a Jungian perspective on the use of 'emotional' language
in descriptions of number theoretical phenomena
"As archetypes of our representation of the world, numbers form, in
the strongest sense, part of ourselves, to such an extent that it can
legitimately be asked whether the subject of study of arithmetic is not the
human mind itself. From this a strange fascination arises: how can it be
that these numbers, which lie so deeply within ourselves, also give rise
to such formidable enigmas? Among all these mysteries, that of the prime
numbers is undoubtedly the most ancient and most resistant."
G. Tenenbaum and M. Mendès France,
Numbers and Their Distribution (AMS, 2000) p.1
The collected quotes (from which the above is taken)
concerning the distribution of
prime numbers (and the related Riemann zeta function and
Riemann hypothesis) contain a remarkable amount of
'emotional-ecstatic-poetic-religious' language. The words
mystery, mysterious and secrets appear numerous times,
but also strange, stunning, astonishing, baffling, bafflement, surprise, endless surprises, exasperating, perplexing, bedevilled, cruel and compelling,
stultifying, fascinating, (strange) fascination, obsession, mysterious attraction, breathtaking(ly), beautiful, most beautiful, incredibly beautiful, immense beauty, beautiful harmonies,
elegant, elegance, gorgeous, glamorous, incredible, exalted, majestic, fantastic, amazed, amazing, absolutely amazing, awed, impenetrable, impenetrability, tantalized, tantalizing,
tantalizingly, tantalizingly vulnerable, unveil, blazed...fearlessly, wreath its conqueror with glory, most ancient, formidable enigmas, great white whale, quest, vast toil,
unthinkable complexity, strange conundrum, profundity, profound mystery, great mystery, magic, aesthetic appeal, works of art,
arcane music, secret harmony, inexplicable secrets of creation, gem, gemstone, jewels, heart, soul, cosmos, abyss(es),
divine, Holy Grail, Lucifer, Devil and God.
William Blake or
John Milton might
feel at home with this. Mathematicians, however, are not ordinarily inclined to use such language so freely.
It is hard not to wonder what it is we are ultimately dealing with here.
I have discussed this matter with a couple of people who are versed in both Jungian/archetypal/transpersonal psychology and higher mathematics: Barry Jeromson (BJ) and Robin Robertson (RR). Jeromson has taught psychology at the University of South Australia, having obtained his Ph.D. with a thesis entitled "Jung and Mathematics in Dialogue: A Critical Study". Robertson is an editor of Psychological Perspectives, the journal of the Los Angeles Jungian Society.
They have offered some interesting insights. Here are some excerpts from our e-mail
discussions. In reading these, it would be helpful to have a basic grasp of such
Jungian concepts as archetypes, anima and animus, the shadow, compensation, projection, the
four-way scheme of the thinking, feeling, sensing and intuition functions, etc.
For these purposes, I recommend Daryl Sharp's online Jung Lexicon.
[BJ - in response to the language collected above]:
"Many of these adjectives are also used to describe a beautiful woman,
especially one who is aloof and remote. Are the mathematicians projecting
their animas onto the mathematics? In the words of the post-Jungian
writer James Hillman, the mathematics 'anima-tes' their imaginations. Incidentally
the use of the term "elegant" to describe a proof lends weight
to this argument."
[MW - commentary]:
Over a year later, I discovered that Andre
Weil, in his review of
Collected Works, wrote:
"Perhaps the best part of [Artin's] career may be described as a love
affair with the zeta function."
In a similar vein, J.-F. Burnol characterises the Riemann Hypothesis as female in the
following quote from his recent paper "Fourier and zeta(s)":
"What is more, Theorem 7.2 has encouraged us into trying to encompass in our speculations
the GUE hypothesis, and more daring and distant yet, the Riemann Hypothesis Herself."
From p.17 of Karl Sabbagh's book Doctor
Riemann's Zeros (Atlantic, 2002):
"Henry Iwaniec of Rutgers University, put matters very simply.
'I just love working in prime numbers', he said, with passion in his voice."
