Ernst Cassirer, The Philosophy of Symbolic Forms, tr. Ralph Manheim (3 vols.: 1
New Haven: Yale University Press, 1955-57), II, 29.
24
Chapter 2
The Symbolism of Science
The awareness of the outer world is not the awareness of reality in
itself. The object of our knowledge is not something transcendent behind our
representations. On the contrary, the outer world as we apprehend it is a
represented world. It obtains within the unity of the form-producing
consciousness that orders the particulars of experience into a pattern of
meaning.
In our day, science is the preeminent mode in which the outer world is
formed within the activity of the human spirit. But there are, and have been,
other modes of apprehension. Myth is one such mode.
There is the mythical consciousness of the object. Again, it must be
emphasized that the object is not given as a thing in itself, but only as it
is constituted the kind of object it is by the act of consciousness. The
German philosopher Ernst Cassirer makes this point:
It is one of the first essential insights of critical philosophy
that objects are not “given” to consciousness in a rigid, finished state,
in their naked “as suchness,” but that the relation of representation to
object presupposes an independent, spontaneous act of consciousness. The
object does not exist prior to and outside of synthetic unity but is
constituted only by this synthetic unity; it is no fixed form that
imprints itself on consciousness but is the product of the formative
operation effected by the basic instrumentality of consciousness, by
intuition and pure thought.1
The perception of sensible contents is to some extent ordered, in the
sense that they appear and reappear in certain combinations and sequences. If
this degree of coherence and sequence were not, as it were, embedded in the
25
stream of sensations, there could be no establishment, on the basis of the
sensory given, of an objective world. Thus this level of sensory experience
is a necessary condition of the object. But it is not a sufficient condition.
What is further required is the act of thought, by means of which the sensory
flux is reformed into a more fully ordered whole in accordance with a law that
governs the series of sense presentations.
Scientific thought fulfills this purely intellectual activity by which
the sensory given is molded to form the object and the object-world in what
for modern man is a preeminent way. For the mythical consciousness and
mythical thought, however, this process is set in motion in a rudimentary and
undeveloped manner.
There are, in this regard, two main characteristics of mythical thought.
In the first place, the various sensory contents given in perception are
placed on the same footing, i.e., they all, without any distinction, possess
the same force of reality. They are all designated as the real object. No
distinction is drawn between items of sensory experience that are essential
and those that are peripheral as regards the nature of the object.
The second characteristic is one that is particularly significant for
this study, and is one to which reference must be made repeatedly. Mythical
consciousness and thought views the object of immediate presentation as the
real object. Its contents are given an object form, as “real contents.” This
form is completely homogeneous and undifferentiated. No distinction is, or
can be, made between different spheres of objects, and no line is, or can be,
drawn between the world of truth and the world of appearances. In this regard
Cassirer notes:
Myth lives entirely by the presence of its object—by the intensity with
which it seizes and takes possession of consciousness in a specific
moment. Myth lacks any means of extending the moment beyond itself, of
looking ahead of it or behind it, of relating it as a particular to the
elements of reality as a whole. Instead of the dialectical movement of
thought, in which every given particular is linked with other particulars
in a series and thus ultimately subordinated to a general law and
process, we have here a mere subjection to the impression itself and its
26
2Ibid., p. 35.
Ibid., p. 38. 3
momentary “presence.”2
There is the view that the contents of the mythical consciousness are
symbolic, that behind them is another, hidden sense to which they mediately
refer. Both medieval and romance thought held, with some variation, this
“allegorical” theory of myth. There is, they affirmed, an ideal content that
can be glimpsed behind the imagery of the myth. But this is to view myth from
the outside, from a reflection that reads into myth such a distinction. If we
examine myth itself, what it is and knows itself to be, we find no such
distinction between the real and the ideal, between immediate reality and
mediate signification. In its fundamental and original form, myth perceives
real identity:
The “image” does not represent the “thing”; it is the thing; it does not
merely stand for the object, but has the same actuality, so that it
replaces the thing’s immediate presence. Consequently, mythical thinking
lacks the category of the ideal, and in order to apprehend signification
it must transpose it into a material substance or being.3
It is for this reason of the identity of image and thing that myth is
not symbolism. And this explains the import of the ritualism of the mystery
cults, which has been dealt with in the preceding chapter. The ritual is not
merely mimetic portrayal of the event; it is the event. When the Eleusinian
initiates participated in the passion play in which the two goddesses, Demeter
and Persepone, find happiness in their reunion, the initiates were not merely
dramatizing the event; they were actually sharing in the reality of the event.
When they in silence beheld the sacred corn, they were not merely dramatizing
the new life; they were actually receiving and participating in that new life.
Scientific thought transforms the object of mythical consciousness. By
a process of analysis the sensuous contents of experience are reduced into
ultimate elements that can be apprehended only by thought. At the purely
sensuous level the contents are a pure flux that have no fixed pattern or
27
order. Thus they must be transformed by thought into conceptual elements.
For example, the laws of motion cannot be formulated simply in terms of the
flux of perceptible contents. Only when atoms are postulated as the true
subjects and ideal elements of motion, can exact mathematical laws of motion
be constructed. It is in this fundamental sense that the concept of the atom
and the mathematical formulation of motion are symbolic: symbolic, not of
“reality” as it is in itself, but of the structure of experience.
