EDGE 37 — March 25, 1998



QUESTION for Pinker: What do you believe consciousness is?

PINKER: Consciousness is a word that refers to a number of different concepts. There's Freud's distinction between the conscious and unconscious mind, which I relate, following a number of other cognitive scientists, to the fact that no computational system can make all its information available to all of its processes. Thus there is a division in the human brain between the kind of information that we can verbally report on and that can affect our day-to-day decision making, and the kind that goes on "beneath the level of consciousness," such as the control of individual muscles in arms and legs or the rules of syntax that govern how we put sentences together.

ROSE: I don't regard consciousness as a property locked inside the brain of an individual. I regard it as a process which emerges in interaction between individuals, particularly humans, during their development, and the society and culture in which they're embedded. Therefore consciousness, in a very interesting sort of way, is not a brain property alone; it involves many many other features as well, and we reduce it excessively—and I don't think Steve is as guilty of this as many of my neuro-scientific colleagues are, in trying to argue that it's simply the reverse of being asleep, or unconscious. Or make the Freudian distinction. I think there are richer meanings; it's a process, not a thing.


Stanislas Dehaene and Margaret Wertheim on Verena Huber-Dyson's "On The Nature of Mathematical Concepts"

Huber-Dyson responds to Herz, Hersh, Dehaene and Wertheim

John Horgan and Stuart Hameroff on Marvin Minsky's "Consciousness is a Big Suitcase"


A-lists: Some lists owners don't care about selling ads or subscriptions, and they don't value volume either. For them, their lists are about density—a tightly-packed nucleus of powerful people. These A-lists are impossible to join unless you have clout in some way. That's because A-lists derive their power from the social network with which they connect. If you're not in that network in real life, you can't get online, either....Dorothy Parker's Vicious Circle without the food and alcohol.

From "The Hot New Medium is ... Email"
by David S, Bennehum
Wired, April 1998

(11,368 words)

John Brockman, Editor and Publisher | Kip Parent, Webmaster



[On January 21st, Steven Pinker and Steven Rose debated each other in an event chaired by Susan Blackmore and held at London University's Institute of Education under the sponsorship of Dillon's and The London Times. Part I of "The Two Steves," was published on EDGE 36 (March 10th) and is available on the EDGE site. In Part II Pinker and Rose answer questions from the audience.]

QUESTION for Pinker: What do you believe consciousness is?

PINKER: There is an extensive discussion of consciousness in the book. Consciousness is a word that refers to a number of different concepts. There's Freud's distinction between the conscious and unconscious mind, which I relate, following a number of other cognitive scientists, to the fact that no computational system can make all its information available to all of its processes. Thus there is a division in the human brain between the kind of information that we can verbally report on and that can affect our day-to-day decision making, and the kind that goes on "beneath the level of consciousness," such as the control of individual muscles in arms and legs or the rules of syntax that govern how we put sentences together. That's, I think, a tractable definition of consciousness, and it can be readily explained by the fact that the particular sequence of muscle movements is not relevant to my global course of planned action, and so therefore should be sealed off and not allowed to interfere with that planning process.

There are other definitions of consciousness, such as the philosophical concept of "qualia," or pure subjective experience: why red looks red to me, or whether my red is the same as your red. I don't have an evolutionary, or neural, or any kind of explanation as to the origin of that sense of consciousness.

ROSE: I don't regard consciousness as a property locked inside the brain of an individual. I regard it as a process which emerges in interaction between individuals, particularly humans, during their development, and the society and culture in which they're embedded. Therefore consciousness, in a very interesting sort of way, is not a brain property alone; it involves many many other features as well, and we reduce it excessively—and I don't think Steve is as guilty of this as many of my neuro-scientific colleagues are, in trying to argue that it's simply the reverse of being asleep, or unconscious. Or make the Freudian distinction. I think there are richer meanings; it's a process, not a thing.

QUESTION for Pinker and Rose: The parts of the brain which distinguish us from the animals are the least modular, and that's the frontal lobes, which take up 30% of the brain. The frontal lobes have the capacity to modulate and even change the physical structure of the brain. Posterior structures, for instance, are extremely flexible; you can cut out quite large chunks of them and they can reorganize. Similarly, the growing evidence for plasticity generally in the cortex, for instance, the use of apparently visual areas ... in blind people who are not using them gives a very different picture of, if you like, culture and society shaping, the brain—particularly, for instance, the growth of intelligence as society has developed over the last 50 or 60 years. It is quite a different picture of determinance of behavior and brain function than the picture of these rather crude and easily overridable systems of ancient structures of evolutionary adapted brain.

PINKER: It is certainly true that the brain has a great deal of plasticity. I think of each one of these subsystems or faculties as systems that are designed to learn, that are designed to shape themselves in interaction with the environment. But it's not true that these faculties are infinitely plastic, and that the brain can do whatever it wants with itself. One example is the difference between spoken language and written language. All children learn to speak without lessons, spontaneously, by exposure to a community of other people, whereas to learn to read requires extensive practice, artificial curricula, and has a high failure rate. If the brain were completely plastic there should be no difference between reading and speech. There is a huge difference, and that is likely to characterize other mental faculties as well. But it certainly is true that they all are designed to learn and interact with the environment.

ROSE: I think the dialogue between specificity and plasticity in the development of the brain is much the most important and interesting thing that we need to understand. Of course the brain cannot be infinitely plastic; our eyes as we develop need to wire up to the visual cortex in the brain in a fairly ordered and systematic sort of way, or we couldn't preserve binocularity, we couldn't have a visual analyzing system of the sort that we've got. At the same time we have to have brains that are modified by experience. That's plasticity, and the capacity for both specificity and plasticity is there genetically to start with, so I entirely agree with you, and I think it's a mistake to have to think in terms of modularity, to an excessive degree, when one's concern is much more complex functions than simply visual analyzing functions.

QUESTION for Pinker and Rose: As our environment is changing by the decade, and our interactions with the environment impact who we are, how can our genetics keep up? Surely we're way out of date genetically, so how are we surviving?

