The Image of a Machine in The Liberation of Life by Granville C. Henry, Jr. Granville C. Henry, Jr. is Associate Professor of Mathematics and Philosophy at Claremont Men’s College, Claremont, California. The following article appeared in In reading A mechanistic perspective entails viewing the world in the image of a machine. Birch and Cobb have defined the image of a machine as "of an entity whose internal structure is given and whose behavior follows from this structure" (LL 79). They acknowledge that machines cannot be abstracted altogether from their environment, but they do affirm that "the relevant features of the environment are conceived as limited and controllable" (LL 79). The image that Birch and Cobb are using for mechanism is of a 1 Turing Machines I propose that the image of a machine, from which one may abstract a more adequate philosophy of mechanism, should be a Turing machine, and in particular, a universal Turing machine. The reasons for choosing Turing machines are twofold. First, a Turing machine is precisely defined. We know exactly what one is. Second, major recent developments in mathematics, logic, and computer science have shown an exact relationship between certain logical notions and Turing machines. Since the goal of this paper is to describe a mechanistic model of explanation, the connection between these logical notions and Turing machines will allow us to establish the relationship between the explanatory model and the image of the machine. Furthermore, we can establish whether an explanation is ultimately a mechanical one or not. An explanation will be considered mechanical if it can be transformed without loss of information, literally, Alan Turing invented the Turing machine as an intellectual tool in the 1930’s in order to solve problems in mathematics. He never made an actual machine, and I am not sure whether anyone has ever constructed one as a machine. It is much easier today to build a digital computer which can function as a Turing machine. But clearly a Turing machine can be constructed as a machine, and must be considered as a machine. It conforms exactly to the definition of a machine given by Birch and Cobb. Its internal structure is rigidly given in the form of a table of ordered quadruples. This table specifies exactly the behavior of the machine. The exactly determined behavior of the Turing machine, however, is conditioned upon its environment. Birch and Cobb acknowledge that all machines react in some sense to their environment. They fail to recognize, however, how complex the relationship between a machine and its environment can really be. In a certain technical sense (explained later), there is no limit to the complexity of the interaction of Turing machines with their environment. A Turing machine in its simplest characterization is a self-contained entity that gives symbolic output information in response to symbolic input information. For the same input information an individual Turing machine always gives the same output information. The symbolic input and output information are normally considered to be words composed of individual symbols that are on the machine s tape, even though a Turing machine can be designed to accept and give information in any standard medium. As normally understood, a classical Turing machine reads one symbol at a time, and in terms of its current state and the symbol read takes an action, which may be to erase, print a new symbol, or move to the right or left one space, after which it goes to a new (or the same) state. The complete action of a Turing machine, in response to its input information, is controlled by the lines in its fixed table each of which has four entries: the current state, the symbol read, the action to be taken, and the new state. Turing machines differ from automobiles and other kinds of machines in that they are designed specifically to manipulate the symbols of some finite alphabet. They are symbol processing machines. They communicate with their environment through the symbols of their input, and output information. Most ordinary machines can function (in part) as Turing machines if there is some way to identify symbolically their input from the environment and their output to the environment. For example, an automobile may act as a Turing machine as it determines a new direction of travel from the angle of the steering wheel and its current direction. Digital computers act as Turing machines as they determine output digital, normally 1-0 (high-low voltage), information in terms of input digital information. We understand very well the effective, and almost intelligent, power that Turing machines can have through their current use as computers. The remarkable thing about Turing machines as proposed in their original archaic form is what can be proved about them. Turing himself proved that a single Turing machine can be constructed which can reproduce the action of any Turing machine whatsoever. This universal Turing machine reads as part of its input data the Turing machine table of the duplicated machine and takes whatever action this machine would have taken on its appropriate input data. A modern digital computer is a universal Turing machine. Its program can be considered to correspond to a duplicated Turing machine. Given enough time and storage space, it can perform the actions and accomplish the results of any other computer whatsoever, whether presently existing or to be made in the future. Turing also established by proof