The second process is that of synthesis. We do, indeed, intuitively
perceive objects in a spatial form. But there is nothing in the contents of
simple sensations that discloses that form. It is established only by the
intellect, by a judgment that situates the contents in a relationship and
system. The sensory data are thus apprehended according to position, size,
and distance. These relationships in which the data are ordered constitute
the fixed structure of “objectivity.”
The synthesis that positions sensory items is also a critical activity.
As the context in which the particulars are ordered is progressively
developed, data that are out of harmony with that context are rejected. What
certifies objectivity is the unity of experience. Objective existence is now
no longer, as it is in mythical thought, identified with the mere givenness of
perception; it is identified with the system in which certain data are
structured. And appeal is constantly made to the whole of experience as the
determining ground of objective reality. The only meaning that can be given
to empirical reality is from within the system of thought. And this system is
a symbolic structure; not, again, of reality in its innermost nature, but as
the reality constituted by and obtaining within the synthesis of judgment.
The symbolism is a formation that elucidates the structure, displayed
incipiently in the flux of the data, of perceptual experience. That is its
symbolism.
Cassirer puts the matter cogently:
28
Ibid., p. 31. 4
A. S. Eddington, The Nature of the Physical World (New York: The Macmillan Co., 5
1929), pp. xiii-xiv.
What distinguishes empirical reality, the constant core of objective
being, from the mere world of representation or imagination, is that in
it the permanent is more and more clearly differentiated from the fluid,
the constant from the variable. The particular sense impression is not
simply taken for what it is and immediately gives; instead we ask: will
it be confirmed by experience as a whole? Only if it stands up under
this question and this critical test can we say that it has been received
into the realm of reality and determinate objective existence.4
Another reference may be helpful in emphasizing the symbolic nature of
science. Professor Eddington writes:
Science aims at constructing a world which shall be symbolic of
the world of common experience. It is not at all necessary that every
individual symbol that is used should represent something in common
experience. . . . It is like our experience in learning to read. That
which is written in a book is symbolic of a story in real life. The
whole intention of the book is that ultimately a reader will identify
some symbol, say BREAD, with one of the conceptions of familiar life.
But it is mischievous to attempt such identification prematurely, before
the letters are strung into words and the words into sentences. The
symbol A is not the counterpart of anything in real life. To the child
the letter A would seem horribly abstract; so we give him a familiar
conception of it. “A was an archer who shot at a frog.” The letters are
abstract . . . . In physics we have outgrown archer and apple-pie
definitions of the fundamental symbols. To a request to explain what an
electron really is supposed to be we can only answer, “It is part of the
A B C of physics.”
The external world of physics has thus become a world of shadows.5
Further, there is the mythical consciousness of causality. The two
considerations that define mythical thought and set it apart from scientific
thought are precisely those that relate to the category of the object. Just
as every object presented to mythical consciousness is taken as real, so any
event can indiscriminately be taken as a cause of another event. Anything and
everything that stands in temporal or spatial contact with something else can
become a cause. This means that for mythical thought the cause is derived
from the mere presence of sensory elements. No intellectual judgment is
required to establish a causal relation and order.
Further, mythical thought views the causal object, indiscriminately
selected, as a real cause. Mythical thinking holds to the principle of
causation, just as does scientific thinking. There is agreement between the
29
two forms of thought at this basic level. The difference between the two
consists in the interpretation of the manner of causation. Science is content
when it demonstrates that an individual event in space and time is an instance
of a general law. It does not ask why the individual event came under the
aegis of the general principle of cause and effect. But myth asks just this
question, i.e., it seeks to explain why the individual event happened to occur
under the province of the general law. It explains this by postulating
individual acts of will. A catastrophe is brought upon the land by the act of
a demonic will. Science explains the death of an individual in terms of
certain universal conditions of “nature.” It leaves unanswered the question
as to why just this particular individual died, or, if it addresses the
question at all, calls it an “accident.” But mythical thought knows no
accident. The death of this individual is the result of the act of a demonic
will. This means, in sum, that for myth causation is purposive action, not an
instance of an event subsumed under a merely formal principle that leaves the
existentiality of the event unexplained as accidental.
Scientific thought rejects the free and unrestrictive selection of the
cause. The temporal and spatial co-presence of events does not signify or
constitute a causal relation. The mere temporal precedence of one event over
another does not mean that the former is the cause of the latter event. When
one eats bananas and gets sick, that does not necessary mean that the bananas
are the cause of the illness. Every wide-awake beginning Logic student learns
about the fallacy, post hoc ergo proper hoc (after, therefore, because of).
Thus common understanding, to say nothing of scientific understanding, turns
away from the presuppositions of mythical consciousness and thought.
In establishing a causal relationship, science resorts to the judgment.
There are two phases of this judgment: analysis and synthesis. These are the
two phases of the judgment that creates the category of the object.
Through the act of analysis, the sensory contents are distinguished from
30
one another and assigned to different sets of conditions. Regardless of how
often an event B follows event A, it does not follow that A is the cause of B.
