Mathematics and the Language of Nature
F.David Peat
Published in Mathematics and Sciences, edited by Ronald E.
Mickens (Word Scientific, 1990)
1. Introduction
"God is a Mathematician", so said Sir James Jeans1. In a
series of popular and influential books, written in the
1930s, the British astronomer and physicist suggested that
the universe arises out of pure thought that is couched in
the language of abstract mathematics. But why should God
think only in mathematics? After all, some of most
impressive achievements of the human race have involved
architecture, poetry, drama and art. Could the essence of
the universe not equally be captured in a symphony, or
unfolded within a poem?
Three centuries earlier, Galileo had written, "Nature's
great book is written in mathematical language" an opinion
that has wholeheartedly been endorsed by physicists of our
own time. Mathematics today occupies such an important
position in physics that some commentators have argued that
it has begun to lead and direct research in physics. In a
frontier field , called Superstrings, some critics are
arguing that mathematics is actually filling in the gaps
left by the lack of any deep physical ideas. But why should
mathematics play such a powerful role in physics? Is its
central position inevitable? And is the present marriage
between physics and mathematics always healthy, or are there
ways in which mathematics may, at times, block creativity?
In this essay I want to explore, in a speculative and free-
wheeling way, some possible answers to these questions and
to make some suggestions as to some radical developments in
a language for the physical world.
2. The Role of Mathematics
While there are exceptions, it is generally true that great
mathematics is studied for its own sake and without
reference to anything outside itself. Mathematics has a
beauty all its own and there is, for the mathematician, an
aesthetic joy that comes from solving an important problem,
no matter what value society may place on this activity. In
this sense, mathematics has constantly sought to free itself
from its practical origins.
Geometry, for example, began with rules for surveying,
calculating the areas of fields and making astronomical
studies and acts of navigation. Probability theory had its
origins in the desire to raise gambling to a high art. But,
very quickly, mathematics shook itself free from such
pedestrian origins. While it is certainly true that some
exceptional mathematicians have begun their studies with a
concrete problem taken from the physical world, in the end,
the mathematics they have developed has moved away from
these specific cases in order to focus on more abstract
relationships. Mathematics is not really concerned with
specific cases but with the abstract relationships of
thought that spring from these particular instances. Indeed,
mathematics takes a further step of abstraction by
investigating the relations between these relationships. In
this fashion, the whole field moves away from its historical
origins, towards greater abstraction and increasing beauty.
The English mathematician G. Hardy2 refused to justify
mathematics in terms of its utility and pursued it as an art
for its own sake. He seem to rejoice in the very abstraction
of his own research and in its remoteness from practical
applications. Indeed, Hardy once spoke of a monument so high
that no one would ever be able to see the statue that was
placed at its pinnacle - a fitting metaphor for his own,
somewhat extreme, view of the role of mathematics.
In von Neumann words, mathematics is "the relation of
relationships." Today it is possible to go further, for a
that branch of mathematics called Category theory is not
concerned with any particular field of mathematics but with
the relationships between the different fields themselves!
Mathematics at this level has the appearance of the purest
and most rarefied thought. It is like a piece of music of
such abstract perfection that the realization of a single
performance would destroy its purity.
But it is exactly at this point that a staggering paradox
hits us in the teeth. For abstract mathematics happens to
work. It is useful. It is the tool that physicists employ
in working with the nuts and bolts of the universe! Indeed,
scientists of the old school referred to mathematics as "the
handmaid of physics". But why should an abstract
codification of pure thought, divorced from any reference to
physical objects and material processes, be so useful in the
daily practice of science? To echo Eugene Wigner's famous
remark, mathematics is unreasonably effective.
There are many examples, from the history of science, of a
branch of pure mathematics which, decades after its
invention, suddenly finds a use in physics. There are also
cases of a mathematical approach, developed for one specific
purpose, that is later found to be exactly what is needed
for some totally different area of physics.
Probability theory, first devised to deal with strategies of
gambling, ends up as the exact language needed to give a
molecular foundation to thermodynamics - the physical theory
dealing with work and heat. But why should this be so? When
Einstein formulated his general theory of relativity he
discovered that the necessary mathematics had already been
developed in the previous century. Similarly the mathematics
required for quantum theory was ready and waiting. Group
theory, the cornerstone of much of theoretical physics of
the last fifty years, had its origins in fundamental
mathematics of the 18th and 19th centuries. And, when it
comes to Superstrings, a topic at the frontiers of
contemporary theoretical physics, the mathematical tools of
cohomology and differential geometry are waiting to be used.
