Tuesday, September 13, 2016

My mathematical thoughts on Institute today

I always try to read my scriptures in both English and Spanish, because it gives me two different perspectives on the same idea.

However, today, being fluent in math, gave me some additional insights into my institute class today.  Today we were reading Moses 1: 24-42, which is basically God's response to Moses's 2 questions: "[W]hy are these things so[?]"  and "[B]y what thou madest them?" (See Moses 1:24)  These are some pretty deep questions, and something that I think we should all spend some time pondering.  (God often provides deep meaningful insight we when ask the right questions.)

God responds by telling Moses that he has made "worlds without number" and that "innumerable are they unto man; but all things are numbered unto me, for they are mine and I know them."  (See Moses 1:33,35)  This is probably because we just had the lecture on countability in Analysis, but I started asking myself what is the order of the set of all worlds.  (Here is the Wikipedia page on Countability)

Mathematicians deal with the infinite all the time.  It is a concept that we are very comfortable with.  Dr. David Brown said recently that an engineer's or a computer programmer's task is to solve one problem at a time, but a mathematician's task is to solve every similar problem at the same time.  In order to do this we work with infinite sets every day.  Because we touch infinity, some mathematicians realized that some infinities are different than others.  The following is some definitions and explanations of mathematics to the lay reader.

When we talk about the cardinality or order of a set, we mean that we want to describe how many objects are in a set.  A set is finite if there are a finite number of things in them (that is there is a N in the positive integers, such that the number of things in the set is N).   A set is infinite, or not finite, if there are not a finite number of things in them (that is there is no N is the positive integers, such that the number of things in the set is N).  A set is countable, if there are the same numbers of things in it as the positive integers (that is that there exists a surjective* function from the positive integers onto the set.  That means that you can line up every element in some order and associate each one with a positive integer somehow.)  A set is uncountable, if there are more things in it than the positive integers (that is there is no surjective function from the positive integers onto the set.)   
These definitions lead to surprising results (at least they were surprising to me).  For example there are the same number of even positive integers as there are positive integers, because you could just use the function f which maps from the positive integers to the even positive integers given by the f(n) = 2n.  Clearly**, this function meets the requirements for countability, and therefore the even positive integers are countable.
Another result from these definitions is that there are more  real numbers than there are natural numbers, and that the real numbers are uncountable.  The proof to this statement was made by Georg Cantor and is called Cantor's diagonal argument.  It is a really cool proof, but quite involved.
So I was sitting in my institute class thinking about the order of the set of all worlds.  I know from my physics classes that the visible universe extends in every direction  for about 14 billion lightyears (this is a measurement of distance, not time), I also know that we have every reason to believe that the universe continues farther than we can see for ever.  I also look at the images taken by the Hubble telescope taken of a single point of darkness in the night sky and they get something like this: 

Hubble Deep Field
Nearly every spec of light in this image is a galaxy.  This leads me to believe that galaxies are dense in the universe (and therefore that worlds are also dense in the universe.)

However this doesn't help with determining if the number of worlds is countable or uncountable (as the rational numbers are dense in the real number line and the real numbers are also dense.  However the rational numbers are countable***, but the reals are not.)   So I am pretty sure that as far as science is concerned we can only guess if the set of all worlds is countable or uncountable.

However God told Moses that they are "innumerable," and since Moses hadn't studied modern Analysis, and wasn't concerned with their countability, maybe God just meant that they are infinite.  But the word "innumerable" sounds a lot like "uncountable" to me.  So maybe God was letting us know that not only is the set of all worlds not finite, but it is also not countable.

So if the set of all worlds is indeed uncountable, then the verse "and innumerable are they unto man; but all things are numbered unto me, for they are mine and I know them." takes on deep significance.
If God can somehow know an uncountable set, and know every member of an uncountable set, then God really is omnipresent.  This deep insight helps me to understand how God can know and answer every single prayer, because somehow he understands every member of an uncountable set, of which I am a part.

Sunday, September 11, 2016

My level of sacrifice

My second mission president made sure that we all knew that we should attend the temple when we returned home.  He even coined a phrase "your level of sacrifice," to describe how often we should attend the temple.  He explained that early in his marriage when he lived far from the temple, he only went to the temple a few times a year, and it was a sacrifice for him.  But later when they built a temple closer to his house, he found that attending the temple once a month had become casual, and then he increased his temple attendance to a weekly activity.  He said that for each individual and at different times of life, the level of sacrifice will be different, but that it is important to attend the temple often enough that it becomes a sacrifice.