A mediating judgment must intervene and isolate the factor a in A that links
with the factor b in B, such that A and B can be linked together in the cause
and effect relationship. It is this analytical, discriminatory, judgment that
constitutes the category of causation, that is prerequisite in determining the
nature of causation.
There is also a synthetic judgment in regard to the causal principle.
The principle of causality demands the independence of a consequent y upon an
antecedent x. The demand is a purely formal one: if x, then y. This is the
form of the synthetic judgment informing the principle of causality. And, as
the above has indicated, it functions subsequent to the judgment that analyzes
the sensory items.
Now, the formal schema, which is definitive of the causal principle, is
applied to a concrete situation only if the values to be inserted for x can be
determined by accurate measurement. The process of measuring utilizes, to be
sure, mathematical values, but those values must be given a unit of
measurement. For example, if the number one is assigned as the value of the
weight of an object, there must first be specified a unit of measurement, say
a pound. Otherwise, the mathematical value is meaningless as pertaining to
fact. The same is true with respect to the formal principle of causality. It
bears upon physical reality only when certain physical magnitudes—such as the
place and momentum of an object—have been determined with accuracy. When the
measured variables have been ascertained, the formal principle of causality
can then be applied to the concrete situation. The predication can then be
made that if x occurs, y will occur. Certainly, if the principle of causality
is applied to matters of fact, observation and measurement of physical
magnitudes are necessary. But the principle of causality that is applied to
physical objects is a purely formal schema. And it is its formal nature that
31
admits its general applicability to physical phenomena.
Now, all of this means that the formal principle of causality is
symbolic in its nature. It is located, not, as in myth, in the material of
perceptual reality, but in the mind, in spiritual consciousness. It is a
symbol for the ordering of the felt continuities within the stream of
perceptual contents. If the question is raised as to whether or not the
causal principle, when impinged upon events, “really” discloses that a
corresponding causal order and relationship “exists” in the “real” world, that
question cannot be answered. It is an irrational question. The causal
principle does not exist in the real world. It is purely and only a formation
of the synthetic judgment. Its form does, indeed, inform and mold the
continuities of sensory experience, and that is the extent of its
applicability. It symbolizes the felt continuities of perceptual experience
and gives a fuller measure of order to them that they, despite an
indeterminate continuity, do not in themselves possess. It leaves unanswered
“how” events produce other events. It is thus limited in its
prescriptiveness.
Considerations pertaining to the physical world fall within the scope of
empirical science. There are other considerations that are comprehended in
formal science. The formal sciences are those of logic and mathematics. They
are distinctive symbolic formations.
Logic is a normative science, i.e., it formulates the rules that are
required for correct thinking. It is concerned with the intrinsic nature of
the procedure of thought, regardless of the particular subject matter, the
material, of thought. Its ultimate appeal is to the canon of consistency.
The “laws of thought” formulate explicitly this requirement of thought.
Traditionally, they are three in number: the law of identity, the law of
contradiction, and the law of excluded middle. Leibniz (1646) formulated a
fourth law, the law of sufficient reason.
The three traditional laws of thought admit of various interpretation
32
Aristotle Catagorae, Chap. 12, in Works, ed. W. D. Ross (Oxford: At the 6
Clarendon Press, 1928), Vol. 1, p. 14 . b
and formulation. These interpretations, however, prejudge the question as to
the nature and status of the laws. In order to avoid any such prejudgment,
they need to be formulated in terms of their abstract symbolic character.
Thus the law of identity reads: A / A, A is equivalent to A. The law of
contradiction reads: A ~A, A is not equivalent to Not-A. And the law of
excluded middle reads: A v ~A, A or Not-A.
A realistic interpretation of the laws affirms that they are, eo ipso
laws of being or existence. They formulate, even are, the basic aspects of
the universe itself. They are, first and foremost, the essential traits of
all things whatsoever. They originate in being as being. And for that reason
they are capable of expressing the essential traits of anything that exists or
may exist. The ancient Greek philosopher Aristotle held this view. He argued
that the general nature of things is the ground for the correctness of
reasoning and that this is expressed in the laws of thought. Since they have
to do with anything that exists or may exist, they are formal in their nature.
They are not restricted to any given subject matter. Instead they hold for
anything whatsoever; they are universally applicable.
Aristotle’s argument for the ontological priority of the laws of thought
is as follows:
The fact of the being of a man carries with it the truth of the
proposition that he is, and the implication is reciprocal: for if a man
is, the proposition wherein we allege that he is is true, then he is.
The true proposition, however, is in no way the cause of the being of the
man, but the fact of the man’s being does seem somehow to be the cause of
the truth of the proposition, for the truth or the falsity of the
proposition depends on the fact of the man’s being or not being.6
Now it is conceivable that the structural laws of thought are grounded
in the structural principles of being. On that assumption, the laws are
capable of disclosing the general properties of reality. But it is not
possible to demonstrate either that thought yields the disclosure of reality
or that reality is the ultimate ground of thought and its structural
33
principles. Aristotle himself may have realized this, in his use of the
adverb somehow: “ . . . the fact . . . does seem somehow to be the cause of
the truth of the proposition.” The ontological view of logic is but an
assumption, and an assumption that reflects a certain metaphysical prejudice,
namely, that reality is ultimately mind-independent and that, as an alien
object, may be known independent of and unmodified by the knower. This
ontological view of logic and its structural principles meets in common with
the realist theory of the symbol, that the symbol is the symbol of an
independent reality, subsisting beyond mind and thought, and that the symbol
function is a literal disclosure of being itself. There is, however, a
difference, in that the ontological theory of logic does not argue that logic
is symbolic of reality. It is, via its structural principles, of the nature of
reality itself.
