The Invalid Results of Paul Davies

The Refutation of Paul Davies' Turing Machine Hypothesis and All That This Hypothesis Implies

by Robert A. Herrmann Ph. D. 28 March 2000, revised 23 DEC 2006.

Introduction
Paul Davies is the author of the [I] "The Mind of God: The Scientific Basis for a Rational World." Davies is a "Templeton Prize" winner. This prize is supposed to be given for a significant "Progress in Religion." However, Davies' work is a "giant leap backwards" rather than an advance. Indeed, this refutation of Davies' hypothesis indicates the gross errors that are continually being made in the awarding of this prize.
In this article, I will refute statements Davies makes in the book [I], and elsewhere, about logic, mathematics and Turing computation, which is related to computer programming notions. This article will also refute his notion of what constitutes the "Mind of God" relative to creation. I mention that, relative to science, I have used the phrase "The Mind of God," with a significantly different meaning, since 1978. It is clear to me that Davies has presented no information that allows one to have any specific knowledge of a God such as that described within the Bible. He also claims "[T]hat maybe the ultimate answer cannot be obtained through reason but only through mysticism . . ." a statement that I have demonstrated to be false. Indeed, Davies apparently has no actual experience with the methods God uses to make His supernatural presence known to His created when he makes such statements as "Science may offer a surer path to God than Religion" ["God and the New Physics"] (Note that this last statement seems slightly inconsistent with his previous thoughts.)
This article will concentrate upon one of Davies' basic blunders, his attempt to correspond the results of natural law with the notion of a "Universal Turing" machine. This will be done explicitly. Further, this will counter Davies' naïve attempt to characterize certain Divine attributes of God - the sustaining and continual supernatural control over all aspects of our natural universe - to standard mathematical structures. This is also an error made by Hugh Ross but, in Davies' case, his contention relative to the natural universe itself can be very specifically refuted. Davies presents the philosophy of the "Prime Mover," in that God may exist but His purpose was only to create the standard mathematical structures that appear to guide all the natural-systems and than simply let the laws "do their own thing" so to speak. In the Davies' writings, I find no indication that he has been presented with any true and personal evidence that an actual supernatural deity exists - for if he had such evidence, he would know immediately that such a hypothesis is false. What Davies does, as with other individuals who only approach God through the essence of philosophy, is to present linguistic descriptions for claimed Divine attributes, a "paper and pencil" description. Although, I also present such descriptions, I also freely admit that without an "indescribable" personal relation with God, as say expressed by 1 John 2:27, one has no means to verify that any such description for a Divine attribute is correct. Without such verification, one can expect error to be prevalent.
It is extremely clear from the Scriptures that the supernatural attributes of the Godhead are, at the most, but partially comprehensible when they are compared to natural attributes. For each of these numerous comparisons, it is stated specifically that the supernatural attributes are very dissimilar from natural attributes in power and divinity. Scripturally, only through the continuous application of supernatural processes does our universe "continue" to exist. Since standard mathematical structures are used to model natural-system behavior within our universe and the natural attributes of God's created, using these same structures to model God's supernatural aspects is closely related to pantheism. The only presently known mathematical approach that avoids this pantheistic correspondence and that does yield, at least, a partial comprehension of God's supernatural attributes as they are compared with of those of His created, is the approach using Nonstandard Analysis. This is the approach used in every article on this Web site that analyzes the attributes of a Deity - a Deity with attributes that will always be far superior to any (corporeal) physical-system either within or exterior to our universe (if such exist). I won't repeat any of this nonstandard analysis in this article, but, in the next sections, I will concentrate on the actual "standard" misunderstandings used by Davies to lead his audience astray. I have written on this previously in reference [1]. [The following counter to the Davies' claim is not related to the simplistic computer windows illustration for event sequence construction that appears in the book "Science Declares Our Universe IS Intelligently Designed." Such a process illustrated in my book yields some additional intuitive comprehension as how event sequences can be pre-designed within the Nonstandard Physical World by a higher intelligence.]

