J. Consc. Studies 6, No. 6/7, June/July 1999, pages 5 - 14

Art and the Brain
Editorial Introduction

Joseph A. Goguen

This essay has three main goals: to introduce and somewhat contextualize the papers that follow; to bring out some implicit (and generally unintended) dialogues among them; and, in order to encourage further discussion in future issues of the Journal of Consciousness Studies, to introduce some new perspectives, including the historical development of Western art, cognitive linguistics, and the sacred. The remarks pertaining to the third goal are necessarily personal, and should not be confused with Journal of Consciousness Studies editorial policy, except insofar as their goal is to promote debate across the gap that separates the sciences and the humanities - the `two cultures'. This gap was perhaps first pointed out in the late 1950s by C.P. Snow, who was noted as both a novelist and scientist (Snow, 1959).

The contributions in this volume divide into four main groups: the focus paper by V.S. Ramachandran and William Hirstein with its associated commentaries; the paper by Semir Zeki, from which the volume draws its title; the paper by Nicholas Humphrey with its commentaries; and the papers by Erich Harth, Ralph Ellis and Jason Brown. The approach of Ramachandran and Hirstein and their commentators generally has a psychological flavour, while Zeki takes a more neurobiological approach, Humphrey combines anthropological and biological ideas, Harth takes a cognitive-evolutionary view, Ellis calls upon Gibsonian affordances and Brown combines process theory with clinical pathology.

Perhaps the largest question confronted in this volume is, what does it mean to be human? Like most big questions, this one is addressed in very different ways by the disciplines of psychology, neurophysiology, anthropology, evolutionary biology and philosophy, to say nothing of art. But by exploring some collisions among these multiple perspectives, perhaps we can learn more than through any single perspective, and we might even begin to see some ways to bridge C.P. Snow's infamous gap.

I: The Focus Paper and Its Commentaries

The focus paper by Ramachandran and Hirstein is firmly on the scientific side of the gap, proposing a bold experimental exploration of the aesthetic response. Their approach is decidedly and unapologetically materialist and reductionist. In his commentary on this paper, the eminent psychologist Richard Gregory says that he expects this paper to `stimulate a lively debate' and, indeed, the commentaries are delightfully energetic and diverse; Gregory also suggests considering this debate a test of whether `there has been any progress towards a rapprochement between [the] opposing camps' of the sciences and the humanities `in the half century since the time of Snow's original complaint'. Readers should form their own view on this, but I would encourage them to consider the entire volume as relevant evidence.

While many readers from the humanities side of the gap will doubtless share the disquiet of several commentators on the focus paper, reductionism certainly has its advantages, even in the humanities; for example, as brilliantly argued by E.O. Wilson in a recent popular book (Wilson, 1999), what we learn from science has a resilient, stable and cumulative quality that seems notably absent in the humanities. Of course, results from science are always open to a variety of philosophical and cultural interpretations, and reductionism itself is a philosophical position. Therefore scientific work in humanistic areas can never eliminate the humanities. Moreover, because the arts play an entirely different kind of role in society, they are in an even more resistant position. But this only means that both sides have strengths from which to make their contributions to the great debate, not that the humanities are necessarily in the stronger position.

Testing the outer limits of logical positivism in the late 1920s and early 1930s, the then well-known mathematician G.D. Birkhoff undertook a programme to reduce aesthetics to mathematics, by defining the aesthetic measure of an object to be the ratio of its symmetry to its complexity (Birkhoff, 1928; 1933). While this possibly could be useful for evaluating simple figures, such as company logos, it has generally been regarded as a failure, because it seems so difficult to apply to more complex objects. Still, it can be said that this brave if misguided attempt succeeded in advancing our understanding of the limitations of approaches of that kind.

It seems to me that this is the very least we should expect from reductionist forays into the humanities, and that in fact Ramachandran and Hirstein, along with Zeki, Humphrey and other authors in this volume, have made contributions that go well beyond this minimum, and will withstand the test of time.

V.S. Ramachandran is a distinguished cognitive neuropsychologist, winning recognition for his work on the so-called `phantom limb' phenomenon, and for a recent book arguing against the metaphysical self, in favour of consciousness as a brain-based biological phenomenon (Ramachandran & Blakeslee, 1998). His contribution to this volume with Hirstein argues that `underlying all the diverse manifestations of human artistic experience' are eight `laws of artistic experience', among which the peak shift effect is prominent, both in the focus paper and its commentaries. A special strength of this paper is the experimental programme that it proposes for validating its eight laws. Because of the experimental methodology, involving physiological measurements such as galvanic skin response, it could be argued that the implicit view of what it means to be human taken here is constrained by this narrow `channel' through which flows a rather small amount of information, having a highly specific character.