On the face of it, this apparently perfect marriage between
abstract mathematics and the study of the physical world is
as improbable as discovering that a piece of modern
sculpture fits exactly as the missing component of some
complex new engine!
How is it possible to account for this unreasonable
effectiveness of mathematics and for the powerful role it
plays in physics today? One approach is to take the hint
offered by Galileo and view mathematics as a language. Just
as natural language is used for everyday thought and
communication, so too, physics has to make use of whatever
mathematical languages happen to be lying around.
Mathematics, in this view, is a tool and, like the hammer or
screwdriver, we select the available tool that best fits the
job.
3. Mathematics as Language
It is common to talk of "the language of mathematics". But
is mathematics really a language? Does it possess the
various properties that are characteristic of other natural
languages? Clearly mathematics does not have the same
fluency as a natural language and, even more obviously, it
is rarely spoken aloud. This suggest that mathematics is
really a more restrictive limited form of language.
Nevertheless, the suggestion is that everything mathematics
can do must ultimately find its origin in language. This
means that the rich and abstract proofs and theorems of
mathematics can ultimately be traced back to thoughts and
arguments that were once voiced in language--albeit in a
long winded and cumbersome way. Now, it is obvious that
mathematics doesn't look anything like natural language.
Mathematics deals with numbers and symbols, it is used to
make calculations and it's form is highly abstract. On the
other hand, all these features may already be enfolded
within natural language. The power of language lies in the
way meaning can be conveyed through form and transformation.
The Ancient Greeks, for example, realized that truth could
be arrived at through various patterns of sentences.
All men are mortal
Socrates is a man
Therefore: Socrates is mortal.
Or, to take another pattern,
Some mathematicians are clever.
All mathematicians are animals.
Therefore: Some animals are clever.
What is striking about these patterns is that the truth of
the conclusion does not depend on the content of the
sentences but on their form. In other words, substitutions
do not affect the validity of the proof:
All [cats] are [wanderers].
[Minou] is a [cat].
Therefore: [Minou] is a [wanderer].
Clearly these patterns and substitutions have something in
common with algebra. Other transformations are also possible
within language.
From:
John shut the door
we get:
The door was shut by John.
These are only a few of the great range of abstract
operations possible within language. Indeed the linguist
Noam Chomsky3 has argued that this ability arises
genetically and is inherent in all human thought. To take
Chomsky's idea even further we could say that mathematics
has isolated and refined several of the abstract elements
that are essential to all human languages. An extreme form
of this argument would be to say that while mathematicians
may make abstract discoveries and develop new mathematical
forms, in the last analysis they are simply representing
something that is inherent in human thought and language.
The normal way we express and communicate our thought is
through language and mathematics becomes an formal extension
of this process. So when physicists seek a rational language
in which to express their insights, they simply take what
happens to be at hand - the best available mathematics. It
is not therefore surprising that mathematics happens to
work.
Mathematics has played a vital role in raising the
speculations an earlier age to the highest peaks of
intellectual enquiry. But I am now putting forward the
hypothesis that physicists have, in fact, no alternative.
Mathematics has been forced on them as the only language of
communication which can also serve to make, with precision
and economy, quantitative predictions and comparisons. And,
when no Isaac Newton happens to be around to develop a new
mathematical language hand in hand with new physical
insights, then physics has to make do with what is
available.
In those cases in which the form of the mathematical
language makes a perfect marriage with to the content of the
physical ideas, then the communication and development of
physics is highly successful. But this may not always be the
case. Sometimes it may turn out that a particular
mathematical language is forced, by physics, to say things
in cumbersome ways. The mathematics actually gets in the way
of further creativity. At the other extreme, it is the very
ease of expression that drives a theory in a particular
direction so that mathematics actually directs the evolution
of physics, even when new physical insights are lacking. In
other words, I want to question Wigner's claim that
mathematics is unreasonably effective. For it could be that
the whole thing is an illusion brought about because physics
has no other language in which to communicate quantitative
statements about the world. In the past decades there has
been much talk about paradigm shifts and scientific
revolutions - yet it is still possible to retain the same
mathematical language after such a radical shift. In short,
the whole baggage of unexamined presuppositions that are
inherent in the mathematics are carried over to the new
physics.