ROSE: I think it's a great mistake to argue about our genetics being way out of date. The point is that it is precisely, if you like, the human capacity given to us by our genome, given to us by environmental and cultural history, that enables us go on creating this changing society all the way along the line. And that is that it's our genetics that enable us to make these transformations. I think it's really a mistake to believe that somehow genes got left behind somewhere in the Stone Age, or somewhere in the evolutionary process and they're running to keep up with the things that we are doing as a result of it. It's that way of thinking that we need to transform if we're to understand the complexity of the processes. Some people get round it by talking about gene environment, co- evolution. I think that's a step in the right direction, but it really doesn't begin to address the complexity of the interactions which you're hinting at there, and which are for the biology of the future, once we've got rid of the sterile dichotomies of gene and environment and understand the richness to try to come to terms with it.

PINKER: Let me answer that in a slightly different way. I don't think that our man-made environment is necessarily running away from us and it's going to be a matter of how the genes are going to keep up. I do think there are some aspects of human nature that are stuck in the Stone Age, and it's BECAUSE our minds are adapted to that period that we change our technology and our environment to make ourselves feel at home. An example is the design of computers—I assume that's one of the things you were referring to as "rapid environmental change." Computers work on ones and zeros. Our minds have not been able to grasp that way of interacting with machines, on their own terms, but there hasn't been a problem of how are we going to cope with all these ones and zeros. Indeed, the brain is not plastic enough to get itself to think that way. Instead we've designed computers so that THEY mesh with OUR way of thinking. We have designed elaborate graphic interfaces that translate quite abstract information into representations of physical objects, in a particular location in space, that can be moved in a particular way, because that's how human intuition works. So I think the answer is: our minds are going to shape the environment in ways that we can cognitively deal with.

QUESTION: It seems to me that the only constant in societies over the last four thousand years has been the presence of some sort of religious suasion forming a moral and ethical framework. Where is the God module in the brain?

PINKER: As I mentioned, I don't think that religion is an adaptation, so I don't think there is a God module. I do discuss at some length in the book how it arises as an interaction of other parts of the minds. One part is an intuitive psychology. Once you have an ability to interpret other people's behavior in terms of unobservable beliefs and desires, that is, a mind. We impute minds to one another; we don't treat one another as wind-up dolls. That faculty can, then, in a sense, run amok, and imagine minds that exist independently of bodies, namely spirits, souls, ghosts, and so on. That's an example of how a part of the mind that evolved for one purpose can give rise to something quite different. I don't think that's the totality of religious belief, and I discuss some of the other components that collectively give rise to it, but that's an example of how a kind of belief can be a major part of human experience, but not necessarily specifically selected by evolution.

ROSE: I'm not sure that I have a mind that deals either in god or in modules, so I'm not sure I can answer the question. I do think it's extremely important to understand the function religion has played through humanity's history and the moral vacuum which is the result perhaps in the loss of the faith and the creation either of a religious society or of a more socially just society, which we're facing at the moment. I would not like to see ultra-Darwinism become the religion of the future.

QUESTION: If history plays such an important part in our development, how come human beings keep making the same mistakes time and time again? How do you predict the future? You've got all that history—can you not see the future with that information?

ROSE: The whole point is the future is radically unpredictable. It's unpredictable because we can only track change. We can't predict futures. Humans can do a little better than other species in predicting futures, but because of the rate of change of technology in human society, constantly throwing out new problems because of the complexity of the social changes that are occurring, then predicting the future becomes extremely hard. That is why I say in many respects it's radically unpredictable. What I do insist is that we have the freedom to make choices about it, which is a different argument. But we don't have infinite flexibility in making those choices. Steve Pinker and I would both agree that we are constrained by our evolutionary past, by our biological givens—none of us can walk on water, any more than we can grow wings. What we can do is find technology that can solve those problems. Those constraints are there. We see and understand the world through spectacles that are given us by our biology—the fact that we are somewhere between one and a half and two and a half meters high, most of us, rather than a couple of centimeters high radically transforms the way that we understand the world. If we were those small creatures we'd see the world—we'd have quite different biological problems and social problems to resolve. So our past is indeed in many ways the key to the present.

PINKER: I have nothing to add to that; I agree with it.

QUESTION for Pinker: I wanted to ask Professor Pinker again about Cartesian dualism. Although your book does argue that you want to approach understanding consciousness in physical terms, in a materialist way, in the book at one point you talk about your materialist work being the project you do during the day, and in the evening when you're talking with your friends and so on you acknowledge that human beings are sentient and have free will and so on. You acknowledge that it's a non- trivial problem to bridge that gap. You say it might not be possible to do that, whereas elsewhere in the book you talk about the computational theory of mind, I assume as a way to bridge that gap. But I wasn't convinced by that, because it seemed to me that it was just relocating the problem. Social categories like desires and beliefs were just being relocated in the heads of individuals. So when your Bill gets on the bus, his belief that the bus is going to his granny's can just be re-represented as a physical symbol in the brain, and that fills that gap. There seems to be some flip-flopping between being a physicalist on the one hand, and on the other saying that you can approach the same subject in two completely different ways.

PINKER: There's no flip-flop in my discussion of mental states such as beliefs and desires, which doesn't call for any kind of substance dualism—the idea that there is some kind of stuff different from neural interactions that accounts for how we behave and how we perceive the world. As a nonreductionist I think there are different levels of analysis, and that the information-processing level of analysis gives rise to psychological regularities and generalizations that can't easily be captured directly in terms of the neurophysiology. Take the simple example that our short-term memory can hold only five or so items. We have no neurophysiological explanation of that, but we can characterize it in computational terms. Eventually it will be tied to the neurophysiology because they're two different levels of analysis of the same phenomenon.