Although I regularly attend/work in the temple, I haven't truly understood what "level of sacrifice" really means.  I have lived extremely close to a temple my whole life, and I have regularly attended the temple at least once a month, sometimes even multiple times in a week.  I love the temple and I love the spirit I feel in the temple.

I have always felt that what I was doing was enough, temple attendance once a week is surely sufficient for a busy college student.  And I have always felt blessings from my regular temple attendance.

For the last several months I have had the opportunity to serve as a temple worker in the Logan Temple, which means that I sacrifice every Saturday afternoon/evening to serve in the temple.  And I love it.  This amount of time is a sacrifice, but it hasn't ever been a hard sacrifice to make.  I love being in the temple, and I couldn't imagine anywhere else I would rather be.  It wasn't really a sacrifice but a privilege.

But this semester has been hard.  I am taking 2 senior level, 1 graduate level, and 1 (I don't know where to put it, but extremely advanced) math classes.  On top of that I have committed to spend at least 16 hours a week doing research for my favorite professor.  (Which is super awesome, bytheway.  And we hope to get a publishable paper out of the experience. :D)  On top of that I am TAing my favorite class, also for my favorite professor, which means hours spent helping students with their homework and even more hours spent grading their papers.  But I love it.  However, it all adds up to I am spending 10 to 12 hours a day doing math.  I have a lot to do, regular homeworks to turn in, and extremely advanced concepts to learn.  (In my graph theory/combinatorics/optimization/whatever it is class we are pushing the frontiers of mathematics and it is super exciting.)

I love my life, and I love doing math and I wouldn't change anything about how much time I have to spend to keep up with all of my responsibilities.

But Friday afternoon, while I was in a study group doing some Modern Algebra, I looked ahead at the weekend, and realized that I needed to complete my Modern Algebra homework, start on my Analysis homework, finish my Linear Algebra homework, and start in on my first mock paper for my Graph Theory class before Monday.  I also had just received a large stack of papers to grade. On top of that I needed to spend 3 more hours doing research, and I also had a new graph theory book I wanted to break into.  Friday afternoon, I looked at everything that needed to be done, and the amount of time required to do it in, and thought it was impossible.

For the first time in my life, going to the temple on Saturday evening was really going to be a sacrifice.  (Not just of play time and free time, but of actual real projects and responsibilities).  I love the temple, but it seemed on Friday afternoon, that there was no way that I could attend the temple and still finish everything that was necessary.

I went home and spent all of Friday evening grading papers.  Then I woke up (early, without setting an alarm, God must have blessed me), finished grading and doing my research.  On top of my busy schedule, I have made a personal commitment to myself not to do work work on Sunday (For me that means grading and research, since I am getting paid for it, but I still allow myself to do homework on Sunday.  It is a personal choice between me and the Lord, and I am comfortable with it.), so that means that I had to finish all of my work work before I went to the temple on Saturday, and hope that somehow all of the homework would find a way to get done.

Then I went to the temple, and it was good.  The peace I felt there was real, and although I didn't really do much besides smile at patrons and point them in the right direction, it was where I needed to be.  I felt immediate results from my sacrifice.  While I was in the temple, I knew that even if I didn't get everything done, even if I don't get high grades in my classes because of the sacrifice I had made, it would be alright.  The temple is good, and being there is what I need to do, and a sacrifice that I need to make.

Now, looking back on the weekend, Sunday Evening, I see that God has multiplied my time.  Not only did I finish all of my homework goals, and my work stuff, but I also had time to listen to the CES devotional.  I even got started on next week's homework for my Linear Algebra class.

For me the biggest miracle was my Modern Algebra homework.  We are learning category theory (if that means anything to anyone reading this blog post), and it is super abstract.  I am having problems understanding the definitions, let alone understanding any method to prove the results that we are expected to prove.  I have had two study groups on the same homework assignment, and I still had no idea (not really) what I was doing.  I came home from Ward Prayer, and started on my Modern Algebra homework, and with relative ease found myself understanding the definitions and the notation.  What once was a fuzzy cloud of intangible ideas has now been penetrated.

(And after I finished my homework, I still had time to socialize for an hour or so with two different social groups.)

I am a math major, and I have been carefully planning and counting my time, and I can't explain it.  I was able to accomplish more in one weekend, learn more, and master more, then should be possible.  And I spent a very large portion of it serving the Lord or worshiping Him at church.  Sacrifice works.
There was an error in this gadget