Any writer knows that language has the power to take over
his or her ideas. Words have their own magic, and a style,
once adopted, will gather its own momentum. It has been said
that a writer is possessed by all the texts that have been
previously written. As soon as we put pen to paper and chose
a particular literary form then what we write is, to some
extent predetermined. I would suggest that the same is true
of physics. That the adoption of a particular mathematical
language will subtly direct the development of new ideas.
Moreover there are times when mathematics may actually block
the operation of a free, creative imagination in physics.
Since mathematics occupies such a prominent place in physics
today, these are vital questions to be explored.
In arguing that mathematical languages direct and influence
our thought in science, we now see that the real danger
arises from always focusing on the physical ideas and not
giving attention to the language in which they are
expressed! As long as physicists view mathematics simply as
a tool then it is possible to ignore the subtle but very
powerful influence it has over the way they think and how
they express their thoughts. In fact, I believe that a good
argument can be made that a particular form of mathematics
has been blocking progress in physics for decades- this is
the Cartesian co-ordinate system, a mathematical form that
has survived several scientific revolutions!
A major problem facing modern physics is that of unifying
quantum theory with relativity. One theory deals with
discrete, quantized processes below the level of the atom.
The other with the properties of a continuous space-time.
While it is certainly true that deep physical issues must be
resolved before significant progress can be made, I would
also argue that the mathematical language in which the
quantum theory is expressed is at odds with what the theory
is actually saying. While quantum mechanics and quantum
field theory are a truly revolutionary approaches, the
mathematics they are based on goes right back to Descartes--
to the same Cartesian co-ordinates we all learned at school.
For three hundred years physics has employed the language of
co-ordinates to discuss the movement of objects in space and
time. Later developments like the calculus also rely upon
this idea that space can be represented by a grid of co-
ordinates. But it is this same mathematical language that is
at odds with the revolutionary insights of quantum theory.
Cartesian co-ordinates imply continuity, and the notion of
space as a backdrop against which objects move. So whatever
new insights physics may have in this area, they are still
being expressed in an inappropriate language. This, I
believe, represents a major block to thinking about space
and quantum processes in radically new ways.
The example of how the Cartesian grid has dominated physics
is rather obvious. But there may be many other, and more
subtle, ways in which particular mathematical forms are
currently directing science and limiting the possibilities
for its development.
4. Mathematics Beyond Language
But is it really true that mathematics is nothing more than
a limited and abstract version of natural language? I would
argue that mathematics is both more, and less, than a
language. Since it involves highly codified forms,
mathematics makes it easy to carry out calculations, to
demonstrate proofs and to arrive at true assertions. But, in
my opinion, this is only a surface difference, a feature of
the convenience and economy of mathematics over ordinary
language. A more significant way in which mathematics goes
beyond language is that it involves a particular kind of
visual and sensory motor thinking that does not seem to be
characteristic of ordinary language. Some parts of
mathematics deal with the properties and relationships of
shapes. While these properties can be generalized to many
dimensions and to highly abstract relationships,
nevertheless, mathematicians have told me that their
thinking in these particular fields enters regions which do
not involve language in any way. It calls upon a sort of
direct, internal visualization and may even involve an
internal sense of movement and of tiny muscular reactions.
This "non-verbal" thinking may also take place in other
fields of mathematics and appears to involve a form of
mental activity that goes beyond anything in the domain of a
spoken or written language. It could be that, at such times,
mathematical thought has direct access to a form of thinking
that is deeper and more primitive than anything available in
any natural language. This pre-linguistic mental activity
may be the common source from which both mathematics and
ordinary language emerge.
On the other hand, mathematics is also less than a language,
in that it lacks the richness, the ability to deal with
nuance, the inherent ambiguity and the rich strategies for
dealing with this ambiguity. In this sense, mathematics is a
limited, technical language in which much that is of deep
human value cannot be expressed.
5. Mathematics and Music
It is possible to explore the nature of mathematics, and its
relationship to physics, in another direction. By comparing
it to music. Mathematics is an abstract system of ordered
and structured thought, existing for its own sake. It is
possible to apply a similar description to music. Indeed the
20th century composer, Edgar Varese, has written that "music
is the corporealization of thought". Listening to Bach, for
example, is to experience directly the ordered unfolding of
a great mind. This suggests that music and mathematics could
be related in some essential way. On the other hand who
would employ music to express a new theory of the universe?