Formal Verses Informal
Davies' basic misunderstanding is relative to "formal" logical deduction as done by a mathematical logician, if at all, and the art of "informal" mathematical deduction which are the usual thinking processes one associates with a "mathematician." It appears as if Davies has never created new mathematics since he would know that it is more of an art form than a science. In [1], I discuss these facts and that there appears to be no fully describable set of rules or processes that I "informally" apply to know in advance that a particular mathematical "theorem" can be successfully established. I simply "know intuitively," in most cases, when a particular statement is probably provable as a mathematical "fact," and whether or not I can successfully establish the result. This I have done a few thousand times. The concepts that Davies uses to come to his conclusion that the "laws of nature" are "Turning-computable" (to be further explained) are relative to formal mathematics. The basic Davies claim that will be refuted is that natural laws are Turing-computable and as such the process of combining together the natural laws to produce the behavior of a natural-system can be represented by what is called a "Universal" Turing machine (a theoretical concept related to our modern computers.) My refutation of this philosophical stance will come when I demonstrate, explicitly, that there is natural-system behavior that does not correspond to the output of a universal Turing machine.
First, I'll briefly demonstrate "formal" mathematics and you'll surely see that it does not correspond to the mathematics most of the world is exposed to in say a first, second, three, etc. course in the calculus. Formal mathematics deals with finite strings of symbols written left to right in a very special and simple manner. The rules that allow you to write down the strings on a piece of paper or a computer screen are very simple to follow, we hope. The strings are called formula if they are generated by a set of specific rules or sometimes we need to select the formula we wish to use from a specific set constructed from a more general set of rules, where many logicians call these selected strings of symbols well-formed formula. No matter what you might call them, they are still but a set of "nonsense" symbols written down. This means they have no actually "meaning." All you are allowed to do is to follow the rules, write symbols in a certain way left-to-right and do nothing else. Now how does one know that you've followed the rules? Well, you must also write down the name or number of the rule you've used so that what has been listed can be checked by others. Your list looks just like an "informal" geometry proof you may have done in school. Below is an actual list using symbols and rules for what is called the formal theory of the natural numbers. One list of formal "axioms" S = {S1, . . . S9} for the natural numbers is presented in a formal "first-order language" in reference [2, 117]. [Any formal theory that has the same theorems as S is often called PA.) These axioms do not include the form a = a, which is a theorem. Using the axioms, one can "formally" prove something. (You don't need to "know" what all the following stuff means. It's just a demonstration of the "form" in which formal mathematics is presented. You also might ask whether this type of "puzzle" is truly a game for those "highly" intelligent mathematicians to play? You'll need to wait before I can answer this.) Now the symbol "a" below, represents a "variable" and, when the axioms or allowed "formal deduction" procedures are applied, then the name of the rule or procedure applied is stated in the second column.

a + 0 = a....................................................Axiom S5.

(a + 0 = a) -> (a + 0 = a -> a = a)........................Axiom S1.

a + 0 = a -> a = a..........................................MP(1,2).

a = a.........................................................MP(1,3).

So what this result means is that we've "formally proved" from our axioms and allowed logical methods that a = a. One might think it very strange to establish by strict formal means that 2 = 2 but the fact is that this is not one of the axioms of S. There are various formal axiom systems similar to S that yield the exact same formal theory. The theory is called Peano Arithmetic or PA
The famous Gödel results deal with "formal" proofs and not with what the majority of mathematicians who specialize in abstract mathematics actually do - the "informal proof." There is a great deal more "to this story", but there seems to be little doubt [1] that the informal methods used are vastly more powerful than formal methods and, probably, there does not exist a specific set of rules that can be given that will allow all individuals to achieve the same results as those obtained by the mathematician. This is the area where the subject becomes more of an art form in that the discipline has no strictly definable complete set of methods such as those called "the scientific method."

The 1977 Paris-Harrington Result

Consider the set of six symbols (i.e. natural numbers) K = {0,1,2,3,4,5}.

Let the "2-graph" of K, G(2,K), be the set that results from writing down all of the 2-element sets that can be constructed from K. This set contains 15 members.

G(2,K) = {{0,1},{0,2},{0,3},{0,4},{0,5},{1,2},{1,3},{1,4},{1,5},{2,3},{2,4},{2,5},{3,4},{3,5},{4,5}}.

A 2-partition of G(2,K) is a re-grouping of these 15 2-element sets into 2 subsets of G(2,K). Below we give five examples where the symbols [ ] are used to indicate these subsets. Reading left-to-right the first [ . . . ] is partition I, and the second one [ . . . ] is partition II.

[{0,5},{0,1},{0,3},{0,4},{2,5}],[{0,4},{1,2},{1,3},{4,5},{1,4},{2,3},{2,4}{3,4},{1,5},{3,5}].

[{0,1},{0,3},{1,3}],[All other 2-element sets.].

[{0,1},{1,4},{1,3},{4,5},{3,5}],[{2,3},{2,4},{3,4},{0,2},{0,3},{0,4},{0,5},{1,2},{1,5},{2,5}].

