EDGE


EDGE 28 — October 27, 1997

THE THIRD CULTURE

"WHAT ARE NUMBERS, REALLY? A CEREBRAL BASIS FOR NUMBER SENSE "by Stanislas Dehaene

Psychologists are beginning to realize that much of our mental life rests on the operation of dedicated, biologically-determined mental modules that are specifically attuned to restricted domains of knowledge, and that have been laid down in our brains by evolution (cf. Steve Pinker's How the Mind Works). For instance, we seem to have domain-specific knowledge of animals, food, people, faces, emotions, and many other things. In each case — and number is no exception —, psychologists demonstrate the existence of a domain-specific system of knowledge using the following four arguments:

- one should prove that possessing prior knowledge of the domain confers an evolutionary advantage. In the case of elementary arithmetic, this is quite obvious.

- there should be precursors of the ability in other animal species. Thus, some animals should be shown to have rudimentary arithmetic abilities. There should be systematic parallels between their abilities and those that are found in humans.

- the ability should emerge spontaneously in young children or even infants, independently of other abilities such as language. It should not be acquired by slow, domain-general mechanisms of learning.

- the ability should be shown to have a distinct neural substrate.


THE REALITY CLUB

George Lakoff, Marc D. Hauser & Jaron Lanier on Stanislas Dehaene

(GEORGE LAKOFF:) Dehaene's work is important. It lies at the center of some of the deepest and most important issues in philosophy and in our understanding of what the mind is and, hence, what a human being is. What is at stake in Dehaene's work? (1) Platonism: The objective existence of mathematics external to all beings and part of the structure not only of this universe but of any possible universe. (2) The correspondence theory of truth, and with it all of Anglo-American analytic philosophy. If the correspondence theory falls, the whole stack of cards falls. And if it fails for the paradigm case of mathematics, the fall is all the more dramatic. (3) Functionalism, or The Computer Program Theory of Mind as essentially brain-free.

(MARC D. HAUSER:) First, and as Paul Bloom and a few others have articulated, one of the central issues that we must grapple with is whether the combinatorial power underlying both number and language are separate systems with separate origins, whether they share one system, and whether the power of combinatorics evolved for language first or number first. In this sense, animals contribute in a fundamental way, assuming of course that we do not wish to grant them the symbolic power of human language.

(JARON LANIER:) One of my formative experiences in understanding the psychology of mathematics occurred in grad school. I noticed that students learning new material would complain about how hard the professor pushed them, how little time they had. But other students who had mastered the same material would show no sympathy at all. "Trivial, trivial, once you've seen it", they would mutter. (Visual metaphors seem to be the most common when mathematicians explain their insights.) I began to wonder if all mathematicians were assholes.

Marc Hauser, David G. Myers, Howard Rheingold, Cliff Pickover, and Lee Smolin respond to "EDGE University: A Proposal"

(MARC D. HAUSER:) Harvard students very much interdisciplinary dialog. Last term I gave a seminar on the biology of morality. It was to be restricted to 20 students, and 150 arrived! I still restricted the course to 20 by giving a first day exam. The course was based on invited speakers giving lectures and students leading discussions, critically attacking some quite spectacular lecturers: Dan Dennett, Howard Gardner, Jerry Kagan, Danny Goldhagen, Dan Schacter, Richard Wrangham, Carol Gilligan....the students were fearless. My guess is that they would very much like to take part in a dialog with EDGE contributors. The question might be then, would EDGE contributors enjoy responding to student queries?? Time is of ....

(CLIFF PICKOVER:) What you should do is bundle Edge into a book. Believe it or not, lots of people who would love this material do not regularly access the web, and even those of us who do access the web find it much more convenient to peruse material in book form.


EDGE IN THE NEWS

"Two Cultures - Never the Twain Shall Meet — Scientists wonder why today the word "Intellectual" is used to describe only those in arts and letters" (Phenomena: Comment and Notes) Smithsonian (October, 1997) by John P. Wiley, Jr.

Text available at:
http://www.smithsonianmag.si.edu/smithsonian/issues97/oct97/phenom_oct97.html


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Esther Dyson: Release 2.0: A Design for Living in the Digital Age; Carl Steadman: Application to Date Carl


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THE THIRD CULTURE

"WHAT ARE NUMBERS, REALLY? A CEREBRAL BASIS FOR NUMBER SENSE"
by Stanislas Dehaene


Stan Dehaene is a thirty-two year old mathematician turned cognitive neuropsychologist who studies cognitive neuropsychology of language and number processing in the human brain. He was awarded a masters degree in applied mathematics and computer science from the University of Paris in 1985 and then earned a doctoral degree in cognitive psychology in 1989 at the Ecole des Hautes Etudes en Sciences Sociales in Paris. He is at present a researcher at the Institut National de la Santé in Paris.

Dehaene claims that number is very much like color. "Because we live in a world full of discrete and movable objects, it is very useful for us to be able to extract number. This can help us to track predators or to select the best foraging grounds, to mention only very obvious examples. This is why evolution has endowed our brains and those of many animal species with simple numerical mechanisms. In animals, these mechanisms are very limited, as we shall see below: they are approximate, their representation becomes coarser for increasingly large numbers, and they involve only the simplest arithmetic operations (addition and subtraction). We, humans, have also had the remarkable good fortune to develop abilities for language and for symbolic notation. This has enabled us to develop exact mental representations for large numbers, as well as algorithms for precise calculations. I believe that mathematics, or at least arithmetic and number theory, is a pyramid of increasingly more abstract mental constructions based solely on (1) our ability for symbolic notation, and (2) our nonverbal ability to represent and understand numerical quantities."

He argues that many of the difficulties that children face when learning math and which may turn into full-blown adult "innumeracy" stem from the architecture of our primate brain, which has not evolved for the purpose of doing mathematics.

