Logic and Christianity

A recent thread on Network Norwich & Norfolk has been created here in which some of the contributors have been wrestling with the science and Christianity problem (a problem only in the sense that they make it one, I hasten to add). I am going to submit a few weekly articles on Christianity and science soon (mostly biological science and evolutionary theory) so I won’t comment on that particular issue here – but the welcome return of non-believers Mike H and Mike 2 has seen the re-emergence of the evidence/proof of God’s existence debate.As I have said from day one, we do not deal with proofs on here. One must realise that God is an infinitely complex personality, so the best we can do is sample Him. The Father has revealed Himself to us in Christ Jesus, and although the Bible is the word of God – one can learn lots more about God both from creation around us, and from having a relationship with Him. I think what the two Mikes (and any sceptics for that matter) should ask themselves is whether the world they see around them does in fact scream out that God exists, and how they might be expected to realise this.There’s a great story often recounted about Elizabeth Anscombe saying to the brilliant Wittgenstein, that she can “understand why people thought that the sun revolves around the earth.” Wittgenstein asks, “why?” - Anscombe says, “Well, it looks that way.” - to which Wittgenstein responds, “and how would it look if the earth revolved around the sun?”In other words, the way something looks from a certain standpoint is, from the individual’s perspective, a direct proprietary fact about the person's perception of that 'something'. ‘How it appears from a certain place’ is, nonetheless, also of interest in its own right and belongs to what is sometimes called ‘the reality of the appearance’. In Biblical times the ancient Hebrews referred to it as 'language by appearance' – so, for example, if something 'filled the earth' it did not necessarily, in the literal sense, fill it entirely.Using my ‘four principle rules of logic’ I can show why the proof arguments are wrong. Godel’s theorem warns us that the axiomatic method of making logical deductions from given assumptions cannot in general provide a system which is both provably complete and consistent. There will always be truth that lies beyond, that cannot be reached from a finite collection of axioms. This also means that no door in the labyrinthe palace of empiricism opens directly onto the ‘Absolute’ - there are hints of the infinite in mathematics (Cantor’s Absolute), but any complete unity must include itself, thus we hit the self-referencing problem of Russell’s paradox. But the way ‘minds’ can attune themselves into the nature of the Absolute gives us the biggest hint that we are truly meant to be here, and that this life is but a shadow of a deeper and more astounding reality.Moreover, there will always be statements which are true but that one cannot prove are true. Take the following statement (call it ‘S’)..S - “Mike cannot prove this statement to be true”Suppose Mike were to arrive at the conclusion that S is true - this means that the contents of S will have been falsified, because Mike will have just done it. But if S is falsified, S cannot be true. Thus if Mike answers ‘true’ to S, he will have arrived at a false conclusion, contradicting its infallibility. Hence Mike cannot answer ‘true’. That means that S is true; but in arriving at that conclusion we have demonstrated that Mike cannot arrive at that conclusion. This means we know something to be true that can’t be demonstrated to be true. Now consider something a little different. Let us say that we meet a man for five minutes that we have never met before and will never meet again. Our job is to find out if he can speak English. If he remains silent throughout the five minutes and he disappears never to be seen again, we cannot prove that he cannot speak English but we have no evidence that he can. If, however, he were to say the words ‘My name is Robert and I can speak English’ we would, of course, have the deductive proof that he can speak English, as the English words themselves would be contained within the statement. This is different from the first example, for the first example is about the axiomatic method of logical proof itself and is not a property of the statements one is trying to prove or disprove. One can always make the truth of a statement that is unprovable in a given axiom system ITSELF an axiom of some extended system. But then there will always be others statements in the self same system that are unprovable. Now we come to something else that is key - in the first example there is no new predicate that can be added to S (without changes to its intrinsic structure) that can alter the fact that S cannot be shown to be true. Now very obviously with the second example that is not the case - if we were told that Robert was an English lecturer and that there was footage of one of his lectures, we could show that ‘Robert can speak English’ is a fact without having to prove it in those five minutes. Moreover, unless we start flirting with nonsense there are a number of things we could find out about Robert that improve the probability that he could speak English.S1 - Robert was born and raised in GhanaS2 - Robert was born and raised in MozambiqueIt is very clear which out of S1 ad S2 is more likely to be suffixed with the statement ‘Robert speaks English’ - S1 because Ghana is a former British colony whereas Mozambique is a former Portuguese colony. Of course S1 and S2 might only improve the probability very slightly, but this is what we do in all walks of life. Knowing, as most mathematicians do, that there are some statements of logic that cannot be proved to be true (also read about the distinctions between realism and antirealism) we use our perceptive and investigative toolkit to reason our way through these things. If Robert lives and works in this country it is much more likely that he speaks English than if he lives and works in Ecuador. Do we completely abandon a mechanical procedure for investigating mathematics because of Godel’s theorem and Turing’s halting system? Of course not!! Those unprovables are rare elements of mathematics and can be sifted out allowing us to continue on a logical trajectory. That’s just one example of how silly it is to stridently decree ‘Unless there is proof of God’s existence, I’m going to carrying on believing that He doesn’t exist’.