[{0,1},{1,4},{1,5},{4,5}],[All other 2-element sets.].

[All other 2-element sets.],[{4,5}].

Now we notice the following facts about these five partitions. For no. 5, let the set H = {3,4,5} and let #H mean the number of elements in H. Then #H = 3. Observe that G(2,H) = {{4,5},{3,4},{3,5}} is a subset of partition II.

For no. 6, let H = {0,1,3}. Then #H = 3. Observe that G(2,H) is a subset of partition I.

For no. 7, let H = {2,3,4}. Then #H = 3. Observe that G(2,H) is a subset of partition II.

For no. 8, let H = {1,4,5}. Then #H = 3. Observe that G(2,H) is a subset of partition I.

For no. 9, let H' = {0,1,2,3,4}. Then #H'= 5. Let H'' = {0,1,2,3,}. Then #H'' = 4. Let H''' = {0,1,2,}. Then #H''' = 3.Observe that G(2,H'), G(2,H''), G(2,H''') are all subsets of partition I. Note that if you have any H such that #H > 3, and G(2,H) is a subset of one of the partitions, then, since we are using 2-element graphs, we can remove members from this set H one at a time to get a set H' such that #H' = 3 and the graph G(2,H') will be a subset of the same partition.

We now repeat the above process for the K' = {0,1,2,3,4}.

G(2,K') = {{0,1},{0,2},{0,3},{0,4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}.

Consider the partition [{0,1},{1,2},{0,4},{3,4},{2,3}],[{0,2},{1,3},{1,4},{0,3},{2,4}].

It is not difficult to trace out all of the possibilities like H = {0,1,x} and show that no such set H such that #H = 3, (that is, of course, a subset of K') exists such that G(2,H) is a subset of either of these partitions. Using the remark in 14, then there cannot exist any subset H of K' such that 5 > (or =) #H > (or =) 3 and such that G(2,H) is a subset of either of the two partitions.

Let n denote the number of members of our of n-element sets that are composed of members of a set of natural number (symbols) K = {0,1,2,3,4,5, . . . k}. Let S denote the number of partitions used and #H = L. What has been demonstrated above?

For k = 6 , S = 2, n = 2, L = 3 and for each of the five two division (S = 2) partitions illustrated, there exists a subset H of K such that #H = L= 3, and G(2,H) is a subset of one of the partitions.

But for the case of k = 5, we found, at least, one of the S = 2 partitions where no such H exists, with #H = L = 3, such that G(2,H) a subset of either of the partitions.

Why are things like this happening? Well, one version of the Finite Ramsey Theorem, that can be formally established using PA [3], shows that the problem is with k . That is if we start with a fixed set of numbers S, n, #H = L, then there is a k such that no matter how you distribute the n-element sets among the S partitions there will always exist an H with #H = L and G(n,H) is a subset of one of the S partitions.

The above result does not refute the Davies hypothesis for there exists a formal proof of the Finite Ramsey Theorem. There is one additional observation about the partitions 5 - 9. Notice that in 10 above the smallest number 3, that appears in the H, is equal to the #H; in 11, the smallest number in H is < #H; in 12, the smallest number in H is < #H; in 13, the smallest number in H is < #H; in 14, the smallest number in H''' < #H'''. So, we have an obvious question. Can the Finite Ramsey Theorem be extended so that the following conjecture holds? (T) If we start with a fixed set of numbers S, n, #H = L, then there is a k such that no matter how you distribute the n-element sets among the S partitions there will always exist an H with #H = L, the smallest member in H < (or =) L and G(n,H) is a subset of one of the S partitions.
Paris and Harrington [3] have shown a startling result by application of allowable mathematical methods. (A) Such a k always exists so that such an H will have the indicated properties; that is conjecture T is a theorem. This theorem is established by human mental processes and an extension of the Finite Ramsey.

