## Saturday, February 25, 2012

### Recursion and The Principle of Sufficient Reason

The principle of sufficient reason (PSR): For every fact F, there must be an explanation why F is the case.

The reason the PSR caused so much trouble in 17th and 18th century philosophy is because it's a recursive function.  It's always possible to run the PSR function on the output of the PSR function.  This results in an infinite regress.  Even if we trace the conditions back to something which has no cause (like God), it's still possible to ask after the reason for the existence of such a thing or the reason it acted to create the world in the first place.

Infinite Regression
One might wonder why an infinite regress is an undesirable consequence.  One way to think of it is in terms of end-conditions.  For analogy, imagine we're trying to convert a decimal number into a binary number.  We know that for every whole number that k = 2n + b, where is a whole number, and b is either 0 or 1.  If you apply this function recursively, you can transform any decimal number into a binary number.

sub binary {

my (\$n) = @_;
my \$k = int(\$n/2);
my \$b = \$n % 2;
my \$E = binary(\$k);
return \$E . \$b;
}

This function takes whatever is passed to binary and tosses it into variable \$n.  \$k is then \$n/2 (discarding the remainder), and \$b is the remainder.  But then we compute the binary expansion of \$k and call it \$E.  Once the recursion has gone as far as it will go, it will come out the way it came (so to speak) and return a string of bits.

Now, there's one problem with this subroutine as its written: we have not specified an end-condition.  If you call this subroutine in a program, you'll never find the binary equivalent of anything.  It will run until it has filled the computer's memory, and then your machine will crash.  We can fix this by adding the following line, right below the declaration of \$n:

return \$n if \$n == 0 || \$n == 1;

If the number passed to binary is 0 or 1, there's nothing further to do, since 0 and 1 are already their binary equivalents.  At that point it's time to stop the recursion, come out the way we came, and string the results together.

Complete Intelligibility
Now, the PSR applied universally is a lot like our first version of binary before we put in an end-condition.  It goes on forever.  The fact that it goes on forever is not the problem per se.  Perhaps we're computing the function on an ideal Turing machine with infinite memory and an infinitely long tape.  The problem is that, like our first draft of the procedure, it won't return an answer.  Positing F' as the cause of F doesn't do the work of explaining F any more than saying that 11 = 2*5 + 1 gives us the binary of 22.  It's just one piece of a string of explanations that, in itself, doesn't tell us anything useful.  It's only when we take all the answers we've gotten already and put them together that we get the explanation.  But we don't get to concatenate our results until we reach the end of the recursion and come back out.

You might wonder why we need to apply the PSR universally, though.  Why not accept some things just are the case, and that's it?

Well, it seems as though we have to do something like that.  After all, the end-condition in the binary subroutine is an example of that.  But we have to distinguish between two kinds of end-conditions: conditions that explain themselves and those which do not.  The end-condition of binary is the former kind.  It follows from the fact that binaries by definition are made up of 0s and 1s that our subroutine treats those as building blocks.

The latter kind of end-condition - what we might call a finite end-condition - is justified neither by itself nor by anything outside itself.  It simply is - and for no reason.  There's no obvious reason why the world can't ultimately be grounded in such a finite condition.  But if it were grounded that way, then once again ordinary things would not be fully intelligible.  It would be like binary stopping when it reaches the number "5".  There would be no rhyme or reason to the program, and we would never get any answer.

The Intellectual Categorical Imperative
There are other reasons a finite condition is undesirable.  It seems to go against our rational instincts, what we might call an intellectual categorical imperative.  For instance, observation tells us that, in our universe, the speed of light is constant, c.  This is treated as an unconditioned in most physics.  Yet there's also an implicit belief that there's a reason it is what it is and not some other number; we just don't know that reason yet.  Or in the standard model, there are various particles, and as we discover them, we find out their properties like spin, mass, charge, etc.  We have no theory right now that predicts these properties before they're discovered.  So for the time being, they're treated as unconditioned.  But it's assumed such a theory exists - which is why many theoretical physicists work on things like string theory.

So even when we're confronted by things or truths which appear to be finite unconditioneds, there's a natural expectation that these things actually are conditioned, we just don't know yet by what.  This is no guarantee that such an explanation exists.  It's no guarantee that our powers are great enough to discover the explanation.  But this drive explains the demand for universal recursion in the PSR, and why nothing short of an absolute application of the PSRappears to satisfy it.

Summary
The principle of sufficient reason (PSR) is the demand, for every F, why F is the case.  It is an attempt to explain why things are the way they are and not otherwise.  It is concerned with the ultimate reasons for existence itself.  It is recursive and universal, meaning that every output of the PSR must become an input for the PSR.  Once the PSR is applied recursively and universally, there are three possible outcomes (what philosopher Paul Franks calls the "Agrippan Trilemma"):
1. The series is infinite.  The PSR function will run on itself forever, and so there is no answer to why anything is the way it is.
2. The series is finite and ends for no reason.  The PSR will return an answer, but the answer will not fully explain anything (if it explains anything at all).
3. The series is finite but terminates in something self-caused/self-grounding/self-explanatory.  It's easy to see what this would mean in the case of computing binary numbers from decimals, but it's less clear what this means in the context of beings as a whole.
Whatever it is that would satisfy #3, it's the only answer that would satisfy a philosopher who thinks about reason in terms of the PSR.  If we're going to use the PSR, then we need #3.  If we don't have it, then the world is unintelligible and ultimately groundless, and reason is meaningless.

Next time I'd like to think about what would fit the bill and how different philosophers and theologians have thought about this problem throughout the ages.  This is usually treated as a problem particular to the rationalist philosophies of the 17th and 18th centuries, but I think the problem is more general than that.  This gets into questions about infinity, time, the Seinsfrage, and even mysticism.