EDGE 34 — February 16, 1998

THE THIRD CULTURE

"ON THE NATURE OF MATHEMATICAL CONCEPTS: WHY AND HOW DO MATHEMATICIANS
JUMP TO CONCLUSIONS?"

by Verena Huber-Dyson

Conceptual visualization is an indispensable attendant to mathematical
thinking. Formalization is only a tool and may encourage lazy thinking!
Look at the freshmen who enroll in math because they assume they
won't be expected to produce coherent arguments or to write grammatical
text, that bureaucratic neatness in "plugging in numbers and turning
the crank" will suffice to pass the course....That practice, familiarity,
experience and experimentation are important prerequisites for successful
mathematical activity goes without saying. But less obvious and
just as important is a tendency to "day dream", an ability to immerse
oneself in contemplation oblivious of all surroundings, the way
a very small child will abandon himself to his blocks. Anecdotes
bearing witness to the enhancement of creative concentration by
total relaxation abound, ranging from Archimedes' inspiration in
a bath tub to Alfred Tarski's tales of theorems proved in a dental
chair.

**EDGE UNIVERSITY**

"E-Mail Messages Of Scientific Stars Are Required Reading In Harvard
U. Course"* ‹ *Article in *The Chronicle Of Higher Education
*(2/12/98) by Lisa Guernsey

Some undergraduates at Harvard University are reading the e-mail
messages of world-renowned scientists and cultural thinkers this
semester as part of an introductory science course.

**THE REALITY CLUB**

Delta Willis on Colin Blakemore's Question

Oliver Morton on Duncan Steel

Howard Rheingold, Kevin Kelly on Pattie Maes

(13,685 words)

**EDGE UNIVERSITY**

"E-Mail Messages Of Scientific Stars Are Required Reading In Harvard
U. Course"

Article in *The Chronicle Of Higher Education *(2/12/98) by
Lisa Guernsey

Some undergraduates at Harvard University are reading the e-mail
messages of world-renowned scientists and cultural thinkers this
semester as part of an introductory science course.

The people behind the e-mail messages are contributors to Edge,
a group of intellectuals who use the Internet to discuss the social
implications of science and technology. The use of the e mail in
the Harvard class is an experiment to test whether their exchanges
can be integrated into college courses.

Anyone on the Internet can get a taste of the Edge discussions
by visiting the group's World Wide Web site ().
The Harvard students, however, are subscribers to the group's e-mail
newsletter, which is distributed every few weeks and contains e-mail
messages, essays, and queries from Edge contributors.

"The idea is to bring students up to speed on the latest thoughts"
of top scientists, says Marc D. Hauser, an anthropology professor
at Harvard and an Edge contributor. He subscribed his students to
the newsletter as part of "Human Behavioral Biology," (http://icg.fas.harvard.edu
/~scib29/) a course he teaches with another anthropology professor,
Irven DeVore.

Using Edge material seemed ideal, Dr. Hauser says, because of the
range of topics that members of the group discuss. A few weeks ago,
for example, the newsletter featured a talk with Patti Maes, an
artificial-life researcher at the Massachusetts Institute of Technology's
Media Lab. Other contributors have included Richard Dawkins, an
evolutionary biologist at the University of Oxford who wrote *Climbing
Mount Improbable *(W.W. Norton, 1996), and Steven Pinker, an
M.I.T. psychologist and the author of *How the Mind Works*
(Norton, 1997).

John Brockman, a prolific author and literary agent for scores
of writers on science and technology, founded Edge after he wrote
*The Third Culture: Beyond the Scientific Revolution *(Simon
& Schuster, 1995). The book argues that today's intellectual luminaries
are "scientists and other thinkers" who are "rendering visible the
deeper meanings of our lives." He decided to open an e-mail salon
for those "Third Culture" thinkers, people who, in his estimation,
bridge the worlds of literary theorists and traditional scientists.

Mr. Brockman says he has wanted to expose college students to
the group's e-mail messages for some time. He has dubbed the concept
"Edge University," and hopes that if the Harvard experiment works,
professors from around the world will come on board. "E-mail lays
bare what people working in these areas really think."

The Harvard biology course enrolls about 500 undergraduates, who
each week attend lectures as a whole and then meet in groups of
20 for discussions. When the Edge e-mail messages relate to course
readings or lectures, they are integrated into those discussions.
Because the course is introductory and interdisciplinary, Dr. Hauser
expects Edge's exchanges to intersect often with what students are
learning in class.

Already, Dr. Hauser says, after mentioning the Edge discussions
in one of his lectures, "I hear students in the coffee shops yapping
away. It is very exciting."

Dr. Hauser acknowledges that he could simply direct his students
to the Web site each week, but he says the students respond better
to the immediacy of e-mail. And, he argues, a thread of e-mail messages
is more likely to be read as a discussion than is the drier, impersonal
content of a book. As he puts it, "E-mail lays bare what people
working in these areas really think." Soon, Dr. Hauser hopes, the
students will be asking Edge contributors about material they have
posted.

"E-mail lays bare what people working in these areas really think."

Creating that level of interaction with the members, however,
will get tricky. Mr. Brockman, Edge's founder, keeps distractions
on the newsletter to a minimum by filtering out irrelevant messages
from subscribers before he distributes them. So instead of posting
messages directly to subscribers of the Edge newsletter, the students
will send their questions to Dr. Hauser and his research assistants,
who will compile and forward them to Mr. Brockman. Mr. Brockman
will then ask contributors for responses, which will be relayed
back to the students through Dr. Hauser.

If Edge University catches on at other colleges, Mr. Brockman
envisions operating it through a network of participating professors,
who would forward the Edge newsletter to their students. He also
says that a parallel e-mail discussion list could be created to
allow students around the world to debate issues raised in the newsletters.
In e-mail messages that Mr. Brockman has posted on the Web site**,
Edge contributors have praised the idea of reaching out to students.

"The Edge material potentially plugs an important intellectual
hole," wrote David Gelernter, a computer scientist at Yale University
who is known for his critiques of technology's influence on society.
Understanding the implications of science and technology "is a tremendously
important topic ... universities know it, students want to study
it," he continued. But, he added, "universities in general don't
have a clue about how to teach it."

Also see: An excerpt from John Brockman's 1995 book, The
Third Culture 6/16/95

Copyright © 1998, The Chronicle of Higher Education. Posted
with permission on EDGE. This article may not be published, reposted,
or redistributed without permission from The Chronicle.

LISA GUERNSEY writes about
information technology for The Chronicle and manages its on-line
information technology section.

**THE THIRD CULTURE**

"ON THE NATURE OF MATHEMATICAL CONCEPTS: WHY AND HOW DO MATHEMATICIANS
JUMP TO CONCLUSIONS?"

by Verena Huber-Dyson

(In early December, the following I received the following email
message from Verena Huber Dyson - JB)

"Quite a while ago my son George Dyson handed me a batch of comments
dated Oct. 29 - ED GE 29 (What Are Numbers, Really? A Cerebral Basis
For Number Sense", by Stanislas Deheane and Nov. 7 - EDGE 30 (the
subsequent Reality Club discussion) by your group of EDGE researchers.
I read it all with great interest, and then my head started spinning.

Anyone interested in the psychology (or even psycho-pathology)
of mathematical activity could have had fun watching me these last
weeks. And now here I am with an octopus of inconclusive ramblings
on the Foundations bulging my in "essays" file and a proliferation
of hieroglyphs in the one entitled "doodles". It is so much easier
to do mathematics than to philosophize about it. My group theoretic
musings, the doodles, have been a refuge all my life.

Although I am a mathematician, did research in group theory and
have taught in various mathematics departments (Berkeley, U of Ill
in Chicago and others), my move to Canada landed me in the philosophy
department of the University of Calgary. That is where I got exposed
to the philosophy of Mathematics and of the Sciences and even taught
in these realms, although my main job was teaching logic which led
to a book on Gödel's theorems. To my mind the pure philosophers,
those who believe there are problems that they can get to grips
with by pure thinking, are the worst.