certain ultimate limitations of Turing machines, and hence of all machines including computers. He proved that there cannot exist a Turing machine which can decide in all cases whether an arbitrary Turing machine will halt or not when given an arbitrary input. Turing’s "halting problem" proof is intimately related to (and can be used to prove) other major limitative theorems in logic and mathematics, including Gödel’s Incompleteness Theorem for Arithmetic and Church’s Theorem for predicate logic. Gödel’s Incompleteness Theorem entails that there cannot exist any well defined set of axioms from which all the true theorems of arithmetic can be derived. Church’s Theorem entails that there cannot exist any well defined procedure that can decide in all cases whether or not an arbitrary sentence does or does not logically follow from an arbitrary set of axioms. I shall use these limitative theorems, considered as extensions of Turing’s halting problem, to show the basic limitations of a philosophy of mechanism. These limitations correspond very closely to those expressed by Birch and Cobb. Aside from the limitations imposed by the halting problem, there are two basic reasons that computers cannot accomplish certain things today -- for example, be programmed to act with human common sense. The first reason is that there may not be enough computer storage space or computer time to solve a problem for which we have an effective procedure. The second, and more likely reason, is that we have not been able to specify as yet the exact procedures which may solve the problem. Most intractable problems are due to conceptual difficulties on our part. They are software problems. The hardware acting as a universal Turing machine is adequate. 2 Church’s Thesis Contemporary computer scientists, mathematicians, and logicians are very confident that if we can specify exactly the procedures for accomplishing something, we can build a Turing machine, or more practically, program a computer, that will do it according to the procedures. This confidence takes a technical name and is called Church’s thesis or the Church-Turing thesis. Church’s thesis is a metaphysical generalization like the law of conservation of energy in physics and has almost comparable authority to this law in computer science. It is backed by massive evidence and the strictest logical analysis. It is one of the most important results of computer science, logic, and mathematics. It is a thesis (to be assumed) rather than a theorem (which can be proved) because of the essentially intuitive and (formally) imprecise nature of the idea of a general well defined procedure. Church’s thesis entails that any algorithm (well defined procedure that eventually terminates) may be translated without loss of information into a Turing machine. For example, a set of well defined axioms with appropriate well defined laws of transformation in logic may be considered to be a well defined procedure for deriving consequences from the axioms. There are technical ways of translating the axioms and laws of transformation in logic into a Turing machine that will do the same thing that can be done with the axioms through the logical laws of transformation. What is normally done with a working axiom system? One brings information to the axiom system and asks what this information entails from the axioms. A constructed Turing machine that represents a set of well defined axioms and effective procedures would accept input information (appropriately coded) and give the entailed consequences in its output information -- the same consequences that would be obtained by using the logical procedures on the axioms. 3 Models Biological models of explanation, such as the DNA model of the gene or the natural selection model of evolution, give pictures of reality that function in practical ways. They may guide new experiments in terms of established, but nonformalized, biological procedures. One may bring experimental information to a picture of a model and ask how the picture clarifies the information. Does the information fit the picture and support the model? If so, how and in what way? What new information is entailed by the interaction of the picture with the experimental data? The processes for deciding the relevance of the experimental information in the light of a picture of a model are not always evident -- but often they are. These procedures are part of the discipline ofbiology and are intimately associated with the model. As the model is clarified and made precise, they must be considered to be part of the model itself. Church’s thesis says that if the model, including its associated procedures, is well defined and the procedures are effective, then the model may be translated into a Turing machine that will accomplish the same thing that the model does. This means that any model, including the natural selection model of evolution or the ecological model of Birch and Cobb, if it is well defined and includes only effective procedures, can be translated into Some pictures of models, such as DNA, seem more mechanical than others, such as natural selection. In terms of the images that we have, it is possible to imagine that DNA is in fact a complicated machine, whereas natural selection as a process resists mechanization. We may seek to find and understand certain mechanical elements that form the explanation of the natural selection model (such as DNA), but the model The discussion which follows is divided into two subsections: that which asks whether certain possible machine-like entities, such as DNA or the cell, can be viewed actually as machines; and that which discusses the nature of machines that are translations of nonmachine-like models or theories. We shall find that the distinction is in no way absolute. Machine-like entities have apparently nonmachine-like modes ofexplanation that are part of the models, and nonmachine-like models have parts that are patently mechanical. Both, however, may be interpreted in terms of Turing machines. 