Are all things possible?

The class ends and students make the usual mad dash to the exits.  Lines of exiting students quickly form as new students flood in.  A student approaches me with a brief question as I desperately try to ready the classroom for the incoming professor.

“Dr. B., why are we only interested in the invertibility of square matrices?”  The student asks.

“Because, it doesn’t make sense to talk about the invertibility of matrices that are not square,” I respond.

In the study of linear algebra, matrices are a common conversation point.  Matrices stem from a wide range of applications.  They can be viewed as linear transformations, that is, they map objects from one space to another space.   They may represent a vector equation or a linear system of equations.  The latter case may be more familiar from your high school algebra training.  In either case, a matrix can be used to store all the relevant information, for which allows for easier manipulation by hand or through a computer.

All matrices have N rows (number of equations) and M columns (number of unknowns).  If N equals M then it is called a square matrix.   When solving a linear system of equations, say A xb,  a unique solution exists if and only if the matrix is invertible, that is, there exists a matrix C such that A C = C A = I, where I is the identity matrix.

This only makes sense for square matrices.  Therefore it is nonsense to talk about the inverse of matrices that are not square (rectangular matrices).  It is a contradiction of terms.  We simply can not have an invertible 2 by 3 matrix. It simply does not make sense.  It is like having a square circle or a married bachelor.

What does this have to do with God?  Everything. Consider a common dialogue that I have been privy to:

Skeptic : If God is All-Powerful then all things are possible.

Me : No, God can do any thing but not anything.

Skeptic : Huh?

Me : God can not do what is logically inconsistent.  He can not make a married person also a bachelor.  He can not make a positive number also be a negative number.  He can not make a square also a circle.

Skeptic : Oh, okay, but I still do not think God is All-Powerful.  Maybe the reason why there is so much pain and suffering in the world is because God is not strong enough to overcome it. Could God create a stone that even He could not lift?

This conversation seems to make a truly profound claim about an attribute of God, that is, that He can not be All-Powerful.  However, the last question by the skeptic is simply a contradiction of terms.  It doesn’t make sense.

If God is All-Powerful, that is, He can lift anything, then it does not make sense to talk about something He can not lift!  This is equivalent to a student wishing to talk about invertible 2 by 3 matrices; be it their wish or not it simply does not make sense from basic principles.

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Finely Tuned Questions – Part I

Matrix“Given these three linear equations with three unknowns you have 30 seconds to guess the answer,” I stated.  The students befuddled and stymied by the request took a few seconds and then formed a guess. The time quickly exhausted itself.

“Please check to see if your guess is the solution and then, through email, indicate to me if you were correct or not,” I said.

Determining solutions of systems of linear equations is at the heart of many applications of mathematics.  In nonlinear problems, a linear system may rear its head as the result of trying to obtain a solution through an iterative method.  Nevertheless, solving a system of N equations for a collection of N unknowns is not a complicated task, rather it is tedious!

A linear system is best viewed as a matrix, hence a linear system of N equations is best viewed as a N x N matrix (A) of coefficients of the unknowns (x) from each of the N equations.  The terms that do not involve any of the unknowns can also be collected (b).  Hence the linear system is written as A x = b.

For a general matrix it takes roughly N cubed operations to directly obtain a solution, provided that it exists. Not a problem you say?  Standard linear algebra textbooks will suggest to a learner that these methods will always work, regardless of the tedium.  However, it if it takes that many operations then what happens if the linear system is extremely large?  In my work I commonly deal with matrices on the order of 10^5 x 10^5.  That is a small matrix relative to the large matrices employed in data analysis of experiments on the large Hadron collider, typically on the order of 10^12 x 10^12.  To appreciate the time and effort for a computer to work through the tedium of determining the solution let’s consider an example.  Say your computer’s time to perform a flop, that is a multiplication or addition, is 10^-10 seconds.  Fast. Now to determine the solution of 10^5 x 10^5 system requires 10^15 flops, that is, 10^5 seconds.  That’s over a day just to solve ONE matrix problem!  In my work I have to solve a comparable system a few million times!  So it would take a few million days to complete!  Yikes!

So what can a person do?  They iterate, that is, they attempt to find a sequence of solutions that converge to the true solution.  The hope is that each iterate is relatively fast and that it converges quickly, lowering the required amount of iterates.  Sounds good and promising!  But where does the information for the first iterate come from?  Hence, you need an initial guess.

Now would you be surprised if your initial guess was the exact solution?  Suppose that I have a system of 3 equations with 3 unknowns.  Only I know the equations, you don’t know them at all.  I want you to guess a solution.  You can email or comment to this post with your solution and I’ll tell you if it is correct.  Would you be surprised to find out that your guess was correct?  In fact, if I gave you 100,000 guesses without telling you the form of the equations, I would be shocked if you could guess it exactly with anyone one of those!  Shocked!

Hence, when I look at the degree of fine-tuning of the cosmos, I am shocked!  What do I mean by fine-tuning?  Think of sequence of knobs that are dialed in to allow a specific outcome.  If these knobs are not within an extraordinarily small range of values then the specified outcome would not occur.  For the cosmos, the outcome is allowing life to be permitted, and the knobs are fundamental constants of nature (weak force, cosmological constant, electric charge size, strong nuclear force, density of the universe, entropy, etc.).  How narrow?  Take the weak force, if this value was off by one part in 10^100 our universe would not be life-permitting.  Period.

This begs an obvious argument for the existence of God as the fine-tuning can only be possible from either physical necessity, chance, or design.  Thus the argument follows:

Premise 1.  The fine-tuning of the universe is due to either physical necessity, chance, or design.

Premise 2.  The fine-tuning of the universe is NOT due to physical necessity.

Premise 3.  The fine-tuning of the universe is NOT due to chance.

Therefore it due to design.

Premise 1 is obvious as this just lists the available options, hence one only needs to provide a plausible argument for premises 2 and 3.  The conclusion then follows naturally.

In my follow-up posts I want to address each individually.

“For since the creation of the world God’s invisible qualities—his eternal power and divine nature—have been clearly seen, being understood from what has been made, so that people are without excuse.” – Romans 1:20

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