In terms of morality, I believe that there is a role in our discourse for moral judgments and for a concept of free will that is not dualistic but that simply is part of a different system of reasoning, in the same way that mathematics is a system of reasoning that differs from science. We don't actually believe that there are perfect circles or infinite straight lines or Euclidean planes, but we can still perfectly well reason within mathematics. Like many moral philosophers I believe that there's a sphere of moral reasoning we can engage in that makes use of idealizations like free will but without making any commitments that there's actually a different kind of stuff in the physical world. It's an assumption that makes that system of reasoning possible. We can't have ethics unless we hold someone responsible for their behavior; we can't hold them responsible for their behavior unless we believe that the behavior is not directly caused. That's how we make moral judgments, but it doesn't obligate us when we shift to a scientific mode of explanation to believing that there's a ghost in the machine.

ROSE: Very briefly, there's a book which has just been published called The Number Sense by Stanislas Dehaene. It makes a very interesting point about this question about whether you can hold more than five things in your mind at the same time, which is a classical piece of data which appears in every student psychology textbook. Dehaene points out that it is entirely culture-bound. Chinese culture, for example, which has a different way of counting and representing numbers, can hold many more than five items in their mind at the same time. So it's got again this beautiful interaction between culture, society, biology and history, which I think we have to again take into account whenever we try to say these are universals about the way the mind works.

PINKER: The Dehaene finding is part of a set of phenomena that's been known for as long as the five-item constraint has been known, namely that a particular item in memory can point to a much larger data structure, a phenomenon called chunking. The difference between the Chinese memory span and the American one is simply a difference in chunking; the underlying constraint in memory, according to my memory of Dehaene's work, is the same as it is in American children.

QUESTION to Pinker and Rose: What experimental scientific procedures would you do to determine which of your theories is correct?

PINKER: For approaches of this magnitude there isn't going to be one experiment that's decisive. The proof is going to come from the entire body of research that's inspired by the general idea: the hypotheses that flow out of the theory and the ability of the theory to make correct predictions in a wide variety of domains that mutually cohere and that wouldn't have been made otherwise. One of the main points in How the Mind Works is that there has been an enormous body of experimental literature that has been generated by the hypotheses that I present and that hang together well. Any one of them could turn out to be false and require reinterpretation, but it's our general understanding of the emotions and memory and visual perception and so on over a long period that will determine whether we hang onto that approach as basically sound.

ROSE: I don't think that theories are ever overthrown by decisive experiments. Their protagonists merely fade away, despite what Karl Popper said. However there are two sorts of experiments or pieces of biological information I would like. One is very specific: I would like to know why it is that although we share 98% of our genome with chimpanzees no one can mistake the phenotype of a chimpanzee with the phenotype of a human. And the second, and it's a much more easy question to answer in some ways, is the information—the understanding that's coming—on mapping mental processes that come out of the windows into the brain which are provided by positron emission tomography, magnetoencephalography, and all the other technologies that there are around at the moment, that are bound to give us a richer understanding than the rather crude mechanistic models that we all share of the way minds and brains work at the moment.

QUESTION to Pinker and Rose: Both speakers espouse the idea that we have active control over what we do and what we don't do. I've got a bit of a problem with that. For myself and what I see in other people, we operate within very strictly controlled parameters. So I just wonder why in both your investigative researches, there hasn't been more emphasis on what we might call simple preference, such as why you've both got different hairstyles and wear different suits.

PINKER: I'm not sure I understand the question.

ROSE: I'm not quite sure why Steve wants (as he was described in The Guardian a few days ago) his hair to look so beautifully like a bouffant rock star . On the other hand I do think you're right to speak about the constraints in which we operate. I've given the impression that we are free agents, but of course we're not free agents; we're bound socially, we're bound economically, we're bound culturally, we're bound historically, and we're bound biologically, so the constraints which all of those provide—and they're much much sharper, despite what Steve says, for unemployed workers than they are for company directors, and much sharper for black footballers than white racists on the terrace, again a point he seems to disapprove of, and a point I made in the book. I think those are the constraints in which we need to operate, and those are the constraints which I think a different sort of science than either Steve's or mine needs to try to understand.

PINKER: The point I made concerning people with different social backgrounds is not that they have equal choices in life, which they obviously don't. I was raising a specific point as whether that affects the scientific metaphors and analogies that they take seriously, and I think that there's no evidence that they do and some evidence that they don't.

QUESTION for Pinker: Steve Pinker talks about the 45% personality variation which is not under genetic control or family influence. I have a question about identical twins. In both of your books, you selected convergent examples of identical twin behavior and did not talk about the divergent behavior, which is so interesting, in identical twins. When one interviews identical twins that are divergent, what one is struck by is the thoughtful way in which they have thought about their differences and come to observe them compared to the extraordinarily boring way in which the identical twins converge. It's almost as bad as memes, as in the time when wearing your baseball hat backwards was a similar piece of behavior many people did. They're like that. Divergent twins seem to have fought their way along different pathlines, and if they end up with a different inner environment, which leaves them freer. Can you say a word about divergent twins?

PINKER: Yes, I talk not only about the extraordinary similarities in quirks of behavior, such as sneezing in elevators; that I mentioned just to illustrate that the mind has a great deal more genetic specificity than we would have naively predicted. But I also talk about more profound similarities and differences between identical twins. The similarities are not just in the quirks; they are in fundamental dimensions of personality, such as whether you're conscientious or sloppy, whether you're anxious or relaxed, and whether you're antagonistic or friendly. Those traits also show a high, though nowhere near perfect, correlation between identical twins.

I also discuss hypotheses about why identical twins, though highly similar, are not identical in personality. One possibility is sibling interaction, in which each twin strives to differentiate herself or himself from the other twin. I also talk about chance factors that occur in an individual's lifetime: perhaps there is some effect of being chased by a dog, or receiving an act of kindness. Also, there are surely many unpredictable factors in the growth of the brain, since the gene can't specify every connection. I think it's an exciting project for psychology to test these hypotheses, and many personality psychologists are engaged in it. We know that one putative factor, namely growing up with a given set of parents, has a surprisingly small effect on long term personality. In general, this research focuses our attention on the factors other than the genes that make us what we are.