It is his view that the human brain does not work like a computer and that the physical world is not based on mathematics — rather math evolved to explain the physical world the way that the eye evolved to provide sight.

JB

STANISLAS DEHAENE is a researcher at the Institut National de la Santé. He is the author of The Number Sense: How Mathematical Knowledge is Embedded in Our Brains (US: Oxford; UK: Penguin Press; France: Editions Odile Jacob - La Bosse des Maths; Italy: Mondadori, forthcoming).


"WHAT ARE NUMBERS, REALLY? A CEREBRAL BASIS FOR NUMBER SENSE"
by Stanislas Dehaene


STANISLAS DEHAENE: In a recent book as well as in a heated discussion at the EDGE forum, the mathematician Reuben Hersh has asked "What is mathematics, really?". This is an age-old issue that was already discussed in Ancient Greece and that puzzled Einstein twenty-three centuries later. I personally doubt that philosophical inquiry alone will ever provide a satisfactory answer (we don't even seem to be able to agree on what the question actually means!). However, if we want to use a scientific approach , we can address more focused questions such as where specific mathematical objects like sets, numbers, or functions come from, who invented them, to what purpose they were originally put to use, their historical evolution, how are they acquired by children, and so on. In this way, we can start to define the nature of mathematics in a much more concrete way that is open to scientific investigation using historical research, psychology, or even neuroscience.

This is precisely what a small group of cognitive neuropsychologists in various countries and myself have been seeking to do in a very simple area of mathematics, perhaps the most basic of all : the domain of the natural integers 1, 2, 3, 4, etc. Our results, which are now based on literally hundreds of experiments, are quite surprising: Our brain seems to be equipped from birth with a number sense. Elementary arithmetic appears to be a basic, biologically determined ability inherent in our species (and not just our own — since we share it with many animals). Furthermore it has a specific cerebral substrate, a set of neuronal networks that are similarly localized in all of us and that hold knowledge of numbers and their relations. In brief, perceiving numbers in our surroundings is as basic to us as echolocation is to bats or birdsong is to song birds.

It is clear that this theory has important, immediate consequences for the nature of mathematics. Obviously, the amazing level of mathematical development that we have now reached is a uniquely human achievement, specific to our language-gifted species, and largely dependent on cultural accumulation. But the claim is that basic concepts that are at the foundation of mathematics, such as numbers, sets, space, distance, and so on arise from the very architecture of our brain.

In this sense, numbers are like colors. You know that there are no colors in the physical world. Light comes in various wavelengths, but wavelength is not what we call color (a banana still looks yellow under different lighting conditions, where the wavelengths it reflects are completely changed). Color is an attribute created by the V4 area of our brain. This area computes the relative amount of light at various wavelengths across our retina, and uses it to compute the reflectance of objects (how they reflect the incoming light) in various spectral bands. This is what we call color, but it is purely a subjective quality constructed by the brain. It is, nonetheless, very useful for recognizing objects in the external world, because their color tends to remain constant across different lighting conditions, and that's presumably why the color perception ability of the brain has evolved in the way it has.

My claim is that number is very much like color. Because we live in a world full of discrete and movable objects, it is very useful for us to be able to extract number. This can help us to track predators or to select the best foraging grounds, to mention only very obvious examples. This is why evolution has endowed our brains and those of many animal species with simple numerical mechanisms. In animals, these mechanisms are very limited, as we shall see below: they are approximate, their representation becomes coarser for increasingly large numbers, and they involve only the simplest arithmetic operations (addition and subtraction). We, humans, have also had the remarkable good fortune to develop abilities for language and for symbolic notation. This has enabled us to develop exact mental representations for large numbers, as well as algorithms for precise calculations. I believe that mathematics, or at least arithmetic and number theory, is a pyramid of increasingly more abstract mental constructions based solely on (1) our ability for symbolic notation, and (2) our nonverbal ability to represent and understand numerical quantities.

So much for the philosophy now, but what is the actual evidence for these claims? Psychologists are beginning to realize that much of our mental life rests on the operation of dedicated, biologically-determined mental modules that are specifically attuned to restricted domains of knowledge, and that have been laid down in our brains by evolution (cf. Steve Pinker's How the Mind Works). For instance, we seem to have domain-specific knowledge of animals, food, people, faces, emotions, and many other things. In each case — and number is no exception —, psychologists demonstrate the existence of a domain-specific system of knowledge using the following four arguments:

- one should prove that possessing prior knowledge of the domain confers an evolutionary advantage. In the case of elementary arithmetic, this is quite obvious.

- there should be precursors of the ability in other animal species. Thus, some animals should be shown to have rudimentary arithmetic abilities. There should be systematic parallels between their abilities and those that are found in humans.

- the ability should emerge spontaneously in young children or even infants, independently of other abilities such as language. It should not be acquired by slow, domain-general mechanisms of learning.

- the ability should be shown to have a distinct neural substrate.

My book The Number Sense is dedicated to proving these four points, as well as to exploring their consequences for education and for the philosophy of mathematics. In fact, solid experimental evidence supports the above claims, making the number domain one of the areas in which the demonstration of a biologically determined, domain-specific system of knowledge is the strongest. Here, I can only provide a few examples of experiments.

1. Animals have elementary numerical abilities. Rats, pigeons, parrots, dolphins, and of course primates can discriminate visual patterns or auditory sequences based on number alone (every other physical parameter being carefully controlled). For instance, rats can learn to press one lever for two events and another for four events, regardless of their nature, duration and spacing and whether they are auditory or visual. Animals also have elementary addition and subtraction abilities. These basic abilities are found in the wild, and not just in laboratory-trained animals. Years of training, however, are needed if one wants to inculcate number symbols into chimpanzees. Thus, approximate manipulations of numerosity are within the normal repertoire of many species, but exact symbolic manipulation of numbers isn't — it is a specifically human ability, or at least one which reaches its full-blown development in humans alone.

