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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Academic Lessons on Pain and Suffering

calvary_hillHow do you deal with temptation and trials? How do you view suffering or pain that you experience? Is the recognition of pain and suffering in the world render your belief in God futile?  Does the suffering in the world seem senseless random?  Why does the ‘good’ suffer while the evil prosper (Psalm 44:9-16)? As a lowly university mathematics professor let me offer an analogy from my regular experiences with students.  I implore the thoughtful reader to read the scripture references; you can access a bible for free at

In the beginning of each semester, during the first class I offer the student a class vision, even if they do not have the ability to understand it.  I ask them, however, to believe and put their trust in my guidance (Isaiah 55:9, John 3:12, John 6:47-51, Psalm 118:81 Pet 2:6-8).  You see throughout the term each student’s vision is incredibly short-sighted and cloudy (1 Cor 13:12).  Indeed, they will all struggle in different ways (1 Cor 8:9-13).  Often it is working with others that can make a big difference in their work (Prov 27:17), and I hope that they allow me to keep their paths straight (Prov 3:6). Oh, as my previous students can attest, they undoubtedly suffer through the work (James 2:14-26) and various tests (James 1:13, 1 Thes 3:3-5, Deut 8:2-3, Exo 20:20, Psalm 66:8-12).  They are all accountable and know this! In fact, they are disciplined/graded (Deut 8:5,  Rom 3:19) to provide additional instruction (Psalms 32:8).  Often, students will cry out for help and I do console them (Psalm 23), however, I assure them that they are never being tested more than they can understand or bear (1 Cor 10:13).  Consequently, each student is built up (2 Tim 3:16) and, thusly, learns endurance/perseverance  (James 1:4).  Unfortunately, some will fail, trusting that their way is simply better (Prov 3:5, Isaiah 8:15, Romans 1:21-32).  They no longer listen to sound teaching  and, in desperation they follow unsound paths (2 Tim 4:3-4).  The take the easy road rather than the narrow path to understanding and truth (Matt 7:13).  Oftentimes, the failing student becomes embittered to the success of other students (Matthew 24:10), they even make fun of the Way (John 15:18-25).  In contrast, to those who have endured (2 Tim 4:7-8) they look back and realize the immeasurable good that came out of their pain, suffering, trails, and temptations (Romans 8:28). Finally, the student comes to the realization the love that I had for them (Matt 6:26) and that I longed for them to allow me to lead them and serve them in this way (Matt 23:37).  In all, the semester starts with a student’s personal decision to receive (John 14:15-18), respond (John 3:16), and, subsequently, trust (John 12:35-36, Psalm 118:8).