( But could this simply be a prejudice that is
characteristic of our earth-bound consciousness? Do beings
in some remote corner of the universe explore the nature of
the universe in music and art?)
Music and mathematics are similar, yet different. Indeed, I
believe that both the strengths and the weakness of
mathematics lie in this difference. Mathematics has
developed to deal with proof and logical truth in a precise
and economical way. Mathematics also makes a direct
correspondence with the physical world through number,
calculations and quantitative predictions.
While it could be said that music is "true" in some poetical
sense and that the development of a fugue has a logical
ordering that is similar to that of a mathematical proof, on
the other hand these are not the primary goals of music.
Music deals with the orders of rational thought, yet it is
also concerned with the exploration of tension and
resolution, with anticipation, with the control of complex
sensations of sound and with the evolution and contrast of
orders emotion and feeling. To borrow a Jungian term, music
could be said to be more complete, for it seeks a harmony
between the four basic human functions; thought balanced by
feeling and intuition by sensation. While mathematicians may
experience deep emotions when working on a fundamental piece
of mathematics, unlike composers, their study, per se, is
not really concerned with the rational ordering of these
emotions or with the relationships between them. The
greatest music, however, moves us in a deep way and leaves
us feeling whole. It engages thought and emotion, it
expresses itself through the physical sensation of sound.
In this sense it could be said that physics, with its
reliance on the language of mathematics, must always present
an incomplete picture of the universe. Its language is
impoverished, for it lacks this basic integration of the
four human functions. It can never fully express the
essential fact of our confrontation with, participation in,
and understanding of nature.
But is it possible, in wonder, that, in the distant future,
science, inspired by the example of music, may develop a
more integrated and versatile language, one which would have
room, perhaps, for the order of emotion and direct sensation
while, at the same time, retaining all the power of a more
conventional mathematics?.
There is yet another significant way in which "the language
of music", and of the other art, differs from mathematics.
While all these languages are concerned with relationships
and rational orders of thought, the arts are able to unfold
these orders in a more dynamical way by exploring the way
order is generated in the act of perception itself. Quantum
theory is also concerned with the indivisible link between
the observer and the observed. And this suggests that it
would be to the advantage of physics to develop a similar
flexibility in its basic language giving it the ability to
explore the rich orders that lie between the observer and
the observed.
Let me explain what I mean. A great work of art possesses a
rich internal order. In music, for example, a theme may be
transposed, inverted, played backwards and otherwise
transformed in a variety of ways which still retain a
certain element of its order. Of course, this is only one
simple example of the sorts of order explored in a musical
composition, indeed the order of great music is so rich as
to defy complete analysis. Likewise, a painting contains
complex relationships between its lines, masses, areas,
colors, movements and so on. In some cases such objective
orders may have much in common with the sorts of order that
are found in mathematics. But what makes any work of art
come alive is its contemplation by the human observer:-
Music played in a vacuum is not music, art that is never
seen is not art. For the work of art arises in that dynamic
interaction between the active perception, intelligence,
knowledge and feeling of the viewer and the work itself.
To take a particular example, some of the drawings of an
artist like Rembrant, Picasso or Mattise or a Japanese
master appear, on the surface, to be extraordinarily simple.
Few marks appear on the paper when contrasted with, for
example, the detailed rendering done by an art student. A
trivial analysis would suggest that the sketch contains
"less information" than the detailed rendering and that its
order is relatively impoverished. Yet the confrontation of a
viewer with a Mattise drawing is a far richer experience in
which complex orders of thought and perception are evoked.
To make the slightest change in position, direction, gesture
or even thickness of a single line can destroy the balance
and value of a great drawing, but may have only a negligible
effect on a student work. In this sense great art has an
order of such richness, subtlety and complexity that it is
beyond anything that can be addressed in current
mathematics. Yet it is something to which the trained viewer
can immediately respond.