2. There are systematic parallels between humans and animals. Animals' numerical behavior becomes increasingly imprecise for increasingly large numerals (number size effect). The same is true for humans, even when manipulating Arabic numerals: we are systematically slower to compute, say, 4+5 than 2+3. Animals also have difficulties discriminating two close quantities such as 7 and 8. We too: when comparing Arabic digits, it takes us longer to decide that 9 is larger than 8 than to make the same decision for 9 Vs 2 (and we make more errors, too).

3. Preverbal human infants have elementary numerical abilities, too. These are very similar to those of animals: infants can discriminate two patterns based solely on their number, and they can make simple additions and subtractions. For instance, at 5 months of age, when one object is hidden behind a screen, and then another is added, infants expect to see two objects when the screen drops. We know this because careful measurements of their looking times show that they look longer when, a trick makes a different number of objects appear. Greater looking time indicates that they are surprised when they see impossible events such as 1+1=1, 1+1=3, or 2-1=2. [Please, even if you are skeptical, don't dismiss these data with the back of your hand, as I was dismayed to discover Martin Gardner was doing in a recent review of my book for The Los Angeles Times. Sure enough, "measuring and averaging such times is not easy", but it is now done under very tightly controlled conditions, with double-blind video tape scoring. I urge you to read the original reports, for instance Wynn, 1992, Nature, vol. 348, pp. 749-750 — you'll be amazed at the level of detail and experimental control that is brought to such experiments.]

Like animals and adults, infants are especially precise with small numbers, but they can also compute more approximately with larger numbers. In passing, note that these experiments, which are very reproducible, invalidate Piaget's notion that infants start out in life without any knowledge of numerical invariance. In my book, I show why Piaget's famous conservation experiments are biased and fail to tell us about the genuine arithmetical competence of young children.

4. Brain lesions can impair number sense. My colleagues and I have seen many patients at the hospital who have suffered cerebral lesions and, as a consequence, have become unable to process numbers. Some of these deficits are peripheral and concern the ability to identify words or digits or to produce them aloud. Others, however, indicate a genuine loss of number sense. Lesions to the left inferior parietal lobe can result in a patient remaining able to read and write Arabic numerals to dictation while failing to understand them. One of our patients couldn't do 3 minus 1, or decide which number fell between 2 and 4! He didn't have any problem telling us what month fell between February and April, however, or what day what just before Wednesday. Hence the deficit was completely confined to numbers. The lesion site that yields such a number-sense deficit is highly reproducible in all cultures throughout the world.

5. Brain imaging during number processing tasks reveals a highly specific activation of the inferior parietal lobe, the very same region that, when lesioned, causes numerical deficits. We have now seen this activation using most of the imaging methods currently available. PET scanning and fMRI pinpoint it anatomically to the left and right intraparietal sulci. Electrical recordings also tell us that this region is active during operations such as multiplication or comparison, and that it activates about 200 ms following the presentation of a digit on a screen. There are even recordings of single neurons in the human parietal lobe (in the very special case of patients with intractable epilepsy) that show specific increases in activity during calculation.

The fact that we have such a biologically determined representation of number in our brain has many important consequences that I have tried to address in the book. The most crucial one is, of course, the issue of how mathematical education modifies this representation, and why some children develop a talent for arithmetic and mathematics while others (many of us!) remain innumerate. Assuming that we all start out in life with an approximate representation of number, one that is precise only for small numbers and that is not sufficient to distinguish 7 from 8, how do we ever move beyond that "animal" stage? I think that the acquisition of a language for numbers is crucial, and it is at that stage that cultural and educational differences appear. For instance, Chinese children have an edge in learning to count, simply because their number syntax is so much simpler. Whereas we say "seventeen, eighteen, nineteen, twenty, twenty-one, etc..", they say much more simply: "ten-seven, ten-eight, ten-nine, two-tens, two-tens-one, etc."; hence they have to learn fewer words and a simpler syntax. Evidence indicates that the greater simplicity of their number words speeds up learning the counting sequence by about one year! But, I hasten to say, so does better organization in Asian classrooms, as shown by UCLA psychologist Jim Stigler. As children move on to higher mathematics, there is considerable evidence that moving beyond approximation to learn exact calculation is very difficult for children and quite taxing even for the adult brain, and that strategies and education have a crucial impact.

Why, for instance, do we experience so much difficulty in remembering our multiplication tables? Probably because our brain never evolved to learn multiplication facts in the first place, so we have to tinker with brain circuits that are ill-adapted for this purpose (our associative memory causes us to confuse eight times three with eight times four as well as will eight plus three). Sadly enough, innumeracy may be our normal human condition, and it takes us considerable effort to become numerate. Indeed, a lot can be explained about the failure of some children at school, and about the extraordinary success of some idiot savants in calculation, by appealing to differences in the amount of investment and in the affective state which they are in when they learn mathematics. Having reviewed much of the evidence for innate differences in mathematical abilities, including gender differences, I don't believe that much of our individual differences in math are the result of innate differences in "talent". Education is the key, and positive affect is the engine behind success in math.

The existence of mathematical prodigies might seem to go against this view. Their performance seems so otherworldly that they seem to have a different brain from our own. Not so, I claim — or at the very least, not so at the beginning of their lives: they start in life with the same endowment as the rest of us, a basic number sense, an intuition about numerical relations. Whatever is different in their adult brains is the result of successful education, strategies, and memorization. Indeed, all of their feats, from root extraction to multidigit multiplication, can be explained by simple tricks that any human brain can learn, if one were willing to make the effort.

