are the ones that deserve it the least, and perhaps the people that desire socialism the most are the ones that deserve it the least.

And I wouldn't be surprised if this desire idea could be generalized to other things as well.

Personally, I'd like to believe that I could accomplish whatever goals I have under either political system.

Serving in Heaven vs. Ruling in Hell

Paradise Lost brings up this debate.  Some would rather rule over chaos than serve under order, and others the opposite.  And for a while I wondered which I preferred.  Then I thought of a scenario that puts this into perspective.  

When I used to compete in lifting meets, I definitely preferred others to be weaker than me.  I'd rather be 1st place with a low combined weight total than 6th place with a much higher total.  Didn't matter if I squatted 315 pounds in the 181 pound weight class, as long as no one else exceeded me. Squatting 700 pounds and being 3rd place just wasn't as good of a prize in my eyes.

And I thought the same initially with regards to intellect.  If I was the smartest in the room, it didn't matter if everyone else WAY below me or just slightly below me.  As long as I was at the top and recognized as the best.

But years later my perspective has changed a bit.  I realized if I was the smartest person ever on earth, the quality of our daily appliances we take for granted would probably be dramatically lower.  Is it worth a lower quality of life for everyone else just to satisfy my own ego?  Can I handle having a lower ranking on the totem pole of recognition if it means a better life for everybody in general?

And it made me realize that it probably is better for me to accept a lower status of governance than to possess total authority over things of inferior quality.  Letting go of a little bit of control can make things go a long way.  And ruling over a bunch of idiots doesn't sound like such a good prize anymore.  Better for us to try to build up those underneath us and accept the benefits of them overtaking us.  

But of course, if those underneath don't want to build themselves up, they are probably better off learning to accept their place.

I Planned a Return at Age Forty - Didn't Happen

I started another blog a few months back called Zen in the Art of Being a Jerk (https://zenintheartofbeingajerk.blogspot.com/), where I talk about how much I like heel wrestling characters.  One of the typical heel characters is often referred to as the closet champion.  This is the champion that refuses to face all challengers, and ducks as many as he can in order to try to artificially increase the length of his often undeserved title reign.

In that spirit, when I was a powerlifter in college, I legitimately feared competing in the very difficult open division.  I was able to compete in the teen division, the junior division, and the collegiate division, and it was much easier to place in those specialized divisions than it was in the open division.  But after college, the number of specialized divisions is slim pickings, and lifters are pretty much forced to compete in the open division.  It isn't until age 40 that you can compete in a masters division for older lifters.  So before I graduated, I joked to my college buddies saying I would avoid competing after college and wait until I'm 40 years old to compete again.

Now at age 44, I'm just too beat up to be anything near competitive.  My digestive issues are the main thing, but my hip flexors bother me on squats too, and last time I tried heavy bicep curls my hand was numb for darn near a month.  I simply can't push myself hard when training anymore without risking injury.  Sure, I could still sign up and lift super low weights in a competition, but that's just no fun.  I guess some guys can keep going strong at that age, but not me.  My back injury in my 20's sealed the deal, and once that finally cleared up a few years later, all my muscle was gone, I had just started a family, and I simply didn't have enough incentive to return.  Then when those digestive issues set in after I hit my late 30's, it was very clear that my competition days were over, never to return.

3rd Grade News

I always like how Trump called the news the Fake News.  Because in some ways he's kind of correct.  I understand not everyone in the population has the mental capacity to read very much (or the free time or interest for that matter), but the choice of articles, and the choice of headlines in particular, are often quite abysmal in order to generate clicks.  They really do seem to be intended, not always to be false as Trump implies, but always to be FAR over-simplified to little more than a catchy slogan that leaves a whole lot of relevant facts out of the account.  Both the left wing and right wing media are very guilty of this.  Probably not EVERY piece of media has this fault, like local newspapers for instance, but online stuff like Yahoo news and the articles it links to often fit this description to a T.  Fox news probably as well, although I haven't paid attention to them in a while.  But I remember in the past they'd take small sound bytes from speeches, ignore the rest, and run with the sound bytes as if those alone said as much as a dissertation.  That is not good journalism.

