## The Omniscient God

So, many say god is an all knowing entity, and with his infinite knowledge, he guides us in his perfect plan through life. In this entry in the series on god, I’ll disassemble The omniscient god, and show why it’s simply not possible to know everything.

First, lets recap, we have three axioms we are operating on.

• Axiom of Incompleteness: There cannot exist a sufficiently powerful logical system that is simultaneously complete and consistent.
• Axiom of Consistency: If we prove something in a consistent logic, then it is unequivocally, irrevocably true. Proof is infallible.
• Axiom of Fair Play: Everything that exists, materially or otherwise, must abide by the rules of logic, and further, anything that interacts with the physical world obeys the the laws of the physical world.

Of these three axioms, the first is actually a theorem due to Kurt Gödel, and the second is actually just a definition of what it means to be a proof in consistent logic. The third is the clincher, it’s the one you actually have to believe in some cases. This is a common theme in mathematics, for instance, in set theory, we have a few axioms which are comparatively “easy” to believe, and one that is somewhat difficult to believe. This usually leads to a set of theorems that are free of the use of the “hard” axioms, and later a set of theorems which include the axiom. We’ll follow the same general format here.

We also admit a heuristic principle, which most people know and use regularly. Scientists typically call it Occam’s Razor, laypersons call it the KISS principle. It’s definition, colloquial and scientific, is as follows:

• K.I.S.S, Keep it simple, stupid.

More formally:

• Occam’s Razor: When in the presence of two or more equally valid explanations for the same phenomenon, the explanation involving the least number, and simplest, assumptions is most likely the correct explanation.

We’ll use this as a guiding principle, when we see many explanations for how something works, we’ll pick the one whose assumptions are both few, and most basic. We’ll also use the principle’s contrapositive[1], that is, if you have to assume many complicated things to explain something, then your explanation is most likely wrong.

One more thing, before we begin. There is a principle in physics, specifically time-travel physics, called the Novikov self-consistency principle, this principle, in effect, states that when one travels through time, you cannot create any paradoxes. It comes up when we think about the grandfather or billiard ball paradoxes, that is, if we shot a billiard ball through a wormhole at just the right angle, and that wormhole sent the billiard ball back through time so that it exited and could collide with it’s younger self so as to knock the billiard ball off it’s original trajectory, so it never goes through the wormhole in the first place, then we have created a logical paradox, where did the new billiard ball come from? This idea is often generalized to mean that as long as one doesn’t create a paradox somehow, then one can do whatever they want. You’ll see that, in effect, this is my third “Axiom of Fair Play”. However, the NSC principle implies something stronger than my Axiom, that one cannot create a paradox, not just that one must not. I mention this in case someone wants to bring up the NSC principle in an attempt to rebut a proof containing my third axiom, I’ve taken it into account, and choose a weaker assumption to boot. That said, on to the proof.

Knowing everything is a tantalizing idea. Imagine, knowing the answer to every question, the proof to every true conjecture, and disproof of every false conjecture. You would be a very famous person if you were able to simply spit out the full history of the planet from it’s beginnings 4.5 billion years ago till now. However, it is fairly easy to see that this idea falls apart under the first axiom. If you had the correct answer to everything, then you would form a consistent logical system, by definition. I would get a consistent answer from you, no matter what I asked, and if you were truly omniscient, that answer would be the same every time, and would be correct every time. However, you would also have to form a complete logical system, in that you have the answer to every question, by definition of omniscience. Therefore, you form a complete, and consistent logical system, which violates the first axiom. So, given our axioms, the naive idea of omniscience fails.

What about a kind of conditional omniscience? That is, you don’t have the immediate answer to everything, but you can acquire an in a constant amount of time? This solves the error with the first axiom, in that you never really contain a complete logical system, you’re just a perfect consistent system, able to answer any question in a finite amount of time with a consistent answer, but not actually knowing the answer until the question is asked. You have now become what computer scientists typically refer to as an “Oracle.” Something with the magical ability to answer some hard problem quickly, in our case, you are really a “superoracle”, capable of answering every question. A common use of oracles is to disprove the existence of a real object which could be that oracle. This is done by constructing a question to ask the oracle which violates the rules of logic or physics. We can do this with our oracle, as well. Consider the following conversation between Kurt, our friendly neighborhood logician, and our restricted omniscience oracle:

• Oracle: What is your question?
• Kurt: Is the statement “This statement is a lie” the truth or a lie?
• Oracle: The statement is the truth.
• Kurt: But if the statement is the truth, then mustn’t it be a lie? Since it asserts such?
• Oracle: Well, yes, so the statement is a lie.
• Kurt: But then isn’t the statement the truth? Since it’s lying about it’s own lack of veracity?
• Kurt: It’s okay, incompleteness can do that to you.