Here is one example: the famous anecdote about Ramanujan and Hardy's taxi number. The prodigious Indian mathematician Ramanujan was slowly dying of tuberculosis when his colleague Hardy came to visit him and, not knowing what to say, made the following reflection: "The taxi that I hired to come here bore the number 1729. It seemed a rather dull number". "Oh no, Hardy", Ramanujan replied, "it is a captivating one. It is the smallest number that can be expressed in two different ways as a sum of two cubes."

At first sight, the instantaneous realization of this fact on a hospital bed seems incredible, too amazingly bright to be humanly possible. But in fact a minute of reflection suggests a simple way in which the Indian mathematician could have recognized this fact. Having worked for decades with numbers, Ramanujan evidently had memorized scores of facts, including the following list of cubes:

1x1x1 = 1
2x2x2 = 8
3x3x3 = 27
4x4x4 = 64
5x5x5 = 125
6x6x6 = 216
7x7x7 = 343
8x8x8 = 512
9x9x9 = 729
10x10x10 = 1000
11x11x11 = 1331
12x12x12 = 1728

Now if you look at this list you see that (a) 1728 is a cube; (b) 1728 is one unit off 1729, and 1 is also a cube; (c) 729 is also a cube; and (d) 1000 is also a cube. Hence, it is absolutely OBVIOUS to someone with Ramanujan's training that 1729 is the sum of two cubes in two different ways, naming 1728+1 and 1000+729. Finding out that it is the smallest such number is more tricky, but can be done by trial and error. Eventually, the magic of this anecdote totally dissolves when one learns that Ramanujan had written this computation in his notebooks as an adolescent , and hence did not compute this on the spur of the moment in his hospital bed: he already knew it!

Would it be farfetched to suggest that we could all match Ramanujan's feat with sufficient training? Perhaps that suggestion would seem less absurd if you consider that any high school student, even one that is not considered particularly bright, knows at least as much about mathematics as the most advanced mathematical scholars of the Middle Ages. We all start out in life with very similar brains, all endowed with an elementary number sense which has some innate structure, but also a degree of plasticity that allows it to be shaped by culture.

Back to the philosophy of mathematics, then. What are numbers, really? If we grant that we are all born with a rudimentary number sense that is engraved in the very architecture of our brains by evolution, then clearly numbers should be viewed as a construction of our brains. However, contrary to many social constructs such as art and religion, number and arithmetic
are not arbitrary mental constructions. Rather, they are tightly adapted to the external world. Whence this adaptation ? The puzzle about the adequacy of our mathematical constructions for the external world loses some of its mystery when one considers two facts.

- First, the basic elements on which our mathematical constructions are based, such as numbers, sets, space, and so on, have been rooted in the architecture of our brains by a long evolutionary process. Evolution has incorporated in our minds /brains structures that are essential to survival and hence to veridical perception of the external world. At the scale we live in, number is essential because we live in a world made of movable, denumerable objects. Things might have been very different if we lived in a purely fluid world, or at an atomic scale — and hence I concur with a few other mathematicians such as Henri Poincaré, Max Delbruck, or Reuben Hersh in thinking that other life forms could have had mathematics very different from our own.

- Second, our mathematics has seen another evolution, a much faster one: a cultural evolution. Mathematical objects have been generated at will in the minds of mathematicians of the past thirty centuries (this is what we call "pure mathematics"). But then they have been selected for their usefulness in solving real world problems, for instance in physics. Hence, many of our current mathematical tools are well adapted to the outside world, precisely because they were selected as a function of this fit.

Many mathematicians are Platonists. They think that the Universe is made of mathematical stuff, and that the job of mathematicians is merely to discover it. I strongly deny this point of view. This does not mean, however, that I am a "social constructivist", as Martin Gardner would like to call me. I agree with Gardner, and against many social constructivists, that mathematical constructions transcend specific human cultures. In my view, however, this is because all human cultures have the same brain architecture that "resonates" to the same mathematical tunes. The value of Pi, thank God, does not change with culture ! (cf. the Sokal affair). Furthermore, I am in no way denying that the external world provides a lot of structure, which gets incorporated into our mathematics. I only object to calling the structure of the Universe "mathematical ". We develop mathematical models of the world, but these are only models, and they are never fully adequate. Planets do not move in ellipses — elliptic trajectories are a good, but far from perfect approximation. Matter is not made of atoms, electrons, or quarks — all these are good models (indeed, very good ones), but ones that are bound to require revision some day. A lot of conceptual difficulties could be clarified if mathematicians and theoretical physicists paid more attention to the basic distinction between model and reality, a concept familiar to biologists.


THE REALITY CLUB

George Lakoff, Marc D. Hauser & Jaron Lanier on Stanislas Dehaene


From: George Lakoff
Submitted: 10.27.97

Commentary on Dehaene

I have been waiting anxiously for Dehaene's book to reach the local bookstores here. I am, however, familiar with his previous work and applaud it. I assume his current book is based on his earlier work and takes the case further. This research, and earlier research on subitizing in animals, has made it clear that our capacity for number has evolved and that the very notion of number is shaped by specific neural systems in our brains.

Dehaene is also right in comparing mathematics to color. Color categories and the internal structures of such categories arise from our bodies and brains. Just as color categories and color qualia are not just "out there" in the world, so mathematics is not a feature of the universe in itself. As Dehaene rightly points out, we understand the world through our cognitive models and those models are not mirrors of the world, but arise from the detailed peculiarities of our brains. This is a view that I argued extensively in Women, Fire, and Dangerous Things back in 1987 and which has characterized Cognitive Linguistics as a field for two decades.

