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anewbeginning
Registered Member
Date Joined Jan 2002
Total Posts : 88
| Posted 2/18/2003 6:05 AM (GMT -6) |   | In the "Open Theism and the Principle of Bivalence" thread we talked a little about Godel's theorem. I would like your comment on this theorem, which utilizes Godel's theorem as part of a proof against humans being deterministic automata. I'll reproduce the main part of it here. The entire proof can be found at:
www.starcourse.org/lucas.htm
Lucas' Theorem
Are we, as far as our thinking processes go, nothing but 'Computers made of Meat'? Is it possible that Scientific Laws will be discovered that totally explain human behaviour? Lucas' Theorem, due to the great Oxford Philosopher John Lucas FBA, explains why both these hypotheses are impossible.
Note for the rigorous/pedantic: if by computers you mean 'any entity that can do a computation' then of course we are, in that rather vacuous sense, computers. (My mother was actually employed as a "computer" in the Admiralty Research Lab in the 50s) What Lucas' Theorem shows is that, unlike artificial computers (or anything equivalent to a Turing Machine), we are not automata.
Background
In the early days of mathematics, the Ancient Greek mathematicians got a shock. They had been investigating rational numbers (numbers of the form a/b where a and b are whole numbers) and had been making great strides in their ability to do calculations with and about them. The puzzling variety (2/3, 4/6 and 6/9 are all the same number) had been tamed by proving that any rational number can be expressed uniquely in its 'lowest terms' (ie when a and b have no common factors) and it seemed that every mathematical problem could be solved by them. Some of their philosophers went so far as to suggest that "everything is a number".
But whilst equations like 3x + 5 = 7 could be solved by rational numbers (x = 2/3) other proved more intractable. Consider x2 = 2. x = 7/5 is fairly close: 49/25 is only slightly less than 2 - 141/100 is even better. Clearly, by using larger and larger denominators you can get closer and closer to the answer.
Then a pesky mathematician showed that, however closely you could approximate the answer by rational numbers, you can never get there. For suppose that there was a rational number a/b such that (a/b)2 = 2. Then there must be a way of expressing the fraction in its lowest terms, as c/d say, where c and d have no common factor. But if (c/d)2 = 2, then c2 = 2d2, and thus c2 is an even number. But the square of any odd number is an odd number, so c must be an even number: 2f say. Hence (2f)2 = 2d2 so 4f2 = 2d2 so d2 = 2f2 so d2 is even, and hence d is even: 2g say. But this is impossible because c/d was a fraction in its lowest terms. Hence there is no rational number which solves the equation x2 = 2.
Note 1 it is not enough to say "ah well, let's take f/g as the new solution" because the proof can also be applied to the 'new fraction' f/g. Because this proof applies to any possible fraction it shows that no fraction has the required property. Changing the fraction half way through to 'avoid the contradiction' does not work.
It is said that the first mathematician to prove this was put to death for his un-orthodoxy, and it was only many years later that other mathematicians brought the result to the attention of the educated public, so that it became an undoubted result.
Now all the 'arguments' for scientific determinism and "the brain is a computer" go along the lines of "of course we don't have anything like a complete theory yet, but we are continuing to make advances and so eventually we will have the full picture". You might have thought that the advocates of this line, who are all (presumably) aware of the above proof, must realise that this argument is completely fallacious, even if the gullible public to whom they offer the pronouncements does not. But in fact, Lucas's Theorem shows that, not only are the 'arguments' for Scientific Determinism and The Brain is a Computer fallacious, but the hypotheses are logically impossible. (In addition, it is now known that even if the components of the brain were completely deterministic, which is far from certain, the brain as a whole would not be. See here for details).
John Lucas FBA was an Oxford Philosopher until his retirement in the 1990s. In the early days of Artificial Intelligence he proved a general result that has fundamental importance to any discussion of the distinction between computers and human beings, which was announced first in a paper and then in his book The Freedom of the Will (OUP).. Investigators of computers have been making great strides in their ability to do calculations which mimic the results of the human intellect. Decision making, calculation, symbolic manipulation, speech synthesis, expert systems, neural networks and the like provide closer and closer approximations to given aspects of human behaviour. Similarly, much trumpeted advances in neuroscience provide increasingly deeper understanding of the mechanisms of the brain. Surely it is "obvious" that eventually computers will be able to mimic human behaviour totally, and that scientists will be able to explain all aspects of the brain, and hence the mind. Far from being uniquely made in God's image, humans are 'nothing but' computers made of meat?
Lucas' Theorem Argument
Essentially it goes like this:
Define a Deterministic Logical System as a set of (at least two) states S, a set of inputs I and a logical system L which uniquely defines, given a state s in S and some inputs i in I, what the next state s' of the system will be.
There is a non-empty set of human beings HFree such that, for any member h of HFree, there does not exist a Deterministic Logical System which would accurately predict all h's actions in all circumstances.
