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Comment on Lefebvre's mathematical models by Richard Sacksteder
Comments
227
systems operate; one might even say it is the ratio of all growth, and even, perhaps, of all evolution. [Could it be also that it is the ratio at which thought adds to itself? And could it be, further, that the amount of time the operation takes is my (and Poppel’s 3 seconds?).] By ‘recursive feedback system’, I mean any simple process of accumulation in which the existing level of accumulation determines the amount of new quantity which is to be added to it. Hence, the spiral structures in nature (Fig. I):
Fig. 1. Example
of a naturally
occurring
spiral structure
In other words, evolution is a reflexive process, and conscious (and unconscious) reflexion are a continuation of it. Interesting, then, that some brain scientists think that memories are stored in gene-like sequences, which are as unique relative to each other as are the genes of different individuals. Lefebvre, it has been pointed out accurately, confounds harmonics and mechanics, ethics with esthetics (in fact, that double pairing is very interesting too). I suppose I am an Ancient in that respect; that is, I think that when we have done our harmonics and mechanics well enough we will find that they have converged to become indistinguishable. This has already happened, I think, in quantum mechanics-indeterminacy is surely a collapse of mechanics and harmonics!-and is happening in molecular biology (are we investigating or creating?) and in the new field of anthropological theater. I would be quite happy with a notion of ethics which rested upon an esthetic foundation if that foundation is rich enough and intelligible enough, as I suspect our new sophistication about reflexive processes may make it. KALOS is both good and beautiful. We might all be more tolerant of each other if we recognized goodness as the beauty of the game. Frederick Turner Founders Professor, School of Arts and Humanities, University of Texas at Dallas, Richardson, TX 75083-0688, USA
Comment on Lefebvre’s mathematical models by Richard Sacksteder
Lefebvre has developed a family of mathematical models that seem to be capable of explaining the results of various psychological experiments, justifying the aesthetic appeal of the golden mean, making ethical comparisons of Kennedy and Nixon (or Hamlet and Claudius), and expressing the essence of the difference between the Western and Soviet ethical systems in terms of Boolean algebra. I must confess that, as a believer in the principle that even the most sophisticated use of mathematics is incapable of
228
Comments
transforming vague and inexact perceptions into conclusions that are more precise, I approach Lefebvre’s work with much scepticism. There is a common thread to all of the diverse applications of Lefebvre’s method. It is roughly as follows: (1) A verbally defined concept appropriate to the problem at hand such as a ‘reflexive hierarchy of images’ is identified with a Boolean function. (2) Probabilistic assumptions about the chance that the Boolean variables will be equal to the (Boolean) identity or zero are made. (In a few cases, the binary operations are also selected randomly.) (3) The probability that the function assumes the identity value is computed. (4) The computed probability is identified as some presumably meaningful and important concept such as an ‘ethical status’. The easiest step to discuss is (3), which is a straightforward application of elementary probability theory. The computations have been done correctly in all of the cases that I have checked. Thus, while the exposition is not always to my taste, I have no fundamental criticism of this part of the method. I believe that in steps (2) and (4), Lefebvre implicitly assumes a burden of proof or at least justification that he neglects to supply. For instance, in step (2), certain probabilities are simply asserted to be independent or to assume a value such as $ without serious discussion. In step (4) results of certain computations are given an impressive name that may or may not conjure up a clear image in anyone’s mind, but no real connection is generally established between the probability computed and the concept named. As I understand it, Lefebvre’s point of view is that the whole process amounts to a definition which captures the essence of the concept named in step (4). However, as I will try to show by discussing some examples below, I am not convinced that the method works in this sense. There is an absolutist tone in Lefebvre’s discussion of step (1). The Boolean identity and zero are identified with, for example, ‘good’ and ‘evil’ and the Boolean binary operations with ‘confrontation’ and ‘compromise’. The binary operation of adding x to the complement of y is interpreted as ‘x is aware of y’, which according to the other identifications is the same as ‘x confronting the complement ofy’. It is observed that such assertions as ‘evil confronting good is good’ and ‘evil aware of evil is good’ (which are presumed to be meaningful and true) are consistent with these identifications and the rules of Boolean algebra. Then an interpretation of a character’s images such as ‘Claudius has incorrect images of himself and Hamlet, who confront each other, and he doubts the correctness of these images’ can be identified with Boolean functions of variables representing Hamlet and Claudius. The description of Claudius’ images suggests that more information is passed on to the Boolean function than is in fact the case. For example, the views that Claudius may have about Hamlet knows or intends to do are completely lost. All that is left is an automaton that will generate outputs of either ‘good’ or ‘evil’ from inputs such as ‘Hamlet is evil and Claudius is good’. The Boolean function is completely determined by its responses to all possible inputs. (In the present example, it will respond good for inputs in which Claudius is good and evil in the opposite case, regardless of Hamlet’s state of virtue.) To compute Claudius’ ethical status, one assumes that both characters (independently) have a fifty-fifty chance of being good and calculates the probability that the output will be ‘good’. I find it hard to believe that this kind of analysis can be of any value. Justification for some models is based on empirical grounds. For instance, in Lefebvre (1987, pp. 129-175), it is claimed that a certain model gives ‘theoretical predictions’ of
