17. Logical Space and World of Analytical Philoso-
phy. Georg Cantor, the creator of set theory, fruitfully
applied his theory to what he termed “point sets,” that
is general aggregates of points in general spaces, and
he somehow began to view such an aggregate of points,
as a “subspace” of the general space and then as a
space in its own right. In consequence of this, in work-
ing mathematics of the twentieth century, the concepts
of “general set,” “general point set,” and “general
space” have gradually become nearly synonymous.
Thus, in the theory of probability and statistics, a
“probability space” is a general set in which, subject
to appropriate rules, certain subsets have been marked
off as “events.” With each event there is associated
a probability value which is a non-negative real num-
ber between 0 and 1, and the total set has probability
value 1. If, in Aristotelian terminology, a general set
is a probability space not actually but only potentially,
that is, if the set is not given as a probability space
but is only supposed and expected to be one and is
analyzed accordingly, then, relative to such an analysis,
the general set is called a “sample space.”
Such developments were not limited to mathematics
and science. Thus the Tractatus logico-philosophicus
(German edition 1918, first English translation 1922)
of the linguo-philosopher Ludwig Wittgenstein makes
statements about a “logical space” (logischer Raum),
and, by the text of the
Tractatus, this space is some
kind of aggregate or congeries of logical entities like
“facts,” “atomic facts,” “states of affairs,” “proposi-
tions,” etc. Some commentators of the
Tractatus ascribe
to this space some specific structural features, but even
these features are not very geometrical in a traditional
sense. The
Tractatus also refers to a “world” or “uni-
verse” (
Welt). This universe has some ontological traits,
and in a sense the logical space is but a background
space to it. Nevertheless, the logical space seems to
be more primary than the world, inasmuch as the
constituents of the world are only some kind of “picto-
rial” representation of the constituents of the logical
space.
The air of indeterminacy and vagueness which ad-
heres to the notions of space and world in the Tractatus
is indicative of the fact that Wittgenstein was never
greatly interested in these notions as such; in his later
work, the linguo-analytical Philosophical Investigations
(1953), these notions hardly occur at all. Also, in the
Tractatus Wittgenstein asserted, quite unnecessarily,
that his logical space is “infinite”; this was simply a
standard philosophers' assertion since Giordano Bruno,
and nothing more. This reduced interest in space was
not an innovation of linguo-philosophers but was a
neo-Hegelian trend in which even “phenomenologists”
like Edmund Husserl shared.
The ontological traits of the world of the Tractatus
could be taken straight out of the universe of
Parmenides, which fused Being with Thought, and
added some dosage of Truth (Aletheia); except that the
Truth in Parmenides, although already “two-valued,”
was still ontological rather than logical. But with re-
gard to the question of the size of the universe,
Parmenides made the splendid assertion, which beauti-
fully conforms with twentieth-century cosmology, that
his “sphere” is both “finite” and “complete” (Bochner,
The Size of the Universe). This assertion of Parmenides
was so subtle that even his leading disciple Melissus
of Samos did not comprehend it at all, and—to
Aristotle's uncontrollable chagrin—made the universe
infinite instead. The great handicap of Parmenides was,
that, as a Greek, he did not have the concept, or even
percept, of a background space in his thinking. There-
fore he could not separate his universe into a “space”
and a “world,” and it is this which makes his fusion
of ontology with physics so puzzlingly “antiquarian.”
In developments since the Renaissance, the first
aspects of a “logical” world are discernible in pro-
nouncements of Leibniz, even in his pronouncements
about a world which purports to be a best possible
one. The innovation of Leibniz was not at all that he
fused physics with metaphysics—this was done by
everybody, even including Kant, his hottest pro
testations notwithstanding—but that he added logic as
well, and that this logic, as a partner of physics and
metaphysics, had an equal standing with them. It is
a fusion of logical ordering with metaphysical being,
and not some specific achievements in logical theory,
which makes Leibniz a precursor of analytical philoso-
phy of the twentieth century, and which makes his
universe of “monads,” however permeated with meta-
physics, congenial and even challenging to many an
analytic “skeptic” of today.
