%This is submitted for publication in ``Mathematics Tomorrow'', a projected
%book by the IMU.
\documentclass[11pt]{article}
\author{by David Ruelle\footnote{IHES, 91440 Bures sur Yvette,
France $<$ruelle@ihes.fr$>$.}.}
\title{CONVERSATIONS ON MATHEMATICS WITH A VISITOR FROM OUTER SPACE}
\begin{document}
\maketitle
\begin{abstract}
While it is probably unrewarding to try to imagine what extraterrestrial mathematics might be like, we may more reasonably try to find out what is peculiar about human mathematics. An investigation of that sort was started by J. v. Neumann in his book {\it The computer and the brain}.
We shall discuss some of the characteristics, and especially deficiencies, of the human brain as revealed by the neurosciences and comparison with the computer. We shall argue that they explain features of human mathematics that we take for granted, but that a mathematical visitor from outer space might find very striking.
\end{abstract}
\paragraph{}
English versions of such compendia as {\it Encyclopedia Galactica}
(Concise Edition), {\it Standard Galactic Dictionary of Mathematics}, and
a few other basic references were made available to me by an extraterrestrial
friend. I can give here only few details of how this happened,
but I must explain that, after extreme initial enthusiasm, browsing
through this material got me perplexed at first and then frankly
distressed. So-called galactic mathematics appears to consist of huge
computer programs which, run on suitable (galactic) computers, deal very
efficiently with all kinds of difficult mathematical problems. As my
friend Pallas explained, such programs are like large mathematical
libraries, but much easier to use. Also, producing such a program
is much more challenging than writing down a human type mathematical theory.
It is, as she told me, like constructing a brain instead of writing a book.
My friend Pallas landed in my garden on a fine morning
of May in her silent spaceship, bright and tall, and departed three months
later. She had convincing and pleasant female human form, to the extent that
I often doubted if she really was extraterrestrial. She told me, however,
that her female guise was only for convenience. I should understand that
mathematicians are sexless, and should be referred to as ``it'' rather
than ``he or she''.
Be that as it may; we had quite nice conversations on all
sorts of topics ranging from poetry to religion and from music to science.
To my relief we abandoned galactic mathematics after a while, and turned
to human mathematics on which Pallas was writing her galactic Ph.D.
thesis. She had the strangest ideas on the subject, but was well
documented, and little by little convinced me. I shall now relate the
results of our conversations. The basic ideas come from her, and my
contribution was mostly to get this material out of her by my questions
and then writing it down.
I had the greatest difficulty with her first statement, that
``\dots to appreciate human mathematics you have to understand
how peculiar the human intellect is, compared with that of the ancient
civilizations of the Galaxy.''
``How on earth can you say such a thing'' I answered, ``and
what right have you to say that the mind of a human is more peculiar than
that of a slimy galactic superoctopus?''
``Why in the galaxy don't you use your brains! Think how
primitive a machine your Personal Computer is, yet it completely outwits
you on simple mathematical problems like deciding if a ten digit number is an
exact square. You can imagine that an ancient civilization would have
fixed such intellectual inadequacies by assisted coevolution, biological
engineering, and so on. I shan't go into details, as they might greatly
upset you.''
``You mean that human civilization too will \dots?''
``Yes, if human civilization survives to become ancient.''
After that we argued, point by point, what made the human
brain so {\it peculiar}. Pallas made some comparisons between humans and
slimy galactic superoctopi, but these comparisons made little sense to
me. So she replaced galactic beasts by human computers, with which I am
more familiar. ``These computers are really very stupid things'', she said,
``but they already give a good idea of features desirable for doing
mathematics, and which are not possessed by the human brain''. Here is
now a list of five or six peculiarities of the human brain that we came
up with, very reluctantly on my part, after many hours of discussion.