Rafael Nunez and I are now in the midst of writing a book on our research on the cognitive structure of mathematics. We have concluded, as has Dehaene, that mathematics arises out of human brains and bodies. But our work is complementary to Dehaene's. We are concerned not just about the small positive numbers that occur in subitizing and simple cases of arithmetic. We are interested in how people project from simple numbers to more complex and "abstract" aspects of mathematics.

Our answer, which we have discussed in previous work and will spell out in our book, is that other embodied aspects of mind are involved. These include two particular types of cognitive structures that appear in general in conceptual structure and language.

(1) Image-schemas, that is, universal primitives of spatial relations, such as containment, contact, center-periphery, paths, and so on. Terry Regier (in The Human Semantic Potential, MIT Press) models many of these in terms of structured connectionist neural networks using models of such visual cortex structures as topographic maps of the visual field, orientation-sensitive cell assemblies, and so on.

(2) Conceptual metaphors, which cognitively are cross-domain mappings preserving inferential structures. Srini Narayanan, in his dissertation, models these (also in a structured connectionist model) using neural connections from sensory-motor areas to other areas. Narayanan's startling result is that the same neural network structures that can carry out high-level motor programs can also carry out abstract inferences about event structure under metaphorical projections. Since metaphorical projections preserve inferential structure, they are a natural mechanism for expanding upon our inborn numericizing abilities.

Nunez and I have found that metaphorical projections are implicated in two types metaphorical conceptualization. First, there are grounding metaphors that allow us to expand on simple numeration using the structure of everyday experiences, such as forming collections, taking steps in a given direction, and so on. We find, not surprisingly, that basic arithmetic operations are metaphorically conceptualized in those terms: adding is putting things in a pile; subtracting is taking away. Second, there are linking metaphors, which allow us to link distinct conceptual domains in mathematics. For example, we metaphorically conceptualize numbers as points on a line. In set-theoretical treatments, numbers are metaphorized as sets. Sets are, in turn, metaphorically conceptualized as containers — except in non-well-founded set theory, where sets are metaphorized as nodes in graphs. Such a "set" metaphorized as a node in a graph can "contain itself" when the node in the graph points to itself. Such sets have been used to provide models for classical paradoxes (e.g., the barber paradox).

We have looked in detail at the conceptual structure of cartesian coordinates, exponentials and logarithms, trigonometry, infinitesimals (the Robinson hyperreals), imaginary numbers, and fractals. We have worked out the conceptual structure of e to the power pi times i. It is NOT e multiplied by itself pi times and the result multiplied by itself i times-whatever that could mean! Rather it is a complex composition of basic mathematical metaphors.

Our conclusion builds on Dehaene's, but extends it from simple numbers to very complex classical mathematics. Simple numeration is expanded to "abstract" mathematics by metaphorical projections from our sensory-motor experience. We do not just have mathematical brains; we have mathematical bodies! Our everyday functioning in the world with our brains and bodies gives rise to forms of mathematics. Mathematics is not "abstract", but rather metaphorical, based on projections from sensory-motor areas that make use of "inferences" performed in those areas. The metaphors are not arbitrary, but based on common experiences: putting things into piles, taking steps, turning around, coming close to objects so they appear larger, and so on.

Simple numeration appears, as Dehaene claims, to be located in a confined region of the brain. But mathematics — all of it, from set theory to analytic geometry to topology to fractals to probability theory to recursive function theory — goes well beyond simple numeration. Mathematics as a whole engages many parts of our brains and grows out of a wide variety of experiences in the world. What Nunez and I have found is that mathematics uses conceptual mechanisms from our everyday conceptual systems and language, especially image-schemas and conceptual metaphorical mappings than span distinct conceptual domains. When you are thinking of points inside a circle or elements in a group or members of set, you are using the same image-schema of containment that you use in thinking of the chairs in a room.

There appears to be a part of the brain that is relatively small and localized for numeration. Given the subitizing capacity of animals, this would appear to be genetically based. But the same cannot be said for mathematics as a whole. There are no genes for cartesian coordinates or imaginary numbers or fractional dimensions. These are imaginative constructions of human beings. And if Nunez and I are right in our analyses, they involve a complex composition of metaphors and conceptual blends (of the sort described in the recent work of Gilles Fauconnier and Mark Turner).

Dehaene is right that this requires a nonplatonic philosophy of mathematics that is also not socially constructivist. Indeed, what is required is a special case of experientialist philosophy (or "embodied realism"), as outlined by Mark Johnson and myself beginning in Metaphors We Live By (1980), continuing in my Women, Fire and Dangerous Things (1987) and Johnson's The Body In The Mind (1987), and described and justified in much greater detail our forthcoming Philosophy In The Flesh.

Such a philosophy of mathematics is not relativist or socially constructivist, since it is embodied, that is, based on the shared characteristics of human brains and bodies as well as the shared aspects of our physical and interpersonal environments. As Dehaene said, pi is not an arbitrary social construction that could have been constructed in some other way. Neither is e, despite the argument that Nunez and I give that our understanding of e requires quite a bit of metaphorical structure. The metaphors are not arbitrary; they too are based on the characteristics of human bodies and brains.

On the other hand, such a philosophy of mathematics is not platonic or objectivist. Consider two simple examples. First, can sets contain themselves or not? This cannot be answered by looking at the mathematical universe. You can have it either way, choosing either the container metaphor or the graph metaphor, depending on your interests.

Or take a second well-known example. Are the points on a line real numbers? Well, Robinson's hyperreals can also be mapped onto the line. When they are, the real numbers take up hardly any room at all on the line compared to the hyperreals. There are two forms of mathematics here, both real mathematics. Moreover, as Leon Henkin proved, given any standard axiom system for the real numbers and a model for it containing the reals, there exists another model of those axioms containing the hyperreals. The reals can be mapped onto the line. So can the hyperreals.

