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.  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.  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.