Refutation of the Davies Hypothesis
Accepted natural laws yield natural-system behavior. The Davies hypothesis implies that the natural laws that the human brain employs to achieve an outcome are representable by Turing-computable functions. Specifically, he claims that these natural laws can be simulated by a universal Turing machine, U. What constitutes a natural law is merely a choice made from among a large collection of "statements." It is self-evident that a finite collection of expressible natural laws is also an expressible natural law. Such a set restricted to a particular physical scenario is, from a statement viewpoint, a natural law that governs the physical scenario and, under the Davis hypothesis, is representable by a Turing computable function that is computable by U; a machine that computes all such functions. [**See statement just prior to references.]
Using the Paris-Harrington theorem T, a 4-ary numerical-relation R can be properly defined such that (S,n,L,k) in R if and only if S,n,L,k satisfy theorem T. (The notation for these natural numbers I use here is not the same as that used by Paris-Harrington.) Consider the characteristic function C defined on all 4-tuples of natural number by requiring C(S,n,L,k)=0 if (S,n,L,k) in R and 1 otherwise. The Davies hypothesis is that human mental processes governed by "natural laws" have lead directly to the definition of the total numerical-function C and, hence, C is computable by a universal Turing machine U. However, application of Corollary 5.9, Proposition 3.23, and then Proposition 3.12 in [2] yields that the relation R is expressible in PA. From this it follows that theorem T, when formally, expressed [3, p. 1135], is "formally established in PA." However, this contradicts the Main Theorem 1.3 [3, p. 1134]. The notion of "cannot be formally established in PA" refers to the specifically defined methods for writing down a formal proof using the axioms of PA and first-order predict logic. Essentially, the Main Theorem 1.3 depends upon the known fact that (*) the simple consistency of PA cannot be formally established using PA and these allowed methods. From the viewpoint of a universal Turing machine, there is a Turing machine "proof" of (*) that relies upon the fact that a special function is not "computable" by a universal Turing machine U. For a somewhat informal proof of this, see reference [4, p. 255]. Reference [3, p. 535-538] is also helpful relative to the modern definition for a Turing machine.
Now one could arrive at the same conclusion that a universal Turing machine cannot determine the consistency of PA by using the self-reference "Halting" problem that shows that a question about the Turing machines themselves cannot be determined by U. But, in practice, this is more of a mental activity rather than something one would actually accomplish physically for the standard interpretation of the Halting problem is based upon a computation "never" stopping and "never" is a very long time to wait. In this Pairs-Harrington case, the expression that we know is "true" is about marks on a piece of paper or your computer screen as in the above example of the Paris-Harrington result - tangible finite physical things since k is an ordinary "finite" natural number that you "know" exists. It thus becomes a question about actual physical objects within the universe, marks on paper or on a computer screen that can be found within a finite time period. This shows that the universal Turing machine cannot produce, at least, one practical result. Such a machine cannot be used to produce all of the results of human thought and its consequences, the human activity demonstrated by the above construction. Of course, the machine can be specifically instructed to make a specific search for an appropriate k given the numbers S, n, L. Although you will know that the machine will find an appropriate k in finite time, the machine cannot compute the fact that such a k always exists for all such numbers S, n, L. Consequently, the basic Davies hypothesis is false relative to an animate object.
Can we apply the Davies hypothesis to inanimate (i.e. lifeless) objects and demonstrate that it remains false? In the above example, consider a set of k boxes with the numbers from 0 to k written on each separate box. The number denotes the number of objects in the like numbered box and you would need numerously many boxes with the same number of members. For our example K = {0,1,2,3,4,5}, you would need five empty boxes, five boxes containing only 1 object, . . . , five boxes contained 5 objects. These boxes can then be considered as arranged into n-element sets. (Note the term "set" signifies that there are no two boxes in these n-element sets of boxes that contain the same number of objects.) Now partitioning simply means grouping the n-element collections of boxes themselves into S collections. The Paris-Harrington Theorem that cannot be established by a U machine can simply be applied to certain collections of such boxes. Note that in certain cases the actual number k may be extremely large, but it is still finite. The actual display of the boxes need only be in the form of n-graph for K that satisfies the conclusions of T. Suppose that the boxes where disjoint 3-dimensional "regions" within our universe at a specific moment. Each region contains "locations" or "positions," termed "balls," of a small macroscopic size determined by the size of a fixed physical object at the universe wide moment under consideration. If the region contains more than one ball, then the distance between each pair of balls is the same as the size of a small macroscopic object. These sizes could be related to such universal notions as the "wave-length" of a specific photon or same other simular physical object. The zero box, would correspond to a region slightly smaller than a ball. Clearly, this would require a universe with a potentially-infinite "number" of balls before the complete collection of the Paris-Harrington physical-configurations, the G(n,K) , could even be considered. Such fixed types of "physical locations" (the balls in this case) can be modified depending upon the natural processes being considered. Except, for one required property that may be altered in the future, it appears that our universe may yield an inanimate example.