If I am right you Reality people all have a definite subject of
research and a down to earth approach to it. That is great.

There are two issues in your group's commentary that I would like
to address and possibly clarify.

For a refutation of Platonism George Lakoff appeals to non standard
phenomena on the one hand and to the deductive incomplenetess of
geometry and of set theory on the other. First of all these two
are totally different situations, to each of which the Platonist
would have an easy retort. The first one is simply a matter of the
limitation inherent in first order languages: they are not capable
of fully characterizing the "intended Models", the models that the
symbolisms are meant to describe. The Platonist will of course exclaim:
"If you do not believe in the objective existence of those standard
models, how can you tell what is standard and what is not ?". The
deductive incompleteness of a theory such as geometry or set theory,
however, simply means that the theory leaves some sentences undecided.
Here the Platonist will point out that your knowledge of the object
envisaged is incomplete and encourage you to forge ahead looking
for more axioms, i.e., basic truths !

Incidentally I consider myself an Intuitionist not a Platonist.

I wonder whether it is appropriate for me to send you my rather
lengthy discourse on non standard phenomena. You may find it tedious.
Yet I believe that the question how it is possible for us to form
ideas so definite that we can make distinctions transcending the
reach of formal languages is pertinent to your topic "what are numbers
really?" It is very difficult to put these phenomena into a correct
perspective without explaining at least a little bit how they come
about.

The other contribution is a simple illustration of the naive mathematical
mind at work on the number 1729! And a remark about a prodigy."

Verena Huber-Dyson

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

ON THE NATURE OF MATHEMATICAL CONCEPTS: WHY AND HOW DO MATHEMATICIANS
JUMP TO CONCLUSIONS?

[Notation: x for products: 2 x 3 =6, ^3 for cubes: 2^3 = 8, ^exponent:
2^11 = 2048.]

While engaged in the mathematical endeavor we simply jump, hardly
ever asking "why" or "how". It is the only way we know of grappling
with the mathematical problem that we are out to understand, to
articulate as a question and to answer by a theorem or a whole theory.
What drives our curiosity is a question for psychologists. Only
after the jump has landed us on a viable branch the labor of proving
the theorem or constructing a coherent theory can set in. The record
of the end result, usually a presentation at a conference, a paper
in a learned journal or a chapter in a book, is laid out in a sequence
of rational deductions from clearly stated premisses and rarely
conveys the process by which it has been arrived at.

The question why we have no other choice but to jump has received
a remarkably precise answer through Gödel's Proof of Incompleteness
in 1931 and Tarski's analysis of the concept of Truth in the thirties
in Poland. Since then the development of a rigorous concept of an
algorithm has led to a proliferation of so-called undecidability
and inseparability results underscoring the limitations of the formal
method.

The question how we jump has many aspects. First: what does the
jumping consist of, what are we doing when we jump, what is going
on in our minds when we are hunting down a mathematical phenomenon?
And then: what is guiding us, how come we jump to CORRECT conclusions?
Even if the guess was not quite correct, it usually was a good hunch
that, properly adjusted, will open up new territory. Where do these
hunches come from? Probably the simplest recorded answer to that
question goes back to Plato and has spawned a school of thought
in the Foundations of Mathematics that bears his name. It puts those
hunches on a par with our spontaneous reactions to physical messages
‹ "smell that? someone must be roasting a lamb in the next
clearing", "there is a storm brewing in the South West, I can feel
it in my bones". According to Plato's view mathematical objects
exist eternally and immutably in a realm of ideas, an abstract reality
accessible, if only dimly, to pure reasoning. That is how we discover
them and their properties. By now, what with 2000 years of escalating
experience with mathematics and painstaking critical analyses of
its tenets, Platonism is no longer the accepted view in the Foundations.
But, if nothing else, it is a wonderful allegory and an extremely
useful working hypothesis.

To put it bluntly, while at work a mathematician is too busy concentrating
on deciphering the hints he can gather from the trail he is following
to stop and bother asking how the trail got here. It is enough for
him to have a good hunch that the trail will lead to the goal.

The following is a slightly polished version of my spontaneous
response to the assortment of EDGE-comments on Stanislas Dehaene's
question "What Are Numbers, Really? A Cerebral Basis For Number
Sense" (click here).
After a simple illustration of how we ponder, jump and then fill
in the steps I address some general considerations raised on EDGE,
which leads me to an exposition of the limitation phenomena.

Although keeping technicalities to a minimum, both conceptually
and typographically, I am careful to be precise and correct. In
our field the smallest inaccuracy can have disastrous consequences
leading head on into contradictions.

1729 AN EXAMPLE OF MATHEMATICAL REASONING

Stanislas Dehaene brings up the Ramanujan-G.H.Hardy anecdote concerning
the number 1729. The idea of running through the cubes of all integers
from 1 to 12 in order to arrive at Ramanujan's spontaneous recognition
of 1729 as the smallest positive integer that can be written in
two distinct ways as the sum of two integral cubes is inappropriate
and obscures the workings of the naive mathematical mind. To be
sure, a computer-mind could come up with that list at a wink. But
what would induce it to pop it up when faced with the number 1729
if not prompted by some hunch? Here is a more likely account:

Confronted with 1729 you will recognize at a glance that:

i) 1729 = 1000 + (810-81) = 10^3 + 81 x (10-1) = 10^3 + 9^2 x
9

= 10^3 + 9^3 = (1 + 9)^3 + 9^3 = 1 + 3 x 9 + 3 x 9^2 + 9^3 + 9^3

= 1 + (3^3 + 3 x 3^2 x 9 + 3 x 3 x 9^2 + 9^3) = 1 + (3 + 9)^3

= 1^3 + 12^3 in view of the pattern

ii) (a + b)^3 = a^3 + 3 x a^2 x b + 3 x a x b^2 + b^3.

Now all those 3's in the above expressions spring to attention,

you fleetingly call up THE EQUATIONS

iii) (a + b)^3 + d^3 = a^3 + (c + d)^3

a^3 + (3 x a^2 x b + 3 x a x b^2 + b^3) + d^3

= a^3 + (c^3 + 3 x c^2 x d + 3 x c x d^2) + d^3

and JUMP to the conclusion that the choice of (1,9; 3,9) for a,b;
c,d will give you the smallest positive integer that can be written
as the sum the cubes of two integers (a+b) and d and also of a different
pair a and (c+d). You have a well trained instinct. But, if called
upon, it will be a simple matter to fill in that jump by a proof,
the fixed coefficients 3 ruling out smaller choices for b,c,d, once
the minimal possible value 1 is chosen for a.

ANALYSIS OF A TRAIN OF THOUGHT

The best way to understand the process encoded above in technical
shorthand is via a metaphor, which should be spun out at leisure.
Say you are driving into a strange town, and, for some reason or
other, a building complex catches your attention. It does not just
pop into your field of vision; at first glance you see it as a museum,
a villa, a church or whatever. And then, depending on your particular
interests and background, you may recognize its shape, size and
purpose, muse over its style, venture a guess as to its vintage,
and so forth.

Upon meeting 1729, your first reaction will probably be to break
it up into the sum of 1000 and 729, because of our habit of counting
in decimal notation. Stop for a moment to consider what would have
been facing Ramanujan if Taxi cab companies were favoring binary
notation! [11011000001 = 11011000000 + 1 = 11^3 x 100^3 + 1^3 =
101^3 x 10^3 + 11^3 x 11^3 = 1111101000 + 1011011001]. On the other
hand, if you are one of those people obsessed with prime factorization
you'll "see" the product 7 x 13 x 19 when somebody says "1729" to
you while a before-Thompson-and-Feit but after-Burnside group theorist
will say "Aha that is an interesting number, all groups of order
1729 are solvable" and anyone with engineering experience immediately
thinks of the 1728 cubic inches contained in a cubic foot 1. But
a historian of Mathematics will see 1729 as the year of Euler's
friend and benefactress Catherine the Great's birth.