...and on p.210, Alain Connes further invokes the feminine:
"I believe I have found a very nice framework [to prove the
Riemann Hypothesis] but this framework is still awaiting the
main actor. So there is the stage - it is perfectly well arranged and
so on - but we are still expecting the heroine to come and complete it."
[notice "heroine" not "hero"]
The great Cambridge number theorist J. Littlewood, in Littlewood's Miscellany,
claimed to have become "infatuated" with the problem of the Riemann Hypothesis.
Voice review of the books by Sabbagh, du Sautoy and Derbyshire contained this:
"In discussing the primes, mathematicians often use the vocabulary of first love.
'They're objects of great beauty, no question,' says AIM's
[Brian] Conrey, explaining how as a teenager he fell for the Twin Prime Conjecture..."
This is from John Derbyshire's 2003 book, called Prime Obsession:
1859 paper], Riemann made an
incidental remark - a guess, a hypothesis. What he tossed out
to the assembled mathematicians that day has proven to be almost
cruelly compelling to countless scholars in the ensuing years...
...it is that incidental remark - the Riemann Hypothesis - that is
the truly astonishing legacy of his 1859 paper. Because Riemann
was able to see beyond the pattern of the primes to discern traces
of something mysterious and mathematically elegant at work - subtle
variations in the distribution of those prime numbers...
It has become clear that the Riemann Hypothesis, whose resolution
seems to hang tantalizingly just beyond our grasp holds the key to a
variety of scientific and mathematical investigations...Hunting down
the solution to the Riemann Hypothesis has become an obsession for
Note the use of the words "cruelly compelling", "mysterious and...elegant", "seems to hang
tantalizingly just beyond our grasp" and "obsession". The
choice of language speaks for itself.
In a similar vein we have:
"Let us now pursue an apparently tangential path. We wish to
consider one of the most fascinating and glamorous functions of
analysis, the Riemann zeta function..." (R. Bellman)
"Those who pursue [the theory of prime numbers] will, if they are
wise, make no attempt to justify their interest in a subject so
trivial and so remote, and will console themselves with the thought
that the greatest mathematicians of all age have found it in it a
mysterious attraction impossible to resist." (G.H. Hardy)
"Hardy grew to love the [Riemann Hypothesis]. He and
Littlewood wrote at least ten papers on the zeta-function." (B. Conrey)
"...Gauss liked to call [number theory] 'the Queen of Mathematics'.
For Gauss, the jewels in the crown were the primes, numbers which had
fascinated and teased generations of mathematicians." (M. DuSautoy)
"Even Alan Turing, the British mathematician who played such an
important part in the British deciphering operation at Bletchley Park
during the Second World War, was seduced by the fascination of the
Riemann Hypothesis. In the midst of laying the theoretical
foundations of what were to become digital computers, Turing designed a
machine to calculate zeros of the Riemann zeta function." (K. Sabbagh)
The word "tantalise" and it's variants appear repeatedly in this context,
which appears highly compatible with the idea of an anima projection.
"A particularly tantalizing aspect of the chaotic scattering
process is that it may connect the mysteries of quantum chaos with the
mysteries of number theory." (M.C. Gutzwiller)
"We have presented several tantalizing connections between xp and
zeta(s). However it is clear that more is required to transform
our hints and guesses into an unambiguous and satisfactory construction of the
Riemann operator." (M. Berry and J. Keating)
"Our purpose is to report on the development of an analogy, in which three
areas of mathematics and physics, usually regarded as separate, are intimately
connected. The analogy is tentative and tantalizing, but nevertheless fruitful.
The three areas are eigenvalue asymptotics in wave (and particularly quantum)
physics, dynamical chaos, and prime number theory." (M. Berry and J. Keating)
"...in one of those unexpected connections that make theoretical physics so
delightful, the quantum chaology of spectra turns out to be deeply connected to
the arithmetic of prime numbers, through the celebrated zeros of the Riemann zeta
function: the zeros mimic quantum energy levels of a classically chaotic system. The
connection is not only deep but also tantalizing, since its basis is still obscure -
though it has been fruitful both for mathematics and physics." (M. Berry)
"The primes have tantalized mathematicians since the Greeks, because they appear
to be somewhat randomly distributed but not completely so." (T. Gowers)
Here's a dictionary definition of tantalise: "To tease or torment by or as if by exposing
to view but keeping out of reach something that is much desired."