Indeed, the rich order of the drawing lies not so much in
some objective order of the surface marks on the paper, but
in the whole act of perception itself and in the way in
which the drawing generates a hierarchy of orders within the
mind. Lines evoke anticipations in the mind that may be
fulfilled in harmonious or in unexpected ways. The mind is
constantly filling in, completing, creating endless complex
orders. A single line may suggest the boundary of a shadow,
the outline of a back or it may complete a rhythm created by
other lines. Indeed the act of viewing a drawing could be
said to evoke an echo, or resonance, of the whole generative
process by which the drawing itself was originally made. The
essence of the drawing does not therefore lie in a static,
objective order--the sort of thing that can be the subject
of a crude computer analysis involving the position and
direction of a number of lines. Rather, it is a rich
dynamical order, an order of generation within the mind.
Through his or her art, the creator of the drawing has
called upon the nature of the subject, the history of art,
and on all the strategies that are employed in perception.
So standing before a drawing involve a deep and complex
interplay between the work itself, the visual center of the
brain, memory, experience, knowledge of other paintings,
and of the human form. The eyes, memory, mind and even the
body's sensory-motor system become involved in the
generation of a highly complex order, an order in which
every nuance of the drawing has its part.
The order within an economical drawing may, therefore, be
far richer than we first suspect. For its power lies not so
much in some surface pattern of the lines but in the
controlled and predetermined way in which these lines
generate, through the act of perception itself, infinite
orders within the mind and body. While attention has
certainly been given, by researchers in Artificial
Intelligence, to what is called the early processes of
vision, it is clear that the sort of order I am talking
about lies far beyond anything that mathematics or
artificial intelligence could analyze or even attempt to
deal with at present.
I feel that the description of complex orders of perception
and generation is a rich and powerful area into which
mathematics should expand. It may also have an important
role to play in physics. Quantum theory, for example, is
concerned with the indissoluble link between observer and
observed and it would be interesting to make use of a
mathematics which can express the infinite orders that are
inherent in this notion of wholeness.
A similar sort of argument applies to music. Some
musicologists have gone so far as to analyze music by
computer, and to calculate its "information content ",
concluding, for example, that "modern music" contains more
information than baroque music! But the essence of music
does not lie in some measure of its objective information
content but in the rich and subtle activity it evokes within
the mind. Music and art are seeds that, in a controlled and
deliberate way, generate a flowering of order and meaning
within the mind and body of the listener.
To return to an earlier point; this generative order
suggests a reason why great music could indeed act as a
metaphor for a theory of the universe. Music is concerned
with the creation and ordering of a cosmos of thought,
feeling, intuition and sensation and with the infinite
dynamical orders that are present within this cosmos. In
this sense, music could be said to echo the generation and
evolution of a universe. Clearly our present mathematics
lacks this essential dimension. But could, in fact,
mathematics move in such a direction? A new mathematics
would not simply offer a crystallization of thought but also
explore the actual generative activity of the orders of this
thought within the body and mind. Such a new formal language
would represent a deep marriage between mathematics and the
arts. It would involve a mathematics that requires the
existence of another mind to complete it, in an ordered and
controlled way, and, in so doing, this mathematics would
becomes the germ of some, much deeper order.
6. Mathematics and the Brain
Let us return again to the question of the unreasonable
effectiveness of mathematics. As we have seen, one answer is
to consider mathematics as a language, indeed the only
available language that can deal, in an economical and
precise way, with quantitative deductions about the world.
Mathematics, in this sense, is a restricted form of natural
language. But, in other ways, it goes beyond language.
Physics, however, is always in the position of being forced
to use mathematics to communicate at the formal level. The
question, therefore, is not so much one of the unreasonable
effectiveness of mathematics, but of physicists having no
real alternatives.
But there may be other ways of looking at this question. One
way is to suggest that mathematics, in its orders and
relationships, is a reflection of the internal structure and
processes of the brain. In moving towards the foundations of
mathematics one would therefore be approaching some sort of
direct expression of the controlling activities of the brain
itself. And, since the brain is a physical organ that has
evolved through its interactions with the material world, it
is inevitable that the brain's underlying processes should
model that world in a relatively successful way. Human
consciousness has developed, in part, as an expression of
our particular size and scale within the environment of our
planet. It is a function of the particular ranges of senses
our bodies employ, and of our need to anticipate, plan
ahead, hold onto the image of a goal and remember. Moreover
consciousness has created, and been formed by, society and
the need to communicate. It has brought us to the point
where we can ask, for example, if we think because we have
language or, if we have language because we think? Or if the
answer could lie somewhere in between.
