**
From the quest for certainty to
the quest for wholeness**

*John Heron,
unpublished paper, 1962*

**
Introduction **

Around about
1927 intellectual events of the greatest interest occurred. In
physics Heisenberg framed his uncertainty principle; in
mathematics a paper by von Neumann foreshadowed results later
made fully explicit by Gödel in his celebrated theorem of 1931.
Physics is one of the positive sciences of nature that has
achieved the highest degree of systematization: it has a great
range of coherent, established and reliable empirical knowledge.
Mathematics is the formal science par excellence with the most
complete range of coherent rational knowledge so far established
by any activity of the human intellect; while it is closely
employed in determining and expressing much of the data of
physics. In mathematics and physics the intellect had, up to
1927, exercised itself in two fields that seemed to contain
within them the promise of an absolute, self-consistent
certainty of intellectual knowledge, to be attained by the high
development of exclusively rational powers of analysis and
synthesis, exercised in relation to number theory and precise
and controlled observation. Certain special problems had of
course begun to emerge since the beginning of the century in
physics with Planck's inauguration of quantum theory, and in
mathematics with the problem of consistency raised by
Burali-Forti's and Russell’s paradoxes. But only in the
uncertainty principle of 1927 and Gödel’s theorem of 1931 was
the demise of the promise of a purely rational certainty made
fully explicit in physics and mathematics respectively.

**Physics:
the goal of total determinism **

The goal of an
absolute knowledge for physics was first expressed in its
crudest and most uncompromising form by Laplace, in the late
18th century, as the doctrine of total determinism. This was an
idea of causal determinism such that "an intelligence which at a
given moment knew all the forces that animate nature, and the
respective positions of the beings that compose it, and further
possessing the scope to analyze these data could condense into a
single formula the movement of the greatest bodies of the
universe and of the least atom; for such an intelligence nothing
could be uncertain, and past and future alike would be before
its eyes" (Laplace, *Essay on the Calculus of Probabilities*).
Here the notion of causality - that there is some law behind
everything - becomes the extreme doctrine of an immanent
mathematical law such that if one could state for a given
instant of time the mathematical relation of functional
dependence between the variables (such as mass, position,
velocity) that affect all particles in the universe at that
time, then from such a formula all past events could be derived
and all future events be predicted.

This great
deterministic myth, or ideal of measurement, of Laplace, which
lasted 150 years, was based on his great success in applying the
principles of Newtonian mechanics to the motion of the solar
system. The idea of an immanent causal determination of this
kind was not present in Newton's mind: there were some
irregularities in the motions of parts of the solar system that
he could not explain, in particular certain anomalies in the
motions of Jupiter and Saturn. Mathematics and astronomers of
his time could still ask whether the solar system was stable;
and Newton himself considered that divine intervention might be
necessary from time to time to correct minor deviations and
irregularities, thus putting the solar system back in order and
securing it from destruction by the collision or scattering of
its members.

Laplace,
however, on the sole basis of Newton's three laws of motion and
the inverse square law, was able to account for all the
irregularities that had troubled Newton, and to show that
despite the fluctuations of the planets' motions the solar
system will retain its stability and structure (the total
planetary invariability of secular inequalities was established
as a general law by Lagrange and Laplace, 1773-84). Thus any
need for divine intervention appeared to be unnecessary, and the
universe appeared in the guise of a self-sufficient mechanism.
The old notion of transcendent law, the imposed command of
deity, still in some measure present in Newton's mind, was
banished in favour of immanent mathematical law. From the fact
that in the solar system if we know at a given moment the
positions and velocities of all the planets in relation to the
sun, we can calculate their motions and positions at an earlier
or a later date, came the idea of causality in its modern,
pre-1927 form, which is that the laws of physics determine the
future of a physical system completely, given all the relevant
information at a point of time (thus "determine" means renders
calculable on the basis of mathematically stated laws).

Laplace
grandiosely suggested that such an idea might be applied to
everything in the universe, to the universe as a whole.
Lamettrie carried the idea through and applied it to the concept
of man as a machine: all atoms whether in an organic or
inorganic setting obey mechanical laws. This broad mathematical
mythology became the common background to much scientific
thought: Ernest Mach stated that the great majority of
scientists at the end of the nineteenth century tended to think
in this deterministic manner. Generally, we may say that the
mythical goal for physics was that of absolute certainty of
prediction in terms of mathematically expressed laws. The
intellect alone could probe, measure, analyze and compute, and
attain absolute powers of quantitative description.

**
Mathematics: the goal of an absolute and finitistic proof of
consistency **

In mathematics
the goal was formally analogous to that in physics. In physics,
ideal rational certainty was to be found in total prediction
arising from a thorough understanding of the quantitative
interrelations between ultimate particles. In mathematics, it
was to be found in a demonstration of consistency arising out of
a thorough understanding of the formal interrelations between
the ultimate elements of a deductive system from which
arithmetic in its entirety could be derived. This goal for
rational certainty in mathematics was first explicitly
formulated by the German mathematician David Hilbert, from the
turn of the century onwards, as the programme of finding an
"absolute" and "finitistic" proof of the consistency of
arithmetic. To show why this programme had become necessary we
must again make a brief historical review.

For centuries
geometry had represented the ideal of certain and consistent
knowledge. The Greeks had developed it systematically by means
of the axiomatic method, which consists of accepting without
proof certain axioms or postulates (e.g. through two points just
one straight line can be drawn) and then deriving from them all
other propositions of the system as theorems. The axioms of
Euclidean geometry were considered to be self-evidently true of
physical space or objects in space; and since true statements
could not be logically incompatible, the whole Euclidean system
was considered to have a built-in and guaranteed consistency.
But the discovery of non-Euclidean geometries after 1825 marked
a certain turning point in the history of mathematics. These had
different axioms to those of Euclid.