Next you decide, more or less deliberately, how to investigate
the phenomenon. Do you drive to the nearest kiosk, buy a "Baedecker",
search for that building and read through all you can find in there
about it, before you make up your mind about what you want to know,
in other words, assuming you have a kiosk full of lists handy in
your own mind, do you run through all the integral cubes smaller
than 1729? If so, why cubes?

If you have that kind of mind you probably would first run through
the squares before getting to the cubes. The less methodical tourist,
eager to enjoy rather than out to complete his (or her) knowledge,
may choose to investigate in a haphazard way, spurred on by curiosity,
guided by experience, using skills automatically while impulsively
following hunches, prowling, sniffing, looking behind bushes, and
then jump to rational conclusions.

Now return to Ramanujan and see how the first thing that springs
to the naive eye beholding the number 729 is that adding 81 = 9^2
turns it into 810, whereupon 10 drops its disguise, shows one of
its true natures as the sum of 1 and 9 and, lo and behold, all those
powers of 3 start tumbling in. All the while you are aware of the
pattern ii), just below the threshold of consciousness, exactly
as a driver is aware of the traffic laws and of the coordinated
efforts of his body and his jeep. That is how you find your way
through the maze of mathematical possibilities to the "interesting"
breakdown of 1729 into two distinct sums of integral cubes.

When you stop to ask yourself what is so great about that, something
clicks in your mind: you are facing a positive integer with a certain
property, you know that

iv) every collection of positive integers has a least member

(in terms of its natural ordering).

That knowledge, always hovering below the threshold of consciousness,
prompts the question whether 1729 might in fact be the LEAST positive
integer expressible in distinct ways as the sum of two cubes. Having
another look at the representation of 1729 as a sum of various powers
of 3 as held in your mind's eye and exhibited in the third line
of i) above, the more or less conscious awareness of ii) invites
you to break up those sums of cubes according to the pattern iii)
where you assume ‹ "without loss of generality" ‹ that
a < d = a + b, and hence c < b. At this point the solution a = 1,
b = c^2 = d and c = 3 surfaces by inspection as "obviously" yielding
the minimal value for (a + b)^3 + d^3.

ABOUT MATHEMATICAL ACTIVITY

I have gone through this simple illustrative example at such length
in order to underscore a few of my pet contentions:

What we sorely need is a phenomenological study of mathematical
practice. Polya and Lakatos had independently started out on that
path, I do not know to what extent it has been followed up. Mathematicians
are well aware of how they work, whether by themselves or in teams.
But their goals are results that must be presented in a conclusive
and "clean" form that makes them publicly accessible, at least within
the profession, a form that necessarily obscures the path that led
to them, just as the most beautiful tombstone will sum up a life
but give no inkling of how it really has been lived, to use an observation
by Claude Chevalley 2.

a) Much mathematical reasoning is done subconsciously, just as
we automatically obey traffic rules and handle our cars, whether
we know why and how they work or not. Symbolic notation is an "artificial
aid" used to secure a hold like a piton, to survey a situation like
a geological map and to encode general patterns for repeated application.
But it is not mathematics. Mathematics can be done without symbols
by a particularly "gifted" individual, like, e.g., Ramanujan. What
that gift consists of is one of the questions raised in the EDGE
piece. Obviously we are not all of us born with it. Nor do I believe
that all people born as potential mathematicians become actual ones.
Tenacity of motivation, an uncluttered and receptive mind, an unerring
ability to concentrate the mind's focus on long intricate chains
of reasoning and relational structures, the self discipline needed
for snatching such a mind out of vicious circles, these are only
a few characteristics that spring to mind. They can be cultivated.
Experience will train the judgment to distinguish between blind
alleys and sound trails and to divine hidden animal paths through
the wilderness.

b) Free association plays an important role, an agility of mind
that allows reasoning to jump ahead with a sure touch, after which
comes the dogged toil of constructing proofs.

c) Conceptual visualization is an indispensable attendant to mathematical
thinking. Formalization is only a tool and may encourage lazy thinking!
Look at the freshmen who enroll in math because they assume they
won't be expected to produce coherent arguments or to write grammatical
text, that bureaucratic neatness in "plugging in numbers and turning
the crank" will suffice to pass the course.

It is fascinating to browse through some of the essays on the
Foundations of Mathematics by the topologist and logician L.E.J.
Brouwer, the father of Intuitionism. You will find very few formulas
in them, and yet they are rigorously reasoned, tightly and succinctly,
more so than many formal texts. 3

d) That practice, familiarity, experience and experimentation
are important prerequisites for successful mathematical activity
goes without saying. But less obvious and just as important is a
tendency to "day dream", an ability to immerse oneself in contemplation
oblivious of all surroundings, the way a very small child will abandon
himself to his blocks. Anecdotes bearing witness to the enhancement
of creative concentration by total relaxation abound, ranging from
Archimedes' inspiration in a bath tub to Alfred Tarski's tales of
theorems proved in a dental chair.

THE EVOLUTION OF MATHEMATICAL CONCEPTS

The tenet that MATHEMATICAL OBJECTS ARE MENTAL CONSTRUCTS conceived
by the human species for the purpose of forging its way through
life and environment is compelling. How could we orient ourselves
in space without discerning dimensions and estimating distances,
how could we keep track of possessions and offsprings without a
sense for numbers (cardinals), groupings and hierarchies (ordinals)?

Maybe the prototypical shepherd just kept a heap of pebbles handy
by his cave, one for each sheep, to make sure by MATCHING that at
the end of the day he had his whole flock together ‹ the first
occurrence of the mathematical arrow. The next guy paid attention
to the pecking order among his charges and chose his pebbles accordingly.
And then ‹ much later ‹ one with a poetic twist of mind
gave individual names to his sheep and picked pebbles to match their
personalities in looks, color, shape and mood so that, if one went
missing, he could tell by looking at the leftover pebble which one
of his flock to search for where, according to the culprit's specific
idiosyncrasies. Finally, with all that time on their hands, some
of the shepherds started creating poetry or inventing music, others
projected and extrapolated their minds into higher realms of mathematics
‹ and started wondering.

Here is the beginning of mathematics, not only arithmetic, the
whole works, structures (you start grouping your flock, and those
groups will interact), mappings and probably even the concept of
infinity, "what if those ewes keep lambing and lambing till I run
out of pebbles...". Pretty soon these concepts become PHENOMENA
and begin evolving in interaction with their creators and with the
uses they are put to.

When the ANTHROPOLOGIST has told his story and the PHENOMENO-

LOGIST has had a look at how a mathematician's mind works it is
for the NEURO PHYSIOLOGISTS to figure out what is going on in the
brain of those shepherds and their descendants. The PSYCHOLOGY of
mathematical activity ‹ and obsession ‹ also deserves
attention and is bound to shed light on the mystery of the prodigy.

The view of mathematical "objects" as mental constructs forever
caught up in a dynamic process of evolution was succinctly articulated
by L.E.J. Brouwer, the Dutch topologist who, during the first quarter
of this century, founded the school of INTUITIONISM as the most
compelling alternative to PLATONISM. Occasionally Intuitionism is
accused of leading into solipsism. But the understanding of mathematical
intuition as a sense for charting one's way around an environment
including fellow creatures implies that its tools, the concepts,
must be evolving by joint and competing efforts of a community.
Very much in keeping with what I understand is Stanislas Dehaene's
view. With Brouwer I believe in preverbal mathematical perception,
where by perception I mean an activity, a process of "seeing as",
picking out of patterns and imposing frames of reference.