According to this general argument, the brain's function is
a direct consequence of, and a reflection of, our particular
status as physical and social beings on this planet.
Mathematics, moreover, is a symbolic expression of certain
of the ordered operations of this brain. It should come as
no surprise, therefore, that mathematics should serve as a
suitable language in which to express the theoretical models
that have been created by this same brain.
This whole question of the formal strategies employed by the
brain is the province of cognitive psychology. One of the
pioneers in that field was Jean Piaget5. Piaget's particular
approach was to suggest that the basis of our thought and
action could be traced to the logic of the various physical
transactions we had with the world during our first weeks,
months and years. Piaget believed that these same logical
operations are also present in mathematics and, in this
respect, he had a very interesting point to make. It is well
known, he pointed out, that mathematics can be arranged in a
hierarchical structure of greater and greater depth. In the
case of geometry, for example, the top, and most
superficial, level is occupied by those semi-empirical rules
for surveying and calculating shapes that were known to the
Egyptians and Babylonians. Below that could be placed the
more fundamental, axiomatic methods of the ancient Greeks.
The history of geometry demonstrates the discovery of
deeper and more general levels, Euclidian geometry gives way
to non-Euclidian, beneath geometry is topology, and topology
itself is founded on even more general and beautiful
mathematics. The longer a particular topic has been studied,
the deeper mathematicians are able to move towards its
foundations.
But Piaget, pointed out, this historical evolution is a
direct reversal of the actual development of concepts of
space in the infant. To the young child, the distinction
between intersecting and non-intersecting figures is more
immediate than between, say, a triangle, square and circle.
To the infant's developing mind, topology comes before
geometry. In general, deeper and more fundamental logical
operations are developed earlier than more specific rules
and applications. The history of mathematics, which is
generally taken as a process of moving towards deeper and
more general levels of thought, could also be thought of as
a process of excavation which attempts to uncover the
earliest operations of thought in infancy. According to this
argument, the very first operations exist at a pre-conscious
level so that the more fundamental a logical operation
happens to be, the earlier it was developed by the infant
and the deeper it has become buried in the mind. Again, this
suggests a reason why mathematics is so unreasonably
effective, for the deeper it goes the more it becomes a
formal expression of the ways in which with interact with,
and learn about, the world.
But, it could be objected, if the history of mathematics
and, to some extent, of theoretical physics, is simply that
of uncovering, and formalizing, what we already know then
how is it possible to create new ideas, like Einstein's
relativity, that totally lie outside our experience? The
point is, however, that this equality or interdependence of
space and time was already present in all the world's
language. Rather than coming to the revelation that time and
space must be unified then have never really been
linguistically separated! According to this general idea,
what may appear to be novel in physics and mathematics is
essentially the explicit unfolding of something that is
already implicit within the structuring of human thought--of
course physics itself also makes use of empirical
observations and predictions. For this reason, the
intelligent use of mathematics as a language for physics
will necessarily make sense.
Piaget's notion, that the evolution of mathematics and
physics is forever reaching towards the deepest structures
of the mind, is certainly interesting. However, I feel that
there is a certain limitation in the approach of cognitive
psychology, with its emphasis upon strategies and programs
of the brain, on successions of logical steps and on
algorithms of thought. There is not sufficient space in this
article to develop any detailed arguments, but I believe
that, while cognitive psychology may produce some valuable
insights, in its present form it does not capture the true
nature of human intelligence in general, and mathematics in
particular. Formal logic is an impoverished way of
describing human thought and the practice of mathematics
goes far beyond a set of algorithmic rules. The
mathematician Roger Penrose7 has, for example, produced
compelling arguments why machine intelligence must be
limited--a Turing machine, or indeed any other algorithmic
device, will never be able to carry out all the sorts of
things that a human mathematician can do. Mathematics may
indeed reflect the operations of the brain, but both brain
and mind are far richer in their nature than is suggested by
any structure of algorithms and logical operations.
7. Mathematics and Archetypes
In this final section I am going to become more speculative
and explore yet another approach to the question of the
unreasonable effectiveness of mathematics. I want to suggest
that mind and matter, brain and consciousness are two sides
of a single process, something that emerges out of a deeper
and hitherto unexplored ground. In this sense the order of
generation that gives rise to the universe has a common
source with the generative order of consciousness. In its
deepest operation, therefore, our intelligence could be said
to mirror the world. But what can one say about the nature
of this source? According to the classical Chinese
philosopher, Lau Tzu, "the Tao which has a name in not the
Tao", which seems to say it all.