This somewhat humorous (I hope) story should illustrate the point, if the oracle can provide a correct answer to every question, that implies that every question must have a correct answer. So by constructing a impossible question like this, we find that this incarnation of our oracle can’t exist. So we must restrict it further. This idea of an “impossible” question lies at the heart of Kurt Gödel’s Theorem of Incompleteness (and our first axiom). Gödel was able to prove the theorem by constructing a sentence like “This statement is a lie” in any arbitrary logic. Interested readers would do well to read “Gödel, Escher, and Bach” by D.R. Hofstadter[2]

So now we must further restrict our oracle to being able to answer all answerable questions quickly, this would still give the appearance of omniscience, but would avoid the incompleteness problem by dodging the impossible questions entirely. However, what good is this kind of oracle? What kind of questions are left when you eliminate those with no answer? This begs the question, “What kind of questions have no good answer?” In general, the ‘important’ questions we might want to ask such an oracle, questions about the meaning of life, or the ethics of abortion or war or any other hot-button issue are impossible to answer firmly in one way or another. We can’t unilaterally say it’s always okay for a woman to have an abortion, what if the child is seconds from being born? What if she wants to abort it simply because it will have the wrong hair color? Similarly, we can’t say it’s always wrong, what if the child will be born with horrible, painful defects? What if the having the child endangers the life of the mother? Who’s life is worth more? These aren’t questions with a blanket answer which an oracle like ours could provide, they are impossible, and must be determined on a case-by-case basis. Sometimes it is necessary for a woman to have an abortion, sometimes it’s not. Sometimes we have to go to war, I like to think that most of the time we don’t. The point is that these deep questions have no “right” answer, there are many answers for them, depending on the exact conditions present, and the beliefs of the people involved. No magic oracle could decide these problems finitely for all people.

Further, there are problems that this oracle would have relating to the third axiom, in that, some problems have an inherent minimal amount of time to answer. For instance, in order to add two numbers, I have to examine every digit of each number at least once. This idea can be proven fairly easily, and is done in first year computer algorithms classes. So, there are some problems which, while the oracle could provide an answer, would take an inordinate amount of time to answer.

So we are left with an oracle which, upon examination, is no better than a very smart, very well read, unbiased human being. Omniscience breaks the rules of logic, and when we remove the parts of omniscience which break those rules, we are left with nothing more than what we already have. I think this shows a real, powerful idea, that we don’t need to invoke the oracle, or god, or the flying spaghetti monster to answer these questions, we are capable on our own of doing so? Why is it that we think so little of ourselves, that we feel we are unable to answer the hard questions, when we haven’t ever tried? I noted that some of the problems with the Omniscience Oracle stem from the time it would take to answer the problem. This idea comes from computer science, where complexity is of paramount importance. Since computer scientists are always aiming to make things run faster, they came up with a particularly apropos idea, parallelism and concurrency. The idea is that, hard problems may not ever be made easy, but when you can’t make the question easier, you can always throw more horsepower at it. I think that what we, as human beings, need to do about these hard problems, is stop turning to dogmatic belief in a single answer, and instead of defending the conclusions of one man, or group of men, throw our considerable collective mental horsepower towards actually coming up with good criteria for when one solution works for a problem, and when it doesn’t. We have a lot of potential, as humans, we are, so far as we know, the only thinking, sentient forms of life on this planet. We have hard problems which need solving, and it is time to set aside the static answers of the past, and learn new, ‘faster’, better ways to solve — or even just approach — those problems today.

I hope you enjoyed the post, this one is a little on the long side (10838 characters…), hopefully the next few will be shorter.

[1] contrapositive : when given a logical statement of the form: $A \Rightarrow B$, then you may deduce $\lnot B \Rightarrow \lnot A$

[2] “Gödel, Escher, and Bach” by Douglas R. Hofstadter, amazon