Friedrich Wilhelm Nietzsche (1844-1900) had a keen understanding
of the anthropological evolution of mathematics and rational thinking.
His *Der Wille zur Macht* (The Will to Power, 1887) contains
poignantly expressed insights into the genesis of the laws of Logic,
many of them anticipating Intuitionism!

George Lakoff's stress on image schemes and conceptual metaphors
is compelling, especially his suggestion of "expansion to abstract
mathematics by metaphorical projections from our sensory-motor experience".
Yes we do have mathematical bodies! On a primordially homogenous
environment we impose a grid commensurate in size and compatible
in shape with our bodies as we know them from direct experience.
One step further, we project our bodies beyond what is immediately
perceivable, spurred on by a tenacious intention "to make sense
of it all". Have you ever noticed how many mathematicians are rock
climbers? The process of mulling over a mathematical problem displays
a striking similarity to that of surveying a cliff before the ascent;
of visualizing and comparing alternate routes, from the big lines
of ridges, ledges and chimneys down to the details of toe and finger
holds, and then weighing possibilities of what might be encountered
beyond the visible; all in perfectly focused concentration, projecting
ahead, extrapolating, performing so-called "Gedankenexperimente"
(thought experiments) and sensing them throughout one's bones and
muscles. And finally setting off to break trail through the folds
of a brain!

Already in 1623 Blaise Pascal articulated in his *Pensées*
(thoughts) the observation that the abstract schemata we impose
on the world in order to interact meaningfully with it are shaped
by the experience of our bodies. 4

During the last half century the evolution of so-called CATEGORY
THEORY out of algebraic topology has developed a dynamic language
of diagrams in which the abstract concepts of universal algebra
find their natural habitat. 5 "Diagram chasing" ‹ a systematic
form of hand waving ‹ is a way of making sense of the abstract
structural and conceptual under-pinnings of mathematics, including
Arithmetic and Geometry, Logic and set theory, as well as of the
juxtaposition between discrete and continuous phenomena. It turns
out that Topoi, a particularly prolific species of categories, have
the structure of intuitionistic Logic ‹ an amazing corroboration
of INTUITIONISM. F. W. Lawvere at SUNY Buffalo, a pioneer in the
field since the early sixties, and his associates are beginning
to make significant contributions to cognitive science.

As to PLATONISM, whether deliberately or inadvertently, most mathematicians
still act and talk as if they were dealing with objects that are
part and parcel of the furniture of their Universe. I do it myself,
and so does George Lakoff when he refers to the straight line and
the reals. It is such a convenient make-believe stance, not to be
confounded, however, with the deep allegorical truths revealed in
the poetry of Plato's dialogues.

But there is more to be said when we stop to contemplate what
we call REALITY. Think how often a writer will create characters
only to find them taking on a life of their own, doing things or
getting into trouble that their creator had not intended for them
at all. So, the positive integers are mental constructs. They are
tools shaped by the use they are intended for. And through that
use they take on a patina of reality! Nor do they rattle about in
isolation. They interrelate, they pick up individual personalities
through interaction, by their position in the natural ordering,
by splitting into primes, by what they are good for, in what contexts
they play what roles.

And before we know it we have a problem on our hands like Fermat's
Last Theorem! Its statement can be explained to every child, using
a bit of hand waving and the ever handy dots. Through generations
the belief in its truth had grown for ever more entrenched. No counter
example was found, but no proof was in sight either until Andrew
Wiles 6 succeeded in blazing the final trail to the goal through
abstract territory, rugged and disconnected in places and prepared
by the toil of his peers in others. To the experts the proof is
illuminating, but not to the ordinary mathematician in the street.
By now our tools are so highly developed that they bring us information
about our own creations that we cannot fathom with the unaided mathematical
senses, even though it may concern situations whose meaning we can
understand perfectly well. In physics and astronomy we are used
to similar situations: our instruments can reach physical phenomena
way beyond the reach of our physical bodies. The interpretations
of these messages from beyond are encoded in theories of our own
construction.

The method of FORMALIZATION is by now widely accepted, used and
discussed. But it has limitations and is trailing some baffling
"non-standard" phenomena in its wake. In order to put these into
proper perspective a technical digression is needed.

FORMAL THEORIES

While mathematics is forging mental tools for charting our way
through the world, our brains playing very much the part of our
senses, things become so intricate that we need artifacts for keeping
track of those constructs. That is where symbols come in ‹
algebraic notation, diagrams, technical languages and so forth ‹
as mechanisms for storing and surveying insights and for communicating
about them. Extension of this method to the analysis of mathematical
reasoning itself leads to so-called meta mathematics and symbolic
logic.

Allowing the articulation of "axioms" and of rules of deduction
governing their use, the systematic construction of formal languages
leads to FORMALIZED THEORIES consisting of theorems, i.e., wellformed
sentences (wfs' for short) obtained from axioms by chains of deductions
according to those rules.

A formal proof is a finite sequence of wfs' starting with axioms,
hanging together by the formal rules and ending with the theorem
proved by it. The formalized theory itself becomes a topic for theoretical
investigation since it is bound to have properties that go beyond
what we put into it. Will it be formally consistent

in the sense that the negation of a theorem will never show up as
a theorem too? Is it formally complete ,i.e., does every sentence
have a proof unless its negation has one? These are typical problems
for the meta-theory.

The choice of axioms is not arbitrary. We are guided by common
sense of mathematical perception, by criteria that deserve investigations
to which the EDGE group seems to be making valuable contributions.
As we acquire and develop intuitive concepts of sets, spaces, geometries,
algebraic structures and all the rest, we try to grasp them by characteristic
properties and are led to basic postulates.

Occasionally sustained experience reveals that the original construction
was not fully determinate, that the axioms are not complete. They
don't suffice to pin down the intended concept uniquely. Some sentence
A ‹ Euclid's fifth postulate for instance ‹ is left undecided
by what was considered an axiomatic characterization of the concept
‹ of, say, a geometry. Both A and its negation not-A are formally
consistent with the axioms. Well, for some purposes it is useful
to assume Euclid's parallel axiom for geometry, or well-foundedness
for sets, at other times it may be handy to deal with bottomless
sets or crooked squares. The tools are evolving as we are using,
refining and adjusting them. Such experiences that at first look
like failures deepen conceptual understanding and expand mathematical
horizons.

The situation of the arithmetic N over the natural numbers 0,1,2,3,...
and that of the ordered field R of the reals are more subtle. In
both cases we "know exactly" what structure we have in mind, there
is no question of bifurcation of concepts. Yet in the case of N
a complete axiomatization founders on the requirement of effectiveness
while, even though completely formalizable, the elementary theory
of R, has so-called non-standard models, as does every theory of
an infinite structure.

ELEMENTARY THEORIES

These phenomena are a manifestation of the precarious balance
between algorithmic precision and expressive power inherent in every
formal language and its logic. The most popular, widely taught formalization
is the first order predicate calculus, also called elementary logic,
a formalization of reasoning in so-called first order predicate
languages. That apparatus leads from "elementary axioms" to "elementary"
theories.

The important requirement for any formalization is the existence
of both a "mechanism" (algorithm) for deciding, given any well formed
sentence (wfs) of the language concerned, whether or not that wfs
is an axiom, and one for deciding of any given configuration of
wfs's whether or not it is an instance of one of the rules. The
resulting concept of a formal proof is decidable, i.e., there exists
an algorithm, which, when fed any finite sequence of wfs', will
come up with the "answer yes" (0) or the "answer no" (1) according
as that sequence is a formal proof in the system or not. The resulting
axiomatizable theory will in general only be effective in the sense
that there exists an algorithmic procedure for listing all and only
those wfs' that are theorems. That does by no means guarantee a
decision procedure for theoremhood. In fact most common theories
have been proven undecidable.