Three of the commentaries are by authors from the humanities side of the gap. The distinguished art historian, Partha Mitter doubts the validity of the `eliding of cognition and pleasure' by Ramachandran and Hirstein, and also questions whether they have taken sufficient account of cultural factors, or used a sufficiently broad sample of art on which to base their conclusions. The curator Julia Kindy, though enthusiastic about the paper, questions the conclusion that `all art is caricature' by giving some fascinating examples, and also questions the notion of beauty used by Ramachandran and Hirstein. The artist Ruth Wallen attacks the focus paper on several grounds, including its lack of attention both to culture, and to academic debates about the nature of art; she also objects to the parallel with Buddhism, in support of which she develops some Buddhist concepts. Jaron Lanier is well regarded as both a musician and computer scientist. In harmony with several other commentators, he suggests that the focus paper might better be entitled `The Science of Design', arguing that crucial aspects of art are overlooked. He also gives a pertinent discussion of Indian aesthetic theory, comparing it to higher mathematics, and once again points to the importance of cultural factors in art, as well as to the basis of culture in learning. Of course, Ramachandran mounts a vigorous defence against all this in his final reply, and readers will no doubt find it more enjoyable if I do not attempt to summarize it here.

There are four commentaries from the science side of the gap. Colin Martindale provides interesting technical details on the peak shift effect, discusses its application to evolution (via sexual selection), and cites some of his own research on the psychology of art, showing that untrained viewers do not like cubist art and do not prefer veiled nudes. Richard Gregory notes that there is `more and more evidence of feedback loops - and an enormous richness of downgoing fibres' and considers that `the role of knowledge - both knowledge of the world and experience of art - is greatly underestimated in this paper'. He also questions whether perceptual grouping and binding are directly reinforcing, suggesting an evolutionary explanation instead. Bruce Mangan argues that, for the study of consciousness, the phenomenology of art is fundamental, particularly its ability to enhance and intensify experience; he also argues against the `all art is caricature' hypothesis of Ramachandran and Hirstein, and against their single dimension hypothesis; in addition, he suggests that the ineffability of art may arise from the fact that most of the work is unconscious parallel distributed processing, and he cites work of Berlyne from 1971 applying the peak shift effect to art. Finally, Bernard Baars joins in calling attention to the role of information in understanding the aesthetic response, noting in particular that emotion can reduce redendancy effects, in which repeated stimuli fade from consciousness; he also claims that the aesthetic `is not just a luxury, but a compelling biological adaptation'. Once again, I encourage reading the commentaries themselves and the responses of Ramachandran before settling on any evaluation of all these positions.

II: Other Papers

Semir Zeki is an eminent neurophysiologist, who takes an orthodox reductionist approach to art when he says that he wants a `theory of aesthetics . . . based on an understanding of the workings of the brain'. However, he employs the unusual rhetorical device of claiming that `artists are neurologists, studying the brain with techniques that are unique to them and reaching interesting but unspecified conclusions about the organization of the brain'. Although he gives several arguments, those that I find most compelling have the form:

Artists who said they are especially interested in X have found ingenious ways to partially isolate X from Y, having a clear basis in known neuroanatomy.

In the case of kinetic art, X is movement and Y is colour, while in the case of fauvism, X is colour and Y is form, noting that in his pioneering prior work, Zeki had already shown that movement, colour and form are processed in different ways, that involve different areas of the brain (see pp. 789 below for references and more detail). Moreover, arguments of the above type would be reinforced if the claim of Ramachandran and Hirstein that art is more effective when restricted to one modality (p. 24 below) were to be established.

An important fact about vision is the massive feedback from `higher' to `lower' centres, including the retina itself; this suggests that vision, far from being a passive reception of `what's out there', is an active search for `what's important' to the organism, based on expectations and prior experience. In particular, work by Zeki, as well as by David Hubel at Harvard, and others, has shown that there is significant feedback among the areas of the brain associated with visual processing. Zeki's approach to art is a natural application of his pioneering discoveries of the spatial modularity of visual perception, and of the temporal asynchrony among various perceptual modalities and submodalities. Perhaps the most basic insight that an outsider can gain from this research is that a great deal of parallel distributed processing is needed in order to create perceptual constancy from the chaos of sensory inputs, and that most of this processing is unconscious. It seems highly likely that we will see further significant insights into the neural bases of various visual phenomena, as the result of future experiments using the rapidly advancing technologies of brain imaging.

