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.


*Surjective is another word that might need some defining.  But if you have taken MATH 1050, then you would have at least been exposed to this concept, and a quick google search should refresh your memory enough that you should be able to work with it.

**I just broke one of the most important pieces of advice I have ever been given when writing math, which is to never ever use the phrases "clearly," "it is clear," or "obviously." And I apologize deeply to the reader.  This statement may in fact not be clear, and it may take some work to understand it.  However further explanation is left out because this is not a math paper, but rather a spiritual paper with some math in it.  If this statement is not clear to you, then I urge you to stop reading, pull out a piece of paper, and work with the definitions that I have given until it does become clear to you.  (Reading math is not like reading a novel, it takes some work).

***Here is another surprising idea, that probably deserves a proof, but I am too lazy to give it to you.  Look it up if you are interested.

1 comment:

  1. I learned a new definition of infinitive today, and I really like it. A set is infinitive if there exists a bijection from a proper subset of that set to itself.

    bijection probably deserves a definition, but I'm too lazy. Also a a proper subset is a subset that is not equal to the original set. A set, A, is a subset of a set, B, if for every element in A, that element is also in B.

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