4 Is DNA a Machine? A machine’s "internal structure is given" (LL 79). It is this fixed or static character of machines that is at the heart of all classical mechanism. Birch and Cobb assert that "the ultimate mechanical model involves ‘disecting the organism down to its constituent controlling mechanisms and building it up from these building blocks" (LL 69). For a philosophy of mechanism, these building blocks are assumed to be static either in being or structure (or at least to have static essential parts), and it is from such fixed structure that the mechanist seeks to predict the nature of the whole, normally by the laws of mechanics. Jacques Monod, for example, is listed as a contemporary mechanist who believes that the DNA molecule is the controlling building block of living systems (LL 70). Birch and Cobb object to Monod’s view because of evidence concerning the nature of DNA: it can self replicate, it assists in its own synthesis outside the cell, it is counterentropic, it determines necessary conditions for the development of organisms and not sufficient conditions (LL 8lf.). In short, it is conditioned (in a very complex way) by its environment as well as conditioning its environment. Birch and Cobb maintain that the ecological model is more adequate than the mechanical model for explaining DNA, the cell, other biological subject matter (as well as subatomic physics), because it holds that living things behave as they do only in interaction with other things which constitute their environment (LL 83) and because "the constituent elements of the structure at each level (of an organism) operate in patterns of interconnectedness which are not mechanical" (LL 83). In response, first, to the issue of interaction with environment, every machine that I know of does interact with its environment. The more complicated the machine, the more complex its potential interaction with the environment. A Turing machine is designed to respond to its environment. It may accept any (finite) magnitude of information, however large, and respond to it. Hence, on this issue, I see no objection to viewing DNA as a type of machine. is a machine, only that there is an image of a machine that can react as complexly with its environment, and in the same ways, as DNA does.The real reason why Birch and Cobb object to viewing DNA as a machine is that they understand it to have "patterns of interconnectedness which are not mechanical" (LL 83). These patterns are "internal relations" of which human feeling and experience are the ideal exemplification. Birch and Cobb, following Whitehead, generalize this experience to all actual entities. This experience "can be used to include not only human and amoebic experience, conscious or non-conscious as the case may be, but also non-conscious taking account of the environment which characterizes molecular, atomic and quantum events as well" (LL 131). What would it mean to understand a Turing machine to have such internal relations? First, we would have to understand a Turing machine to be identified not only with its fixed table, but also with its input and output information. (This is an arbitrary distinction. We can decide what is and is not our basic mechanical image.) A Turing machine would then not only include a part of its environment but would be both constituted by its environment as well as constituted by its reaction to its environment. It could be considered a very dynamic entity, becoming what it is as a synthesis of relations (its response) to other events. Second, we would have to understand internal relations as probably high level, and complicated, software constructions of basic patterns of the alphabet of the Turing machine. Human feelings, in Whiteheadian thought, are quite complex. The internal relations of a Turing machine would have to be similarly complex constructs of the basic alphabet and vocabulary of the machine. Can a human feeling (or idea) be represented by a complicated software construction of symbols? We really do not know. This is 5 Models Translated into Machines A model that is well defined by effective procedures may be translated by the authority of Church’s thesis into a Turing machine that captures the structure of the model. What would such a model look like? Would it have simple mechanical parts or actions that would allow us to interpret more complicated ones, as do simple machines on which a classical mechanism is built? Would the machine that models DNA, for example, be picturable in an intuitive mechanical fashion? Probably not, although the answer to this question depends on our perspective. It is the that each line of a Turing machine table, that fixed structure that determines the nature of the Turing machine, can be considered to be a simple mechanical structure or action on which the complex structure of the machine depends. But to attempt to analyze the function of a complex Turing machine by examining each of the individual lines of its table would be