Despite the variety of disciplines entering into Humphrey's paper and its commentaries, it seems especially important to recognize the biological issues that are involved, among which perhaps the most central is the origin and nature of modern human beings. There is widespread agreement that anatomically modern humans first appeared roughly 100,000 years ago, but there is still debate about when cognitively modern humans first appeared. It has been widely believed that Cro-Magnon humans had a capacity of symbolization and communication that set them apart from the other hominids that they were gradually displacing a few tens of thousands of years ago, and the stunning cave art dating from about 10,000 to 35,000 years ago is usually taken as strong evidence for this belief. But Humphrey argues against this, giving evidence that it may have been the Neanderthals who produced this art.

It is interesting to notice that much of the debate around Humphrey's paper is in fact aesthetic, rather than overtly biological. For example, in the response to his commentaries, Humphrey argues that Spanish-Levantine rock art, Greek vase painting, and Roman murals have an inferior `copy-book' quality, in comparison with the cave art of Chauvet and Lascaux; I wonder though if he would be willing to say the same about Greek sculpture? In any case, this is a fascinating example of the humanities making a substantive contribution to a significant scientific debate.

There is more at stake here than might at first be thought, including the uniqueness of humans, the origin of language, and perhaps even Darwinian evolution. Let's take these one at a time. Although most discussion about the uniqueness of `man' has a more rhetorical than scientific flavour, there are some substantive points, including how the cognitive capabilities of Neanderthals, Cro-Magnons and modern humans (homo sapiens) might differ; Humphrey's paper and the debate around it are directly relevant to this point, with Humphrey arguing against one kind of uniqueness, and most of his commentators disagreeing with him. In a recent provocative contribution, the geographer Jerome Dobson, of the Oak Ridge National Laboratory, has suggested that Neanderthals may actually have been a kind of modern human suffering from chronic iodine deficiency (Dobson, 1998), a view that has been sharply challenged by many anthropologists. So it seems that the debate about the origin and nature of modern man is intensifying, as it continues to twist through a widening range of disciplines.

The origin of language is clearly related to cognitive capability, and among the major issues that have been debated here are Noam Chomsky's dual claims that (1) language has a biological basis, the constraints of which lead to certain universals in human languages; and (2) nevertheless, this biological basis did not evolve to support language, but rather language makes use of cognitive abilities that were already there, perhaps for no purpose at all, or just for their beauty. Chomsky has gone so far as to say that `this poses a problem for the biologist, since if true, it is an example of true "emergence" - the appearance of a qualitatively different phenomenon at a specific stage of complexity of organization' (Chomsky, 1972). By `a problem for the biologist', Chomsky presumably means a challenge to the neo-Darwinian theory of evolution, or at least, to some strict interpretations of that theory. The relevance to this volume is that language requires cognitive capabilities similar to those involved in art and, moreover, the biological origins of art and language are equally mysterious.

The paper by Erich Harth in this volume goes more deeply into several aspects of this area. Harth applies ideas from cognitive science (in particular, his own `sketch-pad' approach), evolutionary biology, neurophysiology and linguistics, to the origins of art. In fact, the argument largely proceeds by discussing the origins of language, and then considering art as a correlative development. This paper includes a nice discussion of the binding and related problems, with a proposed solution. In addition, there is an instructive joke featuring Pablo Picasso.

Ellis also takes a cognitive approach, suggesting that art plays with our expectations, which are based in efferent brain activity, by offering a variety of Gibsonian emotional `affordances'. Meaning is then created in the context of a total ongoing dynamic life process; hence, for Ellis, meaning is far from being the causal result of simple stimuli. This view of art is an application of his general approach to cognition as an expectancy-led interactive and embodied process. Ellis also takes a strong stand for the fundamental importance of emotion in art, and he makes explicit suggestions about its neurophysiological basis.