Could the widespread tendency to use this word be evidence of the collective unconscious
personifying the zeta function and related mathematical structures as 'teasing, tormenting'
entities, keeping something desirable of themselves just in view, but out of reach? Here's one
"Berry and his collaborator Jon Keating used them to show how techniques in number
theory can be applied to problems in quantum chaos and vice versa. In itself such a connection
is very tantalising. Although sometimes described as the Queen of mathematics, number theory
is often thought of as pretty useless, so this deep connection with physics is quite
"Berry is also convinced that there must be a particular chaotic system which when
quantised would have energy levels that exactly duplicate the Riemann [zeros]. "Finding
this system could be the discovery of the century," he says. It would become a model
system for describing chaotic systems in the same way that the simple harmonic oscillator
is used as a model for all kinds of complicated oscillators. It could play a fundamental
role in describing all kinds of chaos. The search for this model system could be the
holy grail of chaos....Berry believes the system is likely to be rather simple, and
expects it to lead to totally new physics. It is a tantalising thought." (Julian Brown)
"When confronted with the following quote from the great Paul Erdös
"It will be another million years, at least, before we understand the primes.",
A friend who more-or-less introduced me to Jungian thought pointed out that this sounds like an exasperated man talking about his wife.
[Here is a humorous piece on this subject, submitted
by Russell Johnston after he read this page.]
Although Erdös was never married nor ever expressed any interest in romantic
relationships, I should clarify here that I am not in any way attempting to draw conclusions
about the individual psychologies of anyone quoted above. I see the issue
here as a collective
phenomenon, and those quoted are simply acting (unintentionally) as conduits for an emerging 'collective
awareness'. The invoking of 'the feminine' is significant NOT in what it suggests
about the individual mathematician's psyche, but as an indication of a collective
encounter with what we could call 'the other', i.e. something fundamentally alien
to the western psyche as it is currently configured. The Jungian analysis of male-female
relationships acknowledges a context wherein the individual experience of the other
person is a particular manifestation of a wider psychic phenomenon of "encounter
with the other'". This way of thinking deals with archetypes and has been
applied to such diverse topics as purported encounters with UFO's, fairies and angels,
the use of racist stereotypes in wartime propaganda, and the analysis of dreams.
The preceding paragraph might sound rather clumsy to anyone properly versed in
psychological theory (any help in improving the text is very welcome). However, I hope
I have conveyed the main point:
Mathematicians have reached a point in their collective
historical exploration of 'the mathematical landscape' where they are now encountering something
fundamentally 'other'. Some mathematicians have been moved to comment
on this emerging situation. As mathematics constitutes the 'core' of western
science and in fact of the entire 'scientistic' civilisation which has almost entirely
excluded 'the feminine' from its considerations, these individuals have
instinctively drawn on the language of 'the feminine', romantic attraction,
etc. in order to convey the sense of 'otherness'.
On a related note, on p.32 of his fascinating book Jerusalem: City of
Mirrors (Fontana, 1989), Amos Elon observes the recurrence of feminine metaphors
associated with Jerusalem through the ages and concludes:
"Language was playing odd tricks on the patriarchal East. Public
statements are often rooted in private dreams. When men are mystified,
they often resort to the feminine gender."
The situation just discussed suggests to me that the scientific
world is on the edge of a collective psychic 'shock' which will be somehow analogous to the
psychic 'shock' that can occur the first time someone falls into a serious romantic
relationship. Could it be that mathematics is about to undergo a similar transition to the
one which occured in physics with the advent of quantum mechanics in the early 20th
I had put forward the idea of the distribution
of primes having some significance in terms of Jung's concept of 'archetypes'.
Following on from this was the idea that the pursuit of the elusive proof
of the Riemann Hypothesis was something like a 'Grail quest' (as in the
Arthurian literature, the multi-levelled symbolism of which has been of great
interest to Jungian/archetypal/transpersonal psychologists):
The Riemann Hypothesis has
been described as "the Holy Grail of Mathematics"
by multiple commentators. For example, on p.90 of Sabbagh's book, we find:
"It was not the first occasion, and would not be the last, on which a distinguished
mathematician believed he had within his grasp the Holy Grail of number theory."