This led to
two developments: firstly, the gradually increasing tendency to
make explicit the axioms of other branches of mathematics - up
to the later nineteenth century only geometry had a fully
explicit axiomatic basis; secondly, and relatedly, a growing
tendency to inquire about the logical consistency of the axiom
systems so developed. This second issue arose partly out of the
nature of the non-Euclidean axiom systems: the axioms were not
in certain cases obviously true of physical space. How, then,
did one determine their consistency, that is, ensure that they
might not at some point produce mutually contradictory theorems?
The problem of consistency for mathematics in general was
heightened by the discovery of a paradox by Burali-Forti (1897)
which indicated that Cantor's arithmetic of cardinal and ordinal
numbers was not consistent, that is, two contradictory theorems
could be derived from it. And the problem of consistency for
cardinal arithmetic was again raised by a paradox concerning the
most general class (Russell, 1901). Meanwhile, even the supposed
self-evidence of the axioms of Euclid was considered to rest on
an inductive argument whose credentials were dubious.

Thus in the
nineteenth century, though mathematics took great strides
forward, expanded in many new directions, became increasingly
emancipated in its methods, explicit and rigorous in its
techniques, yet at the same time serious doubts were raised as
to the guarantees of its absolute consistency. Did the various
mathematical systems harbour hidden internal contradictions? Was
mathematics, in fact, an absolutely certain science? And if so
how could this be demonstrated?

The first
approach to consistency was the relativistic one. Euclidean and
non-Euclidean geometries were shown by Hilbert (1899, 1903) to
be consistent by recourse to arithmetic, that is, by recasting
them in arithmetical terms; in other words, their consistency
was shown to depend on or be relative to the consistency of
arithmetic. So that Hilbert's final programme, in the 1920's,
was to get a consistency demonstration of simple arithmetic
which would be "absolute" - that is, would not be relative to
the consistency of some other system, and "finitistic" - that
is, would involve analysis of only a finite number of structural
features of a fully explicit arithmetical calculus.

In the early
years of this century, then, the rational ideal of causal
determinism lay behind the investigation of ultimate material
particles and the rational ideal of an absolute proof of
consistency lay behind the investigation of the axiomatic
elements of arithmetic, in physics and mathematics respectively.
But the intellect, through the great sharpening of its power and
precision, unexpectedly achieved, not a demonstration of its
ideals, but a demonstration of the impossibility of attaining
them. To these developments and their broad philosophical
implications we must now turn.

**Physics:
the uncertainty principle and the breakdown of determinism **

Heisenberg's
uncertainty principle, arising out of the concepts of quantum
theory and wave mechanics, states that it is impossible to
determine exactly both the position of an object and its
momentum, or any quantity related to its momentum such as
velocity or energy. Since the expected error is of the order of
10^{-27} in the C.C.S. system, the effect is negligible
for bodies of normal mass, but it is pronounced with electrons,
negatively charged atomic particles. For such a particle there
is and can be no physical law in which reference is made to its
exact position and momentum. The product of uncertainty as to
position and uncertainty as to momentum is equal to Planck's
constant (6.625 x 10^{-27} erg sec): the two
uncertainties cannot dwindle to nothing and the ideal of
absolute precision of measurement is unattainable. As one
approaches zero, the other approaches infinity; so that precise
information about the one implies total ignorance about the
other. There is between them a relation of indeterminacy. The
uncertainty relation as defined in terms of Planck's constant
sets a limit to the accuracy of measurement. The limitation is
inherent in the mathematics of the situation and is reflected in
experimental observations: to locate an electron means using
short wave-length radiation whose quanta of energy will change
the electron's momentum in the act of locating it, while any
determination of its momentum means a disturbance of its
location.

The
uncertainty principle, then, states that the causal laws of
physics cannot be applied to atomic events taking place below
the limit set by the quantum constant, since atomic events below
this limit are not observable, in principle and in fact. Does
this mean that below this limit there are no causal laws? This
is the Copenhagen interpretation (Bohr and Heisenberg) of the
uncertainty principle. And it is supported by a theorem of von
Neumann to the effect that any modification of quantum mechanics
in its present shape so as to render it deterministic would make
the theory self-contradictory. Yet the theory has so far proved
potent in rendering coherent a wide range of phenomena. This
interpretation of the principle takes the view that beyond an
explicitly stated limit of measurement, randomness and chance
prevail at the sub-atomic level.

Clearly this
has a bearing on the Laplacian programme for total determinism.
The intellectual quest for certainty in physics was guided for
150 years by Laplace's theory that, in principle, the notion of
chance could be totally ruled out of physical happenings; that,
in principle, all events no matter how minute the scale could be
reduced to mathematical laws of predictable regularity. But the
uncertainty principle involves a renunciation of this ideal of a
total causal description of ultimate mechanical processes. It is
now seen to be impossible, in the nature of the case, to know
the state of an atom with such accuracy that its subsequent fate
would be completely predictable. Quantum mechanics are
essentially different from Newtonian mechanics, they deal with
probabilities not with precise prediction.