To start with, a familiar structure like N or R will serve as
the so-called STANDARD MODEL or INTENDED INTERPRETATION for the
elementary theory meant to describe it. Observe that the notion
of a standard model presupposes some basic concept of mathematical
reality and truth. Gödel talks of "inhaltliches Denken" (formal
thinking) in juxtaposition to "formales Denken" (formal thinking).
His translators use the term 'contentual'. 'Intentional' might be
just as good a choice.

Of course one might dodge the need for a metaphysical position
by using terms like "preverbal" or "informal".

But that does not make the problem go away. If we want to talk
about standard models, if we want our theories to describe something
‹ approximately and formally ‹ what is it that we want
them to describe? A question that would not disturb a Platonist
like Gödel. The formalist's way out is to throw away the ladder
once he has arrived at his construction and to concentrate on the
questions of formal consistency and formal completeness, purely
syntactic notions.

A theory is formally consistent if and only if for no wfs A both
A and not-A are theorems and formally complete if and only if for
every wfs A either A or not-A is a theorem.

To an extreme formalist the existence of an abstract object coincides
with the formal consistency of the properties describing it. If
at all, he will draw his models from yet another theory, most likely
some, necessarily incomplete, formalization or other of elementary
set theory, presumed ‹ but only presumed ‹ to be consistent.
An unsatisfactory strategy.

We, however, are left with the conundrum of Mathematical Truth
and the semantic notions that depend on a "meaning" attached to
the theory. With respect to an interpretation of its language over
a structure S a theory, formalized or not, is

sound (semantically consistent) if and only if

only wfs' true in S are theorems

semantically complete if and only if

all wfs' true in S are theorems.

Granted a clear and distinct idea of the structures N and R we
talk of the sets of all wfs' that are true under the intended interpretations
on N and on R as True Elementary Arithmetic, TN, and as the True
Theory TR of the Reals.

Consider an EXAMPLE: Leaving aside the question where N comes
from, I should think that we all know what we mean by the wfs

(F^3) for ALL positive integers x, y and z: the sum of the cubes
of x and y is not equal to the cube of z.

F is short for FERMAT. To explain it to a naive computer mind,
we would say: "Make two lists as follows; in the left one, L, write
down successively the results of adding the cubes of two positive
integers, 1^3 + 1^3, 1^3 + 2^3, 2^3 +2^3, 1^3 + 3^3, 2^3 + 3^3,
3^3 + 3^3,..., and into the right one, R, put all the cubes 1^3
(1), 2^3 (8), 3^3 (27), 4^3 (64), 5^3 (125) and so on. Now run through
both lists comparing the entries. (F^3) claims that you will never
find the same number showing up both on the left and on the right".
A computer can easily compile these lists in so orderly a fashion
and run through them so systematically that, for each bound N, it
will, after a computable number of steps, say f(N) of them, have
calculated and compared all pairs of numbers in L and in R smaller
than N. You will probably agree that this tedious explanation makes
it sufficiently clear what we mean here by ALL. You may want to
use nicer language like talking about NEVER finding a matching pair.
The purpose of symbolization, however, is not only orderliness,
but clarification. The dual to the so-called UNIVERSAL QUANTIFIER
(for all) is the EXISTENTIAL QUANTIFIER. Just think for a moment,
assuming (F^3) were false, how easy it would be for your patient
computer to prove that. It would only have to go on long enough
until it found a COUNTER EXAMPLE, i.e., a positive integer that
shows up in both lists, R and L. Having done so it would have proved

NOT(F^3) there EXIST positive integers x, y and z, such that

the sum of the cubes of x and y is equal to the cube of z.

In 1753, using clever transformations of the problem, Euler succeeded
in proving the restriction (F^3) of Fermat's theorem to cubes. But
Fermat's general Conjecture

(F) for ALL positive integers x, y, z and n, n greater than 2,

the sum of the n-th powers of x and y

is not equal to the n-th power of z

has only been proved conclusively a few years ago by means of
techniques way beyond elementary arithmetic. It should be noted
here that variable exponentiation is not part of the language of
N but can be paraphrased in it. In the above procedure you will
have to organize your left list according to an enumeration of triples
(x,y,n) and the right one according to pairs (z,n).

The capacity to visualize an ongoing sequence of calculations and
comparisons leads to an understanding of what is meant by the truth
of (F). Yet, in spite of many efforts, its proof had to wait till
algebraic geometry and number theory had achieved the maturity necessary
to allow its construction.

We have a pretty good understanding of what we mean when we claim
that ALL integers ‹ or all pairs or triples of them ‹
have a certain property, provided that we understand the property
itself. The most manageable kind of properties that integers may
have are what we call recursive or computable. They are susceptible
to a decision procedure as illustrated by the example of checking
for fixed n and any given triple of integers x,y,z whether or not
the sum of the n-th powers of the first two is equal to that of
the third.

A property P of triples of numbers is called recursive if and
only if its so-called characteristic function that takes on value
0 at the triple (m,n,q) if that triple has the property and value
1 otherwise (its decision function) is computable by an algorithm
like a Turing machine, or, equivalently, is recursive.

The amazing ‹ often elusive ‹ power of the universal
quantifier brought home by Gödel's incompleteness proof, discussed
in the next section, is again manifest in the intrinsic difficulties
with which the conclusive proof of Fermat's theorem is fraught.

ELEMENTARY ARITHMETIC

Based on a naive concept of Truth, every true theory of a definite
structure is complete and consistent in both senses, a pretty useless
observation. For, a byproduct of Gödel's Incompleteness proof
of 1931 7 is the non-formalizability of elementary arithmetic, TN,
and with it of many other theories.

EVERY SOUND AXIOMATIZATION OF ELEMENTARY ARITHMETIC IS INCOMPLETE.

The most natural candidate for axiomatizing TN goes back to Giuseppe
Peano (1895) and consists of the recursive rules for addition, multiplication
and the natural ordering on the set N of non negative integers built
up from 0 by the successor operation that leads from n to n+1, together
with the Principle P of Mathematical Induction, which postulates
that every set of numbers containing 0 and closed under the successor
operation exhausts all of N, or, equivalently, that every property
enjoyed by 0 and inherited by successors is universal. P is a principle
that adults may consider a definition of the set N, while children
will ‹ in my experience ‹ take it for granted. But, if
you want to articulate it in the language of the first order predicate
calculus you run into trouble. As illustrated in example (F) elementary
languages can quantify over individuals. But quantification over
so-called HIGHER ORDER items like properties is beyond its scope.
P is a typical sentence of second order logic.

PEANO ARITHMETIC, PA, is the first order approximation to second
order arithmetic obtained by replacing P with the following schema
of infinitely many axioms

(PW) If 0 has the property expressed by the wff W and,

whenever a number x has that property, then so does x+1,

then all natural numbers have property W.

one for each wff (well formed formula) W of the elementary language
of arithmetic.

Reformulating (PW) in terms of proofs rather than truth sheds
light on it and illustrates how one might want to go about replacing
the basic concept of truth in mathematics by a primitive notion
of proof. Writing W(x) for "x has property W" the Principle of Proof
by Mathematical Induction reads

(PPW) Given 1) a proof of W(0) and 2) a method for

turning any proof of W(x) into a proof of W(x+1)

THEN there exists a proof of "for all x: W(x)".

Note how appealing this formulation is: Given any number n, you
only have to start with the proof given by 1) and then apply the
method of 2) n times to obtain a proof of the sentence W (n). But
there is a subtlety here. So understood, the principle only guarantees
that, for every number n, a proof of W(n) can be found, a typical
"for all ‹ there exists ‹" claim. Its power lies, however,
in the "there exists-for all ‹" form of the conclusion as exhibited
above.

These may sound like nit-picking distinctions, but they are of
great proof-theoretic significance. For instance, from the consistency
of PA, proved by means transcending PA, follows:

If g is the Gödel number of the Gödel sentence G then:

for each natural number n, the sentence

"n is not the Gödel number of a proof

of the sentence with Gödel number g"

is a theorem of PA.