QUESTION for Pinker: I partly agree with Pinker that procreation is important for loving your partner. But I would argue that procreation actually can much better explain why partners cheat each other, trying to find a higher chance for procreation, but it doesn't necessarily explain why. So: why would a partner stay with their spouse, as opposed to cheating and trying to find higher chances for procreation. Also, you didn't comment on what Rose said about homosexual love.

PINKER: I actually do have an extensive discussion of love as opposed to lust and sexual desire. I think the long-term commitment that you see in a husband and a wife, or in two close friends, and in homosexual lovers—although I don't talk much about homosexuality in the book—comes from a different dynamic. It's analogous to symbiosis in the natural world. You start off with a commonality of interest, that is, what is good for me is good for someone else. In the case of heterosexual marriage that trigger can be the shared genetic interest in the children, but in the case of close friends it could be things like having common interests, having common enemies, having common tastes, and so on. Once what's good for you is good for someone else, that gives you a stake in their well-being, and so you're apt to value them. If you value them, that makes you more valuable to them, and they're likely to value you, and you can get a positive feedback loop where the coalition of two people with common interests can develop into a long-term attachment. We experience this as the emotion of long-term companionate love. I think that's what keeps married couples together, and what keeps close friends together. It's a different emotion than sexual desire, and it's a different emotion than the head-over-heels infatuation that often gets a couple together to begin with. So love is a set of emotions, and I discuss them separately in the book.

ROSE: Steve has provided a neat cost benefit analysis of the merits of love, and it's precisely the point that I was making before about metaphors which he was so uneasy about. Here's a metaphor and a mode of thinking that he's taken over lock stock and barrel from a particular set of economic theories, and applied with enormous energy and ingenuity by evolutionary psychology. I happen to think it's a very impoverished way of trying to describe much more complex phenomena.


From "The Hot New Medium is ... Email"
by David S, Bennehum
Wired, April 1998

[A lenghtly and informative look at communications and information. Currently on the newstands and available on WIRED's site in a few weeks. - JB]


Some list owners don't care about selling ads or subscriptions, and they don't value volume, either. For them, their lists are about density — a tightly packed nucleus of powerful people. These A-lists are impossible to join unless you have clout in some way. That's because A-lists derive their power from the social network with which they connect. If you're not in that network in real life, you can't get in online, either.


A-lists derive power from the social network to which they connect you. As in the real world, it's strictly invitation-only.


A-lists exist all over the world. Usually they're private — the board of directors of a corporation might be on a list, or the clients of a particularly successful consultant. Whatever the membership, A-lists reinforce the feeling of inclusion. It's one of the perks of success.

"People are asked to join the list," John Brockman says of his élite EDGE list, which goes out by email to around 1,000 members two or three times a month. "It started as an outgrowth of what I call 'Third Culture intellectuals.'" Brockman deÞdefines Third Culture intellectuals as "people who are doing empirical work and writing books about it, as opposed to people dealing with opinions. These are people who are creating and changing the world." Brockman, the literary agent known for a client list thick with scientists, pundits, and philosophers, likes to deÞdefine his clientele as a clique that also happens to be changing the world. His EDGE list is an outgrowth of years of tireless networking that began when he ran The Expanded Cinema Festival at Filmmakers Cinematheque in New York in 1965 at the age of 24.

EDGE allows networking among this élite, some of whom were identiÞidentified as the digerati in Brockman's book by the same title. The list has a simple format: a single member is either interviewed by Brockman or asked to write an essay. For instance, Stanislas Dehaene wrote an
essay on numbers and the brain, which in turn was critiqued by EDGE members George Lakoff, Marc D. Hauser, and Jaron Lanier. It's a brilliant format, partly because of who's on the list — Richard Dawkins, Freeman Dyson, David Gelernter, Nathan Myhrvold, and Naomi Wolf, to name a few. And since Brockman's business is brokering book deals, it's an outstanding means to stay on patterns of thought. If an idea hot enough to be a book emerges on EDGE, Brockman has Þfirst-mover advantage.


"The Model is creating reputation," says economist Hal Varian. "Lists are about relative status."


This isn't Brockman's primary motive, however. "The purpose is to create — to arrive at an axiology of the world's knowledge. Get the brightest people in the world in the room and have them ask the questions they are asking themselves. They get to try out ideas on a group of peers who are not in their own discipline. They get to be tested and challenged. It's very vigorous — and very entertaining." The public is permitted to view archives of EDGE on Brockman's Web site (www.edge.org/), which, in turn, allows him to ventilate some of the ideas in the public sphere. But Brockman's list would collapse were the hoi polloi allowed in. It's unlikely that people like Nathan Myhrvold have the time or interest to listen to just anyone with email. The moment EDGE moves away from being the A-list, it collapses and becomes a B-list, otherwise known as a chat room.


List Name: EDGE
Circulation: 600
Owner: John Brockman
Start: 1996
Type: A-list; private
Description: Superagent John Brockman recreates Dorothy Parker's Vicious Circle without
the food and alcohol.
How to Join: Visit www.edge.org/ for digital leftovers.

Excerpted from, "The Hot New Medium Is...Email," by David S. Bennahum
© 1998 Wired Magazine Group, Inc. All rights reserved.


Stanislas Dehaene and Margaret Wertheim on Verena Huber-Dyson's "On The Nature of Mathematical Concepts"

From: Stanislas Dehaene
Submitted: 3.2.98

I find myself in agreement with most of what Verena Huber-Dyson states about the mathematical mind. Non-symbolic processing is clearly crucial, and non-conscious (or subconscious) mental activity plays a considerable part. Jumping to conclusions, only later to go back and work out an exact proof, has been stressed by many mathematicians in the past, including Hadamard, Polya, Einstein and Poincare. For the most part, this conclusion is based solely on mathematicians' intuitions, but my research suggests that at least in the number domain, it can be validated by neuropsychological experiments. Yes, there is a non-symbolic representation of numerical quantities (which can be called "analogical" or "conceptual"). Yes, it plays a crucial role whenever we think about number MEANING, rather that merely do symbolic number crunching. And, yes, it can be activated unconsciously and give us "intuitions" about arithmetical relations among numbers as well as between numbers and space. It even has a specific cerebral substrate, so that in the near future we may hope to image its conscious or unconscious activation during (elementary) arithmetic.