extremely difficult. The table would be entirely too long (millions of lines), too detailed, and too "microscopic" for easy comprehension. Examining such a machine through its table, or as it is operating on its table, would be like examining only the electronic activity of a running computer, or of an operating brain, and trying to decide what the computer is accomplishing or the brain is thinking. The reason that we can understand standard models in biology is that they are already formulated into higher level language and concepts. They are not formulated in the simple actions of a line of a Turing machine table or of the (relatively) simple firing of a brain’s neuron. If however, the higher level concepts are clearly defined by effective procedures, they may then be subdivided into component parts, which parts may also then be subdivided until we finally reach very simple actions that can be described by lines of a Turing machine table. This is one of the insights that tends to confirm Church’s thesis, that well defined procedures may be successively subdivided until the whole can be described in terms of (very complicated) relationships among simple actions -- which have a mechanical counterpart. The human mind, however, finds it very difficult, if not impossible, to understand the whole in terms of the immense complexity of the simple parts. It understands only if the simple actions are grouped successively back towards a higher level language, so that groups of simple actions or instructions may have an intuitive understanding that can be expressed in ordinary language. A Turing machine that represented a complex theory or model could be understood in a simple mechanical way to the degree that its table is grouped appropriately so that each group of lines in the table stood for a readily understandable intuitive idea. The groups would then have to be organized so that they would relate to each other in a recognizable logical fashion. The overall understanding of the machine would then probably be more logical than mechanical, although the logical structures could be translated into ideas that had a more mechanical feel if desired. 6 The Limitations of Machines Machines are adequate to represent any effective well defined procedure. This is Church’s thesis. For example, if there exists an effective procedure that can describe adequately the operation of the human brain, a machine can be built to duplicate the procedure, subject, of course, to time and storage constraints. What then are the limitations of machines? Machines cannot solve problems, or represent things or events, for which there can exist no effective well defined procedures for solving or describing them. How much of a limitation is this really? I think that it is a fundamental one. We may show, from extensions of Turing’s proofs regarding the halting problem, that no effective well defined procedure, no precise axiom system, no metaphysical system, or no clear means whatsoever, can ever describe accurately and completely any individual thing or event in the real world. If one cannot describe completely any individual thing, one cannot describe the whole, which contains the individual things. One way of understanding Turing’s halting problem is in terms of magnitudes of complexity of information. (There are technical ways of assigning numbers to information that express its magnitude of complexity.) The halting problem entails that for any effective scheme that can decide whether Turing machines halt or not, there exists a Turing machine more complex than the scheme -- hence the scheme (or a Turing machine constructed from the scheme) will not necessarily be able to decide its action. Gödel’s Incompleteness Theorem states that no axiom system can ever give us all the true theorems of arithmetic. It entails that for any axiom system for arithmetic, there exist theorems in arithmetic more complex than the axiom system, and hence, the axiom system does not contain enough information to present the theorem completely. Church’s theorem states that there exists no well defined procedure that can tell us in all cases whether a conclusion logically follows or not from premises. It entails that for any well defined logical procedure, there exist axiom systems and conclusions more complex than the logical procedure; hence, the logical procedure cannot guarantee us whether the conclusion does or does not logically follow from the premises. Another way of understanding the limitative theorems is in terms of theories. A theory is a set of sentences that cohere -- in the sense that any sentence which logically follows from sentences in the set is also in the set. How complex can a theory be? The theory of arithmetic (all sentences true about the natural numbers) is so complex, according to Gödel, that no well defined set of arithmetic axioms is sufficient to derive all true theorems. Any set of axioms whatsoever gives only part of the true theorems of arithmetic. How complex is an actual thing or event in the real world? Can we ever describe or understand it completely? To do so would require not only describing all its constituent parts, down to its subatomic particles, but also its relationships to all other things, that is, its relationship to the whole cosmos. We might well expect, therefore, that a complete description of any event would be immensely complex. Does an actual thing or event have a complexity comparable to the theory of arithmetic? Is it so complex that no system, scheme, or