Society for Philosophy and Technology - Volume 2, numbers 3-4  Digital Library & Archives ETDs ImageBase Ejournals News EReserve Special Collections  Society for Philosophy and Technology Current Editor: Davis Baird db@sc.edu Current Editorial Assistant: A Bryant aubreybryant@hotmail.comNumber 3-4Spring-Summer 1997Volume 2 DLA Ejournal Home | Techné Home | Table of Contents for this issue | Search Techné and other ejournals GÖDEL'S INCOMPLETENESS THEOREMS AND ARTIFICIAL LIFEJohn P. Sullins III, San Jose State University In this paper I discuss whether Gödel's incompleteness theorems haveany implications for studies in Artificial Life (AL). Since Gödel'sincompleteness theorems have been used to argue against certain mechanistictheories of the mind, it seems natural to attempt to apply the theorems tocertain strong mechanistic arguments postulated by some AL theorists. We find that an argument using the incompleteness theorems can not beconstructed that will block the hard AL claim, specifically in the field ofrobotics. However, we will see that the beginnings of an argument castingdoubt on our ability to create living systems entirely resident in a computerenvironment might be suggested by looking at the incompleteness theorems fromthe point of view of Gödel's belief in mathematical realism.1. INTRODUCTION For many decades now it has been claimed that Gödel's two incompletenesstheorems preclude the possibility of the development of a true artificialintelligence which could rival the human brain.[1] It is not my purpose to rehash these argument in terms ofCognitive Science. Rather my project here is to look at the two incompletenesstheorems and apply them to the field of AL. This seems to be a reasonableproject as AL has often been compared and contrasted to AI (Sober, 1992;Keeley, 1994); and since there is clearly an overlap between the two studies,criticisms of one might apply to the other. We must also keep in mind that notall criticisms of AI can be automatically applied to AL; the two fields ofstudy may be similar but they are not the same (Keeley, 1994). Gödel himself realized that the incompleteness theorems alone do notpreclude the possibility of a machine mind (Wang, 1987, pg. 197). In factthere is an interesting argument posed by Rudy Rucker where he shows that it ispossible to construct a Lucas style argument using the incompleteness theoremswhich actually suggests the possibility of creating machine minds (Rucker,1983, pp. 315-317). Arguments like Rucker's point out the inadequacy of usingthe incompleteness theorems alone to try to prove the improbability of machineminds. In fact an important part of understanding Gödel's reluctance toaccept the project of AI stems from his belief in mathematical realism (seeTieszen, 1994, for a full discussion of this point). My purpose here is not totry to convince anyone of the validity of the Penrose-Lucas arguments incognitive science, but rather to see how a similar argument might be applied tothe field of AL. I will endeavor to keep my arguments as close to those thatGödel himself might have made if he had been presented with the ideasexpressed in strong AL. It may turn out that the incompleteness theorems haveno relevance to AL, but we must take a closer look before we dismiss them outof hand. AL does not start out with the goal of modeling human intelligence; rather itis interested in studying life at a fundamental level, comparing andcontrasting our knowledge of "life-as-we-know-it within the largerpicture of life-as-it-could-be" (Langton, 1987, pp. 1). AL begins withmodest goals. Examples of AL projects would range from the modeling of actualbiological processes like the life cycles of slime mold (Resnick, 1994, p. 50),to the creation of simple "artificial ecosystems" like Thomas Ray's Tierraprogram, a system entirely resident in a computer which makes few claims tobe an accurate representations of life-as-it-is while still claiming to besome new form of synthetic life (Ray, 1992, p. 371). So we can see that atleast some of the researchers in the field of AL do claim that these creationsare (or could be) in a real sense an actual member of the set of things living(see Emmeche, 1994, p. 3). There are clearly two ways to approach AL models: one is to consider themtools for studying the natural world, and the other is to claim that ALprograms, properly executed, simply are living things (Sober, 1992, p. 749).It seems to me that Gödel's theorems will have little impact on the formerclaim as it already concedes that AL is simply a modeling technique for and notan instantiation of life. Conversely, though, Gödel's theorems probablydo apply to the much stronger latter claim that AL can currently, or willeventually, create artificial living things. This is because the later claimsuggests that an artificially constructed reality can completely capture theminimum necessary criteria for the creation of life and, as we will see later,Gödel's theorems can be argued to imply that this may be problematic.2. GÖDEL'S VIEWS ON MECHANISM IN BIOLOGY I was spurred in the direction of applying Gödel's theorems to AL when Icame upon the following passage in Hao Wang's From Mathematics toPhilosophy, where he is discussing Gödel's views on the relationshipbetween minds and machines:Gödel believes that mechanism in