Brown explores conceptual, intentional, and emotional dimensions of art with some sophistication, drawing on ideas from process theory (in the tradition of Whitehead), clinical neuropathology and phenomenology. Like Ellis, Brown cites the interdependence of emotion and perception, a point also made in several of the commentaries on Ramachandran and Hirstein. But Brown puts greater emphasis on the more general role of knowledge in guiding perception. There seems to be almost a consensus among our contributors on the importance of emotion and knowledge in perception, and especially in art; hence this may well be a fruitful area for future scientific exploration. Readers may also enjoy comparing Brown's clinically-oriented discussion of brain damage (p. 146 below) and art with the various views in the paper of Humphrey and its commentaries; Brown's comparisons of music and visual art are also interesting.

III: Diverse Perspectives

We now consider some themes not represented elsewhere in this volume, but that may nevertheless be relevant. These include the curious parallel between some phases in the historical development of Western art and in our scientific and philosophical understandings of perception, a preliminary view of what might make some works of art great, and the perhaps surprising anticipation of some recent neuroscientific findings by ancient Indian thought. There is also a brief discussion of cognitive linguistics.

Favouring long-term trends, and risking serious over-simplification, we might try to see the recent history of Western art as a gradual exploration of more and more contextual aspects of seeing. Medieval art, consisting primarily of scenes from the stories of Christ, the saints, etc., presents a pure, eternal, self-existing world (much as in traditional Indian religious art); it is non-perspectival in form, and symbolic in content; it does not directly address the realm of human experience - for example, the size of objects tends to correlate with their importance rather than with their location relative to a viewer. Renaissance art introduces perspective, and hence an individual observer, with a much wider range of subject matter, and (passing through the baroque) develops into the classical period, with its interest in realism and its fascination with form, especially symmetry, as perhaps best exemplified in the architecture and music of this period.

Art in the romantic period makes emotion a more explicit focus, while the impressionists and cubists can perhaps be described as interested in the processes of perception, the former more concerned with colour and design, and the latter more concened with form. Even critical social realism can be found, for example in the ashcan school. The expressionists, especially abstract expressionists, explored the emotional content of pure form, while surrealists (like Magritte) and conceptual and pop artists explored the effect of our conceptions and preconceptions on art. More recently still, many performance artists have reached out for more direct involvement with their audiences.

Western theories of art have tended to follow a similar development. Though fixing and comparing their ordering is more difficult, there does seem to be a tendency to take more and more account of context. For example, the realist vision of the early Renaissance tried to confine context to the location of an observer, following the geometrical theory of perspective. This realism later gave way to broader views, e.g., in mannerism. The Renaissance use of perspective also illustrates how scientific theories of art have often been the most prestigious, perhaps because of their precision; another example is the influence of advances in anatomy on Renaissance art. Contemporary aesthetic theories tend to challenge earlier absolutist theories, and emphasize concepts like `intertextuality', which asserts the interdependence of art objects with other objects (Derrida, 1976). Recent progress in neuroscience has inspired a significant increase of interest in theories of aesthetics, with often exciting results, as I hope readers of this volume will be able to affirm, but it is not yet clear that science is able to keep pace with the expanding contextualization evinced by artists and by humanist theories of art.

We can also compare the progression of artistic styles with the various visual centres discussed in the paper by Zeki. The visual processing areas and pathways among them are complex, and it seems fair to say that what is known today is only a very small part of what could be known. Still, much of what is known is relevant to the picture that we are trying to paint in this essay. First, the two retinas and two lateral geniculate bodies (LGBs) in the thalamus seem mainly to be doing low-level processing, using neural structures whose connectivity reflects the contiguity of incoming signals. Subsequent processing is done in the visual cortex, where what we think of as reality begins to emerge from some two dozen areas, of which V1 through V5 are the best understood, due to work of Zeki, Hubel, and others; these centres extract lines, colour, brightness and motion in complex but fascinating ways: V1 and V2 seem to prepare information for V3 to V5, with V3 especially sensitive to lines, V4 to colour (including relative brightness, which is computed by comparing an area with its surround), and V5 to motion; there is also a good deal of feedback to V1 and V2. `Symbolic' recognition, e.g., that a certain pattern of lines and circles is a go board, comes further along in the neural circuitry, and is little understood; but it is natural to suppose that this is where the organism's experience, including culture, is brought to bear on perception.

To summarize, there is a broadening of context as we move from retina and LGB to V1, V2, then V3, V4, V5, and on to the still higher areas, and it should once again be emphasized that more highly contextual information is constantly being fed back to `lower' areas, to help improve their performance. So we seem to have a suggestive parallel between the neural architecture of visual processing and (aspects of) stylistic development in Western visual art.