Diametrically opposed to the realist theory, is the view that the laws
of thought are stipulative conventions, purely man-made prescriptions for the
use of names. As intelligence developed in the slow evolution of the human
race, it was discovered that words had to be used with the same meaning. The
principles of thought were developed to deal with this necessity. They do not
indicate the structure of reality, nor are they derived from any ontological
ground. If there should come a time when a more useful set of conventions is
developed, the traditional laws will become obsolete and discarded.
The main difficulty with this theory is the essential arbitrariness that
it gives to logic and the principles of consistent thought. It overlooks the
circumstance that there is an irrefragable element of compulsiveness about the
demand for consistency and those principles that express that demand. It is,
to be sure, correct in questioning the claim of ontologism, that logic is
grounded in the structure of being. But it leaves logic and its principles
wholly unsupported; it finds nothing substantial in which to anchor them and
thereby remove the caprice and arbitrariness that the theory gives them.
There must be a mediating position between the two extremes. The
34
question becomes, then, “What is the locus of the force, the compulsiveness,
that impinges upon thought and demands the consistency of meaning?
It is impossible to escape the role and force of presuppositions. The
fate, which is our fate, that our systems of thought rest upon presuppositions
has not always been breasted. In the past there have been philosophical
systems whose evidential value have been taken for granted. Metaphysical
prejudices that underpin systems have given them their semblance of authority,
which remains intact as long as they lurk in the background without
recognition and admission. The task, then, is not to find a beginning and
foundation for thought that escapes all presupposition, but to push the
necessary presupposition to the largest degree of generality that will provide
the adequate foundation for logic and the demands of its structural
principles.
The presupposition behind which we cannot proceed further is the
presupposition of communication. The intent to communicate with others, and
the activity of that communication, is the final foundation upon which thought
and the canons of logic rest. This presupposition is not something arbitrary.
It is not a mere man-made device that may at some later time be replaced by
other canons. It is a requirement the necessity of which is disclosed in the
very attempt to communicate our meanings. As long as we intend to
communicate, as long as we engage in that activity, these canons must be
accepted and embraced. They are, therefore, the morality of thought. They
are values that must, in their context, be acknowledged.
Wilbur Marshall Urban was an American value theorist who was active
during the first half of the twentieth century. In an article published in
1927, he insisted upon the view that logic is a normative, not a descriptive
science. It describes neither psychological processes nor the structure of
being. He wrote:
Logic, I should maintain, is not, as in so many modern
conceptions, the ultimate science of existence. It remains what it has
always been, the science of correct thinking and of intelligible
35
7W. M. Urban, “Value, Logic, and Reality, in Proceedings of the Sixth
International Congress of Philosophy (New York: Longmans, 1927), p. 288.
W. M. Urban, The Intelligible World: Metaphysics and Value (London: George 8
Allen & Unwin, Ltd., 1929), p. 103.
expression. Its material is not the existent,——that is the material of
the special sciences,——but rather the meanings of the existent. Logic is
the ultimate science of whatever is the case, but only in the sense that
whatever is the case must be capable of being expressed; must, in order
to enter into intelligible discourse, be capable of statement in logical
form. As such, logic is die Moral des Denkens, the science of those
absolute norms or values that must be acknowledged if judgments of truth
and existence are themselves to have any intelligible and communicable
meanings. As such, moreover, it is above all ontologies and all
ontological prejudices.7
The purely normative character of logic does not imply, however, that it
has no bearing upon reality. If we undertake to think and express reality via
the formations of symbolism (as discussed in the foregoing), it is necessary
to respect the canons of logic. Logic thus sustains this measure of
ontological relevance as it governs the process of thought and expression. In
a later work Urban made this point:
[Logic] is the structural form of whatever we can intelligibly express .
. . . Logic can give us the form of intelligible discourse, but never by
itself determine its intelligibility. It can give us the scaffolding of
an intelligible world, but can never by itself determine the ultimate
character of that world.8
Logic, we have argued, is the demand of consistency. And that demand
rests upon the intent to communicate. That intent forces the acknowledgment
of the principle of consistence, and thus, of the laws of thought that inform
logic. This intent, it must be emphasized, is the ultimate postulate of all
thought and knowledge. It cannot be explained, and this because it is the
presupposition of all knowledge and science. It can only be acknowledged.
Thus the law of identity, A / A, or A is equivalent to, means that our
concepts must have a degree of stability and distinctness of content.