[on Radamacher's failed 1943 proof attempt at the Riemann Hypothesis]
"That's fascinating. I like the idea of a mathematical grail quest.
That really brings in the archetype of the feminine you mention...
I assume you know the Jungian stuff on the grail quest as a search for the missing feminine.
In essence, Parsifal had to find the feminine inside himself in order to find the projected
feminine. And, of course, it was too early in time for him to fully
succeed. Put that bluntly, it sounds trivial, but there's a lot in it."
"Barry Jeromson pointed out that a lot of the emotional adjectives
used by mathematicians in regard to the distribution of primes:
"strange", "mysterious", "astonishing, "breath-taking", etc. suggest
an anima projection. A friend of mine has taken this idea
a bit further, in terms of a compensation from the shadow, due to an
overemphasis of mathematicians on the thinking function. As he put
it, mathematics (and with it science) may be about to be confronted with
its feminine side."
"Your friend sounds right on the money to me. I've been heavily involved
with a chaos theory society for a number of years, always as a friendly
visitor, not one for whom chaos theory was central. Among those folks,
especially the techies, the desire for the feminine is palpable. They're
clearly thirsting for something beyond the dry, arid land of pure logic
and they don't know how to find what they need."
[MW - commentary]:
I would go further than Barry Jeromson. Above, he asks whether
mathematicians are projecting their animas onto the mathematics.
As stated above, I do not believe that this is merely about the cumulative effect of
a number of individual male mathematicians experiencing some personal
psychological reaction to the object of their study. That is, it isn't
just about male mathematicians suppressing their animas and consequently
experiencing some sort of compensation involving a projection of the
anima onto the mathematics. I think we are seeing a collective
compensation involving the collective anima being projected onto the
mathematics associated with the primes and the zeta function.
The entire 'civilisation', increasingly concerned with technologies
and economies to the exclusion of all else, has been suppressing its collective
anima. The compensation phenomenon applies to the whole civilisation,
but is currently being experienced in the heart of number theory, the
'Queen of Mathematics', mathematics itself being acknowledged as the
'Queen of the Sciences'.
The number theorists whose quotations included
the poetic-ecstatic language above belong, it could be argued, to the inner
priesthood of a scientistic civilisation. I'm speculating that they
may be the first to be encountering some collective psychological
reaction brought on by a mass psychic imbalance associated wtih
a collective obsession with quantity.
In January 2006 I received an email from Alex Abercrombie, a retired mathematician
familiar with Jungian thought, in which he stated:
"You've undoubtedly tumbled to something with your idea of a collective anima projection. My belief is that
this has has to do with the anima being part of the interface between conscious and unconscious. The key moment
in mathematics is when you suddenly find that an idea has a life of its own - you define something with a certain
purpose in mind and then you discover other entirely natural properties which weren't intended in the definition. Of
course a prime example (excuse the pun!) of this is the primes, which are boring enough if you just consider them
as generators of the multiplicative semigroup on Z. But throw in the order on Z - not to mention the additive structure
- and suddenly you're in the world of the Prime Number Theorem, Dirichlet's theorem, Goldbach's conjecture and all
that. (Another example along the same lines is partitions of natural numbers, where just addition in N yields a problem
whose solution is that amazing formula of Hardy-Ramanujan-Rademacher.) So the sequence of primes in a way
embodies this business of a limited conscious idea suddenly showing signs of independent life - but at the same time
showing that it still conceals at least as much as it reveals. It is a first approach of the unconscious - therfore definitely
an occasion for anima projection!
It would be interesting to know whether women mathematicians are inclined to project the animus in similar
situations. I rather suspect not, and if I'm right this would tend to support your idea of the anima projection being
collective rather than personal."
My understanding has been that Jung concluded (near the end of his
life) that the set of natural numbers constitutes a single archetype,
the archetype of order.