model can ever describe it completely? Any thing or event has spatial extension or temporal dimension. We may map the natural numbers of arithmetic into such extension or dimension. To give the fill truth about a thing or event, we would have to give the possible numerical relationships of its constituent temporal or dimensional parts. That is, we would have to be able to determine the full truth of arithmetic theorems. We cannot do this by authority of Gödel’s Incompleteness Theorem. Hence, there is no clearly defined method of describing the event or thing including its possible internal and external relationships. Hence, there is no machine -- constructed from such a procedure -- that can reproduce the event or thing. The thing, or event, however, may be a machine. The limitative theorems entail that there is no procedure for understanding completely this machine. 7 Good Models and a New Mechanism A good model is considered to be one that is clearly defined and for which there are effective procedures for deciding the relevance of information brought to it. Such models have some fixed internal structure. In an axiomatic model, for example, the axioms are given and static. Yet a model also functions dynamically. One brings information to axioms and gets new information from the axioms and the old information. Note the similarity between good models and machines. Both have a fixed structure and act dynamically in terms of this structure. An actual machine, however, as part of the physical world is more complex than any good model. No good model (or other well defined method) can ever describe a machine completely. Any good model, however, can be translated into a Turing machine (by Church’s thesis) whose abstracted structure can represent the model. A revised philosophy of mechanism, understood from a contemporary image of a machine, must recognize that a good model is the best vehicle for describing nature, rather than old images of machines, which are not complex enough to represent adequately a reaction of a complex entity with its environment. A model is, of necessity, an abstraction from and a limitation of the reality being expressed. Although any good model can be expressed as a machine, it cannot necessarily be expressed completely in traditional mechanical principles. Its expression as a machine will contain both mechanical principles and logical ones. Nevertheless, since the possibilities of grouping the Turing machine’s primitive table into higher level concepts are quite varied, a new philosophy of mechanism would tend to support an organization of software into a higher level language that is intuitively mechanical. The good model understood as a machine has been, and may continue to be, science’s best heuristic tool. 8 A New Mechanism and the Ecological Model To "trust life," according to Birch and Cobb, "is to allow the challenging and threatening elements in our world to share in constituting our experience." "It is to trust that the outcome of allowing the tension of the old and the new to be felt can be a creative synthesis which cannot be predetermined or planned" (LL 131). This trust in life and the actual experiences which it entails are part of human existence that can be far better explained, as Birch and Cobb affirm, by an ecological model rather than an older mechanical one. A more contemporary mechanism, however, would recognize that since no description of an actual event is ever complete, any actual experience of the event may give novelty, that is, may give an aspect of the event not previously perceived. As Birch and Cobb say, "In every moment there is . . . a new possibility arising out . . . of what is given for that moment" (LL 184). Old structures of existence must give way in creative synthesis to new ones, because, as alive, we experience novelty in the actual world. In my judgment, the ecological model is superior to any present mechanical one, because it integrates this insight into human experience with science, ethics, philosophy, and religion. This does not mean, however, that mechanical philosophies may not be extended to include such integration. I also think that the ecological model of Birch and Cobb is superior in explanatory power to any present mechanical model because the ecological model accepts human feelings as primary givens. These feelings must be higher level constructs of more elementary structures in any mechanistic philosophy. Process thought does, of course, attempt to describe how such feelings arise out of previous ones, but it resists the temptation to claim that human feelings can be explained exclusively by reducing them to simpler lower level events. Any model which relates the wide range of human feelings to the complex human experiences of science, philosophy, ethics, and religion is very valuable, especially in a day when mechanical philosophies cannot effect this relationship. As new and more powerful scientific models are found, especially in the areas of brain research and artificial intelligence, a new mechanism which can show the relationship between primitive biological and logical concepts to higher level concepts may be able to clarify the nature and constitution of human feelings. As the ecological model itself is clarified and made more nearly into a good model, its potential for actual mechanization is enhanced. At this stage a new mechanism may be a very attractive option.
References LL -- Charles Birch and John B. Cobb, Jr.
Notes ^{}
Viewed 7923 times. |