Concepts held before the mind must be retained, produced, and recognized as
the same. The law of contradiction, A ~A, or A is not equivalent to Not-A
is the negative side of the law of identity. It stipulates that ideas held
before the mind must be distinguished from one another. To apprehend A is to
36
know that it is not not-A. The law of excluded middle, A v ~A, A or Not-A,
specifies the relation between a concept and its contradictory. It asserts
that of two contradictory assertions one is necessarily true. There can be no
third possibility. In sum, the laws of logic impose the element of
consistency on our concepts and judgments, and this only on the presupposition
of the intent to communicate. But, given that intent, they are obligatory and
binding.
Granted, now, that logic is a normative discipline, that its principles
are obligatory in that they are necessarily grounded on the presupposition of
intelligible communication. They are neither ontological nor merely
conventional. What then, is the status of logic and its normative
prescriptions?
It cannot be said that the principles of logic are symbolic of reality
itself. Symbolism in empirical science, we have previously noted, does not
function as a disclosure of the inner nature of reality as it is in itself.
Its symbolic function consists strictly in its reference to perceptual
experience. Now, if logic is symbolic, its reference must be restricted to
the givenness of experience. That givenness, however, cannot be the
psychological process of thinking. Logic is not symbolic of any psychological
process or fact. Were it so, it would be but a descriptive science and lose
its normative character. Logic does not describe how thought occurs; it
prescribes how thought ought to occur.
Nevertheless, there is a sense in which logic functions with symbolic
reference. The question thus becomes: “What is the nature of that reference?”
The principles of logic express the demand for consistency of thought.
We have developed this point at some length in the above. It is this feature
of thought that creates and governs the symbolism by means of which the
sensory items of experience are formed into the object of experience. The
judgment of identity situates the perceptual items into a context of
relationship and system. It is this system of relationships in which the data
37
9For a detailed account of the theory of objectives, see Alexius Meinong, Über
Annahmen, 2 ed. (Leipzig: Barth, 1910), pp. 52, 60-61, 69, 73. nd
are ordered that constitutes the fixed structure of “objectivity.”
Objectivity is certified in the unity of experience.
Now, this system in which objectivity obtains is, as we have seen, one
of symbolic form. Since the formal demands of logic create and empower the
symbolism in which objectivity is secured, in that respect logic and its
principles possess symbolic reference and significance.
The discipline of logic contains certain principles other than the
traditional laws of thought. Among them is the logical relation of antecedent
and consequent. This is the logical principle that constitutes the symbolic
form of causality according to which the merely sequential items of perception
are resolved into a system of cause and effect. Attention has been given to
this subject in the foregoing. Here it is sufficient to indicate that in this
bearing of the formal principle of antecedent and consequent lies the symbolic
reference of the principle.
There remains, however, the question as to the status of the principles
of logic. What is their mode of being. In some sense they do have reality.
But their reality, we have pointed out, is not of an existential or
ontological order. This order of being is antithetical to their function as
norms. And that they are normative principles is without question.
These principles are, however, ideally objective. In the language of
phenomenology, they are objectives. The idea of a principle of logic has a
“content-element” that is the reference that the idea bears to a definite
object, e.g., the law of identity. This reference is not intrinsic to the
content-element itself, but is established by means of judgment. The judgment
gives direction to the object. The judgment is a “that,” that there is the
law of identity. The is is not the is of existence or reality. Its mode of
being is that of subsistence. It possesses a strictly ideal mode of being.9
38
Mathematics is the second of the formal sciences. We are here concerned
with the concept of space and number, which fall, respectively, within the
rubric of geometry and arithmetic. We turn first to the subject of space and
geometry.
The designation of the object is brought about by the judgment that
fixes it as an identity above and beyond the flux of sensory items. This
situation of the object, however, is within the framework of space.
Now the term space has at least three possible meanings: intuitive
space, expressive space, and geometrical space. Here we must consider these
meanings, if we are to come to some understanding of the symbolic nature of
space.
Intuitive space is the space of perception. It is bound up inextricably
with one’s awareness of the body and the activity of speaking. The dimension,
position, and direction of objects are determined in reference to the body.
The object is large or small in comparison with the body, is here or there in
reference to the body, is in this or that direction from the standpoint of the
body. The parameters of the intuitive manifold are constituted from the
center of the situation of the body. Further, when a person speaks to another
person, the intuition of position and direction accompanies that act of
language. I am “here,” the other person is “there”, and “there” is in that
direction. Even here there is the beginning of a new standpoint. A bond is
established between the subject and the object. It is a bond that brings the
two together and yet keeps them separate. The positing of this twofold
relationship yields an intuitive space where the factors of separateness and
juxtaposition, of discreteness and combination stand in tensional equilibrium.
Precisely this is the beginning of the representation of space as a schema of
ideal relationships. Even in its incipiency the schema functions as the
symbol of the real relationships in which things stand. The beginning of a
spatial, a measurement, relation between them has been established, albeit but
rudimentary and tentative.
39
Cassirer, op. cit., p. 95. 10
Expressive space is the space of myth. It, like intuitive space, is
bound to the object-world. The postulates of Euclidean space are continuity,
infinity, and uniformity. None of these are realized in the character of
perception. There is no perception of infinity, since it is necessarily
confined within spatial limits. The same holds true for homogeneity.