It is
important to examine in more detail what is implicit in the old
ideal of thoroughgoing causal determinism. For there seems to be
latent within it an unacknowledged element of primitive animism,
linked with the acknowledged mathematical notion of causal laws,
introduced by Galileo, as functional relations between
measurable variables. This is the concept of immanent law
explicitly inaugurated by Laplace to exclude any need for the
invocation of transcendent law imposed by the fiat and command
of deity (to Napoleon's comment that his book about the universe
made no mention of its Creator, Laplace replied: "Je n'avais pas
besoin de cette hypothèse-là”). But it is secretly welded to a
vague animism, and its success in routing the need for imposed
laws rests on the assumption, to put it crudely, that there is a
kind of animistic mathematical formula lodged in the
submicroscopic heart of matter whence all its subsequent
mechanical motions are determined. The alternative to law
imposed from above becomes the idea of precise causal law built
in to the ultimate particles of matter and driving the whole
universal mechanism. It is this reaction against the
authoritarian mediaeval concept of divinely imposed fiats that
has by its implicit assumptions guided physicists into the realm
of ultimate particles. But having arrived, the uncertainty
principle shows, on one of its interpretations, that no causal
laws are operative where the assumptions which guided the search
required that they should be found. There is no totally
deterministic causal principle of mechanical action that can, as
it were, be dug out of the heart of matter. Events below the
limit of the quantum constant fall outside the scope of causal
laws; and the laws of quantum physics themselves are laws of
probability, are statistical in nature. There is a discontinuity
between the principles of celestial mechanics and the principles
of quantum mechanics. The causal concept of Laplace collapses at
the spot where it should be enshrined.

**Statistical
average vs. form **

But if there
is a play of randomness and chance among sub-atomic particles,
how do we account for the regularity and relatively reliable
causal order evident at the macroscopic level? Is it simply that
out of sub-atomic randomness, macroscopic order arises purely on
the basis of statistical theory? It may be sufficient to say,
when considering only the behaviour of, say, a million
electrons, that the laws of chance combined with the quantum
laws of probability as expressed in Schrödinger’s wave
equations, allow predictions with all the accuracy we normally
require. But it seems highly improbable that such sparse
concepts can account for the whole highly articulate order of
nature. What we must protest against, therefore, is the notion
now being put forward (cf. F. Waismann, *Turning Points in
Physics*, p.140) that what appears at the level of normal
human perception and measurement is simply the result of a
statistical knockabout of elementary particles whose random
irregularities become somehow smoothed out into a regular
picture because of the enormous numbers involved. "Certainly,
what God meant, he did. When he said, Let there be light, there
was light and not a mere imitation or a statistical average.
Thus the statistical notion, though it may explain some facts of
our confused perception, is not applicable to the ultimate,
imposed laws." (Whitehead, *Adventures of Ideas*, p.118.)

What the
uncertainty principle rather suggests is that the old notion of
a mathematical determining principle built into the heart of
matter and running it from the microscopic to the macroscopic
scale, can be replaced by the broad concept of a causal order
organizing matter from *other levels of being*. This causal
order becomes manifest at the macroscopic level but allows of a
free play at the sub-atomic and atomic level. The randomness of
ultimate particles is constrained, as it were, within the limits
set by causal laws that organize macroscopic events according to
certain formal principles of regularity. Causal laws, on this
hypothesis, represent forces working from *outside* matter
*into* it, and their penetration falls short of subatomic
events by an amount set by the quantum constant. Order does not
just emerge out of chaos by pure chance; order is imposed on
chaos in accordance with principles that allow of a chance play
within certain limits.

Further, we
must distinguish between laws of process and laws of structure.
The electron may move about randomly, so to speak, below the
limit set by the quantum constant, yet above that limit its
probable position is expressed by the Schrödinger wave
equations. Now in a sense these equations reflect probabilistic
laws of process, they give a statistical view of processes going
on in the atom. Yet in another sense they are related to quite
clear laws of the structure of the atom, for they reflect the
organization of the atom into specific electron shells where the
number of orbitals per shell is the square of the shell number,
and so on. They suggest, then, that there are archetypal laws of
form and structure that *rule over* or *work into* the
atom and so determine the limits within which the probabilistic
sub-atomic processes occur; while wider laws of macroscopic
structure constrain atomic events within processes that reveal a
determinate causal order. Certainly the current mathematical
account of the atom is a very transcendent conception: it is a
partial differential equation in an abstract multi-dimensional
space.

A certain
minimal randomness also occurs at the atomic and molecular as
well as at the sub-atomic level. The atoms and molecules of any
substance, according to the well-substantiated kinetic theory of
matter, are in a state of haphazard agitation. The Brownian
movement, of a large particle (e.g. a pollen grain) suspended in
a fluid, was observed in 1827. Its final explanation, in terms
of haphazard bombardment by the molecules of the fluid, was
secured by Perrin's experiments about 1910. We assume, then,
that the play of chance at the sub-atomic level is cumulative so
that there is a minute random motion of atoms and molecules.

Entropy
essentially measures the degree of disorder in a physical
system. Now entropy or disorder must increase in random
processes. Why then does not nature as a whole increasingly
resolve itself into total disorder and chaos, move to an entropy
death? This is the essential problem which arises once it has
been established that a deterministic causal order does not
prevail among ultimate material particles. The problem comes
into clearest focus in the case of living organisms. These
exhibit a high degree of orderliness; yet to live and grow they
must metabolize, and the processes of metabolism "raise the
total entropy to a larger degree than corresponds to the growth
of orderliness they achieve" (K. Mendelssohn, *Turning Points
in Physics*, p.52). Yet the random molecular agitation in the
inorganic realm combined with metabolic increase of entropy in
the organic realm do not lead to increasing disorder, but are
constrained within the limits of highly articulate principles of
structure.