\paragraph{Slowness and high parallelism.}
The human brain is a highly connected net of several times
$10^{10}$ neurons. Local characteristic times are at least 1 millisecond,
and the speed of propagation of the nervous influx is from 1 to 100
meters/second, so that times of the order of 100 milliseconds are easily
reached. By comparison, the processor of your PC has a speed measured in
millions of instructions/second. The high speed of computers permits
repetitive calculations where each loop provides an updated input for the next
loop. The slowness of the brain is compensated by a high parallelism of
operations. An example of this parallelism occurs in the sensory pathways
to the central nervous system: they preserve the spatial relations of the
receptors. In the visual system this is called {\it retinotopy}. At a higher
level, the visual system also uses parallel processing of different
aspects of visual information (like color, motion, \dots \cite{3}).
The computer and the brain are both information processing
systems, and functional requirements imply some structural analogies
such as existence of input, output, memory. A detailed comparison shows
huge differences, which were first analyzed by J. von Neumann in {\it The
computer and the brain} \cite{5}. This was his last book. He wrote it as his
own brain was being destroyed by cancer.
\paragraph{Deficient memory.}
Just as your laptop has several memories (RAM, floppy drive, hard
disk, CD-ROM) with different characteristics, the human brain has
several functionally different memories. Short term memory allows us to
repeat immediately a random sequence of letters or digits, but is
typically limited to about seven items. This limitation is a nuisance
when dialing a number that you just read in the telephone book, and it
also has consequences for the way humans do mathematics. If the
solution of a problem depends on having ten items of data (say) readily
available, it is necessary to put these items in long term memory,
({\it i.e.}, somehow learn them by heart) or to create an external memory
in the form of a sheet of paper on which the data are written. What
corresponds to short term memory on your laptop is the random-access
memory (RAM) of 8 or 16 megabytes. The difference in favor of the
laptop is huge. If natural selection had favored immediate memorization of telephone numbers, we would have a much better short term memory.
Our long term memory is more satisfactory, but wouldn't it be nice if
your brain could store a cool 500 megabytes of error-free data in as
little space as a CD-ROM (a compact disk that can hold the 46 million
words of the {\it Encyclopedia Britannica}, and has room left for a
number of other things).
\paragraph{Search for regularities.}
A ten digit number may be hard to remember, but with 9876543210 we have no difficulty because it is just ``the ten digits from zero to nine written in reverse order'', and 3141592653 is also easy because it is ``the first digits of pi''. In the same way we find hidden regularities in telephone numbers, and if we face a wall with cracks we shall often interpret the cracks as human profiles.
In general we seem to make up for inadequacies of the human mind, like poor memory, by a search for ``order'' or ``meaning'' often pushed to absurd limits. The unceasing, obsessional search for regularities is certainly fundamental to human intelligence, and in particular to the human mathematical genius.
\paragraph{Importance of visualization.}
Evolution has given a lot of weight to the visual system, which
occupies a big part of our brain. Most mathematicians are thus happy
when they can make use of visual intuition in their mathematical
work. For instance, it is considered a success if one can geometrize a
theory, interpreting its concepts in terms of points, spaces, topology
and the like. There are however non visual mathematicians. Laurent
Schwartz for example boasts a complete lack of geometric intuition.
While some of his colleagues have difficulty in believing him, it must
be admitted that the use of geometric intuition has no logical necessity
in mathematics, and is often left out of the formal presentation of
the results. If one had to construct a mathematical brain, one would
probably use resources more efficiently than creating a visual system.
But the system is there already, it is used to great advantage by human
mathematicians, and it gives a special flavor to human mathematics.
\paragraph{Lack of formal precision.}
One wrong step in a long proof is sufficient to make the proof
worthless. The ability to check mechanically that the rules are
respected would thus seem to be an important mathematical asset.
Our computers are good at doing this sort of mechanical work without
error, but are not good at doing creative mathematics. The human brain,
by contrast, is not very good at long complicated logical tasks which
have to be performed without error, yet can do very difficult
mathematics. Human mathematics consists in fact in talking about formal
proofs, and not actually performing them. One argues quite convincingly
that certain formal texts exist, and it would in fact not be impossible
to write them down. But it is not done: it would be hard work, and
useless because the human brain is not good at checking that a formal
text is error-free. Human mathematics is a sort of dance around an
unwritten formal text, which if written would be unreadable. This may
not seem very promising, but human mathematics has in fact been
prodigiously successful.