Proof: Let HLog be the non-empty set of mathematical logicians capable of understanding Godel's theorem (see Defn 7 for a formal definition), and let h be a member of HLog. Suppose there exists a specific Deterministic Logical System (S1,I1,L1) which would accurately predict all h's actions in all circumstances. Since h can do elementary arithmetic, L1 must be rich enough to contain elementary arithmetic and thus by Godel's theorem there exists a proposition GL1 in L1 whose interpretation is "GL1 cannot be decided by L1". Consider now the circumstance where h (while rational) is asked: "is GL1 true?". h will rationally answer "yes", by reasoning along the following lines: "L1 cannot be contradictory, because if L1 were contradictory then (since in a contradictory logical system, if you can prove P you and prove not-P) the next state of the system would not be uniquely defined. But if GL1 were false, then it would mean that "GL1 cannot be decided by L1" can be decided by L1, which is a contradiction. Hence GL1 is true." However, by definition, L1 cannot decide GL1 and thus will answer "don't know". QED.
Note 2: Some people try to avoid this proof by arguing that "You've just given me a procedure that allows me to program a computer to answer appropriately" and so all that would be necessary would be to extend L1 with this procedure (making L2 say). This is exactly analagous to the logical mistake explained in Note 1. Because this proof applies to any possible Deterministic Logical System it shows that no Deterministic Logical System has the required property. Changing the Deterministic Logical System half way through to avoid the contradiction does not work.
Note 3: Clearly the term "capable of understanding Godel's theorem" is a trifle informal. The specific property we need is that h is capable, for any Deterministic Logical System (S,I,L) of recognising (and verifying, if necessary with the aid of a computer) that GL is a Godel proposition in L, and reasoning rationally about it along the lines indicated. If you want a formal proof, one is given here.
Corrollary 1: All rational human beings are members of HFree. Proof: it is absurd to suppose that there is anything so epistomolgically unique about mathematical logicians, and in principle all rational human beings are capable of being taught to understand Godel's theorem, .
Corrollary 2: No computer/automaton can competely model the mind of a human being. Proof: a computer/automaton is a Deterministic Logical System, and even if there are random factors in the program, these are ultimately either 'pseudo-random' or depend on random inputs, which can be made inputs to the Deterministic Logical System.
Corrollary 3: No scientific theory can ever be discovered that completely accounts for human behaviour. Proof: Any scientific theory that can be discovered which completely accounted for anything must be a Deterministic Logical System. But we have shown that no Deterministic Logical System can completely account for human behavior. Hence QED.
Note 4: none of these corrollaries deny that you can make increasingly accurate models or theories: they only show that the process will never finish. Similarly, you can make increasingly accurate approximations to the solution of the equation x2 = 2 in rational numbers, but you will never get there completely.
Note 5: It seems that the connection with freewill is fundamentally that a human, unlike an automaton, is able to choose in which logical system to work. The human logician, faced with a contradiction if they were limited to the logical system L, chooses to step outside it to decide the question. An automaton cannot do this: even adding a "rule" which says " when faced with a contradicion in a logical system then step outside it" does not rescue the automaton from the proof, because it simply extends the automaton's Logical System by this rule, making a new logical system L' - and recall that the proof applies to any logical system.
Amazingly The Freedom of the Will is out of print. But similar arguments were made by the outstanding mathematician Sir Roger Penrose FRS in The Emperor's New Mind and Shadows of the Mind. There is no doubt that the proof is correct, however uncomfortable and counter-cultural it may be to many in the Western intelligensia.
I just wondered if you felt this argument holds water.
Thanks.
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Alan Rhoda
Registered Member
Date Joined May 2002
Total Posts : 680
| Posted 2/18/2003 2:00 PM (GMT -6) | | I read Lucas's "Freedom of the Will" about 12 years ago. I was impressed with his Godelian argument that humans are not "computers made of meat" then and am still impressed now.
While I'm no expert on Godel's theorem and its implications, my understanding is that Godel proved that any logical system complex enough to contain the basic axioms of arithmetic (Peano's axioms) allowed the formation of statements in that system that could not be decided (as to their truth value) in that system. One could only decide them by changing the logical system--adding more axioms, but then the same result applies to that system, and so on.
Lucas's application of Godel's proof is truly ingenious. He argued that any being capable of proving Godel's theorem could not be adequately represented by a logical system. The basic reason is this: to show that any logical system is incomplete one has to transcend the system. But Godel's theorem shows that all logical systems are incomplete. Therefore, one can prove Godel's theorem only by transcending all logical systems. If human's were just deterministic logical systems then we could never achieve the perspective to see that Godel's theorem is true. Therefore, humans are not deterministic logical systems. There can in principle be no complete lawlike theory of human behavior.
IMHO this is a sound argument. It doesn't completely refute determinism (theological determinism could still be true), but it does refute the idea that we are just complex machines. As for theological determinism, I believe that Lucas also argues against that in his book. He was a kind of proto-open theist, I believe.
God bless,
Alan
Anyone who wants to drag in the irrational where the lucidity and acuity of reason still must rule by right merely shows that he is afraid to face the mystery at its legitimate place. --Karl Mannheim | | Back to Top | | |
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