The similarity between some cave art and some autistic art noticed by Humphrey fits into the physiological and historical themes being developed here, in that the `jittery' outlines seen in both kinds of art are in some ways closer to what our eyes actually take in, due to the rapidity of saccadic eye movements, of which we are generally unconscious. But what this tells us about (so called) primitive man and (so called) primitive art is not necessarily very clear, since it can be argued that the art of pre-colonial Africa, Australia and India is in some ways more sophisticated than much art in the formal Western sense, e.g., because it takes more account of cultural context (other arguments are given in the paper by Ramachandran and Hirstein).

An embodied, enacted view of perception is also taken by work in the relatively new field of cognitive linguistics, which considers metaphor as a key to understanding conceptual aspects of the human mind, thus placing it squarely in the contested borderland between the sciences and the humanities. Here metaphor is defined as `mapping across conceptual domains', in which `the image schemata structure of the source domain is projected onto the target domain in a way that is consistent with inherent target domain structure' (Lakoff, 1993), and image schemata are defined as `recurring structures of, or in, our perceptual interactions, bodily experiences and cognitive operations' (Johnson, 1987). Work by Lakoff, Johnson, Fauconnier, Turner and others demonstrates that (contrary to Brown) metaphor has deep logical structure (Lakoff & Turner, 1989), involving the newly discovered fundamental cognitive operation of blending (Turner & Fauconnier, 1995); this logic can even be expressed precisely in the language of modern mathematics called category theory (Goguen, 1999). Masako Hiraga has successfully applied blending to classical Japanese haiku (Hiraga, 1999a,b). One example of an image scheme is HIGHER IS BETTER, which enables us to understand sentences like `his art reached new heights'. Such image schemas are closely bound up with the contexts provided by the embedding of our bodies in our physical and social worlds.

With an appropriate humility and trepidation, let us now approach the difficult question of what distinguishes great art from merely good art. I suggest that it may be helpful to take a spiritual perspective on this question, and would like to begin by recalling from classical Buddhism the three marks of existence, which are suffering, impermenance, and non-ego. Here `suffering' refers to a general sense of the unsatisfactoriness of human existence, that our hopes and expectations are often frustrated, our goals often unachieved, and even when they are achieved, there is a tendency to escalate them, raising the stakes, and thus increasing the likelihood of later frustration. `Impermanence' refers to the temporary nature of all experience, achievements, and relationships; eventually, we will all die; eventually, even the sun will die. The most difficult of these three concepts is `non-ego', which (in part) refers to the constructed nature of the objects of our attention, including ourselves. For example, a large square rock could be a bench, a coffee table, or an altar, depending on the context; the situation is similar for all the objects of our cognition they take on a certain conceptual identity by virtue of their context, though in general we simply take this for granted. Non-ego is also one aspect of so called `emptiness' (sunyata in sanskrit and muku in Japanese - there is no good western equivalent for this word); here can be found several deeper aspects of non-ego.

Among the three marks of existence, we can perhaps consider non-ego to be the abode of the sublime, in the sense of Kant, a kind of profound aesthetic experience that goes beyond beauty, inspiring awe, perhaps even fear. This can be explained by noting that the inner meaning of emptiness is a vast potential for freedom and creativity, which at the same time is threatening, because of its denial of ego. See also the last section of Brown, especially its lovely final sentences, and the profound paragraph on great art in Ellis (p.172 below). Symbols of impermanence are often found in art; for example, flowers play this role in much oriental art (for some reason, I am particularly thinking of cherry blossoms falling into a stream, in one of Kurasawa's films), and much recent art questions our easy presupposition of everyday reality, in various ways (e.g., dadaism, surrealism, pop art and conceptual art). Of course, we should not neglect what Kant called `the beautiful', which is characterized by stillness, peacefulness and harmony, since the ability to appreciate it properly serves as a ground for appreciating the sublime properly.

When deeply understood, the three marks of existence manifest as their polar opposites: bliss, permanence and self-existence. Within the Western tradition, these qualities seem to be most clearly expressed in medieval art, and they are abundantly present in what the Western tradition has disdainfully called `primitive' art.