Of course, the idea of an unknown, unconditioned source
which is the origin of matter and consciousness may seem far
fetched to many readers. But it is, after all, simply
another way of accounting for the unreasonable effectiveness
of mathematics. Our own age is out of sympathy with such
sweeping assertions as "God is a mathematician", but suppose
one suggests that mind and the universe have an common order
and that the source of material and mental existence lies in
a sort of unconditioned creativity, and in the generation of
orders of infinite subtlety and complexity8? While the
nature of such an order may never be explicitly known in its
entirely, it may still be possible to unfold certain of its
aspects through music, art and mathematics. The great
aesthetic joy of mathematics is not, therefore, far from
the joy of music or any great art, for it arises in that
sense of contact with something much greater than ourselves,
with the heart of the universe itself. Mathematics is
effective when it becomes a hymn to this underlying order of
consciousness and the universe, and when it expresses
something of the truth inherent in nature.
This idea has been expressed in other ways. Carl Jung, for
example, spoke of the archetypes. This is a difficult
concept to convey in a short definition but, very roughly,
the archetypes could be taken as those dynamical orders,
unknowable in themselves, that underlie the structure of the
collective unconscious. The archetypes are never seen
directly but their power can be experienced in certain
universal symbols. In his more speculative moments, Jung
also hinted at something that lay beyond matter and mind,
but included both. This psychoid, as he called it, is
related to the archetypes and suggests that the same
underlying ordering principles give birth and structure to
both matter and mind. Just as human consciousness arises out
of the collective unconscious, so too the universe itself
arises out of something more primitive. Again we meet this
notion that the same underlying order gives rise to both
matter and mind.
Of particular interest is the importance that Jung placed
upon numbers. Numbers, according to Jung, are direct
manifestations of the archetypes and must therefore be
echoes of the basic structuring processes of the universe
itself. It is certainly true that numbers are mysterious
things. To return, for a moment, to the connection between
mathematics and language. When it comes to language, it is a
basic axiom of linguistics that "the sign is arbitrary". In
other words, the meaning of a world does not lie in how it
sounds or the way it is written but in the way it is used.
If you want to know the meaning, the philosopher
Wittgenstein said, look for the use. By contrast, the basic
units of mathematics, the numbers, are totally different,
they are not arbitrary but have a meaning and existence of
their own. While the names given to the numbers may be
arbitrary, the numbers themselves are not, 0, 1, 2, 3, are
not symbols whose meaning changes with time and use but are
the givens of mathematics. In a sense they are almost
platonic. It has been said, for example, that God made the
numbers and the rest of mathematics is the creation of human
intelligence. It is these same numbers that, Jung claims,
are manifestations of the archetypes. Indeed Jung's argument
does have a ring of truth about it for numbers are certainly
curious things and the unfolding of their properties remains
one of the most basic forms of mathematics. Could it be
true, as the Jungians suggest, that the numbers are
expressions of the archetypes or orders that underlie the
universe and human consciousness?6
Curiously enough, this idea may have found favour with one
famous mathematician. One of the most brilliant pure
mathematicians in this century, S. Ramanujan, gave little
value to mathematical proof but appeared to arrive at his
remarkable theorems in number theory by pure intuition
alone. Ramanujan himself, however, believed that these
profound results were given to him by a female deity. In
Jung's terminology, this deity would also be a manifestation
of the archetypes.
So, to Ramanujan, the whole order of mathematics, with its
underlying truth and beauty, essentially lies in a domain
beyond logical truth and rational argument. It is something
which can, at times, be touched directly by the
mathematician's intuition and in a way that appears almost
sacred. As to the nature of this domain, we can call it the
archetypes, psychoid, ground of being or unconditioned,
creative source. But what does it matter? What counts is
that a remarkable mathematician bypassed rational argument
and the need for vigorous proof and picked out outstanding
theorems out of the air. And what is equally staggering is
that, in all likelihood, these symphonies of pure thought
may one day have totally practical applications in the real
word.