\paragraph{}
``I am glad to hear that human mathematics is prodigiously
successful'', I said. ``But I thought that in discussing peculiarities
of the human mind you would address the problem of consciousness,
which seems so fundamental to us.''
``Consciousness is a much disputed topic among humans'', Pallas
answered ``and I didn't want to base my discussion on questionable premises.
I really don't think the question is all that important, but let us
discuss it if you wish.''
\paragraph{Consciousness and attention.}
Consciousness is an introspective concept, and difficult to
approach scientifically. It has nevertheless attracted the attention of
brain specialists (see for instance \cite{1}). Consciousness is related to
attention, which is our ability to direct intellectual resources to a
specific task. (Attention is demonstrably correlated with greater blood
irrigation of areas of the brain concerned with the task performed).
It may be that consciousness arises naturally in a highly parallel system
like the human brain, where some coordination of activities is needed to
avoid chaos. Note however that many tasks are performed unconsciously.
It is a natural idea that mathematical work should require conscious
attention. For instance when Polya (in his book {\it How to solve it} \cite{7}) describes how one should attack a mathematical problem, he uses phrases like
``to understand the problem'', ``to have the data well in mind'', which have
absolutely no mathematical meaning, but reflect the importance of conscious
attention. Nevertheless, unconscious mathematical work may play an
essential role, as stressed by Poincar\'e\footnote{See ``L'invention
math\'ematique'', which is Chapter 3 of \cite{6}.}, once all the aspects of a
problem have become familiar by conscious study. Perhaps the conscious
study is needed to put all the useful information about the problem in
long term memory, so that the combinatorial task of actually finding a
solution can then be managed unconsciously.
\paragraph{}
``Now that you have reluctantly agreed that the human brain is
a bit peculiar'' Pallas said ``I would like to convince you that the
peculiarities we discussed have a profound impact on the human mathematical
output.''
``Some of our social scientists make similar statements, to
the effect that scientific theories are shaped by social forces, and
that what is true today may be false tomorrow when the political power has
changed hands. A mathematical text would be a narrative like any other
narrative, and literary criticism could reveal its true social content:
racist, male chauvinistic, and the like. We, human mathematicians, have
a very different view of the nature of our art. We believe in absolute
truth: 137 is a prime number, and no social event will change this fact.
We use the name {\it Teichm\"uller space} to denote some mathematical
object, even though Teichm\"uller was a Nazi, and most of us abhor the
Nazi ideology. It is our pride that that we have access to a world of
ideas were political ideologies do not count. Would you now replace social
relativism by another kind of relativism where truth would not be the
same for humans and for slimy galactic superoctopusses? Would you\dots''
Pallas interrupted me there: ``No! no! no!'' she said.
``Logical truth is absolute. It is not determined by social
circumstances or by the particular structure of the mind of a galactic
species with mathematical abilities. But mathematical style
depends enormously on the structure of the mind which produces it. I
leave you till tomorrow to find an example of that yourself, and explain
it to me.''
Of course the mathematical discussions which I have been relating
were spread over many days, and we did other things than talking about
mathematics. In June-July in particular, we had splendid weather and took
long walks through fields and forest. ``Coming from an advanced
civilization'' she said ``it is a bit of a shock to be exposed to all
these creepy-crawlies that you take for granted: mosquitos, spiders,
slugs, and the ever-present flies eating from your food and shitting on
it. Yet I have come to love the sort of nonchalant absurdity of your
primitive culture, its art, and its archaic science. I am even
starting to enjoy being a female human\dots''
``That is called a girl, or a woman.''
``I know, but I don't like this sexist language.''
Anyway, the next day I presented my homework to Pallas, a double
homework actually.