To deal
specifically with this problem, we may elaborate our hypothesis
as follows. Laws of structure, form, orderliness are imposed on
the random play of atoms and molecules through the intermediary
of a paraphysical, non-material substance, the ether. In the
ether forces interact, under the influence of the organizing
power of mind, and in a mode expressible in precise mathematical
relations, to produce formative lines and planes that constitute
the archetypal form field or matrix pattern for the organization
of material substance. Randomness at the material level, no
matter how much it may apparently increase, is contained within
the bounds imposed by a paraphysical invisible matrix pattern.
The notion of non-material, etheric patterns or form fields,
certainly seems a relevant hypothesis for the development of
organic forms, crystal formation, snowflake formation, and so
on. Of course, the hypothesis raises elaborate problems of its
own: (1) the nature of the mind or minds working through the
ether; (2) the constitution of the ether itself, and
experimental detection of its existence; (3) the interaction of
material substance and the ether, how the former is obedient to
the controlling influence of the latter; (4) the genesis of
material substance in relation to the ether; (5) the mathematics
of form fields and etheric processes generally. (Some pioneer
speculations in this realm were exercised by Sir Oliver Lodge in
*Beyond Physics*, 1930).

The
statistical concept of truth, current in modern physics, that
laws of nature are statistical averages, is a view of causal law
taken, as it were, from inside matter, from within the notion of
random molecular motion. But it clearly betrays paucity of
imagination, for it by no means accounts for the factor of the
organizing law that shapes the statistical averages into the
determinate forms of perceptual experience: it cannot do justice
to the clarity and differentiation of form that exists at the
macroscopic level and that is especially evident in the
biological realm.

The notion of
imposed law was, of course, the doctrine both of Newton and of
Descartes. Without such a doctrine science would scarcely have
been born. It was connected on the one hand with deism: for
Newton, laws of the solar system expounded in his *Principia*
were themselves to be explained by, and to him made obvious the
need for, the concept of a transcendent purposive deity. The
doctrine was linked on the other hand to the expression of law
as mathematical relations of functional dependence. Laplace, as
we have seen, took over the notion of law as a functional
relation, but dropped the notion of law as transcendental
intention. For him law became entirely immanent, a species of
animated built-in mathematics. This view, always rather
short-sighted, required as its least justification evidence that
Newtonian mechanics prevailed among ultimate particles as among
the celestial bodies. The discovery of random motion among
ultimate particles and the resultant problem of entropy
justifies the reintroduction of the hypothesis of purposive law.

We are not
suggesting, however, that the doctrine of purposive law should
be revived simply in a crude and vaguely deistic form (although
ultimate deistic notions can always be retained), but in a
manner which describes clearly and explicitly the mode of its
execution; and this via the hypothesis of intermediary
archetypal force fields and matrix patterns of an ether that is
conceived as the vehicle of purposive mind. Thus the quest for
certainty among ultimate particles is transposed into a quest
for the hidden origins of total form at the macroscopic level.
Moving outside physics, we may note the great scope for far
reaching hypotheses related to morphology in the biological
sciences; and even here mathematical treatment is highly
relevant (cf. the application of polar-Euclidean geometry to the
form of plant growth by G. Adams and O. Whicher, *The Plant
Between Sun and Earth*).

**The knowing
of form **

What is,
perhaps, significant about this suggestion is that it implies a
general reorientation in our mode of knowing whose effects may
be considered under three heads - psychological, epistemological
and moral. Psychologically, the study of macroscopic form
involves something other than the purely intellectual analysis,
the sharpened rational penetration, required for the close study
of sub-atomic events and structure. For the observer needs to
participate intuitively in the unseen matrix pattern and the
processes that work through it, by an imaginative structuring of
the total physical form and of the sequence of forms and
physical processes that develop through time. Psychological
functions of empathy and identification - with active
imagination and intuition - are employed, as well as purely
intellectual grasp and ability and, of course, careful and
controlled observation. Explicitly and consciously, then, a more
comprehensive range of cognitive powers is involved. This
suggests, firstly, that there is active a richer faith than the
normal faith of the scientist that there is an intelligible
order in nature that the intellect can abstract by reasoned
observation. This wider faith is a faith in a sustaining realm
of ordering forms and processes transcending nature that can be
entered into in an experiential mode beyond pure abstraction and
analysis. Secondly, in order to bring into exact and effective
balance the wider range of cognitive functions, there is
implicit in this approach a harmonious balance and integration
of the personality. Integrated cognition is a function of
integrated being.

Epistemologically there is entailed the view that the range of
phenomena capable of precise scientific description is not
restricted to sense data alone, but that an
imaginative-cum-intellectual grasp of the wholeness of sense
phenomena can lead beyond them to a participation in and
understanding of their immediately transcendent matrix forms and
processes. This species of exact, careful observation allied
actively to imagination and intuition as well as to intellectual
analysis, leads to a consideration of moral issues.

The moral
factor may be stated in terms of values. The exclusive use of
intellectual analysis allied to observation as a mode of
cognition tends to leave underdeveloped those functions of
feeling and intuition that are particularly responsive to
values. There is a certain neurosis of the intellect, a certain
ruthlessness in the quest for certainty that seeks to strip
matter to its bare subatomic bones of immanent mathematical law.
This puts aside as irrelevant the role of these other
psychological functions that contribute to comprehensive human
cognition. The resultant insensitivity is reflected in the old
deterministic myth which excludes from the economy of the
universe any teleological role for values, and in the new
statistical myth which reduces the purposiveness implicit in
form to the mere play of chance. Our technological civilization
based on science has among its characteristic motives those of
dominance and acquisition. This is a consequence of the moral
implications of the scientific mode of cognition, which
acquisitively seeks an intellectual dominance over facts, to the
exclusion of an imaginative and intuitive attunement to and
through the facts. There is a price to be paid for probing
nature with the sharpened tool of the intellect alone, for the
knowledge that is won backfires in a way that draws dramatic
attention to the practical relevance of values and modes of
cognition that have been put aside (cf. the A and H bombs whose
development was unforeseeable in the early days of Rutherford's
work on the atom). It is an approach that disregards the role of
transcendent functions in nature, and is piratical in the sense
that it appropriates and applies its discoveries disregarding
influences from, and effects of, the unacknowledged realms
intimately involved in the zone of operations concerned.