However G itself, namely the sentence

"for all x: x is not the Gödel number of a proof

of the sentence with Gödel number g"

is not a theorem of PA.

G is the sentence that truthfully claims its own unprovability.
Much deep work is required to establish this result rigorously.

Occasionally proofs by mathematical induction are confused with
arguments based on so-called 'inductive reasoning', a term used
in philosophical discussions of logic and the sciences ‹ yet
another reason for all these elaborations.

Axiomatized but undecidable theories are a fortiori incomplete.
In 1939 Tarski proved that

THERE IS A FINITELY AXIOMATIZABLE FRAGMENT OF PA

ALL OF WHOSE CONSISTENT EXTENSIONS ARE UNDECIDABLE. 8

And yet, if we accept an intentional concept of truth, we seem
to obtain a complete theory from Peano's mere handful of axioms
together with that one marvelous second order tool P on which so
much of our mathematical thinking hinges. For:

ALL MODELS OF PEANO'S SECOND ORDER AXIOM SYSTEM ARE ISOMORPHIC.

Still, any attempt to formalize second order arithmetic is again
doomed to founder on the cliffs identified by Gödel and Tarski.
The juxtaposition of these claims is and ought to be baffling. In
fact they bring home the discrepancy between the naive and the formalist
concept of a model. From the naive point of view they mean that
higher order logic cannot be completely formalized. Even so completeness
proofs for it are widely hailed ‹ at the price of allowing
all sorts of non-isomorphic models even for second order Peano Arithmetic.
Enough of that for now. Fermat's last theorem may well be beyond
the scope of elementary Peano Arithmetic. In other words, (F) is
presumably left undecided by PA. A few mathematically interesting
theorems expressible in the language of PA with that property are
already known. They are embeddable in stronger but still convincing
first order theories, some elementary set theory or other.

After all that we are faced with the question where new axioms
come from, in other words with THE PROBLEM OF THE NATURE OF MATHEMATICAL
TRUTH. To declare "OK, as of October 27, 1995, the day that Wiles
was awarded the Prix Fermat by the town of Toulouse, (F) shall be
added to the list of axioms for elementary arithmetic" would seem
quite inappropriate. We want more intuitively obvious first principles.

NON STANDARD MODELS

In amazing contrast to TN the first order theory TR of the ordered
field of the REALS has been successfully and completely formalized
‹ starting with Euclid's axioms, improved by Hilbert just before
the turn of the century and completed as well as proved complete
by Tarski about the time of the Second World War. My immediate reaction
when I first heard of this feat was shock and distrust of those
Berkeley logicians. "How could that be? The reals are so much more
complicated than the integers. Aren't the natural numbers defined
as the non negative integral reals?" Well, the solution of that
conundrum lies in the

LIMITATION OF EXPRESSIVE POWER INHERENT IN FORMAL LANGUAGES.

As a matter of fact, the natural numbers are not "elementarily
definable" among the reals; there is no wff of the language of R
that picks out the natural numbers among the reals.

Moreover, in spite of its completeness, TR has non-isomorphic models!
It has countable models, uncountable ones, Archimedean as well as
Non-Archimedean ones; some harbor hyperreals, others only standard
reals... What is going on? First the chicken-or-egg question must
be faced: what comes first, the model or the theory? Ever since
the elaborations by Tarski in 1934 and by Mal'cev in 1936 of the
results by Löwenheim of 1915, and by Skolem of 1920 (a brief
exposition will follow below) we understand that first order chickens
are prone to lay a medley of eggs, some "real" in the Platonic sense
of being standard and others weird, artificial, substitutes, freaks,
in short non-standard. The Ur-hen, the axiomatization, originated
from a standard egg, the "intended interpretation", a natural mathematical
construct like our everyday arithmetic of the positive integers,
or, more sophisticated, the real number system of the 19th century.
After the chicken has grown to maturity it starts laying models,
and, roaming through the virtual reality of model theory instead
of free ranging in Platonic realms, it comes up with non-standard
eggs. The only constraint on those is consistency and the verification
of the axioms, i.e., the genetic chicken code. These models are
hatched within the confines of some entrenched formalization of
set theory.

What really lies at the basis of non standard objects like hyper
reals is ‹ again ‹ the limitation inherent in first order
languages. In the elementary language of real number theory we cannot
distinguish between Archimedean and non Archimedean orderings and
that opens the door to constructions that were scorned by my teachers
although they might use infinitesimals as a handy figure of speech
the way we still talk Platonically. We thought that Cauchy and Weierstrass'
arithmetization of analysis had done away with that alleged abuse
of language, but now it is back en vogue again and very useful too
(see below).

NON STANDARD PHENOMENA are closely connected with the

SEMANTIC COMPLETENESS OF ELEMENTARY LOGIC, first proved by Gödel
in 1930 9 and extended in many ways since, in particular by Henkin
who also dealt with formalizations of higher order logic. The underlying
meta theorem rests on two facts, one inherent in the finitary nature
of a formal deduction, the second involving non- constructive instructions
for building a model

1) WHENEVER ALL FINITE SUBSETS OF A SET OF WFS' ARE CONSISTENT
THEN SO IS THE ENTIRE SET and

2) EVERY CONSISTENT SET OF WFS' HAS A MODEL.

By definition Semantic Completeness of a formal calculus means
EQUIVALENCE BETWEEN FORMAL DERIVABILITY AND SEMANTIC VALIDITY where
validity stands for truth under all interpretations, i.e. in all
models.

At first this looks like an amazing result especially in view
of currently rampant incompleteness. It is unfortunate that popular
literature so often fails to make a clear distinction between the
two concepts of semantic and of syntactic completeness (pp.10,11).
Only the experienced reader will automatically know from the context
which notion is at stake.

As a matter of fact the completeness of first order logic is achieved
at a price: the expressive poverty of the formal language. Completeness
proofs for higher order logic are ensnared in the same kind of bargain.
They are based on a concept of model that to the naive mind seems
contrived. Elementary languages are incapable of distinguishing
between arbitrarily large finiteness and infinity, and so are forced
to tolerate the infinitely small. Consider the infinite set of wfs'
0 < a < 1, a + a < 1, a + a + a < 1,..,a + a + a +...+ a < 1,...
and let U be its union with TR, the set of all wfs' that are true
in the field R of the reals. Every finite subset V of U has a model:
just take R and interpret a by 1/n, where n is the number of symbols
occurring in that finite set V. By 1) then the whole set U is consistent
and so, by 2) it and with it the elementary theory of the reals
has a model which harbors an element satisfying all these inequalities,
i.e., a non- Archimedean, non-standard, or hyper, real a. It is
positive and yet smaller than any fraction 1/n, n a positive integer.

Ruled by its logic, the language cannot prohibit such anomalies.
But there is a silver lining to this shortcoming: Because of the
consistency of infinitesimals with TR every truth about the reals
that can be expressed in the elementary language of R holds for
all reals ‹ standard or not ‹ and so, by Gödel's
completeness theorem, it has a formal proof. And if the approach
via infinitesimals is smoother that is just great. One cannot help
but marvel at the native instinct with which the seventeenth century
mathematicians went about their work

Similarly, any first order theory of N, including TN, has models
that contain infinitely large integers. The elementary theory of
finite groups has infinite models and so in fact does every first
order theory of arbitrarily large finite models.

All this is meant to explain that these non standard phenomena
have no bearing on the question whether Platonism is an appropriate
view of the origin of Mathematics. I am deliberately not using the
word "correct". Whether Platonism is "true" seems an ill posed question,
luring into vicious circles. How can we contemplate the truth of
this, that or the other "ism" before we have a clear and distinct
idea of what ‹ if anything ‹ we mean by the Truth of a
theory?

