21st-century Christian Philosopher

Bertrand Russell posed that the word “bird” had meaning because you can say “x is a bird”.

Because we could find members to fit the label, “bird” had meaning. The only thing left as a claim to meaning were the definitions of mathematical and logical constructs. For example, it is true that a triangle is the shape formed by three points not lying on the same line, because that’s basically what a triangle is.

So what do we do with the Halting Algorithm?

Because it is not at all useful for a machine to loop forever, when trying to arrive at a product, mathematicians became interested in the ability to tell a loop-forever process from one that arrives at an answer in a finite number of steps. The hypothetical algorithm that does this is called the “Halting Algorithm”. The machine that computes this algorithm would be able to take the specification of any machine and any input and tell you whether or not the algorithm is defined with that input.

What Alan Turing showed is that this machine could never exist. That is the point of his proof. Thus, it is true of no computable algorithm that “x is a Halting Algorithm.” Thus the non-existence of a machine that computes halting must some how be axiomatic of mathematics, or we’re running out of Russell’s classes of meaning.

But the possibility of such an algorithm is not axiomatic, in fact it only shows itself paradoxical in its use as a component of a larger paradoxical machine, shown at Wikipedia as:

function trouble(string s)
if halt(s, s) == false
return true
else
loop forever

http://en.wikipedia.org/wiki/HaltingProblem#Sketchof_proof

What that means is that the algorithm “trouble” takes an encoding of another algorithm (TM) and checks to see if it halts on that same encoding as input. So far, nothing insurmountable has happened. But if we compute trouble( t, t ) (where t encodes trouble), we get an unresolvable paradox. Only if trouble does not halt on itself, will it halt on itself. And conversely when it does halt on its own encoding, it won’t. Thus halt() cannot compute trouble( t ) correctly. So there is no one algorithm that computes the Halting Problem in all cases. And by subsequent proofs, we can show that this is just the wedge that widens the crack. But we can see that there neither are such things, nor are their provisional forms be considered in any way axiomatic, yet when this placeholder is used to represent an impossibility, we gained real knowledge of the limits of computation.

But the cluster grinds against positivism in two more ways. But that is the subject of my next post.

This post continues on from Godless Heuristics. Here I look at a specific case. It is a puzzle, despite that it can be “resolved” by a parallel to another godless rule. This deals with noted, agnostic, philosophy professor, Michael Martin and his TANG argument (The Transcendental Argument for the Non-existence of God (q.v.). Although, I am no professor and Martin is, I’m going to pretend to take him to task on his misunderstanding of how necessity works in a natural system.

Martin argues that if Logic is contingent upon God, then it cannot be “necessary”. Necessary is here used in the logical sense, meaning a required condition. I think that this misses how necessary conditions work. Something can be both contingent upon one thing and necessary to something else.

Let’s take the Anthropic Principle used to explain why it is not special that we might be improbable existences but not in anyway designed or special. It also is used to explain why we may see a “fine-tuned” universe. This idea, that the universe is “fine-tuned” to allow carbon-based lifeforms, is dismissible because only where the physical properties allow carbon-based lifeforms are such lifeforms likely to exist to notice that the cosmos is “just so”.

Reasonable enough, I agree. But notice what it says, “tuning” is necessary for life, life is necessary for observation about the universe that conceived the observer. But it balks at suggesting that such tuning is “necessary” in an absolute sense. In fact, often used with this is the multi-verse theory. This says that there are so many universes that are taking various shapes that some just might come to such a balance that appears “tuned”. Thus because all processes are taking place in all configurations, tuning is contingent upon which universe you are in and what has been it’s process to point P.

Nonetheless, we identified “tuning” as necessary first, and now it is contingent upon the development of the pocket universe. And this is supposed to be all scientific.

Thus we have that Martin’s heuristic that what is “necessary” cannot be “contingent” asks a little too much in the second part of his TANG argument (on Science). I also believe that Martin misconstrued the TAG argument (Transcendental Argument for God) in this case. TAG-ers argue that Logic is necessary for the orderly world. So in seeing this, God brought Logic into being for this purpose. To then argue that Logic is then contingent upon God’s design and so not necessary for what he intends, misses part of the framework.

To one sense, to view God as “perfect” means that we accept that all His steps were “necessary” to “work to the good”. To then say, that they were “contingent” upon God’s intent, and thus not necessary, likely throws an equivocation in there somewhere.

However, I can go further in examining Martin vs. the Anthropic Principle. I can say that only in a world where Logic divides a proposition from its contradiction would it be enough of a thing to remark that it acts in such a way. Because were it not to act in this way, in the local condition, we could not say that it definitely does not act the other way as well. Thus we would not see the face of Logic as non-contradictory, but as incoherent mixture.

Of course this makes no sense, but only in a world where the forces combine to make sense can we observe what makes no sense. Without that pre-condition, we’re in a world that makes no sense, and regard proposition that both make sense and do not make sense.

If Logic is in fact contingent, I wonder how it was that Martin saw Logic as necessary at all. The world just does not produce “necessary Logic”.

Also it would be hard to argue that, across one of these impermeable walls which separate the multiverse (an idea in which I invest only curiosity), a universe so inept that it cannot make simple gravity, let alone the observers that could note that it is inept at making gravity, saw “Logic”.

Likewise, without the multiverse dodge, left alone with just our simple improbable, conveniently “tuned” universe to observe, the same holds true–it would seem. Given all those other possible universes that could not evolve life, in what sense would “Logic” still be brooding over the waters of Chaos?