We may
suggest, then, that there is a morality of our modes of knowing,
whose principles are concerned with the range of, and the
interrelation between, our different cognitive functions. The
more comprehensive the range of functions, the more integrated
and mutually fructifying their working, the greater the
interrelation of fact and value in the knowledge gained, the
direction of research, the applications achieved, and in the
characteristics motives of the culture in which they are
applied. The study of form, its origin, processes and
metamorphoses requires an intuitive-aesthetic as well as a
purely rational grasp; and it can never be far removed, via a
general metaphysic of form, from a consideration of the
purposive role of values.

We may
summarize this section by saying that the knowing of form
involves (1) a joint consideration of fact and aesthetic value;
(2) a wider range and integration of cognitive functions within
the psyche; (3) an understanding of the integration between the
phenomenal realm and the transcendent matrix fields that
interpenetrate it with formative influences; (4) in cognizing
this integration, an active attunement to the fields and
processes concerned. The intellectual quest for certainty, then,
becomes transformed in this alternative approach into the quest
for wholeness and depth within the observer, wholeness and depth
within the observed, and an experiential unity as between the
two. And this leads quite naturally on to the unfolding of a new
descriptive metaphysic of the most general characteristics of
form and structure as such, and of the most basic principles
involved in their transformations.

(NB: The above
discussion is not intended to suggest in any way that the
investigation of ultimate particles should be abandoned, but
simply that it should be complemented by and subsumed within the
deeper and wider macroscopic approach outlined).

**
Mathematics: Gödel’s theorem **

Whatever
uncertainty may have arisen out of quantum mechanics and the
formulation of Heisenberg's principle, physicists themselves
took for granted the complete reliability and certainty of the
mathematical tools by means of which the principle itself and
the facts underlying it could be made so explicit. A certain
small group of research mathematicians, however, were less
content to rely solely on the apparent pragmatic evidence of the
reliability of mathematics. And we have seen how, in the light
of the disturbing paradoxes mentioned earlier, they were
concerned to establish an absolute, "finitistic" proof of the
consistency of arithmetic. At this point we come to Gödel's
theorem.

Gödel's
theorem is certainly one of the outstanding intellectual
achievements of the present century and marks a high point in
the development of rational skill, ingenuity and inventiveness.
Nor is it likely that its broad philosophical implications have
as yet been fully fathomed. The theorem is conducted in the
abstruse realm of metamathematics. Mathematics proper embraces
the formal deductive systems - algebra, geometry, arithmetic,
etc., - that mathematicians construct; metamathematics deals
with the description, discussion and theorizing about such
systems. Thus two main metamathematical issues are: (1) is a
mathematical system consistent? (2) are its axioms independent,
such that no one can be derived as a theorem from the others?
Hilbert's programme for an absolute and finitistic proof of
consistency was, first, completely to formalize a deductive
system into a calculus consisting of a set of signs, and rules
showing how the signs are to be combined and manipulated. The
combination rules show how the signs may be arranged so as to
give intelligible axioms and theorems. Theorems are derived from
axioms in accordance with transformation rules, or rules of
inference. The calculus, then, consists of certain signs
combined according to certain rules into a set of axioms from
which, in turn, theorems are derived by rules of inference.
Secondly, Hilbert hoped that such a calculus could be shown to
be a "geometrical" pattern of formulae standing to each other in
a finite number of structural relations examination of which
would establish that contradictory formulae cannot be obtained
within the calculus. What is clearly essential to such a
metamathematical demonstration of consistency is that it should
not involve an infinite number of structural properties of
formulae, nor an infinite number of operational procedure on
formulae. One mast be able to make fully explicit all the axioms
and all the transformation rules to be applied to them.

Until Gödel's
theorem it was taken for granted that a complete set of axioms
for arithmetic or any given branch of mathematics could be
assembled: there was the apparent evidence of the axiom sets of
the various geometries, and Peano's apparently complete
axiomatization of the arithmetic of cardinal numbers (five
axioms formulated with the help of only three undefined terms -
"number", "zero", and "immediate successor of"). Now Gödel
proved, by a highly ingenious technique of mapping
metamathematical statements about a formalized calculus onto
arithmetical formulae within that calculus, that
arithmetic can never be fully axiomatized: one can never set out
the complete set of axioms of a deductive system from which all
true arithmetical theorems could be derived. He showed that any
system within which arithmetic can be developed is essentially
incomplete; for given any set of arithmetical axioms, there will
be true theorems that cannot be derived from the set, and no
matter how the set may be augmented by additional axioms, there
will always be further true theorems not derivable from the
augmented set. There is thus an inherent limitation in the
axiomatic method as a technique for encompassing and sustaining
the whole of arithmetical truth. This is the "incompleteness"
aspect of the theorem.