Such an approach to art from religion is by no means incompatible with science, because its fundamental paradigm is to examine human experience in a state of meditation; the resulting philosophy is no more than an attempt (or really, many different attempts) to bring some coherence to the results of this examination. For example, the various philosophical assertions of Buddhism are not seen as objective truths, but rather as a kind of phenomenological classification of data gathered through meditation experience. Perhaps one day science will be able to say some interesting things about this kind of experience, much as it is now beginning to say some interesting things about art. It should also be noted that the purpose of such philosophy is soteriological, that is, aimed at improving human beings, rather than at obtaining some kind of disembodied knowledge; I like to think this is also the purpose of many great works of art.

From this perspective, it can be argued that there has been a progressive diminution of the sacred in Western art from medieval times into the twentieth century, with some recent attempts to rediscover the sacred in ways that differ greatly from traditional Christianity, such as the close observation of the textures of natural objects; a revival of medieval musical forms can also be seen in the work of Arvo Part and others. However, most contemporary artistic production is in the service of commercialism and consumerism, much as most European art up to the nineteenth century was in the service of the Church or the nobility.

In a different kind of connection between art and religion, many have been surprised to learn that the recently discovered and much debated delay between perception and consciousness was noticed and explained centuries ago by Buddhist meditators, who further noted that an emotional evaluation preceeds conscious experience; see for example the lucid discussion of the five skandhas (form, feeling, perception, concept and consciousness) in Varela et al. (1991). This tradition also recognized that we do not perceive an `objective world' that is `out there', but rather, we construct our own world, based on our values and expectations. When that `world' is transformed through the practice of meditation, it becomes `pure appearance', also called mahamudra, which is experience liberated as `self symbolic', luminous and transparent, without our usual overlays of projection and attachment; in this way, all experience can become heightened aesthetic experience. All the world's great religions seem to have produced mystics who have described roughly similar experiences, for example, Meister Eckhart (see, e.g., Forman, 1991).

The view that great art is related to the sacred stands in sharp contrast to reductionist views. For example, Richard Taylor and others (1999) claim that fractal dimension (Mandelbrot, 1977) is `an essential tool for determining the fundamental content of the abstract paintings produced by Jackson Pollock in the late 1940s', and have shown that this metric gradually increases, `from close to 1 in 1943 to 1.72 in 1952'. Their claim that `fractal analysis could be used as a quantitative, objective technique both to validate and date Pollock's drip paintings' seems perfectly valid, but it also seems clear that poor art could easily be produced having any desired fractal dimension, e.g., by choosing ugly colours and textures. It seems to me that the phenomenological dimension of great art is of key importance, and that this can never be captured by any simple metric, whether fractal dimension, galvanic skin response, or Birkhoff's aesthetic ratio.

Finally, let us return to Richard Gregory's question about whether there has been any progress towards closing the gap between the `two cultures' since the time of C.P. Snow. One salient fact is that the disciplines of sociology, linguistics and anthropology have each achieved a blend of scientific and humanistic methods. After going through periods in which strongly reductionist approaches were tried and then found wanting, it seems fair to say that contemporary humanistic methods now predominate in these fields. Of course methodological debates still rage, but it is also clear that some genuine integration of the two cultures has been achieved in these fields. This essay has also pointed out several interconnections among papers in this volume that bridge the gap in various ways.

On the other hand, the presupposition seems widespread that science has a privileged position, in the sense that scientific assertions automatically dominate assertions from other fields. While this is valid enough for those assertions that have a thoroughly scientific character, it tends to obscure the fact that knowledge obtained by the humanities often has a different quality; for example, it may be phenomeno- logical, i.e., data of the `how it feels' kind. This privileging of science, which can be seen in the editorial policies of many journals, seems to me especially damaging to genuine dialogue between the sciences and the humanities, and if anything, seems to be on the increase, presumably due to the ongoing amazing successes of science and technology. But by abandoning such artificial constraints, I believe that discourse like that found in this volume can flourish, and demonstrate clearly that in some important respects the gap between the two cultures has been significantly narrowed.

Joseph A. Goguen, Dept. of Computer Science, UCSD, La Jolla, CA 92093-0114, USA


I wish to thank Keith Sutherland and Anthony Freeman for their hard work on this volume, without which it could never have come into being, to the extent that they could well be considered its co-editors, though I carry full responsibility for this essay. I also wish to thank Ruth Wallen for valuable comments, and the Japan Society for the Promotion of Science for support during the period in which this was written, mainly in various cafes and trains around Kanazawa, Kyoto and Tokyo.


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