Comments
229
some experimental results. The rationale for the second character used in this model seems especially unconvincing. For instance, the condition 6, = 0 does not correspond at all to my view of bad characters, most of whom seem to rate themselves far too positively. Moreover, the random selection of the binary operations that is part of the description of the A, character in effect amounts to making a random choice between two characters. The first one, viewed as an automaton, responds ‘good’ if and only if the variable a, is ‘good’, regardless of the values assumed by the other four variables. The other will respond ‘good’ under the same condition or if one of the 16 possible combinations of values for the other variables holds. Despite the justifications offered, this model seems extremely arbitrary. The problem with this as well as to most of the models given in Lefebvre (I 987) is that they have been selected from a large family of available models, many of which could be given equally persuasive justifications. Thus, the fact that at least one can be found to agree with the results of an experiment or to yield a number such as the golden mean is not particularly astonishing. The whole technique is reminiscent of the Ptolomaic system, which could explain any observed planetary motion by adding more epicycles. It is probably unreasonable to expect mathematical models of ethical or psychological phenomena to measure up to the standards set in such fields as geometry or mechanics. A fairer sort of comparison would be to models of idealized economic systems such as the free competitive market, n-party bargaining, or the monopolistic firm. Perhaps these cannot claim to correspond exactly to things that exist in the external world, but at least one can recognize in them idealized limiting cases of economic phenomena. I do not find any of Lefebvre’s models similarly compelling. I would like to believe that the use of mathematics could add depth, rigor, and clarity to psychological and ethical studies, but here I see the main contribution of mathematics as merely adding pretentious language and symbolism that will impress only the untutored.
References Lefebvre, Ethical Lefebvre,
V. A. (1982). Algebra of Conscience: A Comparative Systems. Boston: Reidel. V. A. (1987). J. sot. biol. Strucr. 10, 129-175.
Anal.vsis of Western and Soviet
Richard Sacksteder Graduate School and University Center, CUNY, !3 West 42nd Street, New York, NY 10036, USA
227
systems operate; one might even say it is the ratio of all growth, and even, perhaps, of all evolution. [Could it be also that it is the ratio at which thought adds to itself? And could it be, further, that the amount of time the operation takes is my (and Poppel’s 3 seconds?).] By ‘recursive feedback system’, I mean any simple process of accumulation in which the existing level of accumulation determines the amount of new quantity which is to be added to it. Hence, the spiral structures in nature (Fig. I):
Fig. 1. Example
of a naturally
occurring
spiral structure
In other words, evolution is a reflexive process, and conscious (and unconscious) reflexion are a continuation of it. Interesting, then, that some brain scientists think that memories are stored in gene-like sequences, which are as unique relative to each other as are the genes of different individuals. Lefebvre, it has been pointed out accurately, confounds harmonics and mechanics, ethics with esthetics (in fact, that double pairing is very interesting too). I suppose I am an Ancient in that respect; that is, I think that when we have done our harmonics and mechanics well enough we will find that they have converged to become indistinguishable. This has already happened, I think, in quantum mechanics-indeterminacy is surely a collapse of mechanics and harmonics!-and is happening in molecular biology (are we investigating or creating?) and in the new field of anthropological theater. I would be quite happy with a notion of ethics which rested upon an esthetic foundation if that foundation is rich enough and intelligible enough, as I suspect our new sophistication about reflexive processes may make it. KALOS is both good and beautiful. We might all be more tolerant of each other if we recognized goodness as the beauty of the game. Frederick Turner Founders Professor, School of Arts and Humanities, University of Texas at Dallas, Richardson, TX 75083-0688, USA
Comment on Lefebvre’s mathematical models by Richard Sacksteder
Lefebvre has developed a family of mathematical models that seem to be capable of explaining the results of various psychological experiments, justifying the aesthetic appeal of the golden mean, making ethical comparisons of Kennedy and Nixon (or Hamlet and Claudius), and expressing the essence of the difference between the Western and Soviet ethical systems in terms of Boolean algebra. I must confess that, as a believer in the principle that even the most sophisticated use of mathematics is incapable of
228
Comments