The existence of non standard models should NOT be confounded
with the occurrence of incomplete concepts like that of a geometry
or that of a set. In the case of hyperreals we are running into
limitations of the formal language while dealing with complete theories,
in the second case we are simply facing the fact that the intuitive
concept, say of a geometry or a set, that we had in mind when setting
up the formalization is not completely fathomed yet, in both senses
of completeness. Of course the easiest reaction is to say, "that
concept is out there, let us go look more closely and we shall eventually
find its complete characterization". In this frame of mind Gödel
is reputed to have been convinced that we shall eventually understand
enough about sets to come up with new axioms that will decide the
continuum hypothesis. But in other cases the expedient policy will
allow a concept to bifurcate ‹ sailors have no trouble with
non-Euclidean geometries.

The big question is where our standard concepts come from, how
do we all know what we mean by the Standard Reals? How can we distinguish
between Archimedean orderings and non Archimedean ones, when we
cannot make the distinction in first order language? Well we can
always resort to hand waving when words fail. We can indeed communicate
about them beyond the confines of formalism. They are conceptions,
constructions, structures, figments of our imagination, of the human
mind that is our common heritage. Other creatures may have other
ways of making sense of and finding their way in a Universe that
we are sharing with them.

This century has seen the development of a powerful tool, that
of formalization, in commerce and daily life as well as in the sciences
and mathematics. But we must not forget that it is only a tool.
An indiscriminate demand for fool proof rules and dogmatic adherence
to universal policies must lead to impasses. The other night, watching
a program about the American Civil Liberties Union I was repeatedly
reminded of Gödel's Theorem: every system is bound to encounter
cases which it cannot decide, snags that will confront its user
with a choice between either running into a contradiction or jumping
out of the system . That is when, with moral issues at stake, cases
of precedence are decided by thoughtful judgment going back to first
principles of ethics, in the sciences alternate hypotheses are formed
and in mathematics new axioms crop up.

Returning to my question, think of mathematics as a jungle in which
we are trying to find our way. We scramble up trees for lookouts,
we jump from one branch to another guided by a good sense of what
to expect until we are ready to span tight ropes (proofs) between
out posts (axioms) chosen judiciously. And when we stop to ask what
guides us so remarkably well, the most convincing answer is that
the whole jungle is of our own collective making ‹ in the sense
of being a selection out of a primeval soup of possibilities. Monkeys
are making of their habitat something quite different from what
a pedestrian experiences as a jungle.

To sum it all up I see mathematical activity as a jumping ahead
and then plodding along to chart a path by rational toil.

The process of plodding is being analyzed by proof theory, a prolific
branch of meta mathematics. Still riddled with questions is the
jumping.

NOTES:

[1] cf. Charles Simonyi on EDGE (/digerati/simonyi/simonyi_p1.html)
quoting Richard Feynman

[2] cf. D. Guedj: Nicholas Bourbaki, collective mathematician,
an interview with Claude Chevalley, *The* *Mathematical Intelligencer*
vol. 7, no. 2, 1985.

[3] Cf. the two anthologies *Philosophy of Mathematics: selected
readings* edited by P. Benacerraff and H.Putnam, Cambridge University
press 1964 and reprinted with some deletions 1985, and J. van Heijenoort's
*From Frege to Gödel; a source book in mathematical logic
1879-1931* Harvard University Press 1971.

[4] In a fine essay, entitled "Bemerkungen zu zwei wenig beachteten
'Gedanken' Pascals" ("Remarks on two of Pascal's 'thoughts' that
have hitherto received little attention") in his collection *Ausgewählte
Vorträge und Aufsätze* * *(selected talks and
essays), Bern 1955 pp. 226-234, the Swiss psychoanalyst and phenomenologist
Ludwig Binswanger elaborates on Pascal's "pensées" (thoughts
or reflections) on this topic and substantiates them with clinical
experiences pertaining especially to the role played by symmetries
in Rorschach tests.

[5] cf. chapter III of my *Gödel's Theorems; a Workbook
on Formalization*, Teubner Texte zur Mathematik, Stuttgart-Leipzig
1991.

[6] *Modular elliptic curves and Fermat's Last Theorem*,
Ann. Math. 141, 1995, 443 551.

[7] *Über formal unentscheidbare Sätze der Principia
Mathematica und verwandter Systeme*, Monatshefte für Mathematik
und Physik 37, 1931,173-198. For a translation see *Kurt Gödel,
Collected Works*, Volume 1, ed. S. Feferman et al, Oxford U.
Press 1986.

[8] cf. p. 39 ff. of the excellent and very readable monograph
*Undecidable Theories * by Tarski, Mostowski and Robinson,
North Holland Press 1953.

[9] cf. the collected works cited before.

**THE REALITY CLUB**

Delta Willis on Colin Blakemore's Question

From: Delta Willis

Submitted: 1.31.98
Colin Blakemore's Question

Regarding Colin Blakemore's question: "Most human beings perform
effortlessly a variety of tasks that are computationally extremely
difficult (such as seeing, holding objects and understanding speech)
but they are generally poor...in tasks that are computationally
easy (such as solving puzzles, doing mathematics and science)....Does
this disparity reveal a fundamental limitation of the human brain?"

Seems to me the disparity is a result of the more difficult tasks
being done more often and longer, by at least a couple of millionyears.
In evolutionary terms, we humans have had a great deal of practice
at seeing, holding objects and trying to understand language, and
our day is more likely to be fuller with these tasks, while solving
mathematical or scientific puzzles are tasks very recent on the
horizon, done by relatively few people, for shorter periods of time.
The limitations may be with the computer, not our brains, if I am
understanding the term "computationally" correctly. If this term
merely means calculation, then again, the other tasks may have simply
become second nature because of their tenure and practice. -

Delta Willis

DELTA WILLIS is a writer; author of *The Sand Dollar & the Slide
Rule; The Hominid Gang.*

Oliver Morton on Duncan Steel

From: Oliver Morton

Submitted: 1.23.98
Reply to Duncan Steel

Duncan Steel wrote (EDGE 33): "Thus [Lewis] Wolpert's question
for me has a corollary: 'Why do people NOT believe in things for
which there IS evidence and is it a mistake to try and persuade
them to do so?'"

Like Duncan, I've thought about this one too, though not as much
and not necessarily in the same way. It seems to me that people
shirk responsibility when they want to deny that they have it and
when they think they can get way with. If Duncan and his fellow
travellers fail to persuade someone to do something about it, then
the question of whether we can get away with the denial becomes
an empirical one.

But the reasons for the denial are still interesting. If we had
the sense of deep time that Stewart Brand and Greg Benford asked
questions about, then we'd have no trouble taking impacts seriously
(though I suppose that's not necessarily the case ‹ we might
take the falls of civilisations less seriously instead...). As an
aside, it strikes me that a trivial way of increasing people's sense
of deep time would be through simulations, perhaps using simple
screensavers, that show extremely slow processes speeded up to be
merely slow by everyday standards: glaciers at one month:one second,
impacts at one year: one second, the hawaiian island chain at 1,000
years: one second, etc. I'd buy one.

But even without a developed sense of deep time, the belief that
impacts actually happen is already out there. It's widely accepted
as a truth about the world, not just by scientists, but by the public
and by those politicians who have thought about it. When there was
an impact report from Greenland just before Christmas it made the
news all around the world, as did the questions of what would have
happened if it had hit a city or if the next one was bigger. Impact
stories that put the facts more or less accurately have been on
the cover of almost every newsmagazine and are staples of the science
pages; the spaceguard report that Duncan worked on was the subject
of congressional hearings. Intellectually, it is an idea people
can accept. Its just that there is a huge distance between accepting
an idea intellectually and behaving accordingly ‹ ask anyone
who's worked on AIDS prevention programmes.

