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

**BILLBOARD**

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

**EMAIL**

Maria Lepowsky; Mark Stahlman

(10,334 words)

**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.