biology is a prejudice of our time whichwill be disproved. In this case one disproval, in Gödel's opinion, willconsist in a mathematical theorem to the effect that the formation withingeological times of a human body by the laws of physics (or any other laws of asimilar nature), starting from a random distribution of elementary particlesand the field, is about as unlikely as the separation by chance of theatmosphere into its components. Mechanistic or closely related reductionistic theories have been part oftheoretical biology in one form or another at least since Descartes. I do notwant to give the impression that I believe that mechanistic or reductionistictheories form some kind of monolithic doctrine. I realize that there areprobably as many different versions of these arguments as there are theoristsin the field of biology. Later in this paper I will specify which brand ofmechanism and reductionism is employed in strong AL arguments. The various mechanistic and reductionistic theories are historically opposedto the much older and mostly debunked theories of vitalism (see Emmeche, 1991).These theories (the former more than the latter), along with formism,contextualism, organicism, and a number of other "isms" mark the major centersof thought in the modern theoretical biology debate (see Sattler, 1986). It occurs to me that AL falls curiously on many sides of these debates in thephilosophy of biology. For instance AL uses the tools of completemechanization, namely the computer, while at the same time it acknowledges theexistence of emergent phenomena (Langton, 1987, p. 81). Neither mechanism norreductionism is usually thought to be persuaded by arguments appealing toemergence. Facts like this should make our discussion interesting. It mayturn out that AL is hopelessly contradictory on this point, or it may providean escape route for AL if we find that Gödel's incompleteness theorems dopose a theoretical road block to the mechanistic-reductionistic theories inbiology which I will outline later. What I will attempt to do now is to take a look philosophically at how ALrelates to a specific form of the mechanistic and reductionistic philosophiesof biology and then apply Gödel's incompleteness theorems to that specificview in an attempt to determine if the project of AL can avoid the problemsexperienced by AI in its encounters with Gödel. 3. MECHANISM AND REDUCTIONISM IN BIOLOGY In this paper I will be discussing only two of the above mentioned worldviews, namely, mechanism and its closely related theory reductionism.Furthermore, I will be concerned only with specific formulations of mechanisticand reductionistic theories. This means that we need to be very clear indescribing just what we mean by the terms "mechanism" and "reductionism," asthey are often used in many different contexts and their meanings can changesubtly depending on their use. After we have an adequate understanding of thebasic assumptions found in the various mechanistic and reductionisticphilosophies of biology, we can then determine if the underlying metaphysicalassumptions in AL theories should be placed under this heading. The history of the idea of mechanism is an interesting one, but I will notretell it here. We should understand, however, that it received its greatestboost in popularity in the seventeenth century as a reaction to the new scienceof physics on the part of those studying natural philosophy. As we all know,the capacity of physics to explain, model, and predict things like planetaryorbits astounded the scientific community in the seventeenth century. Itoccurred to many thinkers of that time that many biological things might alsobe explainable, modelable, and predictable using the basic laws of physics asthey relate to machinery. After all, if one looks at a body it does seem to bea machine of some sort with, for instance, lever actions explaining theworkings of muscles and limbs among other things. Descartes was willing todescribe all animals as simple machines; possibly even the human body could bereduced this way. But he was not willing to so describe the human mind.Bolder thinkers such as De la Mettrie (1748) were willing to push the metaphorto the limits, describing humans completely as machines. The metaphor of themachine or "clockwork body" is still prevalent today. In this period of rapiddiscoveries in physics and mechanics we find wonderful early AL experimentsconsisting of clockwork people and animals which were built as objects ofamusement in the seventeenth and eighteenth centuries (see Emmeche, 1991, andLangton, 1987). Over the centuries, the mechanistic and the closely related reductionistictheories of biology have keep pace with current discoveries in science untiltoday a mechanist can be thought of as one who believes "that an organism is inreality nothing more than a collection of atoms, a simple machine made oforganic molecules" (Emmeche, 1991, p. 12). We should note that mechanism, likeall theories, changes over time. To be fair, we should realize that themechanistic theories in biology that Gödel would have been referring to(in the quote above) have changed and are slightly different today. In thelate sixties one could have found many mechanistically inclined theorists whowould claim that it was self-evident that, since biological entities