But the
theorem also showed that the consistency of arithmetic cannot be
demonstrated by any argument that can be represented in the
formalized arithmetical calculus. That is to say, it is
impossible to prove the internal logical consistency of a
deductive system from which all arithmetic can be derived,
unless the proof employs rules of inference wider than and
essentially different from the transformation rules used to
derive theorems within the system. But then the consistency of
the assumptions implicit in the wider transformation rules is as
open to doubt as the consistency of arithmetic itself. The long
and short of these two results of the theorem is that Hilbert's
programme for an absolute and finitistic proof of the
consistency of arithmetic has to be abandoned. The intellect has
demonstrated that it cannot, as it were, encompass a finite,
impeccable guarantee that many significant branches of
mathematics are entirely free from internal inconsistency. The
search for absolute intellectual certainty in mathematics thus
finally arrives only at the certitude of its unattainability.

Some
qualifying statements must here be introduced. (1) Gödel only
showed the impossibility of a consistency proof that can be
represented within arithmetic. There may be a finitistic proof
that cannot be so represented; it is difficult, however, to
conceive what such a proof could be like. (2) The theorem does
not exclude the possibility of a metamathematical proof of the
consistency of arithmetic, one that is not finitistic and that
cannot be mapped onto formalized arithmetic. Thus its rules of
inference lie outside the impeccable guarantees of a finite
consistent system. An example is Genzen's proof (1936) which
organizes all arithmetical demonstrations in linear order
according to degree of simplicity. This order has the pattern of
a "transfinite ordinal type". A proof of consistency is got by
applying to this order a rule of inference called the "principle
of transfinite induction". (3) Similarly, the several
mathematical theorems or general statements which have so far
evaded proof, that is, which it has not been possible to derive
from a given set of axioms (e.g. Golbach's theorem that every
even number is the sum of two primes), may be established by
metamathematical proofs; but again the rules of inference used
(and the consistency of their implicit assumptions) will lie
outside those contained within a formalized calculus.

**The
implications of Gödel’s theorem **

We have seen
that a full account of mathematical inference cannot be given in
terms of the axiomatic method. We cannot, that is, make fully
explicit all the rules and principles involved in valid
mathematical demonstrations. The whole logical form of
mathematical proof cannot be set forth in any self-contained
deductive system. There will always be new principles of
mathematical inference awaiting discovery. This gives rise to
two problems: (1) How do we justify the use of, that is, what
are our criteria for the soundness of, metamathematical
principles of inference that fall outside the scope of a
formalized axiomatic system? (2) What sort of an account can we
give of the nature of mathematical and logical truth? In
approaching an answer to these two questions, two suggestions
may be put forward.

(1) If we
cannot forever go on codifying rules of inference within
self-contained consistent deductive systems, then we must appeal
in the case of new and wider rules of inference to a general
aesthetic criterion of elegance, simplicity, cogency. Such a
rule of inference will be justified by its elegance, by its
range of unifying, cohesive power upon the system to which it is
applied. We are again, in the realm of metamathematics, brought
face to face with the notion of form, here pure abstract form,
to which in the last analysis aesthetic criteria alone apply.
(Cf. L.L.Whyte, *Accent on Form*: "All intellectual
processes depend on the operation of the aesthetic sense that
recognizes an elegant ordering when one is presented to it. This
sense is prior to reason and cannot be justified by analysis or
interpreted by definition".) This does not mean, of course, that
in this realm we discard the rational processes of cogent proof,
but that the rules of inference employed which achieve the
cogency find their ultimate sanction in pure intuitions of
formal elegance: the rational or deductive content is subsumed
within the aesthetic criterion.

(2) In the
light of Gödel's theorem, mathematical and logical truth may be
conceived as a world of abstract formal relationships spreading
out beyond the horizons of present intellectual advance, and
awaiting discovery. This, of course, revives a kind of Platonic
realism, a doctrine of subsistent universals: mathematicians
discover the objects which in some sense subsist prior to their
discovery. And it appears to be Gödel's view: "Classes and
concepts may be conceived as real objects existing independently
of our definitions and constructions. It seems to me that the
assumption of such objects is quite as legitimate as the
assumption of physical bodies and there is quite as much reason
to believe in their existence" (Gödel on Russell's mathematical
logic in *The Philosophy of Bertrand Russell, Schlip*p,
1944). It might be more apposite to speak of a universal and
subsistent mind whose content of intelligible relationships
extends beyond both the confines of concrete actuality and the
areas of formal possibilities so far mapped out by human
inquiry.

The
mathematician's intellectual faith in the possibility of a
consistency demonstration for arithmetic has to be replaced by a
faith that in subsistent, universal mind there are wide
principles of inference which can be discovered and whose
ultimate justification resides in an aesthetic criterion of
unifying elegance. And it is at this point that we may move
forward toward a more general philosophy of symbolic forms.

**
Towards a philosophy of symbolic
form**

**Intellectus
archetypus**

What is
remarkable about the two events we have been discussing is that
they both point to the capacity of the intellect to determine
its own limitations. In physics the intellect has established
what it cannot measure, in mathematics it has demonstrated what
it cannot prove. If we examine this peculiar ability of the
intellect we may see in it evidence of the implicit application
of a criterion that transcends purely rational demonstration and
analysis as such. The intellect is able in a rational mode to
reflect principles that transcend its own nature by virtue of
their aesthetic content. To establish what you cannot measure is
to reflect a subliminal awareness that spatial physical form is
transcendentally organized by the action of forces flowing
through creative symbols or patterns or ultimate matrix
structures. To prove what you cannot prove is to appeal to
standards, ultimately, of cogency, fittingness, elegance which
subsume yet still transcend rational principles of inference. It
is a process, perhaps, in which the intellect reflects its
access to realms of purely symbolic thought that lie behind all
rational demonstrations.