transforming vague and inexact perceptions into conclusions that are more precise, I approach Lefebvre’s work with much scepticism. There is a common thread to all of the diverse applications of Lefebvre’s method. It is roughly as follows: (1) A verbally defined concept appropriate to the problem at hand such as a ‘reflexive hierarchy of images’ is identified with a Boolean function. (2) Probabilistic assumptions about the chance that the Boolean variables will be equal to the (Boolean) identity or zero are made. (In a few cases, the binary operations are also selected randomly.) (3) The probability that the function assumes the identity value is computed. (4) The computed probability is identified as some presumably meaningful and important concept such as an ‘ethical status’. The easiest step to discuss is (3), which is a straightforward application of elementary probability theory. The computations have been done correctly in all of the cases that I have checked. Thus, while the exposition is not always to my taste, I have no fundamental criticism of this part of the method. I believe that in steps (2) and (4), Lefebvre implicitly assumes a burden of proof or at least justification that he neglects to supply. For instance, in step (2), certain probabilities are simply asserted to be independent or to assume a value such as $ without serious discussion. In step (4) results of certain computations are given an impressive name that may or may not conjure up a clear image in anyone’s mind, but no real connection is generally established between the probability computed and the concept named. As I understand it, Lefebvre’s point of view is that the whole process amounts to a definition which captures the essence of the concept named in step (4). However, as I will try to show by discussing some examples below, I am not convinced that the method works in this sense. There is an absolutist tone in Lefebvre’s discussion of step (1). The Boolean identity and zero are identified with, for example, ‘good’ and ‘evil’ and the Boolean binary operations with ‘confrontation’ and ‘compromise’. The binary operation of adding x to the complement of y is interpreted as ‘x is aware of y’, which according to the other identifications is the same as ‘x confronting the complement ofy’. It is observed that such assertions as ‘evil confronting good is good’ and ‘evil aware of evil is good’ (which are presumed to be meaningful and true) are consistent with these identifications and the rules of Boolean algebra. Then an interpretation of a character’s images such as ‘Claudius has incorrect images of himself and Hamlet, who confront each other, and he doubts the correctness of these images’ can be identified with Boolean functions of variables representing Hamlet and Claudius. The description of Claudius’ images suggests that more information is passed on to the Boolean function than is in fact the case. For example, the views that Claudius may have about Hamlet knows or intends to do are completely lost. All that is left is an automaton that will generate outputs of either ‘good’ or ‘evil’ from inputs such as ‘Hamlet is evil and Claudius is good’. The Boolean function is completely determined by its responses to all possible inputs. (In the present example, it will respond good for inputs in which Claudius is good and evil in the opposite case, regardless of Hamlet’s state of virtue.) To compute Claudius’ ethical status, one assumes that both characters (independently) have a fifty-fifty chance of being good and calculates the probability that the output will be ‘good’. I find it hard to believe that this kind of analysis can be of any value. Justification for some models is based on empirical grounds. For instance, in Lefebvre (1987, pp. 129-175), it is claimed that a certain model gives ‘theoretical predictions’ of
Comments
229
some experimental results. The rationale for the second character used in this model seems especially unconvincing. For instance, the condition 6, = 0 does not correspond at all to my view of bad characters, most of whom seem to rate themselves far too positively. Moreover, the random selection of the binary operations that is part of the description of the A, character in effect amounts to making a random choice between two characters. The first one, viewed as an automaton, responds ‘good’ if and only if the variable a, is ‘good’, regardless of the values assumed by the other four variables. The other will respond ‘good’ under the same condition or if one of the 16 possible combinations of values for the other variables holds. Despite the justifications offered, this model seems extremely arbitrary. The problem with this as well as to most of the models given in Lefebvre (I 987) is that they have been selected from a large family of available models, many of which could be given equally persuasive justifications. Thus, the fact that at least one can be found to agree with the results of an experiment or to yield a number such as the golden mean is not particularly astonishing. The whole technique is reminiscent of the Ptolomaic system, which could explain any observed planetary motion by adding more epicycles. It is probably unreasonable to expect mathematical models of ethical or psychological phenomena to measure up to the standards set in such fields as geometry or mechanics. A fairer sort of comparison would be to models of idealized economic systems such as the free competitive market, n-party bargaining, or the monopolistic firm. Perhaps these cannot claim to correspond exactly to things that exist in the external world, but at least one can recognize in them idealized limiting cases of economic phenomena. I do not find any of Lefebvre’s models similarly compelling. I would like to believe that the use of mathematics could add depth, rigor, and clarity to psychological and ethical studies, but here I see the main contribution of mathematics as merely adding pretentious language and symbolism that will impress only the untutored.
References Lefebvre, Ethical Lefebvre,
V. A. (1982). Algebra of Conscience: A Comparative Systems. Boston: Reidel. V. A. (1987). J. sot. biol. Strucr. 10, 129-175.
Anal.vsis of Western and Soviet
Richard Sacksteder Graduate School and University Center, CUNY, !3 West 42nd Street, New York, NY 10036, USA