Excerpts from "The Concept of Group and the Theory of Perception,"

by Ernst Cassirer, *Philosophy and Phenomenological Research*,

Volume V, Number 1, September, 1944.

(Published in French in the *Journal de Psychologie*, 1938, pp. 368-414.)

The group-theoretical interpretation of the fundaments of geometry is,

from the standpoint of pure logic, of great importance, since it enables us to

state the problem of the "universality" of mathematical concepts in simple

and precise form and thus to disentangle it from the difficulties and ambigui-

ties with which it is beset in its usual formulation. Since the times of the

great controversies about the status of universals in the Middle Ages, logic

and psychology have always been troubled with these ambiguities….

Our foregoing reflections on the concept of *group* permit us to define more

precisely what is involved in, and meant by, that "rule" which renders both

geometrical and perceptual concepts universal. The rule may, in simple

and exact terms, be defined as that *group of transformations* with regard to

which the variation of the particular image is considered. We have seen

above that this conception operates as the constitutive principle in the con-

struction of the universe of mathematical concepts….

…Within Euclidean geometry,

a "triangle" is conceived of as a pure geometrical "essence," and this

essence is regarded as invariant with respect to that "principal group" of

spatial transformations to which Euclidean geometry refers, viz., displace-

ments, transformations by similarity. But it must always be possible to

exhibit any particular figure, chosen from this infinite class, as a concrete

and intuitively representable object. Greek mathematics could not

dispense with this requirement which is rooted in a fundamental principle

of Greek philosophy, the principle of the correlatedness of "logos" and

"eidos." It is, however, characteristic of the modern development of

mathematics, that this bond between "logos" and "eidos," which was indis-

soluble for Greek thought, has been loosened more and more, to be, in the

end, completely broken….

…This process has come to its logical

conclusion and systematic completion in the development of modern group-

theory. Geometrical *figures* are no longer regarded as fundamental, as

date of perception or immediate intuition. **The "nature" or "essence" of a**

figure is defined in terms of the *operations* which may be said to

*generate *the figure. The operations in question are, in turn, subject to

certain group conditions….

…What we

find in both cases are invariances with respect to variations undergone by

the primitive elements out of which a form is constructed. The peculiar

kind of "identity" that is attributed to apparently altogether heterogen-

eous figures in virtue of their being transformable into one another by means

of certain operations defining a group, is thus seen to exist also in the

domain of perception. This identity permits us not only to single out ele-

ments but also to grasp "structures" in perception. To the mathematical

concept of "transformability" there corresponds, in the domain of per-

ception, the concept of "transposability." The *theory *of the latter con-

cept has been worked out step by step and its development has gone through

various stages….

…By the acceptance of

"form" as a primitive concept, psychological theory has freed it from the

character of *contingency* which it possessed for its first founders. The inter-

pretation of perception as a mere mosaic of sensations, a "bundle" of simple

sense-impressions has proved untenable….

…In the domain of mathematics this state of affairs mani-

fests itself in the impossibility of searching for invariant properties of a

figure except with reference to a group. As long as there existed but *one*

form of geometry, i.e., as long as Euclidean geometry was considered as the

geometry *kat' exochen* this fact was somehow concealed. It was possible

to assume *implicitly* the principal group of spatial transformations that lies

at the basis of Euclidean geometry. With the advent of non-Euclidean

geometries, however, it became indispensable to have a complete and sys-

tematic survey of the different "geometries," i.e., the different theories of

invariancy that result from the choice of certain groups of transformation.

This is the task which F. Klein set to himself and which he brought to a

certain logical fulfillment in his *Vergleichende Untersuchungen ueber neuere*

geometrische Forschungen….

…Without discrimination between the

accidental and the substantial, the transitory and the permanent, there

would be no constitution of an objective reality.

This process, unceasingly operative in perception and, so to speak, ex-

pressing the inner dynamics of the latter, seems to have come to final per-

fection, when we go beyond perception to enter into the domain of pure

thought. For the logical advantage and peculiar privilege of the pure con –

cept seems to consist in the replacement of fluctuating perception by some-

thing precise and exactly determined. The pure concept does not lose

itself in the flux of appearances; it tends from "becoming" toward "being,"

from dynamics toward statics. In this achievement philosophers have

ever seen the genuine meaning and value of geometry. When Plato re-

gards geometry as the prerequisite to philosophical knowledge, it is because

geometry alone renders accessible the realm of things eternal; *tou gar aei*

ontos he geometrike gnosis estin. Can there be degrees or levels of objec-

tive knowledge in this realm of eternal being, or does not rather knowledge

attain here an absolute maximum? Ancient geometry cannot but answer

in the affirmative to this question. For ancient geometry, in the classical

form it received from Euclid, there was such a maximum, a *non plus ultra*.

But modern group theory thinking has brought about a remarkable change

In this matter. Group theory is far from challenging the truth of Euclidean

metrical geometry, but it does challenge its claim to definitiveness. Each

geometry is considered as a theory of invariants of a certain group; the

groups themselves may be classified in the order of increasing generality.

The "principal group" of transformations which underlies Euclidean geome-

try permits us to establish a number of properties that are invariant with

respect to the transformations in question. But when we pass from this

"principal group" to another, by including, for example, affinitive and pro-

jective transformations, all that we had established thus far and which,

from the point of view of Euclidean geometry, looked like a definitive result

and a consolidated achievement, becomes fluctuating again. With every

extension of the principal group, some of the properties that we had taken

for invariant are lost. We come to other properties that may be hierar-

chically arranged. Many differences that are considered as essential

within ordinary metrical geometry, may now prove "accidental." With

reference to the new group-principle they appear as "unessential" modifica-

tions….

… From the point of view of modern geometrical systematization,

geometrical judgments, however "true" in themselves, are nevertheless not

all of them equally "essential" and necessary. Modern geometry

endeavors to attain progressively to more and more fundamental strata of

spatial determination. The depth of these strata depends upon the com-

prehensiveness of the concept of group; it is proportional to the strictness of

the conditions that must be satisfied by the invariance that is a universal

postulate with respect to geometrical entities. Thus the objective truth

and structure of space cannot be apprehended at a single glance, but have to

be *progressively* discovered and established. If geometrical thought is to

achieve this discovery, the conceptual means that it employs must become

more and more universal….