arephysical they must obey the laws of mechanics, and that meant that livingsystems were simply matter in motion obeying the laws of classical mechanics(Sattler, 1986, p. 216). But physics has gone far beyond classical mechanics,and many biological mechanists would now agree that it is not possible toaccurately describe a living system using only classical mechanics (Sattler,1986, p. 216). This is perfectly reasonable. If it is generally acceptedthat classical mechanics is unsuitable for a complete understanding ofnonliving matter, then how can it be expected to be sufficient for explainingthe much more complex actions of living matter (Sattler, 1986)? So it is safeto say that most theorists have outgrown the idea that life can be explainedwholly in terms of classical mechanics. Instead, what is usually meant is thefollowing (paraphrased from Sattler, 1986): 1) Living systems can and/or should be viewed as physico- chemical systems. 2) Living systems can and/or should be viewed as machines. (This kind ofmechanism is also known as the machine theory of life.) 3) Living systems can be formally described. There are natural laws whichfully describe living systems. Now it is not necessary for one to hold all three of the above statements inorder to be a biological mechanist. All one has to do is believe at least oneof the above statements. So a mechanist believes, basically, that livingsystems can be completely explained by the operation of the physicallaws of matter, such as classical mechanics, quantum mechanics, complexitytheory, etc. Any particular mechanist may think that we do not yet have withinour grasp all of the laws we need to understand life, but no mechanist will saythat we cannot theoretically discover them in a reasonable amount of time. Reductionism is related to mechanism in biology in that mechanists wish toreduce living systems to a mechanical description. Reductionism is alsothe name of a more general world view or scientific strategy. In this worldview we explain phenomena around us by reducing them to their most basic andsimple parts. Once we have an understanding of the components, it is thenthought that we have an understanding of the whole. There are many types ofreductionist strategies. To help clarify the different categories ofreductionism I will turn to the work of John R. Searle. Searle lists fivedifferent reductionist strategies in his book, The Rediscovery of theMind. These are Ontological Reduction, Property Ontological Reduction,Theoretical Reduction, Logical or Definitional Reduction, and Causal Reduction(Searle, 1992). And to this list we should also add Epistemological andMethodological Reduction (see Bonabeau and Theraulaz, 1994, and Sattler, 1986).This complexity causes much confusion when one tries to discuss the concept ofreductionism, so we should briefly describe each of these strategies. Ontological reductionism in theoretical biology occurs when a theory statesthat a living system is nothing but a collection of physical parts (atoms)being acted upon by the laws of physics. This can be abstracted further bysaying that the laws of physics are nothing but a set of formalizable axiomswhich can be understood separate from physical matter. "Hence, a completeknowledge of the physics and chemistry of life would entail a fullunderstanding of life" (Sattler, 1986, p. 218). This concept applies to ALtheories that promote the belief that, "Since we know that it is possible toabstract the logical form of a machine from its physical hardware, it isnatural to ask whether it is possible to abstract the logical [form] of anorganism from its biochemical wetware" (Langton, 1987, p. 21). Property ontological reduction can occur in theoretical biology and in AL whenone attempts to describe a property or behavior of a living thing by appealingto low-level phenomena or rules which dictate the behavior. An example ofproperty ontological reduction in AL would be if some one claimed that theflocking behavior of birds could be completely reduced, for instance, to theworkings of Craig Reynolds's famous boids program. Theoretical, or, as it is sometimes called, epistemological, reductionism isthe belief that the theories of one science can be reduced to the theories ofanother. "In biology the central question of epistemological (theoretical,explanatory) reductionism is whether the laws and theories of biology can beshown to be special cases of the laws and theories of the physical sciences"(Dobzhanaky, et al., 1977, p. 491, as quoted in Sattler, 1986, p. 221).In AL this brand of reductionism appears when the claim is made that the lawsof nature might be reducible or capturable in the laws surrounding theinformation processing of computation. Logical or definitional reductionism "is a relation between words andsentences, where words and sentences referring to one type of entity can betranslated without any residue into those referring to another type of entity"(Searle, 1994, p. 114). This occurs in AL when we use terms usually used inbiology to describe events that occur in our computer simulations, notmetaphorically but descriptively. For instance, the words "population,""organism," "fitness," etc., are all used interchangeably in AL when describingreal and artificial life forms. Causal reductionism "is a relation between any two