I enjoyed Verena Huber-Dyson's dissection of the mental processes going on in her head as she was reflected over Ramanujan's finding (that 1729 is the smallest integer that can be expressed as the sum of two cubes in two different ways). Obviously, there are many different ways to "jump to this conclusion" (although a formal proof is relatively long). All of them seem elementary once they have been found! I still think, however, that my explanation is simplest because it only appeals to VISUAL recognition. Presumably, any mathematician knows that 1, 1000, 1728 and 729 are cubes (1728, I know realize, is obvious to any Englishman because it is the number of cubic inches in a cubic foot; 729 may not be so well known). Once you know this, it is VISUALLY obvious that 1729 is 1000+729 and 1728+1. No conceptual activity is required, although of course higher mathematics may be used afterwards to deepen understanding of this fact, as shown by Verena Huber-Dyson.

The whole point for which I used that particular example is that, at first, Ramanujan's feat looks super-human—but under my explanation, it is a feat that anyone with some interest in arithmetic facts can understand and could have performed! I thus fully agree with the following passage from Verena Huber-Dyson, which I think is worth emphasizing:

"Mathematics can be done without symbols by a particularly 'gifted' individual, like, e.g., Ramanujan. What that gift consists of is one of the questions raised in the EDGE piece. Obviously we are not all of us born with it. Nor do I believe that all people born as potential mathematicians become actual ones. Tenacity of motivation, an uncluttered and receptive mind, an unerring ability to concentrate the mind's focus on long intricate chains of reasoning and relational structures, the self discipline needed for snatching such a mind out of vicious circles, these are only a few characteristics that spring to mind. They can be cultivated. Experience will train the judgment to distinguish between blind alleys and sound trails and to divine hidden animal paths through the wilderness."

Stanislas Dehaene

STANISLAS DEHAENE, researcher at the Institut National de la Santé, studies cognitive neuropsychology of language and number processing in the human brain; author of The Number Sense: How Mathematical Knowledge Is Embedded In Our Brains.

From: Margaret Wertheim
Submitted: 2.28.98

As in other various other EDGE postings on the subject of mathematics V.H-B talks about math being an embodied phenomena. She noted that counting stems from physically embodied beings enumerating physical objects (such as pebbles). Lakoff has also stressed the importance of embodiment in areas like spatial perception, and Dehaene has noted that our brains seem to be physically wired for some sort of mathematical perception. All this suggests the beginning of a post-Platonist conception of mathematics, and potentially even of numbers. Perhaps, as Huber-Dyson hinted, even the integers are not the work of God (as Kronecker so famously remarked) but are intimately bound up with embodiment itself. I find it so provocative that many of the posters in the EDGE discussion on mathematics have been sidling up to the edge of Platonism, and looking beyond pure abstraction towards the idea of an embodied explanation of mathematics.

Apropos of this, I would like to note that mathematician/philosopher Brian Rotman has already formulated a powerful post-platonist and inherently embodied conception of numbers that meshes beautifully with the issues we have been discussing. Rotman has taken the implications of embodiment for mathematics very seriously indeed and has worked out a truly comprehensive post-platonist conception of "what numbers really are". His work goes far deeper than the Intuitionist or the Constructionist approaches and has considerable philosophical consequences for our thinking about mathematical objects.

This post-platonist conception of numbers is presented in his book Ad Infinitum: Taking God Out Of Mathematics And Putting The Body Back In.. The subtitle immediately signals the relevance to our discussions. In a nutshell, Rotman suggests that the integers do not have a separate Platonic existence, but only emerge from the (necessarily physically embodied) act of counting. In the absence of an embodied counting being, Rotman suggests that integers have no ontological validity. From this rather simple observation, he goes on to outline what amounts to a semiotic theory of mathematics. In his account, mathematics consists only of that set of objects and theorems that can be realized through a finite set of procedures, in a finite space and time, using a finite amount of energy, by a finite (i.e. physically embodied) being. Mathematics becomes, then, just like the other sciences, inseparable from the physical world. And NOT, as Platonists believe, a separate "transcendent" reality.

Quite apart from the major philosophical issue at stake here—Platonism having such a powerful psychological grip on western culture—Rotman's work has immediate consequences for our thinking about mathematical objects. In particular, if Rotman's vision of mathematics as INHERENTLY embodied is correct (as I believe it is, and as the work of Dehaene, Lakoff etc suggests), then mathematics CANNOT contain by definition any infinitistic objects—including (most importantly) the real numbers. Thus Rotman's philosophy of numbers provides an answer to the question raised by Reuben Hersh in response to Huber-Dyson. In his posting in EDGE 35, Hersh wrote as follows: "The rational number line vs the real number line--how do you envision the difference? Does the rational line have a lot of little holes scattered everywhere?"

Rotman specifically answers Yes to this question. Yes, he says, the real number line is not a continuum—it is, in effect, peppered with holes. One consequence of his post-platonist philosophy of number is that ALL infinitistic objects (including the irrationals) are idealizations that do not have any ontological reality. Since we cannot get to them by ANY realizable constructivist method in a finite amount of time, then they cannot be said to "exist". According to Rotman, such concepts as the irrationals (and as infinity itself) ought to be regarded as theological abstractions. As Rotman's subtitle suggests, his aim is to strip mathematics of illegitimate "theological" woolyness and ground it firmly in the physically embodied world.

As a corollary to the above, I note that the issue of embodiment is cropping up in a number of other EDGE discussions—Rodney Brooks, for example, insists (rightly I believe) on the necessity of physical embodiment for the realization of an "artificial intelligence". The COG project is founded on this premise. Embodiment, it seems, is a hot issue. It may therefore be of interests to EDGE readers to know that feminist philosophers have long been insisting on just this point—that knowledge and understanding of the world requires not just purely "mental" processes, but must also be grounded in the reality of bodily experience. As a feminist and a lover of science, I find it most interesting to see these two strands meeting up—though I wonder if many scientists are aware that they are now supporting a major claim of feminist philosophy?