The slight shift in relatively small patterns of technoscience
spending required to meet the threat Duncan warns of might seem
rather easier to accomplish through rational argument than a personal
change in sexual behaviour. But political changes need constituencies,
and "people who will be harmed by an impact" simply do not make
up an identifiable constituency, while people who will benefit directly
from a search programme and protection mechanisms make up a small
constituency in the already politically-very-well-served space sector.

However, I don't think this political problem is quite the whole
story. I think the denial is rather deeper ‹ and it's a denial
not of the threat, but of our powers. I first became aware of the
impact problem when I read Niven and Pournelles *Lucifer's Hammer*,
part of the message of which was that if the impacting comet had
come a decade or so later mankind would have just pushed it aside.
When I came to write about the subject years later I found that
this was largely true ‹ the problem of finding most potential
impactors and dealing with any direct hazards is relatively simple
(though the long period comet envisioned by N&P would be much harder
to deal with). And now I think that that very fact contributes to
the denial. It's not that people worry about the ability to divert
asteroids being misused in the ways that Carl Sagan warned of: its
just that the sheer power involved seems too much. In this regard
people are more worried about our capacity to disturb the universe
than they are about the universe's capacity to destroy our civilisation.
I wrote about this for *The Independent* in London last February
‹ the piece may still be on their website. It ended like this:

Perhaps people do not want to see themselves connected to the
universe in this sort of way. The geologists who for years resisted
the impact explanation for the dinosaurs' death simply didn't want
asteroids to play as big a role in the history of the earth as,
say, the wanderings of one of its own tectonic plates. Tough: they
do. Humans and the earth they live on are linked to the universe
in all sorts of strange, indirect, unsettling ways - and, worse
yet, humanity now has the power to change these connections. We
can empty seas and denude vast forests, we can warm an entire planet
and now, given just a little warning, we can push aside flying mountains.
It's genuinely frightening to contemplate such power, especially
when you realise how poorly decisions about using it are made or
not made. Better to deny the risk of asteroid impacts than to accept
the fact the humans can redirect the stars in their courses. It's
a delusion, in this case a slightly dangerous one ‹ but you
can understand it.

Best, o

OLIVER MORTON is a freelance writer, and a contributing editor
at *Wired* and *Newsweek International*. He used to edit
*Wired UK*, and previously worked at *The Economist*,
spending almost five years as Science and Technology Editor.

Howard Rheingold, Kevin Kelly on Pattie Maes

From: Howard Rheingold

Submitted: 1.23.98

Comments on Intelligent Agents

I commend Pattie Maes for not just raising but acting upon the
social implications of intelligent agent technology through her
support of the Open Profiling Standard ‹ before I raise a few
questions about the wider implications of the technology. I believe
that it is not just proper but imperative for the people who discover
and apply knowledge to ask about what their applications might do
to us. I have three questions about nomenclature, privacy, and potential
systemic social effects of hypersegmentation of groups of people
according to what they buy.

It is non-trivial to continue questioning the use of both words,
"intelligent" and "agent" because of the anthropomorphic associations
each word evokes. This question has come up before, but I am more
interested in what kind of world ‹ and what kind of human ‹
those associations imply than in whether the terms are properly
used. Words used to describe technology wield considerable power
of political persuasion. Consider how the word "progress" is almost
universally regarded as an unequivocally beneficial description
of the way people or societies should move, but the question, "progress...toward
what?" is almost never asked when this potent word is invoked (usually
as the argument for designing, deploying, distributing new technology.)

I spent the last five years dealing with the implications of the
term "virtual community," which I helped inject into popular culture.
Some of the critical attacks on my use of this phrase to describe
online discussions caused me to change my thinking about the assumptions
and associations the word evokes. I know Jaron Lanier and I have
both spent years answering questions about whether "virtual reality"
is oxymoronic. It's too late to withdraw any of these phrases from
popular parlance. But it is not too late to think and discuss and
perhaps act upon the changes in our world and ourselves that these
phrases imply.

I'd love to have the use of more sophisticated search software and
interest-matching utilities. The advent of internet search engines
totally transformed my daily info-hunting-and-gathering experience.
I would pay for search engines that could react more precisely to
my needs as it compiled and indexed a database of my past searches,
and traded meta-info with the databases of others on this mailing
list. But this is a sophisticated computer program for information
finding and matching via the Internet, not a generalized intelligence
amplifier, the way alphabets and GUIs are. Compare intelligent agents
as intelligence augmentation with Doug Engelbart's ideas about "augmenting
human intellect" ‹ still worth reading in this regard, thirty
five years after it was published.

The privacy implications of intelligent agents are important now
because the business model that could drive widespread diffusion
of this technology derives primarily and overwhelmingly from the
value of collecting detailed information about people's behavior
and interests in order to sell them goods and services. The service
this technology provides the people who use it ‹ precise and
accurate matching of people with information, commodities, and other
people ‹ is potentially a far smaller source of revenue than
the service this technology provides to marketers. Intelligent agentry
is more than a service for people looking for books or CDs or soul-mate
s or good conversations. It's a potential revolution in the way
marketers can identify, understand, persuade, and manipulate customers.
The longer-range implications of intelligent-agent hypersegmentation
are worth thinking about now.

A market for privacy can be conducted with decent ethics, as the
Open Profiling Standard attempts to enable. If people derive value
from the use of information and communication technologies, and
the information collected about the way they use these tools can
be used to sell them goods and services, then the individual ought
to be able to benefit from the value created by computer profiles
of their transactions and interactions. If you want inexpensive
high-speed Net access and state-of-the-art affordable intelligent
agentry, perhaps you would consent to let consumer electronic companies
track your web-surfing for a few months? But this is a question
about the technical application of a tool for identifying people
by their interests and purchases. There is another question to be
asked, about where the collection and use of such information might
be pushing society.

For example, consider this excerpt from an article in the November,
1997 issue of AMERICAN DEMOGRAPHICS: "Breaking Up America: The Dark
Side of Target Marketing" by Joseph Turow:

"With the triumph of target marketing in the last decades of the
20th century, the United States is experiencing a major shift in
balance between society-making media and segment-making media. Segment-making
media are those that encourage small slices of society to talk to
themselves, while society-making media have the potential to get
all those segments to talk to each other. In the ideal society,
segment-making media strengthen the identities of interest groups,
while society-making media allow those groups to get out of themselves
and talk with, argue against, and entertain one another. The balance
can lead to a rich and diverse sense of overarching connectedness
or understanding: what a vibrant society is about.

"As with most ideals, this one has never existed. It has been
far too easy for both segment making and society-making media to
lapse into stilted stereotyping of many groups rather than to act
out the complex, fascinating texture that is America. But as marketers
get better at targeting desirable customers in media environments
designed for them, even the possibility of the ideal is fading.

"The hypersegmentation of consumers into specialized media communities
is transforming the way television is programmed, the way newspapers
are 'zoned,' the way magazines are printed, and the way cultural
events are produced and promoted. Advertisers' interest in exploiting
differences among individuals is also woven into the basic assumptions
about media models for the next century ‹ the so-called 500-channel
environment. In the next century, it is likely that media formats
and commercials will reflect a society so fragmented that the average
person will find it impossible to know or care about more than a
few of its parts."

Intelligent agents are what Seymour Papert or Sherry Turkle would
call "objects to think with." We can think about what the tool might
do if we use it, and how we might try to design the tool to minimize
negative effects, but we also need to think about what kind of world
tools like this will be used to create. Perhaps new technologies
ought to have societal impact reports, not as an attempt at political
regulation, but as a way of thinking systemically instead of just
instrumentally. Do we know where we are going? Do we want to go
there? Is there anything we can do about it?

Howard

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

From: Kevin Kelly

Submitted: 1.24.98
(Howard Rheingold wrote:) the United States is experiencing a
major shift in balance between society-making media and segment-making
media.

I find that duo and distinction pretty provocative. Thanks for
the pointer!-

kk

KEVIN KELLY, executive editor of *Wired* magazine, is the
author of *Out of Control.*