One must also bear in mind that there are many axioms or regularities that only become that way by our adding something to facts. Look at this set of six numbers (all single integers used are under the value of six) 234232 - 344232 - 121232 - 523334 - 552555 - 122311. A Turing machine can show which number sets are computable given a set of rules. If the rule is, say, take the first number of the first set and add 1 (giving us 3), do the same to the second number in the second set (giving us 5), the third in the third (giving us 2) , and so on, we find that with that rule in place we have the number 352462. Unlike the “Mike cannot prove this statement to be true” example, this time we have created a rule or procedure and shown that logical deductions can be reached without messing around with the axioms in logic. For example, given this rule, I know that if I have the answer 352462 then none of the sets will contain the exact sequence 352462. Last one….Chaitin’s theorem. Having shown you with Turing that there are mathematical problems that cannot be proved by any fixed heuristic procedures, we now move on to how we know if what we know (or contend) is right or whether further compressibility is required. You also ought to remember that one can compress something too much into logical nonentity - the biggest example being with self-referencing paradoxes such as ‘This statement false’ - here we have something that is too compressed to be logical, because if it’s true then it’s false and if it’s false then it’s true. It is nonsensical because there is no subject or predicate pointer extended to ‘false’. The statement “3 + 5 = 9 is false” is true because ‘false’ has mathematical integer subjects extended to it - it has the ideal compressibility for deductive analysis. Now in Chaitin’s theorem, a computer is given this command - “Search for a string of digits that can only be generated by a program longer than this one”. Now obviously if the search succeeds the search program itself will have generated the digit string. But then the digit string cannot be “one that can only be generated by a program longer than this”. This obviously leads to the fact that the search must fail, even if it is an infinite search. The search was intended to find a digit string that needed a generating program at least as big as the search program, which is to say that any shorter program has to be ruled out. But as the search fails, we cannot be sure there is no shorter program, as we do not know whether a given digit string can be encoded in a program shorter than the one we happen to have discovered. Now here’s the rub - a random sequence is one that cannot be algorithmically compressed - but as I have just shown, you cannot know whether or not a shorter program exists for generating that sequence. The cardinal point in these algorithmic programs is that you never know if you’ve unturned every stone in trying to shorten the description. Therefore you cannot prove that a sequence is random, although you could disprove it by actually finding a compression. If you are paying close attention you will see that this is isomorphic with the Robert speaking English model - here you have much greater access to empirical evidence (certainly less complex than algorithmic mathematics) but you can find the shortest compression (so to speak) by hearing him speaking English. The practical conclusion here is twofold. In the first place, one can prove mathematically that almost all digit strings are random, but one cannot know precisely which. But more essentially for everyday purposes, taking the cosmos as an algorithmic whole, events or activities that appear random may not be random at all - even things like the indeterminism of quantum mechanics. The cardinal point here is not that something like, say, Heisenberg’s Uncertainty Principle might belong to uniform laws of which we as yet know nothing, because I am sure that that is the case. The cardinal point is that we might never be able to know - in fact, Chaitin’s theorem ensures that we can never PROVE that quantum mechanical measurement outcomes are random.

But how does that work for the theist - after all, how does God load the quantum dice to have a random yet uniform world as well as having the difficulty of human minds that can affect things in the cosmos with God kindled free will? A stochastic system can produce the ‘random yet uniform’ cosmos, but what about free-thinking minds? Think about it like this; regarding an electron in quantum mechanics a position measurement is indeterministic because it is free to choose among a limited range of possibilities. On the other hand the measuring itself is contingent on whether one is measuring ‘position’ or ‘momentum’ - that is to say, the nature of the alternatives is fixed by an external agent - the one doing the measuring. To make this clearer, imagine a game in which you have to guess which whole number I am thinking of between 1 and 250 - you only have 20 questions or guesses and I can only answer ‘yes’ or ‘no’. imagine your questions begin as follows - Q1 - Is it bigger than 125? A - Yes; Q2 - Is it even?A YesQ3 - Is it divisible by 7?A -Yes Q4 - Is it 232?A - Yes. Now here’s the trick; let us say that I hadn’t chosen any number in advance - I had agreed to answer purely at random subject to consistency only with the previous questions. The answer was not determined in advance, but it was not arbitrary either - its nature was decided particularly by the questions asked but partly by chance (depending on what was being asked, and when the final guess was offered and the final answer given). After question 1, my answer was limited to 126-250, after the second question my answer was limited to all even numbers between 126 and 250, after the third question the range was lessened to those divisible by seven, thus preventing me from answering, say, 233 but giving me several other options. If the fifth question to which I randomly answered ‘yes’ was ‘Is it greater than 230?’ - then the only remaining number that would fit the five ‘yes’ criteria would be 238. In the same way, the reality of quantum mechanics is decided in part by the question - whether one is asking about the position of the electron or its momentum and in part by the uncertain nature of the values obtained for these quantities. The Divine analogue here is that God determines certain uniform facts about creation and what other options are available in any given system, but a variety of subsystems in which randomness plays a part too in choosing among several alternatives. The freedom of the individuals’ choices is much the same, it is freedom within a set of ordinances that are statistically part of a uniform plan - an end result that has already been predetermined by God outside of creation. Just because God already knows which way the coin will land before I throw it, it doesn’t alter the fact that for a free-thinking mind like my own the result is an unpredictable 50-50. So, regarding this subject, these are the four basic principles of logical analysis - and one must be mindful of the above when contemplating the big question of whether God exists or not.