Prime Movers

Brandon roused me from my mathematical slumber with a post on one of my hobby-horses, prime numbers.  Just enough impetus to put down a thought I was mulling over last week.

Primes are the delightful irreducibles of the number world.  As a kid I thought of them as weird exceptions to good, common-sense mathematics–the kind of things you memorized and played goofy games with.  But that has things almost backwards.

The Fundamental Theorem of Arithmetic says that any integer can be expressed as a unique prime factorization (this is what Brandon was posting about, so I shan’t repeat him).  Said in reverse, primes are the building blocks of the number world.  Everything traces back to them.  That got me thinking about primes as an example of first movers like one would use to explain the First Way or STA’s explanation of the act of the will.  

We could consider the primes as the principle or cause of the composites (and indeed this is what the name “prime” is getting at).  In this order, they are uncaused–they simply are, the first movers of the other numbers.  Imagine an empty number line stretching to infinity, a la Donald Duck’s Mathmagic Land.  Then populate it with just the primes.  Look at all that empty space stretching out to infinity!  How dreadful!  But by combining primes, you can fill in all that space.  Primes as prime movers.

But primes are not absolutely first.  Primes are first with respect to composites but are in turn composed, in a different order, by the successor operation in Peano Arithmetic.  Don’t run away screaming from the lingo: here I just mean that every prime is composed of repeated addition of the number 1.*  That is to say, the primes are in turn “moved” by, or find their principle in, one.**

Now let’s turn all this high-falutin’ math to something concrete and practical, like medieval philosophy!

When Aquinas wants to explain how the will is moved by objects but is still an intrinsic principle of action, he compares it to conclusions of one science serving as principles of another.  One goofy thought is to compare it to one, primes, and composites, as I’ve (very roughly) sketched above.

We could also use the sequence one-prime-composite as a way to think about instrumental causes that the Thomists like to go on about when they discuss the First Way.  One is the Prime Mover, primes the instrumental movers, of numbers.  Or kick it into Second or Third way on causes and necessity.  I think it works there too, and yes I realize that in my flight of fancy I’ve been shamelessly drifting between language of cause, mover, etc.

Here’s a goofy thought, then: primes and composites are two different orders in a hierarchy of numbers.  One generates the primes, which in turn generate the composites.  Neo-platonist overtones not intended but embraced with a hearty chuckle.

I’ve been thinking about ways to use my students’ existing mathematical knowledge more explicitly when going over the philosophical stuff we do in class.  May come back to refine this idea later, after I’ve tossed it to some kids to chew over.

*Nowadays mathematicians take 0 as their starting point for number theory, but Peano himself followed the ancient tradition of using 1.  Mathematicians claim the choice is arbitrary.

**We would say “the number one,” but we are bouncing around the reasons why the Greeks didn’t regard “one” as a number, but as the principle of numbers.

BONUS: A few days after writing this up, I ran into this fun Numberphile video that talks about primes as building blocks of other numbers:

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