types of things that canhave causal powers, where the existence and a fortiori the causal powers of thereduced entity are shown to be entirely explainable in terms of the causalpowers of the reducing phenomena" (Searle, 1994, p. 114). This seems to occurin biology when one describes phenotype as being nothing but the actualizationof the genotype. And this occurs in AL when we say, unarguably, that theobserved behavior of a program is nothing more than the implementation of itsprogram code. Finally, methodological reductionism in biology is the claim that livingsystems should be studied at their most basic level, either the actual atomsand molecules or their theoretical interactions (Sattler, 1986, p. 224).Clearly this occurs in AL when it is suggested that we can gain understandingof the real world by seeing it as the interaction of "information" at eitherthe cellular level or at the level of the patterned interaction of electrons incircuit boards (see Rucker, 1987, for an example). So we can see that reductionism is a tool or strategy for solving complexproblems. There does not seem to be any reason that one has to be a mechanistto use these tools. For instance one could imagine a causal reductionisticvitalist who would believe that life is reducible to the elan vital orsome other vital essence. And, conversely, one could imagine a mechanist whomight believe that living systems can be described metaphorically as machinesbut that life was not reducible to being only a property of mechanics. 4. MECHANISM AND REDUCTIONISM IN STRONG AL As this paper is concerned with strong AL arguments, I will narrow down ourdiscussion of the various reductionistic and mechanistic theories of biology tothe specific types commonly found in strong AL claims. The strong argumentclaims that AL simulations are, or can be, complete in their formalization ofthe basic laws describing living systems. Now since Gödel's incompleteness theorems apply specifically to systemswhich attempt to completely and consistently axiomatize arithmetic, andgenerally only to systems which attempt to completely and consistentlyaxiomatize their subject (Nagle and Newman, 1958, p. 100, Braithwaite, 1962, p.1). So If we refer to the three mechanistic theories of life listed above wecan begin eliminating the ones that do not apply to the strong AL conception ofliving systems. With this in mind we can eliminate number 1 from the listabove, as the strong variety of AL does not believe that living systems shouldonly be viewed as physico-chemical systems. AL is life-as-it-could-be,not life-as-we-know-it (Langton, 1989, p. 1), and this statementsuggests that AL is not overly concerned with modeling only physico-chemicalsystems. Postulates 2 and 3 seem to hold, though, as strong AL theoriesclearly state that the machine, or formal, theory of life is valid and thatsimple laws underlie the complex, nonlinear behavior of living systems(Langton, 1989, p. 2). As far as reductionism is concerned, AL theories taken as a whole clearly fitinto all the above categories of reductionism (for some discussion of thispoint see Bonabeau and Theraulaz, 1994, p. 314). But the strong claim in ALclearly relies heavily on property reductionism, causal reductionism, andmethodological reductionism, so we can remove the other types of reductionismfrom our discussion. Having clarified what we mean by the terms mechanisim and reductionism, we cannow formulate a concise statement of the general beliefs of strong AL theoriesas follows:1. Living systems are properly reducible to the laws described in thetheories of complex adaptive systems.2. Since a complex adaptive system is causally and methodologically reducibleto the mechanistic processes involved in the computation ofinformation at the fundamental level in nature, it is then conceivablethat one could completely formalize all of the laws operating in such asystem.3. These laws can be implemented on the proper type of computing machinery.Conclusion: A properly conceived AL program running in a complex enoughcomputer or robot can correctly be said to be alive. Now that we have a clearly stated expression of the strong AL claim, we are atthe point where we can apply Gödel's incompleteness theorems to theargument. I believe that Gödel's incompleteness theorems have somebearing on the question of the validity of the strong claim in AL since thesecond premise just listed makes a claim to a level of formal completeness thatmay be subject to the limitations of formal systems described by Gödel.5. GÖDEL'S INCOMPLETENESS THEOREMS APPLIED TO AL In order to show that Gödel's incompleteness theorems have a bearing onAL, we have to prove that it is necessary for strong AL to hold to postulatenumber 2 as I have stated it above. In order to achieve this I will use SteenRasmussen's (1992) article, "Aspects of Information, Life, Reality, andPhysics" (p. 767), as it does a wonderful job of laying out the logical stepstaken in the strong AL argument. Briefly stated, his argument goes likethis:1. A universal computer at the Turing machine level can simulate any physicalprocess (Physical Church-Turing thesis).2. Life is a physical process. Corollary: 1, Hence life can besimulated on a universal computer.3. There exist criteria by which we are able to distinguish living fromnon-living objects. Corollary 