\paragraph{Greek geometry.}
A contemporary mathematician leafing through Euclid will find
absolutely nontrivial theorems even if they are well-known to it (following
the advice of Pallas I am not saying ``to him or to her''). Greek
geometry is early but in a sense completely modern mathematics. It does
however show more clearly than later mathematics two peculiarities of
the human brain that produced it:
(1) it uses the human visual system, in fact geometry is
directly derived from visual experience and intuition,
(2) it uses an external memory in the form of a drawing formed
of lines and circles, with points labeled by letters\footnote{It
seems that the drawings of Greek geometry have not been much studied
(historians of science study {\it texts}). See however \cite{4}; I am indebted
to Karine Chemla for this reference and for an enlightening conversation
on the subject.}.
Combining these two tricks permits elaborate logical
constructions which the Greeks rightly considered as prodigious
intellectual feats. Hilbert's version of Euclidean geometry without the
help of (1) and (2) shows how hard the subject really is.
\paragraph{Parsimony.}
Instead of asking how the characteristics of the human brain
influence human mathematics, one could ask what kind of humans are
likely to do good mathematics. A productive artist might be insane or
a heavy drug abuser, but a working mathematician must be relatively
normal. Some level of paranoia is however acceptable, and not uncommon.
But the most widespread feature is an obsessive disposition associated
with the characteristic traits of {\it order}, {\it parsimony}, and {\it
stubbornness}. Freud has interpreted those traits as leftover from the
so-called anal-sadistic stage of the child's libido evolution. Whatever
the interpretation, it is clear why order and stubbornness should be
assets for a scientist in general. But parsimony is particularly
meaningful for human mathematics. We know indeed that mathematical
proofs tend to be long (this is related to G\"odel's theorem), and that
the human ability to check the correctness of formal texts is limited.
One has thus to cut proofs into segments which make sense ({\it i.e.}
make sense for the easily tired attention of a human mathematician). The
compromise between long proofs and short attention span calls for an
ungenerous, parsimonious attitude. Many mathematicians, of course, have
grown up to be generous in social relations where systematic parsimony
would be crippling.
\paragraph{}
Pallas had remained silent during the presentation of my
homework, doodling as it seemed on a large sheet of paper. As I looked
at it I saw that she had drawn a kind of octopus, a rather noble-looking
creature with a high forehead and intelligent eyes.
``This is a slimy galactic superoctopus'' she said. ``Notice
that each arm trifurcates: it has twenty-four fingers, an important number.''
I reflected that the slimy galactic superoctopodes must have
discovered arithmetics by counting their fingers and toes, exactly like
humans.
``But let us return to your homework. I am glad that you
recognized that Greek geometry is human mathematics at its most human,
and I would say at its most beautiful. Concerning the psychological
setup of mathematicians, I did not know Freud's ideas on the anal-sadistic
stage, and I have no personal insight into the matter, but I shall
mention your remark in my PhD thesis. What you say about parsimony goes
to the heart of trying to characterize human mathematics, but does not
solve the problem. And the problem, as far as I am concerned remains
unsolved.''
We continued our discussion on other days, during the long
walks that we made. I wanted her to help me guess at the future of
human mathematics, which she was quite reluctant to do, ``not knowing''
as she said ``if mankind will still exist in a few decades''. She
preferred to analyze the forces which would drive mathematics. Here
again I shall put in my own words the tentative conclusions that we
reached after hours of conversation.
\paragraph{Towards formalized mathematics.}
Formalization is one of the great dreams of mathematicians. But
they have been content with mathematics that could {\it in principle}
be formalized, so that the correctness of the formalized text could {\it in
principle} be checked mechanically. The progress of computers should
lead in due time to the possibility of translating human mathematics into
formal language so that proofs can be checked mechanically. Such an
enterprise appears much less difficult than producing interesting original
mathematics by computer. Computer formalization, however, may bring
surprises, like any mathematical endeavor. Here, interestingly, the
principle of parsimony for the length and simplicity of proofs could be
largely relaxed: checking by computer that a formal text is correct would
be extremely fast, and wasting a factor of ten in the length of the proof
would not change things much.