The
justification, then, for accepting the hypothesis of a
transcendent function of mind, an "intellectus archetypus" as
Kant called it, is that this function demonstrates its existence
at the rational level through the activity of reason proving the
impossibility of certain kinds of proof. A mind that can
describe its own rational limits is implicitly revealing at a
rational level its transrational insights, and is implicitly
calling for a wider and deeper faith to prompt the search for
knowledge.

We may state
the broad philosophic hypothesis as follows. The intellectus
archetypus has access to subsistent, universal mind whose
content of symbolic forms is independent of human minds and
awaits their discovery. The active intellectus archetypus is a
heightened function of mind in which intuition and intellect,
feeling and will are in harmonious integrative balances. As a
result the symbolic forms that it apprehends are not simply
mentally observed, but are inwardly experienced: they are passed
through like cognitive gateways to a living reality. These
symbolic forms are on the one hand the ultimate justification
for the intelligible content of all human speculation,
reflection, contemplation and demonstration; and on the other
hand they are the ultimate origin for the formal properties of
form fields and matrix patterns in the ether, on which material
entities are built up.

**Paralogic
**

If the
discursive intelligence, Kant's "intellectus ectypus", finds a
purely rational justification for its products in the formal
principles of logic, then the intellectus archetypus finds a
transcendent justification (wider than the rational) for logic
itself in the symbolic forms the study of which we shall
designate as "paralogic".

It is clearly
the case that paralogic will have a language of its own in which
the interrelations within and between symbolic forms will be
expressed in a series of symbols or glyphs arranged together in
accordance with principles that are wider end deeper than is
evident to purely rational inspection and analysis. This is not
to assert, however, that there is an unbridgeable gulf between
logic and paralogic. For once the intelligible content of a set
of paralogical glyphs is grasped, or rather, entered into and
experienced at the level of the intellectus archetypus, then its
rational content will be susceptible of consistent and coherent
explication. In the same way, while paralogic itself may be
expounded in its own paralogical terms, yet it is still
susceptible of reflection in the explanatory terms of the
discursive intelligence. There is, however, a leap involved in
the passage from a discursive account of paralogic to the *
experience* of its intelligibility. For there is more than a
rational dimension to a symbolic form, whose general properties
we must now consider.

It is also
clear that the ultimate study for paralogic will be the symbolic
form of a symbolic form. A symbolic form has both a rational and
an aesthetic component. It is, as we have said, susceptible of
rational explication and interpretation for there is a
correspondence between its elements and their interrelation and
certain rational concepts and their interrelation; logical
relations reflect paralogical relations. Yet in its own nature
it has a cohesion, unity, elegance, fittingness, cogency,
compactness, simplicity that is susceptible of purely aesthetic
appreciation. It is this aesthetic component, involving the
grasp of a whole as a whole, that raises cognition of symbolic
form to the level of the intellectus archetypus.

The two
components are, of course, interrelated. A symbolic form
deficient in rational potential is not likely to persuade
aesthetically; while one that is aesthetically disingenuous is
not likely to sustain adequate rational development. Yet the
aesthetic component subsumes the rational, and in transcending
it, acts on it, unifies it, and transforms it, creating out of
the whole a well-formed cogency. But we cannot reduce the
rational to the aesthetic (or vice versa); each has its own
distinct mode of mental being. We may, then, symbolize a
symbolic form by a horizontal line for the rational dimension of
its intelligible content, a vertical line (intersecting the
horizontal at right angles) for the aesthetic dimension that
interacts with the rational, and a circle for the well–formed
cogency of the result. The lines extend beyond the circle since
we cannot suppose that the experienced cogency of a symbolic
form totally exhausts the intelligible possibilities of the
interaction of its aesthetic and rational components. This
symbol might be better conceived with the two lines of
indefinite length and a series of circular ripples proceeding
out continuously from the central point of interaction. In
graphically symbolizing this mental symbol of a symbolic form,
we present one circular ripple or wave on a portion of the
lines.

It can scarcely be argued that this symbolic form of a symbolic
form has a solely subjective reference to the mode of human
understanding, that it simply depicts intelligent understanding
and nothing more. For it seems more reasonable to suppose a
consonance between our understanding and that which is
understood, and to argue in the light of our general
hypothesis that it is a symbol of the intelligibility inherent
in universal mind as such: it depicts the prior substrate of all
knowledge. If so, we shall expect to find contained within it a
correspondence between ultimate principles in diverse realms of
knowledge. We shall therefore call it the basic symbol of the
intelligible.

**Aesthetic
validation through the paralogical principle of correspondence
**

Systems of
formal concepts, belonging to diverse domains of enquiry, that
can be seen to be subsumed under a symbolic form of wide
application will thereby tend to be validated by the aesthetic
cogency of the correspondences established between them. While
conversely the symbolic form itself will receive an aesthetic
validation by the extent to which its rational component is
explicated in terms of the varying sets of formal concepts that
can be developed under it.

For example,
we can take the basic symbol of the intelligible given above and
seek to subsume under it an explication of key concepts
descriptive of metamathematical reasoning on the one hand, and
of the (meta)physics of spatial form on the other. The symbol
will then elucidate a correspondence, having a purely aesthetic
cogency, between formal elements in these two realms of inquiry.
If we consider metamathematical demonstration, the vertical line
represents what we have seen to be the transcendent rules of
inference, interacting with the horizontal line representing the
ordered elements of the deductive system to which the
transformation rules are applied; the circle represents the
well-formed cogency of the result. Similarly, in a general
metaphysic of material forms, the vertical line represents the
transcendent laws of change or process working through the
matrix patterns, interacting with the horizontal line that
represents the elements of the material system affected; the
circle represents the well-formed organism that results.