So I think I'd rather call it 3rd Grade News than Fake News, because it simplifies things too much, where maybe some of the facts are accurate, but they leave out a lot of relevant information to paint a false impression.  I think Lisa Simpson said it best when she said, "As usual, the playground gets the facts right, but misses the point entirely."

I guess the news does what it can to generate revenue and keep afloat.  I know I sure as heck don't like to think deeply about politics much in my time away from work, and I don't blame anyone else for not wanting to either.

I Usually Vote Republican, But Understand Why Many People Don't

I believe in the last general election I voted in, I chose one Democrat, and everyone else I chose on the ticket was Republican.  

As much as I dislike progressivism, I can't deny that there are some things moderate and even staunch Democrats stand for that have genuine merit.  Showing concern for the environment and making sensible efforts to control pollution is certainly not a bad idea.  Showing compassion for refugees attempting to escape from an oppressive government to start a new, honest life looks like the right thing to do as well.  Increasing the tax burden on those who have benefitted greatly from living here and can easily afford the increased burden is difficult to argue against too.  So I certainly understand why people often vote against Republican stances and don't blame them for doing so.  Particularly when economic conditions for the general population are not doing so well under the Republican watch and when a rude fellow with a questionable record and a lot of inherited money is put in charge of the land.

In fact, most of the time I vote, it isn't because I love the Republican stances at all.  I just can't stand the progressive stances.  The ones that look for fault in every darn thing and refuse to be satisfied.  They rarely give those that they are against a clean slate and a chance to start over.  In their eyes, America is forever a land of bigots and will always be racist, sexist, xenophobic, and whatever accusation you want to throw against it - because it was without a doubt a major part of the Atlantic slave trade hundreds of years ago - long before its birth and all the way until the mid 1800's.  

Now, SHOULD progressives hate America forever because of this glaringly awful part of its history?  Don't they have a point?  Maybe so.  But man, if you're going to hate a country, or anything else, that much - you know, FOREVER, no matter what it does in the future - then just don't even be a part of it.  Go to Europe (even though it doesn't have anywhere close to a clean slate as far as its history of human rights goes), or start your own country.  It makes no sense to hate something so much that it can't ever redeem itself, yet to still stay associated with it.  

So until the forever unsatisfied progressives exit the picture, it will be difficult for me to consider lending my support to anyone - even a moderate - who fails to strongly differentiate himself from such a negative and hopeless group that prides itself on unachievable yearnings.  Even if you give this progressive group everything it wants, that's still not good enough because it cannot erase the past.  And the best way to deal with anyone who can't be satisfied with anything you ever do going forward is to simply ignore them.  Why waste effort to appease those who can never be appeased?  

But American progressives do not have to stay in this forever-unsatisfied purgatory.  They can climb out if they so wish without sacrificing much, if any, of their push for improved human conditions.  I think they just need to realize that a big part of changing the future is to accept that the past cannot be changed and to go from there.

That said, do I blame people for picking permanent America-haters over convicted felon Trump?  Do I blame people for picking convicted felon Trump over permanent America-haters?  No.  I don't blame either one.  Why?  Because some decisions are tough and prioritizing can be difficult.  I'm not even proud of my own decision.  But I think I'd be even less proud if I chose otherwise.

Trump is really doing the exact same thing I would do if I was in his position regarding the progressive extremists.  He's completely ignoring them, and even going actively against them.  I have never seen a politician before him have enough brash guts to do that.  So that's one thing about the man I can admire and support, even if his personal life isn't much different than Bill Clinton's.  It really does get exhausting seeing moderate politicians trying to appease the unappeasable, which I witnessed for many years.  FAR too long.

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.