2: From this postulate it follows that it ispossible to determine if some specific computer process is alive or not. 4. An artificial organism must perceive a reality R2 , which, forit, is just as real as our "real" reality, R1 , is for us (R1 and R2 may be the same).5. R1 and R2 have the same ontological status. Usingpostulate 5 and Corollary 1 we can say that the ontological status of a livingprocess is independent of the hardware that carries it. Since R1 and R2 are ontologically equal, that is, oneis not more real than the other, then actual living systems can be created ina digital computer.6. It is possible to learn something about the fundamental properties ofrealities in general, and R1 in particular, by studying the detailsof different R2's. An example of such a property is the physics ofa reality. Postulates 1, 2, and 3 are not completely unproblematic but I willnot take that up here; rather we will jump to postulates 4 and 5. In postulate4 Rasmussen rightly claims that in order for an AL program to be alive it hasto create an environment that is as real to its inhabitants as nature is to us.In explaining this idea he appeals to a concept called a"Meaning Circuit." The basic idea behind this concept is that the world is aself-synthesized system of existence. On the one hand, physics provides themeans for communication (light, sound, etc.). Reality can, thereby, acquireits meaning through a conscious conception of the world, via an organization ofthe information we get from our senses. On the other hand, physics also givesrise to chemistry and biology, and through them, an observer participation,namely the emergence of life and later the evolution of man (Rasmussen, 1992,p. 769). So what postulate 4 is saying is that the living systems in an artificialreality must have some form of robust interaction and awareness of thatreality and this interaction, this "meaning circuit," is what makes theartificial reality real. In postulate 5 an interesting jump is made.He claims that, "In postulate 4 we argued that a reality obtains its meaningthrough the existence of an observer" (Rasmussen, 1992, p. 770). He then goeson to explain that the artificial reality is a real reality whenever it has aliving agent interacting with it. If this is achieved then R1 andR2 have equal ontological status (Rasmussen, 1992, p. 770). The problem with this argument so far is that it seems to be circular. It ismaking the claim that an artificial reality created in the computer is able tocapture all of the essential qualities of our reality (R1is equal toR2) as long as living agents are interacting with the system, butthe artificial reality must already be ontologically equivalent to our realityin order to produce truly living artificial life forms. So in a sense theargument is saying that in order to create artificial life one needs to haveartificial life to create the proper artificial environment with the rightontological status. Which comes first? I believe that this is a serious flawin the strong AL argument, and it may be much more difficult to get around thanany of the arguments which will be posed below. Let us assume that we can get around the circularity of the argument justdescribed. According to postulate 4, the artificial reality experienced by theartificial life agent must be as real to it as our reality is for us. Usingthe concept of the meaning circuit as described above, it is necessary, inorder to capture the essential qualities of the reality we perceive, for an ALprogram to have some form of internal logic equivalent to the physics weperceive in nature so as to provide the artificial organisms with the same kindof meaningful interaction with their world which organisms in our realityexperience. This physics can be a simplified version of the one we experiencein our reality (Rasmussen, 1992, p. 769), but it must be a completeformalization of a certain number of basic physical laws required for theexistence of life. For instance, there must be some way for the agents and theenvironment to interact. Since we are programming a computer to invoke thisenvironment, then this set of basic physical laws must be one that can beformed into specific statements in which the program will mechanically deducethe environment and the agents in that environment. We can state this as apostulate:There exists a minimum set of formal axioms which can be used to create acomplete artificial physics capable of sustaining artificial life. Now here is the tricky part. One of the main differences between an actualliving organism and its potential AL counterpart is that the AL entity existsin a computer. Also a living creature is presented with the physics of thenatural world, where an AL entity has to have its physics provided by thecomputer. So in accord with the above postulate, a programmer must code into acomputer system the minimum set of formal axioms needed to create a completeartificial physics capable of sustaining artificial life. In order to become aproper artificial physics capable of sustaining life the program used wouldhave to be able to simulate a reality that is as real to its inhabitants asours is to us. Now if we hold to a level of mathematical reality as strictlyas Gödel does, then concepts like arithmetic are as real an entity asanything else we