So given an arbitrarily chosen line L, does every point on L correspond to a real number? Or does every point on L correspond to a hyperreal number? (If the answer is yes to the latter question, it cannot be yes to the former question — not with respect to the same correspondence.) This is not a question that can be determined by looking at the universe. You have a choice of metaphor, a choice as to whether you want to conceptualize the line as being constituted by the reals or the hyperreals. There is valid mathematics corresponding to each choice. But it is not a matter of arbitrariness. The same choice is not open for, say, the integers which cannot map onto every point on a line.

Mathematics is not platonist or objectivist. As Dehaene says, it is not a feature of the universe. But this has drastic consequences outside the philosophy of mathematics itself. If Dehaene is right about this-and if Reuben Hersh and Rafael Nunez and I are right about it-then Anglo-Ame rican analytic philosophy is in big trouble. The reason is that the correspondence theory of truth does not work for mathematics. Mathematical truth is not a matter of matching up symbols with the external world. Mathematical truth comes out of us, out of the physical structures of our brains and bodies, out of our metaphorical capacity to link up domains of our minds (and brains) so as to preserve inference, and out of the nonarbitrary way we have adapted to the external world. If you seriously believe in the correspondence theory of truth, Dehaene's work should make you worry, and worry big time. Mathematics has been, after all, the paradigm example of objectivist truth.

Dehaene's work is also very bad news for the theory of mind defended in Pinker's How The Mind Works (pp. 24-25), namely, functionalism, or the Computer Program Theory of Mind. Functionalism, first formulated by philosopher Hilary Putnam and since repudiated by him, is the theory that all aspects of mind can be characterized adequately without looking at the brain, as if the mind worked via the manipulation of abstract formal symbols. This is like a computer program designed independent of any particular hardware, but which happened to be capable of running on the brain's wetware. This computer-program mind is not shaped by the details of the brain.

But if Dehaene is right, the brain shapes and defines the concept of number in the most fundamental way. This is the opposite of what is claimed by the Computer Program Theory of Mind, namely, that the concept of number is part of a computer program that is not shaped or determined by the peculiarities of the physical brain at all and that we can know everything about number without knowing anything about the brain.

Challenging the Computer Program Theory of Mind is not a small matter. Pinker calls it "one of the great ideas in intellectual history" and "indispensable" to an understanding of mind. Any time you hear someone talking about "the mind's software" that can be run on "the brain's hardware," you are in the presence of the Computer Program Theory.

Dehaene is by no means alone is his implicit rejection of the Computer Program Theory. Distinguished figures in neuroscience have rejected it (e.g., Antonio Damasio, Gerald Edelman, Patricia Churchland). Even among computer scientists, connectionism presents a contrasting view. In our lab at the International Computer Science Institute at Berkeley, Jerome Feldman, I, and our co-workers working on a neural theory of language, have discovered results in the course of our work suggesting that the program-mind is not even a remotely good approximation to a real mind. Among these are the results mentioned above by Regier and Narayanan indicating that conceptual structure for spatial relations concepts and event structure concepts are created and shaped by specific types of neural structures in the visual system and the motor system.

Dehaene's work is important. It lies at the center of some of the deepest and most important issues in philosophy and in our understanding of what the mind is and, hence, what a human being is. What is at stake in Dehaene's work? (1) Platonism: The objective existence of mathematics external to all beings and part of the structure not only of this universe but of any possible universe. (2) The correspondence theory of truth, and with it all of Anglo-American analytic philosophy. If the correspondence theory falls, the whole stack of cards falls. And if it fails for the paradigm case of mathematics, the fall is all the more dramatic. (3) Functionalism, or The Computer Program Theory of Mind as essentially brain-free.

I can barely wait for his new book to get to my local bookstore.

GEORGE LAKOFF previously taught at Harvard and the University of Michigan and since 1972 has been Professor of Linguistics at the University of California at Berkeley, where he is on the faculty of the Institute of Cognitive Studies. He has been a member of the Governing Board of the Cognitive Science Society, President of the International Cognitive Linguistics Association, and a member of the Science Board of the Santa Fe Institute. He is the author of Metaphors We Live By (with Mark Johnson), Women, Fire and Dangerous Things: What Categories Reveal About the Mind, More Than Cool Reason: A Field Guide to Poetic Metaphor (with Mark Turner), and most recently, Moral Politics, an application of cognitive science to the study of the conceptual systems of liberals and conservatives. He has just completed (with Mark Johnson) Philosophy In The Flesh, a re-evaluation of Western Philosophy on the basis of empirical results about the nature of mind, and is now working with Rafael Nunez on a book tentatively titled The Mathematical Body, a study of the conceptual structure of mathematics.


From: Marc D. Hauser
Submitted: 10.26.97

Having worked on the problem of numerical representation in animals, I would like to raise a few quick points. First, and as Paul Bloom and a few others have articulated, one of the central issues that we must grapple with is whether the combinatorial power underlying both number and language are separate systems with separate origins, whether they share one system, and whether the power of combinatorics evolved for language first or number first. In this sense, animals contribute in a fundamental way, assuming of course that we do not wish to grant them the symbolic power of human language. When Stanislas says that animals lack the symbolic power of human number systems, this is an assumption. There is not proof of this yet, because the only data we have come from massive training programs. and yet, some of the non-training studies that we have begun to run (not yet published) suggest that rhesus monkeys and tamarins may go beyond the low level numerical discriminations that Stanislas has in mind.