Geometric space is homogeneous in that its associated elements are mere
determination of positions that have no independent content of their own
outside of that relationship. The reality of the elements consists strictly
in their reciprocal relation. In mythical space position and content cannot
be separated. Position is always the position of a definite individual
sensuous or intuitive content.
Although the space of myth is always object oriented, it is analogous to
geometrical space in that it operates as a schema. It provides a schema in
which to bring otherwise opposed elements into a relationship. In so doing,
it marks off zones into which to place various groups of items or elements.
The articulation is complex and lacks any selective and ordering principle.
Totemism is a rudimentary form of this articulation. The Zuûis divide space
into seven zones: north and south, east and west, the upper and lower worlds,
and, finally, the center of the world. Every reality occupies its respective
zone.
In science the principle of articulation is made on the basis of ideal,
logical form. It “. . . establishes a determinate spatial order by relating
the sensuous diversity of impressions to a system of purely logical, purely
ideal, forms. In mythical space, the principle of articulation is feeling. 10
Feeling provides a value accent according to which the sacred is separated
from the profane. The fundamental spatial zones are the sacred and profane.
And these zones are not abstract zones; on the contrary, they are the abode of
good and evil forces, of gods and demons. Here, as in intuitive space, space
40
is bound to the substantive.
There is a foundation in perception that gives rise to the mythical
opposition of the sacred and profane. That foundation is the perception of
light. Thus light and space are intimately associated together. The
directions of east, west, north, and south are not ideal forms, but entities
with a life and a value of their own. The gods are the gods of direction: of
north and south, of east and west, and of the lower and upper world. Each
zone has a specific value quality: divine or demonic, friendly or hostile,
holy or unholy. The east is the source of light and thus of life itself. The
west, where the sun sets and darkness sets in, is the place of terror and
death.
Geometrical space is the full development of representative space, whose
beginning, we have earlier noted, lies in intuitive space itself. Geometrical
space is posited when definite perceptions are singled out and selected as
fixed points of reference. The flux of sensory items is halted and certain of
these elements are placed in an enduring pattern. The pattern, when viewed as
pattern, is the abstract schema of geometrical space, and, unlike intuitive
space, assumes the attributes of continuity, infinity, and uniformity. It
functions to represent, for example, the real form of an object, which until
fixed by the analytic and synthetic power of thought is but a kaleidoscopic
flux of sensations. It functions, also, to represent the actual relations
that obtain between those objects. It is thus that geometrical space is
representative. In short geometrical space is a symbolic construction. And,
as elsewhere, it does not symbolize an absolute, ontological reality. It
holds only for the formation of experience. Its form is subjective.
Some attention should be given to the subject of Euclidean and non-
Euclidean geometries. Both orders of geometry are symbolic in nature. The
differences that separate the geometries do not affect this factor. The basis
upon which the differences rest is the frame of reference in which they
function. Euclidean geometry rests on the presupposition of a flat surface.
41
Cassirer, op. cit., I, 229. 11
Given this presupposition, the postulates and theorems of Euclidian geometry
“fit,” i.e., symbolize, the world of intuited space. But they do not entirely
fit the presupposition of a spherical space. Riemann’s geometry, for example,
redefines Euclid’s definition of a straight line. In spherical space
straight lines are the “greatest circles,” which are, therefore, the shortest
distance between two points. Since they all intersect, there are no parallel
lines in this geometry. This geometry, too, symbolizes, not a lived intuition
of spatiality, but a constructed spatiality. The final step in thinking about
the nature of geometrical concepts was taken by David Hilbert. He proposed a
formalism according to which all basic concepts are nothing but symbols that
may designate anything from spatial entities to, as he put it, “angelic
holiness.”
The second branch of mathematics to which we turn our attention is that
of number and arithmetic. As in the case of space and geometry, there are
both sensuous and mythical beginnings to number and arithmetic.
For primitive man, the body is the basis and foundation of enumeration.
In this regard, Cassirer notes:
The differentiation of numbers starts, like that of spatial relations,
from the human body and its members, thence extending over the whole of
the sensuous, intuitive world. Everywhere man’s own body provides the
model for the first primitive enumeration: the first “counting” consists
merely in designating certain differences found in external objects, by
transferring them, as it were, to the body of the counter and so making
them visible.11
Among these primitive peoples, there are various counting gestures. For
example, the Ewe count on their outstretched fingers, beginning with the
little finger of the left hand, then continuing with the fingers of the right
hand. They then squat on the ground and count on their toes.
Language reflects the bodily gesture of counting. The Klamath Indians
employ a variety of numerals formed from verbs of various bodily actions. One
class of objects is placed on the ground to be counted; another class, piled
42
Ibid, pp. 241-49. 12
in layers, and so forth.
Now what is significant of this bodily, intuitive method of counting is
that a prescribe order, according to which objects are placed, is instituted.
While the order is an intuitive schema, it nevertheless is the indispensable
groundwork for the purely intelligible schema of number.