Apparently, under the notion of the expanding universe, a natural process exists that appears to be continually creating a potentially-infinite collection of Paris-Harrington physical-configuration termed "space" as cosmic-time progresses. (An infinite spatial universe is also potentially-infinite.) Together, this yields a potentially-infinite space and time. The natural constituents that yield this behavior have recently been termed as "dark energy." Dark energy is, of course, a natural process. There are various proposals as to the physical behavior and what are the physical constituents of such dark energy. It appears that there are equally possible ultimate effects produced by such dark energy. One possibility is that "expansion" will never cease. Consider this possibility. The ball locations are not to be considered as "empty" but rather are composed of the physical constituents that yield this expansion effect. On the other hand, if physical processes are capable of producing a potentially-infinite amount of matter such as the (1) Quasi-Steady State creation of matter model, or the potentially-infinite collection of universes such as postulated by the (2) Everett-Wheeler-Graham many-worlds model, or (3) various cyclic universe proposals, where universes would continually be produced by quantum-field fluctuations, or (4) many other multi-universe models, then a modification of the description as to what physical constituents are being considered as the physical composition of the ball locations is rather easily made. Relative to "expansion" observations, it is obvious that some of the proposed natural processes being considered are designed to eliminate the notion that our universe is a one-time unique event. Since it is more a matter of choice than hard observational evidence, I have no doubt that the secular scientific-community will eventually accept a model for such expansion effects that will satisfy the observed data and lead to a cosmology composed of a potentially-infinite collection of universe-like objects. Whether or not the constituents of the dark energy lead to a potentially-infinite universe actually does not affect the following falsification of the Davies hypothesis. All one needs is that the parameters used to determine the effects of the dark energy constituents could lead to such a potentially-infinite possibility. From the philosophy of quantum mechanics, I have no doubt that this will be a necessary conclusion for any dark energy theoretical constructs. Dark energy is but one example, however, since any physical process for which there is a statistical probability that it is capable of producing any potentially-infinite collection of Paris-Harrington physical-configurations would suffice. The notion of "location" can have non-spatial interpretations.
Thus, using the material discussed in the last paragraph, for a given S, n, L, and for each k , proposed natural processes can produce the required collection of Paris-Harrington physical-configurations now or in the future. Evidence that the Paris-Harrington Theorem conclusions hold for some k comes from the notions of inductive quantification and the uniformity of nature assumption, where both notions are essential ingredients within modern scientific methods. The Paris-Harrington Theorem T conclusions that cannot be completely established by a universal Turning machine, can be partially established by explicit paper-and-pencil activity and computer generated combinations of natural numbers and these combinations can then be related specifically to Pairs-Harrington physical-configurations that satisfy theorem T. From this, the usual methods of scientific generalization and application of the uniformity of nature leads to the Pairs-Harrington conclusions being accepted, in general, in that for each triple (S,n,L) there is a Pairs-Harrington physical-configurations that satisfies T. This is one of the most obvious uses of the notion of scientific generalization which is often used with much less evidence than would be displayed by these concrete examples. Under this scientific generalization process, which some term induction, there are physical consequences of the natural process that yield a potentially-infinite collection of Paris-Harrington physical-configurations that satisfy T, and these consequences cannot be Turing computed by U. This falsifies the Davies hypothesis.
For all of the following examples, the evidence that such Paris-Harrington physical-configurations exist that satisfy T is accepted and our desire is to determine what types of physical processes lead to such configurations. Unfortunately, due to the requirement that one major type of dark energy be homogeneous, this type of dark energy does not give a true completely inanimate example. The example is but an extension to the potentially-infinite of the human construction via paper-and-penile or computer activity that yield Paris-Harrington physical-configurations that satisfy T. However, since any finite combination of physical processes is a physical process, it is an example of a combination of actual inanimate physical notions and brain processes that would falsify the Davies hypothesis. If dark energy can be statistically considered as actually composed of random collections of constituents with slightly varying strengths and the selected parameters allow for continual expansion, then there would be a probability that such Paris-Harrington physical-configurations exist that satisfy T, where the members of each "box" would be identified by the "strength" of the expansion effect. This would be sufficient to also falsify the Davies hypothesis.