8. Conclusion
The unreasonable effectiveness of mathematics remains an
open question, although I have given some suggestions as to
why it appears to work. I have also argued that mathematics
may not always be as effective as we suppose, for physical
ideas are sometimes forced to fit a particular mathematical
language, in other cases the very facility of the language
itself may drive physics forward, irrespective of any new
physical ideas!
I have also suggested ways in which improvements in the
formal language of physics could be advanced. A major area
would be to discover a mathematics of complex and subtle
orders, a formal way of describing what seems, to me, to be
an essential feature of the universe. There have recently
been several attempts to describe complex orders--
Mandelbrot's fractal theory is capable to describing and
generating figures of infinite complexity; David Bohm's
notion of the implicate order is a powerful concept but has
yet to find an appropriate mathematical expression.9
Finally, I have also argued that there are times when the
mathematical language of physics fails to capture the
essential fact of our being in the universe. And here I must
reveal another prejudice. Physics, to me, has always been
concerned with understanding the nature of the universe we
live in; a way of celebrating and coming to terms with our
existence in the material world, rather than a matter of
discovering new technologies and accumulating more
knowledge. In is in this light that I have criticized the
role of mathematics in physics and have hinted at the way
new language forms could be developed. Of course I
acknowledge the great service that mathematics has done for
physics, how it has lifted it from speculation to precision,
and, of course, I recognize the great power and beauty of
mathematics that is practiced for its own sake. But here, at
the end of the 20th century we must not rest on our laurels,
the whole aim of our enterprise is to penetrate ever deeper,
to move towards a more fundamental understanding and a more
complete celebration of the universe itself. In this
undertaking in which prediction, calculation and control
over the physical world also have a place but they do not
become the whole goal of the scientific enterprise. It is
for this reason that I am urging physicists to play closer
attention to the mathematical language they use every day.
This whole concern with discovering and portraying the
complex orders of nature, was also a preoccupation of the
writer Virginia Woolf. Virginia Woolf was concerned with the
order of the moment, with crystalizing, in language, the
complex sensations, experiences and memories that make up
each instant in a persons life. She recognized that, in the
last analysis, the success of this enterprise depends on
creating a fitting means of expression, on language and on
words. Her own observations on this process convey precisely
what I have been attempting to say in this essay
"Life is not a series of gig-lamps symmetrically arranged;
but a luminous halo, a semi-transparent envelope surrounding
us from the beginning of consciousness to the end. Is it not
the task to the novelist to convey this varying, this
unknown and uncircumscribed spirit, whatever aberration or
complexity it may display, with as little mixture of the
alien as possible?"
For James Joyce it is the epiphanies or transcendent moments
of life that have a special richness. They can occur at any
instant and it is the business of language to capture these
, even "transmuting the daily bread of experience into the
radiant body of evolving life". For Virginia Woolf this
radiant force of the moment must be captured by language "it
is or will become a revelation of some order; is a token of
some real thing behind appearances; and I make it real by
putting it into words.
9. References
1. See, for example, J. Jeans, The Mysterious Universe,
Cambridge University Press, 1930.
2. G.H.Hardy, A Mathematician's Apology, Cambridge
University Press, 1967.
3. N. Chomsky, Syntatic Structures, Mouton Pubs, The Hague,
1957.
4. For a preliminary discussion on the role of language in
science see, A.J.Ford and F. D. Peat, Foundations of Physics
18, 1233-1242, 1988.
5. See, for example, J. Piaget, Structuralism, Harper
Torchbooks, Harper & Row, N.Y. 1971.
6. M-L von Franz, Number and Time, Northwestern Universty
Press, Evanston, 1974.
7. R. Penrose, " Question Physics and conscious thought" in
Quantum Implications: Essays in Honour of David Bohm, eds.
B.J.Hiley and F. David Peat., Routledge and KeganPaul,
London and New York, 1987.
See also Penrose in Mindwaves ed Colin Blakemore,
Blackwells, Oxford, 1988erence to be added in proof).
See also R. Penrose, The Emperor's New Cloths, Oxford
University Press, Oxford 1989.
8. See, for example, F. David Peat, Synchronicity: the
Bridge Between Matter and Mind, Bantam, N.Y., 1987.
9. A discussion of complex orders is given in, D. Bohm and
F.D. Peat, Science, Order and Creativity, Bantam, N.Y.,
1987.
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