There is a more basic and essential role of the body in the development
of the concept of number. Cassirer states that the idea of number does not
originally arise from the perception of the placement of objects. It arises
from the awareness of the opposition of the self and the other, the “I” and
the “thou.” Many primitive languages indicate that this opposition advances
quickly to enumeration: from “one” to “two.” The number three develops with
the “person spoken of” is added. But beyond the number three, there are no
additional numbers in the number “schema.”12
For the mythical consciousness, mythical number, like mythical space, is
attached to sensuous content. Number is not yet a universal specification
applicable to any content whatsoever. To note an example: we have earlier
seen that the expressive space of myth articulates space into zones, the
fundamental zones being those of the sacred and profane. On the basis of the
experience of light, the directions of east, west, north, and south become
entities with their own life and value. Each zone has its own specific value
quality. Now, these four directions, or spatial zones, are given a numerical
value, the value of the number four. The four cardinal points of the world
embrace the structure of the world and the world process. Thus the number
four becomes the sacred number par excellence, expressing the fundamental form
of the universe. And any particular thing that has a four-fold organization
becomes a sacred object. Thus the number is not an abstract schema that may
be applied to anything whatsoever, but is rather an intuitive schema that,
expressing the nature of the universe, attaches to specific entities. And
43
Ibid., II, 141. 13
there are other numbers that likewise relate to the content of perception.
For the theoretical consciousness, number is disassociated from the
given content of sensuous experience. It becomes an abstract and intelligible
schema, an entity that is absolutely homogeneous and uniform. It embraces
various and dissimilar things and connects them withing the unity of the
concept.
And just as number here serves as the true logical instrument for
creating a homogeneity of the contents of consciousness, so number itself
develops more and more into an absolutely homogeneous and uniform entity.
The particular numbers disclose no differences over against one another,
other than those arising from their position in the system as a whole.
They have no other being, no other character and nature, than that which
comes to them through this position, in other words through the relations
within an ideal aggregate. . . . for mathematical thought numbers are
nothing but an expression of conceptual relations; only in their totality
do they represent the self-enclosed and unitary structure of number as
such and of the realm of number.13
Number is constituted through the synthetic activity of thought. This
synthetic activity, through which number and the number series is constituted,
means that ordinal number is logically prior to and thus the foundation of,
cardinal number. We arrange the objects of our perception in a certain order:
a first “something,” a second “something,” and so on. There is then a “one”
and a “two,” and so on.
Now the question becomes: granted that there are two orders of numbers,
ordinal and cardinal, and that they are created by a synthetic and irreducible
act of the mind, in what sense do numbers “exist”? The answer to this
question is largely framed in terms of two opposing views: formalism and
intuitionism.
Formalism is associated with the German mathematician David Hilbert
(1862-1943), who was Professor of Mathematics at the University of Göttingen.
According to him, numbers are but signs. They do not represent anything
beyond themselves. They do not signify determinate ideal relationships. Pure
mathematics is like chess, nothing but a game with arbitrarily devised rules.
44
David Hilbert, Neubegründung der Mathematik, Abhandlungen zus dem 14
mathematischen Seminar der Hamburgischen Universität, 1, 162. Quoted from Cassirer,
op. cit., III, 380.
He writes:
I find the objects of the theory of numbers in the signs themselves,
whose form we can recognize universally and surely, independent of place
and time and of the special conditions attending the production of the
signs as well as of insignificant differences in there elaboration. Here
lies the firm philosophical orientation, which I regard as requisite to
the grounding of pure mathematics, as well as to all scientific thinking,
understanding, and communication. “In the beginning,” we may say here,
was the sign.14
The function of the mathematician is to insure, so far as possible, that
the formal system of signs is free from internal contradictions. In this
respect, and only in this respect, is a mathematical system true. Beyond
this, the mathematician and mathematics have no further role and import.
There is nothing beyond the system of signs to which that system can orient
itself.
Hermann Weyl proposed a remedy for the defect within a purely formal
mathematics. Weyl was born in Germany in 1885 and died in Switzerland in
1955. He had received his doctorate in mathematics under Hilbert, and taught,
among other places, at Göttingen and Princeton. He advanced a theory of
intuitionism.
He argued that mathematical signs have transient meaning, i.e., they
refer to realities beyond themselves. The signs, he says, are symbols. Their
symbolic reference is twofold: they function as symbols of the order in which
physical events subsist and as symbols of a metaphysical world. Of the
physical application of mathematical symbols, he states:
I do not find it [symbolism] unless I let mathematics fuse entirely with
physics and assume that the mathematical concepts of number, function,
etc. [or Hilbert’s symbols] fundamentally partake in the theoretical
construction of the real world, in the same way as do the concepts of
energy, gravitation, the electron, etc.
Yet Weyl believed that something more is required than the physical
reference of mathematical symbols. The components of mathematics must have
independent meaning beyond their physical application. Without that meaning
45
Cf. Weyl, “Philosophie der Mathematik,” pp. 53 ff.; Symposion, I, 30, ff. 15
Quoted from Cassirer, op. cit., III, 382.
they could not even function in reference to the physical. They must have a
life of their own. But what that life is, he said, is something that we
cannot see or know. It is something in which we must believe:
In theory, consciousness succeeds in “jumping over its own shadow,” in
leaving given matter behind it and in representing the transcendent; but
it goes without saying that this can be accomplished only in the symbol.