MARGARET WERTHEIM is an Australian science writer; author of Pythagoras' Trousers: God, Physics, And The Gender Wars.

Huber-Dyson responds to Herz, Hersh, Dehaene and Wertheim

From: Verena Huber-Dyson
Submitted: 3.25.98

Response to J.C. Herz, Reuben Hersh, Stanislas Dehaene and Margaret Wertheim

Are receptivity to mathematical insights as well as creative powers to forge concepts and generate ideas special gifts or is everybody born a potential mathematician? Which ever is the case, motivation, the intellectual curiosity that drives it, an uncluttered mind and a tendency to abstract introspection are prerequisites to the development of an actual mathematician.

Brain research like Dehaene's and his colleagues may eventually show that we are all born with a capacity to project our minds out to infinity and to forge abstract concepts. We probably are all born potential mathematicians to the same extent that we are all born potential musicians, athletes, painters, novelists, even politicians. But some more so than others. The question for the teaching profession is not how to turn out mathematicians, how to stimulate and coax these native talents, but how NOT TO STIFLE THEM. I like Hersh's story of sitting at the back of his class and waiting for the students to do their thing instead of doing it for them. What we can do is teach skills, and inform them of what we know. The wonderful thing about mathematics is that it is cumulative. Once a theorem is proved it remains a theorem, no matter how drastically society may be changing.

The psychological prerequisite for success which most working mathematicians take for granted, but the public may not always be aware of is patience, a readiness to devote untiring painstaking attention to detail. Why should Math's be made "user friendly"? The current trend of making a subject interesting by illustrating its "usefulness" is rather demeaning to the subject. Isn't Euler's insight that the reciprocal squares 1, 1/4, 1/9, 1/16, 1/25 and so on add up to one sixth of the square of the area of the unit circle a most astounding gem in its own right? Whoever beholds that result is not likely to ask for its applications. (There are more efficient ways of approximating pi squared). How often, by the way, will a practical use surface as a total surprise only ages after the result has become what we call mathematical folklore!

No amount of philosophical acumen could have given Euler criteria for summing his infinite series. On the other hand I am sure that a basic conviction, a trust in the rational behavior of infinity did give Euler and his peers the confidence needed for trusting their instincts. Goedel explicitly stressed the importance of philosophical attitudes when pointing out that Skolem had had all the technical tools to anticipate Goedel's famous result but did not "see" it because he lacked the philosophical motivation.

A few days ago I received a letter from a colleague in Switzerland, a 93 year old Euler scholar and group theorist, who is currently reviewing a paper on hidden lemmata in Euler's summation of reciprocals of squares. Leonhard Euler, 1707-1783, was the prototype of a prolific and creative mathematician with a great gift for JUMPING TO CORRECT CONCLUSIONS. The authors are able to justify Euler's dubious procedure of factoring an infinite series by what we now know of nonstandard analysis! The theory of nonstandard models, evolved from research in mathematical logic, puts the age old tool of reasoning by analogy onto a precise basis. It actually gives criteria under which such reasoning leads to correct results. No such criteria were available 200 years ago. Even criteria for convergence of series were far and few between. When you think of how many foolish results could be obtained by unreasonable handling of the tool, you appreciate that gift.

To Reuben Hersh's question about my use of the term "CONCEPTUAL VISUALIZATION" let me first say that he is right in assuming that I mean it in a broader sense than the mere contemplation of spatial pictures. How often do we say "yes I see " when we mean "I understand" or even "I hear or I smell what it is", "I feel it in my bones", "yes it's twitching under my skin", "indeed I am straining my muscles hopping along the arrows of that diagram trying to reach a conclusion". Category theory brings home the insight that mathematics is not about static situations calling for the label "Truth" but rather about PROCESSES.

The whole question of the so-called "Ontological Status" of anything seems to me kind of moot. In a nutshell: CATEGORY THEORY is the theory of processes, depicted by mathematical arrows and their interactions. Objects can be defined as a special kind of arrows that behave pretty much like what we have come to expect of objects. The categorical way of thinking and arguing proceeds in diagrammatic form, which is where my way of visualizing mathematics comes in. Arrows pointing this way and that, connecting, commuting, generating structures and creating connections between them.

Let me address Reuben Hersh and Margaret Wertheim together on the subject of the IRRATIONALS. If you wish you can very well visualize the rationals sprinkled as individual numbers densely over the real number line. But it seems rather artificial to single out the rationals. A conceptually more appropriate distinction is that between what intuitionists call "lawful choice sequences" and lawless ones. There are only countably many of the first kind, to each one of them Ms. Wertheim ought to bestow the distinction of existence, for they are all characterized by a finite description. They are the CONSTRUCTIVE REALS, in technical terms they are the limits of recursive sequences of rationals. Such a recursive sequence can be described by a finite law. So, for instance, one sixth of the square of pi is the limit of the sequence of successive sums of reciprocal squares, 1, 1 + 1/4, 1 + 1/4 + 1/9, 1 + 1/4 + 1/9 + 1/16, 1 + 1/4 + 1/9 + 1/16 + 1/25 and so forth (the recursive recipe for that sequence is easy to state with a pencil on the back of a mere grocery receipt, but a nuisance on the e-mail circuit). Why should the ratio between the circumference and the diameter of a circle have less existence (whatever that commodity is) than the number that Hardy happened to notice on a London Taxi plate many decades ago ? The amazing thing is that hundreds of years ago people already figured out how to calculate pi up to any desired degree of accuracy. Even closer to home is the length of the diagonal of a unit square, to which the so-called "Athenian ladder" has been leading us by rational approximations for over two millennia. The Greeks already knew how to prove its irrationality and that meant they proved conclusively that their ladder was and is infinite. INFINITY is here to stay.