experience; specifically, a mathematical realist likeGödel believes that our intuitions, expressed by mathematics, are about,"abstract, mind-independent meanings and objects, including transfiniteobjects" (Tieszen , 1994). As we know, Gödel's incompleteness theoremsseem to have proven that building a consistent formalized system of proving allarithmetic truths is highly unlikely (Gödel, 1962, p. 77, Nagle andNewman, 1958, p. 99). Simply put (if that is possible), Gödel'sincompleteness theorems suggest that there exist sentences which can beformulated in a specific formal system called Peano-Arithmetic which are truebut nonetheless not deducible from the axioms of that system. It follows fromthis that it is unlikely that we currently have a complete formal system whichcan grasp the entirety of even simple mathematical systems. This means (aslong as you are a mathematical realist) that at least one of the basicqualities of our reality will always be missing from any conceivableartificial reality, namely, a complete formal system of mathematics. Thisargument tends to make more sense when applied to strong AI claims aboutintelligent systems understanding concepts (see Tieszen, 1994, for a morecomplete argument as it concerns AI). Still, I feel that it has relevance to AL for two reasons. The first is thateven though the intelligence of a typical postulated AL entity is small, it ishoped that greater intelligences will evolve in time from these modest roots.So, if we are to believe that AL can eventually evolve higher intelligences,we need to know how it can avoid the typical arguments deployed against strongAI claims such as the Gödel argument. Secondly, while one might also askwhat possible effect these postulated mathematical realities have on livingsystems, real or artificial, I believe that it can be argued that some form ofmathematical realism is not unthinkable and that this condition of our reality,coupled with Gödel's theorems, casts doubt on our ability to render anartificial reality which would be equal to our own reality in its ability tosustain life. To illustrate this idea let us look briefly at a quote from Johnvon Neumann regarding mathematics and AI:When we talk mathematics, we may be discussing a secondary language,built on the primary language truly used by the central nervous system.Thus the outward forms of our mathematics are not absolutely relevantfrom the point of view of evaluating what the mathematical or logical languagetruly used by the central nervous system is (quoted by Weizenbaum,1976). It seems that one could broaden the scope of von Neumann's observation fromthe specifics of a living central nervous system to life in general withoutharming the intent of the original comment. I feel that this is the positionthat a mathematical realist like Gödel would take because a mathematicalrealist would believe that there exist mathematical realities which are thefoundations of the reality we experience and that these realities are describedby concepts like Peano-Arithmetic, but that these realities are uncapturable inany complete way by entirely mechanical processes. Thus it would seem that itis impossible to completely formalize an artificial reality that isequal to the one we experience, so AL systems entirely resident in a computermust remain, for anyone persuaded by the mathematical realism posited byGödel, a science which can only be capable of potentially creatingextremely robust simulations of living systems but never one that canbecome a complete instantiation of a living system.6. OBJECTIONS The argument that I have presented above is admittedly brief. In ashort paper such as this it is hard to adequately defend a theory that makesuse of Gödel's theorems as seen from the perspective of his mathematicalrealism. Both of these subjects would take up the better part of a book tothoroughly explain. My purpose here is only to open a discussion of this topicin the hope that others agree that it is a worthwhile subject for furtherstudy. In fact I hope to collect many objections to the argument so that I canattempt to answer them later in a more thorough way. Still it would be helpful here to look at the most common objection that Ihave received to this argument and attempt to begin a counter argument. Those to whom I have shown earlier drafts of this paper usually point out anobjection similar to this. Our reality (R1 ) is a reality in whichthe incompleteness theorems hold. So why does it matter that theincompleteness theorems hold in an artificial reality (R2)? All theabove argument has accomplished is to point out that Gödel's theorems arevalid in both R1 and R2. Also, computers already do someamazing things none of which requires the strict formal completeness andconsistency that Gödel is worried about in his famous theorems. It is true that the incompleteness theorems hold to our perceived reality andthat they point to a fundamental limit in our ability to formalize all of ourmathematical intuitions. I do not believe that Gödel meant to suggestthat mathematics as a separate entity is fundamentally incomplete. Rather, histheorems prove that our understanding of that mental object known asmathematics can not be completely and consistently mechanized. So what I amsaying is this: given Gödel's mathematical realism, the