Theoretical formulation is something other than intuitive insight; its
aim is no less problematic than that of artistic formation. Over the
idealism that is destined to destroy epistemologically naive, there rises
a third realism. . . . If I designate phenomenal insight as knowledge,
then theoretical insight rests on faith——faith in the reality of one’s
own ego and that of others, or in the reality of the outside world or of
God.15
But now what of this “third realism,” according to which mathematical
signs are symbolic representations of transcendent objects? Is this the
symbolic import of mathematics? At this point, Weyl is correct. If this
theory of realism be true, its truth is beyond the certification of reason.
It is a matter of faith only.
Now, mathematical signs are symbolic, but not in a realistic sense. A
theory that makes the symbol symbolize something absolutely and utterly
transcendent, beyond the pale of human ken, collapses upon itself. It is
self-stultifying. For there is, as we have earlier argued, no way in which it
can be known that there is indeed a reality beyond the symbolic function or
that the symbol does indeed symbolize the alleged transcendent reality.
This brings us, again, to the question of the basic nature of symbolism.
Language, physical science, logic, and mathematics are constructions of
meaning developed in accordance with formative laws of their own. The
traditional metaphysics propounds a dualism of two worlds: the world of
“immanence” and the world of “transcendence.” It defines the symbol as
belonging exclusively to the immanence of consciousness. Its function is to
mirror an independent, transcendent reality. But that is not, and cannot be,
the role of symbolism. Instead, the symbol informs its content in virtue of
46
its distinctive formative principle, and that content is, in its own
subjective form, the representation of reality. That is the reality. That is
the only reality with which we have to do. Reality is available only as
disclosure within the symbolic formation, and not as an external something to
be mirrored. In the language of Plato, the symbol is a mode of “growing into
being” (ãÝíåóéò åÆò ïÛóÝáí).
The objects of mathematics are neither free-floating signs nor
transcendent objects. Rather they are objects of thought-formation. They
have their place in the formative process of thought. The logical world
consists of the categorical determinations of unity and otherness, identity
and difference. It determines a world of pure, self-identical ideas
distinguishable from one another. The mathematical world carries these pure
logical determinations further, constituting the relations through and in
which the objects of experience are ordered. Mathematical objects,
accordingly, constitute the transition from the world of pure thought to the
world of the empirical, physical object. Behind and within these activities
is the fundamental unity of thought itself, which brings out of its depth the
relational structures that yield the various orders of reality with which we
have to do. And it is precisely this “yielding” in which symbolism consists
and in which, accordingly, the world is presented as a knowledgeable world.
The careful reader will be troubled with this view of the nature of
symbolism in science. Does not this view of the subject reduce
objectivity——the objectively real world——to but a factor of our own subjective
experience? Is reality just what we experience and apprehend? Is there
nothing more?
This consequence is not forthcoming from the circumstance that thought
yields the relational structures of symbolism. Now, indeed, thought is of an
object. It is intentional in its nature; it intends an object. Thus a
presupposition of reality is a demand of thought. The presupposition of
47
Arthur Eddington, The Philosophy of Physical Science (New York: The Macmillan 16
Co., 1939), p. 159.
Ibid., p. 3. 17
reality as such, or überhaupt, is a necessary presupposition of all thought
and communication. But that presupposition cannot be realized in terms of any
definite formation of experience. Our human predicament is that we are
confined to the “community of subjective form.” That is, we are held fast to
the synthetic and relational forms in which our experience and experienced
world are ordered. The notion that these symbolic forms somehow grow beyond
themselves and literally disclose the inner nature of physical reality is
untenable. But what is tenable is that it is these very forms, constructed
from the spirituality of thought, that yield the ordering of experience that
we call understanding and knowledge. And that very ordering is the
symbolization that we call science. Beyond this symbolic function we cannot
proceed. The truth that we obtain is found and certified, not in any
isomorphism between symbol and presupposed reality, but in the continuing and
on-going building up of a theoretical structure in which its elements are
bound by the demands of consistency. For, again, it is in that structure that
the empirical object-world is located. That, and that alone, is the object
world of empirical science.
The English philosopher-scientist Arthur Eddington expresses the
symbolic nature of physical science in the following manner:
The physical universe is the world which physical knowledge is formulated
to describe, and there is no difference between the physical universe and
the universe of physics.16
Again:
The few who have attempted to give it [“really exists”] a definite
meaning do not always agree on the meaning. By defining the physical
universe and the physical objects that constitute it as the theme of a
specified body of knowledge, and not of things possessing a property of
existence elusive of definition, we free the foundations of physics from
suspicion of metaphysical contamination.17
In this or another of his books, Eddington constructs what may be termed
48
a “Robinson Crusoe” illustration. A man is cast on an island that he
initially takes to be uninhabited by other human beings. He wanders along the
shore and sees no other human individual. Soon he comes on human footprints
in the sand. He is elated, believing that he has found evidence of the
existence of another person. Some time later, after recognizing the
reappearance of familiar surroundings, he realizes that he is walking in his
own footprints. “. . . there is no difference between the physical universe
and the universe of physics.”