Now about the NON-CONSTRUCTIVE REALS, the first observation is that there are incommensurably many more of those on the real number line than of the constructive ones. For, being characterized by finite rules, there are only countably many constructive reals (think of them listed as you list their generating rules) while the set of all reals is uncountable, which means that we have a method for constructing to any given list counting reals (the first, the second, the third and so forth...) a real that is left out by that list.

How then can we visualize the set of all reals ? Here the Intutionist or topological visualization comes in handy. We "see" the continuum as made up of blobs of open neighborhoods. A lawless choice sequence can be thought of as obtained by a process as follows: choose an interval, then 1) split it in half, and 2) choose either the right or the left half, go on repeating steps 1) and 2) . That sounds like instructions from a knitting manual. But we are not given any instruction when to stop. In fact we are expected to keep going. In the case of a constructive real there is a rule that tells which of the two halves to choose, while in the lawless case there is not. Here are two complementary vistas of the continuum, one as an assemblage of individual points, in which only a mere handful (countably many) can actually be specified, and a truly continuous expanse. Leibniz already was both puzzled and fascinated by that elusive property of the continuum, the age-old two-faced wave-particle nature rising up to defy its own creator, man. The story of the sorcerer's apprentice once again. Mathematics is full of this occurrence.

Stop to think about all the abstract concepts that are named by nouns: the government, the law, the student body, the morale, you name them. Each of those nouns stands for a PROCESS or for rules that govern a process. I believe that we can throw out the whole ontological bickering by finally understanding that mathematics is about functions, rules, constructions and projections. Understanding this opens up a grasp of Infinity that takes the mystery out of it without tarnishing its glow.

Let me illustrate this by the two most commonly known irrational but constructive reals. If you want a rough and practical approximation to the square root of 2, you draw a square as big as you find comfortable, draw its diagonal and impose a decimal scale (a binary one would be even easier to handle) on the side, refined as far as you want it, copy it and lay it against the diagonal of the square. The Greeks already were able to prove the irrationality of the square root of 2: the observation that the square of any positive integer is divisible by some even power of two yielding an odd quotient, while the double of such a square leaves an even quotient when divided by an even power of two proves all the infinitely many simple assertions of the form: "the square of k/q is not equal to 2 " where k and q range over all positive integers. For an approximation to the square root of 2 they had the famous Athenian ladder that gets closer to the value the higher you climb it. An infinite series for it is

1 + 1/2 - 1/2x4 + 1x3/2x4x6 - 1x3x5/2x4x6x8 + ........

The next familiar irrational is pi, the area of the unit circle. Here we can work with Gregory's (a Scot of the 17th century) series 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...... which converges (adds up to) a quarter of pi. It converges very slowly, it crawls steadily but very slowly closer and closer to its goal. But that does not matter here. I just want to use it to show what I mean by saying that pi is a process rather than a thing, and to illustrate the concept of a choice sequence mentioned earlier. If you want to generate pi, take a comfortably large line segment and declare it as four units long. On the left hand side, keep dividing it into an odd number of equal parts and mark off on the left hand side the resulting lengths 4/3/, 4/5, 4/7, and so forth. All you need is a square, a straight edge and a compass. Now start at the right end point of your line segment, mark off 4/3 to the left, the number you are zeroing in on lies in the interval between the two marked points, now from the left end point of that first interval mark off 4/5 to the right and you get the next interval then again mark off to the left 4/7 and so forth. You will get a so-called nest of intervals. You will never get just one point, but the intervals are nestled and the longer you continue the smaller the interval gets, the closer it narrows down the further choices, the intersection of the unbounded (infinite sequence ) of these intervals is the point that is at a distance pi from the left end point of your first interval.

IN A NUTSHELL: starting with simple finite steps it is observed that they are repeatable indefinitely with clearly determined modifications, next that process is summed up and becomes part of a higher level process to which they same method of 1) unbounded repetition and 2) summing up in a rule can be applied. I really believe that we have wired into our brains the ability for 1) the mental projection of an unbounded process and 2) the jump from there over the edge to a finitary law or concept, a higher point of view, that will again serve as a stepping stone for a new progression 1).

Even accepting Reuben Hersh's conception of mathematics as a cultural phenomenon, there still remains the question where this phenomenon comes from, why exactly this form, just as one asks what the mechanism is that drives the dancing language of bees.

Back to Reuben Hersh and Margaret Wertheim. Yes, I believe most of the MONSTER GROUPS have been hunted down by investigating geometries. Often the geometry would surface long before the conviction that the group ruling it was simple. In general when dealing with groups, (re)presentations, structure problems and so on, one usually has some, often very personal, physical image, or experience of the situation, even connected with muscular sensations experienced directly as that of reaching, intertwining, twisting, reflecting, extending, canceling out and, of course, repeating a process forever. (Multiplication by a negative number effects a stretching or shrinking, depending on whether the number is greater or less than one, followed—or preceded—by a reflection).

But I cannot understand why some people insist on lumping any sort of abstraction with theology and identifying the use of the concept of infinity with a belief in eternity. Any useful infinite series has a finite sum. I do not want to get involved in a debate about the ONTOLOGICAL STATUS OF INFINITE TOTALITIES any more than about that of abstractions. To my mind such questions are straw-puzzles as much as Platonists are strawmen. Any mathematician who is afraid of the accusations of mysticism that have become so popular, now that mathematics has lost its remoteness from the media, will point out that infinity is a "figure of speech". To say the Zeno's infinite series over the reciprocals of all integral powers of 2 adds up to 2 means that the partial sums 1, 1 + 1/2, 1 + 1/2 + 1/4, 1 + 1/2 + 1/4 + 1/8, ..., 1 + 1/2 + 1/4 + 1/8 + ... + 1/2048 (the 12th partial sum, which is in fact exactly 1/2048 short off the mark 2), and so on, can be brought arbitrarily close to the value 2 by adding up sufficiently many terms. The pair of terms "arbitrarily close" and "sufficiently many" explains the concept of a limit as that of the results a hypothetical sequence of processes. Namely it means that "to every number e, no matter how small, a number N large enough can be found so that the distance between 2 and the Nth partial sum is less than e.