This raises yet another questions. Of course language contributes in some way to our numerical sense, but in precisely what way? I have only just dipped into the new book, so have no sense of whether Stanislas speaks to this issue, but it is critical, not only for our general interest in the organization of domains of knowledge, but the kinds of selection pressure that ultimately allowed us to leave animals in their dust. For example, it is conceivable that the key pressure for fine level discrimination of number (i.e., beyond more versus less) evolved when our system of social exchange emerged, when change of a $1 bill was critical? In most animal interactions, there is never really a situation where a more versus less distinction fails. In cases where it does fail, animals clearly have the capacity to solve the problem...these are small number situations, for example, fights between two allies against a third, or an assessment of fruit in a pile, or individuals in one group versus a second. What ecological or social pressure created the push for more powerful symbolic systems, dedicated to number juggling?

Marc


From: Jaron Lanier
Submitted: 10.26.97

Some quick thoughts on Stanislas Dehaene's presentation:

1) One of my formative experiences in understanding the psychology of mathematics occurred in grad school. I noticed that students learning new material would complain about how hard the professor pushed them, how little time they had. But other students who had mastered the same material would show no sympathy at all. "Trivial, trivial, once you've seen it", they would mutter. (Visual metaphors seem to be the most common when mathematicians explain their insights.) I began to wonder if all mathematicians were assholes. Then I realized that these students were simply reporting their experiences honestly, if not courteously. Mathematical ideas are easier in hindsight. The language of mathematics is awkward and seems ill suited to learning. But mathematical ideas can seem simple once internalized. Mathematicians often report that they struggle to "see" their ideas in the right way, so that they become simple.

We have here a case of trains passing in the night. Dehaene highlights developmental links between numeracy and a variety of things; language, abstraction, logic skills, real world problem solving activities, money. He suggests that these might be vital to fighting innumeracy. Yet many educators trying to reach math-phobic kids have been moving in the direction of visual, somatic, and musical representations of math. (For a great visual/kinetic example, see Jim Blinn's work.) Many kids seem to be allergic to abstractions, and to real world problems (the dreaded "word problems"), but can relate to math ideas presented in a more experiential modality. There is not necessarily a core disagreement here. It could be the case that a connection between innate numeracy and language, logic, and/or problem solving was critical in evolution, but that other modalities are nonetheless useful in education.

Historically, the good counters of Western civilization (the Mesopotamians) arose quite distinctly from the good logicians (the Greeks, who were poor counters). This seems to me to be a bit of anecdotal evidence that the two skills are not as innately connected as Dehaene suggests, but were connected together by cultural development.

2) Let's not deify money just yet. There are other things we do that animals don't. Music is an intriguing alternative. There seem to be a disproportionate number of musical mathematicians, and music has a numerical quality. All human societies make music, even though no definitive purpose for it has been identified. There are, however, human societies without money. Mathematics seems more likely to have sprung from music than from money. (Of course it need not have sprung from any one thing at all.)

3) As I commented earlier in response to Hersh's presentation, finding a non-platonic basis for the ability to count doesn't make math any less objective. In this respect numbers are different from colors. While certainly numbers and colors evolved similarly, as "local variables" in the brain, what we must do with them at this time is quite different. We don't have any reason to treat the experience of "yellow" as undesirable in the conduct of our lives. Instead we must design user interfaces for our computers and the rest of the human-made world to work as well as possible with colors as we see them. While there is nothing universally right about yellow, there is also nothing wrong with it. Indeed we celebrate our yellows; Van Gough's studio comes to mind. (Yellow is not, however, isolatable and abstract in its meaning. Peter Warshall has done some wonderful research on cross-species functions of colors in the natural world.) On the other hand, we have no reason to retain or celebrate our innate/naive sense of numbers. We find no occasion when it is desirable, or even acceptable, to confuse 1006 with 1007, or pi with 3.

4) Dehaene's extrapolations of his work might at times be a little too simple and linear. Perhaps there is some hidden benefit to awkward number names like "seventeen", since they force the young brain to do a tougher, more painstaking job of learning. Japanese kids can be said to be "behind" English speaking kids in language skills at certain ages because of the difficulties of the Japanese language, but that does not mean that less is being learned, or that the kids will turn into less able adults. I am thinking of Einstein, who commented that he was rather slow as a child and considered that to be an advantage, in that he developed basic skills and intuitions in a more considered way at an older than usual age.

A related point: Perhaps Einstein started an alternate neural numerical scratch pad. Might it not be possible that people develop additional representations of numbers in their brains? Why must the inborn representation remain the only one.

JARON LANIER, a computer scientist and musician, is a pioneer of virtual reality, and founder and former CEO of VPL.


Marc Hauser, David G. Myers, Howard Rheingold, Cliff Pickover, Lee Smolin respond to "EDGE University: A Proposal"


From: Marc D. Hauser
Submitted: 10.14.97

Harvard students very much interdisciplinary dialog. Last term I gave a seminar on the biology of morality. It was to be restricted to 20 students, and 150 arrived! I still restricted the course to 20 by giving a first day exam. The course was based on invited speakers giving lectures and students leading discussions, critically attacking some quite spectacular lecturers: Dan Dennett, Howard Gardner,Jerry Kagan, Danny Goldhagen, Dan Schacter, Richard Wrangham, Carol Gilligan....the students were fearless. My guess is that they would very much like to take part in a dialog with EDGE contributors. The question might be then, would EDGE contributors enjoy responding to student queries?? Time is of ....-

Marc

MARC D. HAUSER, is an evolutionary psychologist, and an associate professor at Harvard University where he is a fellow of the Mind, Brain, and Behavior Program. His research focuses on problems of acoustic perception, the generation of beliefs, the neurobiology of acoustic and visual signal processing, and the evolution of communication. He is the author of The Evolution of Communication (MIT Press), and What The Serpent Said: How Animals Think And What They Think About (Henry Holt, forthcoming).


From: Dave Myers
Submitted: 10.14.97
Subject: A Kindred Venture?

Hi John,

I'm enjoying receiving the Edge/Reality Club mailings. You've managed to engage an impressive group of minds!