incompletenesstheorems suggest that it is not possible to capture this one aspect of ourreality in any artificial reality. If one assumes that our universe isinfinite,"then it embodies the full set of natural numbers, so Gödel'stheorem seems to say that for any given finite theory of the universe, thereare certain facts having to do with sets of physical objects that can not beproved by the theory" (Rucker, 1982, p. 141). Now any AL program that isattempting to entirely create an environment separate from our own which iscapable of sustaining life is attempting to capture the sufficient conditionswhich make life possible here. I am claiming that Gödel's theoremsuggests that any such program might be missing an important essential portionof our reality, namely, its fundamental mathematical reality, so that theartificial reality (R2) would not be ontologically equal to ourreality (R1). And since this is a requirement for creating trulyliving artificial life entities, the artificial reality could not sustain life. 7. SO WHAT Now I will try to mitigate some of the consequences of the above argument andsuggest ways that AL can avoid the argument or change to accommodate it. We should not feel that AL is diminished if it proves to be impossible tosynthesize living systems in the manner described above. AL in its so called"weak" form is still a challenging new science which promises to completelyalter the way we practice the study of biology by giving us powerful new toolsand metaphors for looking at and discussing living systems (Emmeche, 1994, p.156). Secondly, the argument given above only applies to AL experimentscompletely carried out within a computer. When we look at the argument above we can see that all it suggests is thatthere is not a complete one-to-one correspondence between nature and asimulated nature. Remember that the artificial organism must perceive areality that is as real to it as our reality is to us (Rasmussen, 1992, p.769). Since there may be some problem with a simulated reality, then thatproblem can be solved by allowing the artificial organism to interact with ourreality. This can be done through robotics. In this scheme the robotic artificial organisms are operating in an unarguablyreal environment. If a way could be found to give the robots complex adaptivebehavior and self reproduction then we might be on our way to creating trueartificial life. It may be possible, but certainly not easy, to evolve livingorganisms from robots.8. CONCLUSION We have seen that due to a specific interpretation of the implications ofGödel's incompleteness theorems it may not be possible to create a trulyliving system which is entirely resident in a computer. We were not able toadvance very far Gödel's claim that mechanism in biology can be disprovenmathematically. We have only proven that life may not be reducible to acertain type of mechanical implementation on a computer. This modest resultmay lead to a more complete refutation of mechanism, but that question is leftopen for now. It may be that studies in AL itself will lead to themathematical proof that Gödel postulated in the quote above. The value of this finding is not to discourage certain types of research inAL, but rather to help move us in a direction where we can more clearly definethe results of that research. In fact, since one of the above arguments restson the assumption that the universe is infinite and that some form ofmathematical realism is true, if we are someday able to complete the goaladvanced in strong AL it would seem to cast doubt on the validity of theassumptions made above. So succeed or fail AL gives us much to ponder. It may be that AL is still a long way from capturing completely the answer tothe question "what is life?" It may be that this question is unanswerable orthe wrong question to ask. But every attempt at answering that question, frommodest attempts in AL at the explication of life, to extreme attempts in strongAL to synthesize life, helps us move closer to an understanding of the world wefind ourselves in.ACKNOWLEDGMENT I would like to thank the staff and faculty of the San Jose State Universityphilosophy department for their support of my studies. I am also indebted toDr. S. D. N. Cook for his critique and support of this project, Dr. R. Ruckerfor his scathing criticisms, and Dr. R. Tieszen for his comments on earlierdrafts of this paper. This work has been partially supported through a grantfrom the National Science Foundation.REFERENCESBonabeau, E. W., and G. Theraulaz. 1994. "Why Do We Need Artificial Life?"Artificial Life 1:3 (Spring).Emmeche, C. 1994. The Garden in the Machine: The Emerging Science ofArtificial Life. Princeton, N.J.: Princeton University Press.Keeley, B. L. 1994. "Against the Global Replacement: On the Application ofthe Philosophy of Artificial Intelligence to Artificial Life." InArtificial Life III, ed. C. G. Langton. SFI Studies in the Science ofComplexity, vol. 17. Reading, Mass.: Addison-Wesley.Langton, C. G. 1987. "Artificial Life." In Artificial Life, ed. C. G.Langton. SFI Studies in the Science of Complexity, vol. 6. Reading, Mass.:Addison-Wesley.de la Mettrie, Julien Offray. "L'Homme Machine, 1748." In Man aMachine, ed. G. C. Bussey. La Salle, Ill.: Open Court.Gödel, K. 1962. 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