The concept of an unbounded sequence and of endless iteration is certainly basic and might lend itself to a neurophysiological investigation. If you are bothered by the thought of an actual completed infinite totality, well you can get very far in mathematics by restricting yourself to the concept of potential infinity. But strict finiteness leads you crashing with your nose into artificial walls wherever you turn.

Finally a cautionary remark to those who talk about "scandals of IGNORANCE". We cannot explain the "unreasonable effectiveness of Mathematics", we don't know what "philosophy" we are basing our work on, we let accusations of a grotesque version of Platonism drop off our hide without remonstrating. We are criticized for not being motivated by any philosophy at all. In fact we do not have sufficient reason to favor one choice over another and are faced with the alternative: either rashly declare an allegiance to some "—ism", possibly of your own creation, or else sincerely admit ignorance. What will you do, will you commit yourself prematurely to a Position, or will you leave the door open to some one more astute than you who may end up showing that ignorance is inherent in the problem?

VERENA HUBER-DYSON is a mathematician who received her PhD. from the University of Zurich in 1947. She has published research in group theory, and taught in various mathematics departments such as UC Berkeley and University of Illinois at Chicago.She is now emeritus professor from the philosophy department of the University of Calgary where she taught logic and philosophy of the sciences and of mathematics which led to a book on Gödel's theorems published in 1991.


John Horgan and Stuart Hameroff on Marvin Minsky's "Consciousness is a Big Suitcase"

From: John Horgan
Submitted: 3.3.98

Marvin Minsky is the most entertaining self-help guru to come down the pike in a long time. I can't wait to read his book, and to see him dispense his cyber-counsel on Oprah.

A couple of nits. First, his defense of the strong AI position neglects a rather basic fact of neuroscience. Minsky seems to believe any machine engaging in complex information-processing must also be conscious, by definition. But as he surely knows, even humans can cogitate without any subjective awareness. I'm not talking about zombies or other esoterica but about blindsight, which is caused by stroke or other brain damage. A man with blindsight has no subjective, visual awareness; he insists that he can see nothing. But if you put a cartoon drawing of a lion in his hands and insist that he guess what it shows, he will guess correctly. If you throw a ball thrown at him, he will catch it. Perception and awareness seem to be to some extent distinct functions, depending on different neural regions.

Also, Minsky's confessed fondness for Freud seems to undercut his predictions about all the wonderful things that AI and neuroscience will surely accomplish in the future. Are we really going to have autocerebroscopes and intelligence-boosting implants and all this other sci-fi stuff if psychoanalysis is still the best theory of mind we can muster? To broaden the question a bit here, what does it say about modern science's grasp of the mind when people as smart as Marvin Minsky and Steve Pinker are besotted with theories—psychoanalysis and evolution, respectively—that date back to the last century? Is this progress?

JOHN HORGAN, science writer; author of The End of Science : Facing the Limits of Knowledge In The Twilight of the Scientific Age, has also written freelance articles for The New York Times, The New Republic, Slate, The London Times, Discover, The Sciences and other publications.

From: Stuart Hameroff
Submitted: 3.3.98

Minsky's Big Suitcase is Big Sandbag

Marvin Minsky's recent attempt to explain away consciousness makes me wonder if my Samsonite is feeling distended, or still angry at being lost at Heathrow. OK, I know its a metaphor, but that's just the problem. Consciousness may indeed be like a theater spotlight, neural net computer, nonlinear attractor such as the Great Spot on Jupiter, or a suitcase. But we need to ask what consciousness actually is, rather than merely what it is like.

What is consciousness? There have always been two types of answers. Socrates argued that conscious experience was something created by the cerebrum, whereas Thales, Plotinus and other ancient "panpsychists" saw conscious experience as a fundamental feature of reality.

Professor Minsky and other "computationalists" follow Socrates in that consciousness is seen as a property of complex activity in the brain's neural networks (and will eventually occur in electronic computers). However others find this view alone unable to accommodate subjective experience—the explanation seems too much like "and then a miracle happens".

Could proto-conscious qualia actually exist as fundamental properties, like spin, or charge? At very small scales spacetime geometry is not smooth, but quantized. Granularity occurs at the incredibly small "Planck scale" (10^-33 centimeters , 10^-43 seconds) which Roger Penrose portrays as a dynamical spider-web of quantum spins. Experiential qualia as well as Platonic values could exist in Planck scale geometry of quantum spin networks. How did they get there? How did anything get there. In this view qualia ensued (directly or indirectly) as particular patterns and dynamics in spacetime geometry from the Big Bang ("...a miracle DID happen").

How could the brain access this supposed "funda-mental" spacetime? Roger Penrose and I have developed a model of consciousness based on quantum computing in protein structures called microtubules inside the brain's neurons. The proposal ("orchestrated objective reduction - Orch OR") involves sequences of pre-conscious superpositions of information ("qubits") which reduce to classical "bit" solutions. Reduction occurs (non-computably) by Roger's quantum gravity threshold—instability in superposed (separated) Planck scale geometry. The Orch OR model thus portrays consciousness as brain processes connected to self-organizing ripples in the basic makeup of reality. (I'd rather be a ripple than a suitcase.)

Regardless of whether the Orch OR model pans out (and unlike other theories it is testable), computer technology seems to be evolving toward the quantum computer. As the mind has always been viewed as contemporary information processing technology, the 21st century metaphor for consciousness (and AI) may well be self-organizing quantum computation.

STUART HAMEROFF, MD is Professor, Departments of Anesthesiology and Psychology, University of Arizona, and a collaborator with Roger Penrose in proposing a specific model (orchestrated objective reduction). In 1996 he coorganized an international, multidisciplinary conference "Toward a Scientific Basis for Consciousness" held at the University of Arizona. He is coeditor of Toward a Science of Consciousness — The First Tucson Discussions and Debates.

Copyright ©1998 by Edge Foundation, Inc.


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