If you haven't heard of it, I thought you might be interested in the efforts of a new media company, Peregrine Publishers (working in partnership with its part owner, Scientific American, Inc.) to deliver resources for teachers and learning activities for students in the introductory college courses for biology, chemistry, and psychology. I've helped them launch their psychology site and manage its "Op-Ed Forum."

The audience an anticipated discourse is lower level than your own, but the spirit of harnessing the Web to build intellectual community seems kindred. The Psychology Place (ww w.psychplace.com) is, for now, free and available for exploring.

All best,

Dave

DAVID G. MYERS is a professor of psychology at Hope College and the author of Psychology (5th ed.) and The Pursuit of Happiness.


From: Cliff Pickover
Submitted: 10.15.97

What you should do is bundle Edge into a book. Believe it or not, lots of people who would love this material do not regularly access the web, and even those of us who do access the web find it much more convenient to peruse material in book form. A recent article in Communications of the ACM notes that when university course material is offered on the web, students always print out the material on paper and bring it home to read, to underline, and organize. If this is so, why not make Edge a book? The book is still a better medium for information exchange than the web — because printed pages are easier to read, to take with you anywhere in the home or on vacation, to annotate, to flip back and forth through...

Regards, Cliff

CLIFFORD A. PICKOVER, research staff member at the IBM Watson Research Center, received his Ph.D. from Yale University and is the author of numerous highly-acclaimed books melding astronomy, mathematics, art, computers, creativity, and other seemingly disparate areas of human endeavor. Pickover holds several patents, and is associate editor for various scientific journals. He is also the lead columnist for the brain-boggler column in Discover magazine.


From: Howard Rheingold
Submitted: 10.15.97

This is a great idea. As you know, one of my areas of interest and expertise is in the art of hosting online conversations. Ultimately, what you want is a platform that combines web publishing, web conferencing, chat, instant messages, and email newsletters. And I believe you will also need some online facilitation.-

HOWARD RHEINGOLD, founder of Electric Minds, is the author of Tools For Thought; Virtual Reality, and Virtual Communities


From: Lee Smolin
Submitted: 10.15.97

I did talk about your idea of a course based on the EDGE site to the administrator here in charge of "computer-assisted education", I am waiting for him to get back to me. I do think its a good idea, and I would be very happy to do it.

LEE SMOLIN is a theoretical physicist; professor of physics and member of the Center for Gravitational Physics and Geometry at Pennsylvania State University; author of The Life of The Cosmos (Oxford).


EDGE IN THE NEWS

"When I was young, it was understood that an "educated person" would know the classics; history; literature, art and music, and be at least generally familiar with the sciences. No one could know everything, of course, but it was possible to have a frame of reference. Standards were high: an educated person, we were told in high school, never reads something in translation. Needless to say, I never made the grade, but as the years went by some of those same criteria became part of my own definition of an intellectual: being aware of the intellectual trends of the day, reading in several languages, having a familiarity with literature and music. In my mind this person lived in a large city, was not affiliated with a university, and spent at least some of her time in coffeehouses reading obscure publications. (Until recently I always pictured her framed in a curl of cigarette smoke.) She didn't have much money, but she always vacationed in Europe."

From "Two Cultures - Never the Twain Shall Meet - Scientists wonder why today the word "Intellectual" is used to describe only those in arts and letters" (Phenomena: Comment and Notes) Smithsonian (October, 1997) by John P. Wiley, Jr.

Text available at:
http://www.smithsonianmag.si.edu/smithsonian/issues97/oct97/phenom_oct97.html


BILLBOARD

Esther Dyson: Release 2.0: A Design for Living in the Digital Age; Carl Steadman: Application to Date Carl


From: Esther Dyson
Submitted: 10.25.9

My new book is called Release 2.0: A Design for Living in the Digital Age.
It is from Broadway Books in the US, Viking/Penguin in the UK and various other publishers in other locations, supported by a Website -- http://www.release2-0.com.

Esther

PC Forum in Tucson, Arizona, 22 to 25 March 1998

ESTHER DYSON is president of EDventure Holdings and editor of Release 1.0. Her PC Forum conference is an annual industry event.


From: Carl Steadman
Submitted: 10.17.97
Subject: Application to Date Carl

To be spread far and wide, John.

http://rhumba.pair.com/carl/date/

CARL STEADMAN, the Cofounder of Suck, is coauthor of Providing Internet Services via the Mac OS . He is a Producer for the HotWired Network, and a contributor to CTHEORY.


EMAIL

Maria Lepowsky; Mark Stahlman


From: Maria Lepowsky
Submitted: 10.20.97

I'm one of your lurkers at the edge (a small part of the answer to the recent query, where are all the women?), and have been greatly enjoying the digital conversations. I do promise to become a more responsible, and responsive, cybercitizen. Thanks for putting it all together!

Regards,

Maria Lepowsky

MARIA LEPOWSKY is Associate Professor, Department of Anthropology, University of Wisconsin. She is author of Fruit of the Motherland: Gender in an Egalitarian Society (Columbia University Press), and Dreaming of Islands (Knopf, forthcoming)m based on her research on the island of Vanatinai.


From: Mark Stahlman
Submitted: 10.24.97

John:

I'm curious about why you guys called the "Reality Club" that. I would seem to me that most people involved were (are) much more utopians than realists. Is it the same perverse reason that the Huxley bros, Heard and Co. called their magazine "The Realist" (I just found the first five issues for $80, BTW)? Or, is it the interest in altering reality (by altering the way that we experience it) that justifies the name. Just asking.

Best,

Mark Stahlman

MARK STAHLMAN is president of New Media Associates in New York City.



Copyright ©1997 by Edge Foundation, Inc.

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