\paragraph{Towards intuitive mathematics.}
We know that theorems that are simple to state may have very
long proofs, and we are seeing more and more examples of this,
particularly in algebra (beginning with Feit and Thompson \cite{2}, later
followed by the classification of finite simple groups). It is thus
hard to exclude that some very interesting results are inherently
off limits to the unaided human brain, and might only become accessible
when sufficiently intelligent computers take over. As long, however, as
humans use their own brains to do mathematics, some areas will be
privileged. Our visual intuition of space and our intuition of time
make for instance the theory of dynamical systems particularly
attractive, and this has indeed been a flourishing domain of research
in recent times. The human brain is also in a favorable position in
branches of mathematics related to physics, not just for reasons of
intuition, but because the physical world itself proposes a wealth of
facts expecting theorization.
\paragraph{Towards natural mathematics.}
Here are two facts:
$$ 3^2+4^2=5^2\qquad(a)\qquad,\qquad3^3+4^3+5^3=6^3\qquad(b) $$
Many things can be said about (a), for instance it ``means'' that if
the sides of a right triangle are 3 and 4, the hypotenuse is 5. It is much
less clear what (b) means. In fact, it can be argued that many
properties of integers occur, in some precise sense, at random. In
apparent contradiction with this randomness, we see mathematicians put
much effort to organize what they know in neat natural structures. It seems indeed natural to use compact sets, groups, or functors because it makes human mathematics efficient in its use by human mathematicians.
But how much of what we consider natural follows from the specific structure of the human mind? How much is in some sense universal? My galactic friend was not very helpful in finding an answer to thes questions which she found ``a bit imprecisely formulated''.
\paragraph{}
In the discussions with Pallas which I have just reported, on the role of formalization, intuition and naturalness, she made rather less provocative remarks than in our earlier discussions. I was almost disappointed.
``Have you then rounded up your study of human mathematics
with what we have just discussed?''
``Not at all. I have a new project for after my thesis. When I have looked into it some more, I want to discuss it with you.''
``What is it about?''
``The creative role of error and intellectual confusion in
human mathematics.''
``Oh, boy!''
As I got up one fine morning of August, I did not see Pallas.
Looking through the window, I found that her spaceship had gone. There
was a note on the breakfast table:
\paragraph{}
{\sl To my favorite human mathematician\footnote{\rm In fact I am a mathematical physicist, as I told her several times}.
They have scheduled my thesis for very soon, and I must leave in
a hurry. Also, I am a bit homesick, and I want to get away for a while
from your crazy world and all its flies. As you must have guessed, I am
really a slimy galactic superoctopus, and at times a bit tired of my female
human guise. What I want now is to relax in a little pool of clean warm
water, and make bubbles pink and blue (as we say in our language). But
as soon as I have passed my thesis, I shall apply for a travel grant,
and be back.
So long,
$\qquad\qquad$Pallas.}
\begin{thebibliography}{9}
\bibitem{1} F. Crick. {\it The astonishing hypothesis: the scientific
search for the soul.} Touchstone, New York, 1994.
\bibitem{2} W. Feit and J.G. Thompson. ``Solvability of groups of odd
order.'' Pacific J. Math. {\bf 13},755-1029(1963).
\bibitem{3} E.R. Kandel, J.H. Schwartz, and Th.M. Jessel. {\it
Essentials of neural science and behavior.} Appleton and Lange, East
Norwalk, CT, 1955.
\bibitem{4} R. Netz. {\it The shaping of deduction in Greek mathematics. A study in cognitive history.} Cambridge U. P. to appear.
\bibitem{5} J. von Neumann. {\it The computer and the brain.} Yale U.
P., New Haven, 1958.
\bibitem{6} H. Poincar\'e. {\it Science et m\'ethode.} Ernest
Flammarion, Paris, 1908.
\bibitem{7} G. Polya. {\it How to solve it.} 2-nd ed. Princeton U. P., Princeton, 1957.
\end{thebibliography}
\end{document}