As mentioned, identifiable physical clumping for some physical objects, if they also all possible collections of Paris-Harrington physical-configurations, from the potentially-infinite viewpoint, would yield inanimate examples. All that is actually needed is one physical process that has the possibility of producing such identifiable variations. The process need not be attenuated by any other physical process. A viable interpretation of a quantum-physical process, accepted by some physical scientists, has this capability and is the bases of model (2). For model (1), there is also a statistical possibility that in this cosmology, that has no beginning and where matter is being continually created, such identifiable clumps of matter would form the required Paris-Harrington physical-configurations that satisfy T. The scalar field proposed for (1) has similar properties as other accepted quantum-fields. Models for (3) and (4) have been proposed using accepted and new natural processes. What is somewhat significant here is the even if the cosmologies proposed use new natural processes, the theoretical aspects of these new natural process are produced in the exact same logical manner as the properties for similar and accepted natural processes. Consequently, even if these new processes are rejected there consequences, according to Davies, should be "computable" via a universal Turing machine. But, this would not be the case.
If such natural processes as discussed in the previous paragraph are capable of producing such configurations, then would this provide examples of "true" inanimate representations for the Paris-Harrington Theorem? (A) Some might claim that such examples still appear to require a "counting" or "cataloging" by an intelligent biological creature before the pattern would be recognized. (B) Further, unless direct verification is possible these examples appear to come about, at least in part, by human mental constructs and are mainly animate examples that need not correspond to reality. But, for (A) would not the potentially-infinite collection of configurations that establishes the Paris-Harrington Theorem still exist whether or not an intelligent being recognized the pattern? For (B), this is difficult to answer since it refers mostly to future determinations. However, scientific communities accept many "natural processes" as existing in reality only based upon indirect evidence and explanatory power. Hence, for case (B), the choice that is made does influence whether the example is inanimate or not. Unfortunately, there can be no actual direct sensory evidence that exactly verifies that any physical objects form a potentially-infinite collection. Moreover, this is an old philosophic problem as to the assumed necessity of human observation and human mental activity. But if as Davies contends, "that we are truly meant to be here . . . " [8], then all of these would be Davies allowable contradictions to his universal Turing machine hypothesis since we cannot "be here" and then discount the human brain activity that has been accorded us. Since these examples are based upon individual choice and that actual potentially-infinite physical constituents are necessary, then the conclusions that any one of these examples (1) - (4) is a true completely inanimate falsification of the Davies hypothesis must wait until the majority of physical-science community accepts one of these scenarios as fact. However, we have presented example(s), that imply that there are human brain processes that cannot be "computed" via a universal Turing machine.
If none of the proposed inanimate examples are accepted as fact, then Davies can preserve his hypothesis by simply conceding that there are aspects of human thought that can be rationally modeled [7] and that do not correspond to any accepted physical laws. That is, he can restrict his hypothesis.
From a theological point of view, a potentially-infinite collection of Paris-Harrington configurations does not contradict the existence of a creator deity. Such possibilities are predicted by the GGU-model and none of these GGU-model predications contradict the existence of a God that can create them [5,6]. However, this requires consideration of nonstandard mathematical modeling. Among other things, this contradicts Davies' contention that standard mathematical notions and standard science allows us to truly see into the mind of God.
[Note: Although not as practical or explicit as the above examples, Roger Penrose also concludes that results of human thought cannot be reproduced by a universal Turing machine. The Penrose physical examples are all related directly to the Halting problem and do not appear to effect adversely the Davies hypothesis with respect to inanimate objects. ("The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics," (1989)).]
**Intuitively, for the Turing computable hypothesis, one assumes that a fixed set of natural laws applied to a fixed physical scenario should produce the same describable results. Such a situation is intuitively algorithmic in content due to the usual requirements for exact repetition. Through proper encoding one would conclude that the description for such an algorithm might be Turing computable. In this article, I describe animate and inanimate scenarios that are assumed to follow from a set of natural laws but do not satisfy such a Turing computable conclusion.

References
[1] Incompleteness, Creation-Science, and Man Made Machines, Creation Research Society Quarterly, 31(1994):148-152.
[2] Mendelson, E. Introduction to Mathematical Logic, Wadsworth & Brooks/Cole, Monterey, CA (1987).
[3] Handbook of Mathematical Logic, Ed. Barwise, North-Holland, New York, pp. 1133-1142.
[4] Kleene, S. , Mathematical Logic, John Wiley & Sons, New York, (1967), p. 255.
[5] Herrmann, R. A. Solutions to the General Grand Unification Problem . . . ., (1994):13 http://arxiv.org/abs/astro-ph/9903110
[6] Herrmann, R. A. Science Declares our Universe Is Intelligently Designed, Xulon Press, Longwood, FL. (2002)
[7] Herrmann, R. A., The rationality of hypothesized immaterial mental processes, C. R. S. Quarterly 43(2)(2006):127-129.
[8] Kirkus Reviews, Kirkus Associates, LP, (1991).

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