We are able,
then, to suggest a significant correspondence between the
aesthetic component of a symbolic form, the rules of inference
employed in a metamathematical demonstration, and the
transcendent formative processes acting on any physical system.
Laws of inference are like laws of process: they tend to weld
horizontally disposed units into a whole, just as the aesthetic
component of a symbol welds its rational potential into a cogent
and fitting whole. The transcendent nature of metamathematical
rules of inference and of the aesthetic component of a symbolic
form enables us, via the paralogical principle of
correspondence, to suggest the aesthetic reliability of the
notion of the transcendence of formative processes in nature.

But where does
this paralogical method stand in relation to an empirical
methodology? Are we seriously suggesting that such a principle
of correspondence can be a substitute for empirical
investigation? Certainly not. What we are suggesting is that it
is a way of cohering and organizing our intuitions prior to such
investigation. The paralogical method introduces determinate
principles into the hitherto mysterious process of insight that
precede scientific method. The sets of correspondences which it
establishes serve on the one hand to unify, deepen and enrich
knowledge by establishing harmonic relations between concepts,
and on the other to provide by analogy and the structuring of
operational concepts fruitful hypotheses to guide empirical
research. Certainly, if it fails to do the latter it can
scarcely claim comprehensive validity as a method.

**Symbolic
forms **

By way of
conclusion, this final section seeks to set out some general
features of the hypothesis of symbolic forms.

(1) Symbolic
forms in their pristine, original, subsistent state will of
course be quite distinct from, although they will participate
in, any two dimensional graphical representation. We may perhaps
conceive of the origin of a symbolic form as a particular focus
or area of intelligibility in the universal mind, a zone of
realization that is in its ultimate nature perhaps never
entirely attainable, but which springs into determinate form in
the mind of an interpreter attuned to it. So according to the
nature of the interpreting mind and the level of being at which
its understanding is being exercised, different facets, aspects
and dimensions of the symbolic form will disclose themselves.
Thus we have the original focus of intelligibility in universal
mind, its many-faceted representation in diverse interpreting
minds, and its basic graphical or embodied representation
outside a mind. The graphical expression will be to a
considerable degree adequate as an anchorage for a set of
correspondences. But reflection on symbolic forms can go beyond
the purely graphical expression or glyph to experience the
unrestricted thought-form in which the symbolic form discloses
itself in the multi-dimensional space of the interpreter's mind
or creative imagination. Out of such a thought-form a richer and
more complete set of correspondences can perhaps be developed.

(2) Need
symbolic forms be represented to the interpreter as obvious
geometrical forms? No, but we may suggest that any explicit
representation to the mind must have spatial characteristics.

(3) Symbolic
forms, as we have already indicated, may be considered not
simply as transcendent foci for the unification of basic
concepts in diverse realms of knowledge, but also as focal
points that radically influence the nature and activity of
formative processes. Individual sciences will seek the key
symbolic forms relevant to their special domains. Metaphysics
will seek those that are relevant to many or all domains. If
symbolic forms are generically related to the formal properties
of etheric matrix patterns, etc., then paralogical intuition may
fruitfully precede the formation of working hypotheses in the
new physics.

(4) Symbolic
forms, as distinct from their point of origin in universal mind,
may also be considered as the unifying focus for distinct groups
of transcendent, discarnate minds, which relay the formative or
conditioning influences characteristic of the forms to which
they are particularly attuned. Here again is an obvious
connection with the new physics.

(5) Comparable
with what we said about participation in unseen matrix patterns
of macroscopic forms, the psychological significance of the
paralogical method is that a symbolic form invites total
participation: we need to experience its aesthetic component by
empathetic identification. Symbolic forms subsist in a zone of
intelligibility and invite us to enter into an inner rapport
through them with this zone and to experience its intelligible
life as a sustaining reality. The intellectus archetypus
achieves this by the integration of diverse psychological
functions in one experiential mode of knowing.

(6) We may,
finally, suggest the view that a knowledge of symbolic forms is
received by a rapport with the minds referred to in (4) above.
This raises the question of the conditions under which the
symbolic forms are received. We here come again to a deep
interior connection between morality and epistemology. We do not
of course mean morality in any narrow puritanical sense; and it
is prefereable to use the term metamorality for a fusion of
psychological wisdom, creativity, experimentation and ethical
values, that is more subtle, adventurous, far-reaching and
imaginative than any rigid, authoritarian prescriptions
governing correct behaviour. What we are suggesting, of course,
is that a knowledge of symbolic forms is not simply the fruit of
intellectual speculation or rational analysis. It is rather that
which is bestowed upon the mind when it enters into and acts on
comprehensive attitudes of faith which we have broadly outlined;
and when as a result of this it cultivates in a deliberate
manner transrational modes of knowing. There is no
recommendation in all this that rational modes of knowing as
such are to be discarded; far from it. But the cultivation of
transrational insights is closely related to the integration of
behaviour through its interaction with deep metamoral values:
the insight involved is the fruit of integration occurring
within deep levels of the personality. Metamoral values interact
with total behaviour to produce well-formed insights, a thesis
which can also be explicated under the basic symbol of the
intelligible given above. Receptivity is allied to rational
judgment, intuition to discrimination, and the whole is an
inquiry guided by a deep and aspiring faith in the reality of a
transcendent knowledge. Its results are justified aesthetically
by the cogency with which symbolic forms unify metaphysically
different areas of experience, rationally by the extent to which
the symbolic forms are capable of rational explication in
diverse realms, empirically by the fruitfulness of the
paralogical method in guiding research and objective inquiry.