Further, he thought that in modern western
humans, this archetype had become conscious. The current western
obsession with quantity and quantification could best be understood in
these terms - the psychological relationship with number has become
dangerously unbalanced. Jung's student Marie-Louise von Franz developed these
ideas and wrote with wonderful lucidity about them - see in particular
Number and Time: Reflections Leading Toward a Unification of Depth
Psychology and Physics (Northwestern Univ. Press, 1974).
Von Franz died a few years ago, and neither
she nor Jung (as far as I know) had a chance to explore the archetypal
significance of the mysterious, irregular sequence of prime numbers
which reality presents to us, embedded within the sequence of natural numbers.
It would have been fascinating to have been able to discuss this with either of them.
the "number theory and physics" content of
this website leaves me with a distinct feeling that, historically, we are on the verge
of some major, extraordinary discovery which could fundamentally alter our perception
and understanding of number and the way we relate to number concepts. This can lead to some interesting speculation within a Jungian
but it is wise to keep in mind that it may all be unfounded and based on wishful thinking.
"I have some problems with this piece of Jungian rhetoric [the 'becoming conscious'
of the natural number archetype of order]. More
likely, large chunks of the collective western consciousness are gripped
by the number archetype. It is still unconscious and driving the ship. If
the number archetype became conscious, then it would lose its power and
the widespread urge to quantify everything would evaporate."
[MW - commentary]:
I am becoming increasingly fascinated by, and concerned about, "the widespread urge
to quantify everything" which now seems to characterise the Western world. This
manifests in two (related) forms, both often associated with the word 'materialism':
(1) the ideological belief in quantitative experimental science as the only valid route to truth
(2) greed-fueled capitalism, rampant consumerism, and high-level decision-making which will affect
communities, populations, environments, etc. being made solely in terms of quantifiable
material or financial gain.
I have recently become aware of the philosopher René Guénon's
major work The Reign of Quantity and the Signs of the Times in which the author considers
the origins and significance of this phenomenon. Information on Guénon and his ideas can
be found here.
Marie-Louise von Franz, mentioned above, suggested that the western obsession with
quantity and quantification indicates a fundamental imbalance in our relationship with
number concepts. She repeatedly stressed the importance of the 'qualitative' aspects
of number which appear to be of considerable importance to all non-Western cultures.
The awareness of these aspects has largely disappeared in the West, in a gradual process which began
when mathematics bifurcated from 'number mysticism'. This does not, of course, mean that
indiscriminately grasping at any number mysticism will somehow improve our collective situation.
However, striving to keep our minds open to new possibilities and maintaining a certain humility
quite possibly will.
More information on von Franz and her writings can be found here.
An article by Robin Robertson entitled "The evolution of number:
the archetype of order" is available here.
J. Aveleira has brought to my attention his recent (08/04) article from
The C.G. Jung Page,
It Be: How many pieces are there in a bit of reality?". It concludes:
"The knowledge we may gather on issues of being, consciousness and reality, possibly
is delineable by means of a single, universal, fractal-holographic and evolutionary process
based on the unfolding and interaction of unsettled prime numbers of identifiable classes,
components or polarities.
Why prime? Because prime numbers are not factorable. In a quest for absolute models,
one tends to look for comprehensive, essential concepts, and likely shall find them
following structures which would not be reducible into others. Why fractal-holographic?
Because the structure of absolute models of reality probably shall surface as a whole in
every part, at every instance and scope of actuality. Why evolutionary? Because that
process of change appears purposeful, directional, aimed to growth and to the solution of
conflicts. Why unsettled? The ultimately detailed and all-encompassing structure of reality
possibly shows an undeterminable prime number of classes arranged in unfathomable nuance.
Good, veritable representations may show 1 or 2 or 3, 2x2, 5, 2x3, 2x3x2, etc. classes or
components and perhaps a little further ones. Those representations may be valid and
functional to a remarkable extent. I believe, however, that the infinite complexity
possibly extant in ultimate reality would not be touchable by minds subject to any
degree of limitation."
I would be very interested to receive further thoughts on these
matters, from any number of differing perspectives. Please use the
e-mail link below.
number theory and Taoist aesthetics