Carnap’s early dissertation on the three meanings of space
A Dissertation Spanning Three Disciplines
Space — Der Raum: Ein Beitrag zur Wissenschaftslehre, or “Space: A Contribution to the Theory of Science” — is Rudolf Carnap’s doctoral dissertation, written at Jena under the neo-Kantian philosopher Bruno Bauch and issued in German as a supplementary monograph in the Kant-Studien series. It belongs to the philosophical foundations of geometry, but its reach is deliberately wider: Carnap set out to supply a logical basis for a theory of space and time in physics, and the work is best understood as an attempt to build a bridge between three fields that had grown used to talking past one another — geometry, physics, and philosophy.
The problem Carnap confronted was as much terminological as mathematical. Mathematicians, physicists, and philosophers all reasoned about “space,” yet the concepts they attached to that single word were not the same. Carnap’s central objective was to show that the apparent conflicts between their theories of space did not reflect genuine disagreement about the world, but rather the fact that each side was discussing something different under a shared name.
Three Senses of a Single Word
The argument turns on a threefold distinction. Formal space is space in the mathematical sense: an abstract system of relations, a purely formal-logical order structure that carries no intrinsically spatial meaning at all. Intuitive space is the space of perception and intuition — something akin to Kant’s form of intuition, and one for which Carnap also drew on Husserl’s notion of essential insight. It consists of contents of intuition that are independent of any single experience, and unlike formal space it does possess an intrinsically spatial, perceptual meaning. Physical space, finally, is the space of actual spatial facts given in experience: the space that physics describes, exemplified in Carnap’s day by the space of Einstein’s general theory of relativity, which is no longer treated as Euclidean.
Because these are three distinct notions, they carry three distinct kinds of knowledge and demand three distinct kinds of investigation. The confusions that had grown up around the topic, Carnap argued, dissolve once the ambiguity of the word is exposed.
Measurement as Convention, Structure as Given
Within physical space Carnap drew a further line between what experience supplies and what is chosen. Following Poincaré and others, he held that the metrical structure of physical space — the system of measurement that fixes distances and curvature — rests on a stipulation or convention rather than on something forced upon us by the facts. That convention is not, however, arbitrary: it is constrained by a perceptually given factual basis of spatial structure that is purely topological, the intuitive facts of order, continuity, and dimension that any metric must accommodate.
This leads to the dissertation’s sharpest formulation: the topological structure of intuitive space constitutes a necessary form, while the metrical structure of physical space is an optional form. The conclusion Carnap drew is that only the necessary form can serve as a condition of the possibility of experience — and that this form is the topological intuitive space of arbitrarily many dimensions, taken together with the corresponding type of formal space.
The First Appearance of a Philosophical Programme
The dissertation is more than a contribution to the space debate. It is the point at which the guiding themes of Carnap’s later philosophy first surface, above all the conviction that many philosophical contradictions arise from a misuse of language, and the corresponding insistence on distinguishing formal from material modes of speech. In treating the quarrels among mathematicians, physicists, and philosophers as a confusion produced by vocabulary rather than by nature, Space already displays the diagnostic style that would define his mature work.
Carnap attended the University of Jena from 1910 to 1914, intending to write a thesis in physics. There he intently studied Immanuel Kant’s Critique of Pure Reason in a course taught by Bruno Bauch, and he was one of the very few students to attend Gottlob Frege’s courses in mathematical logic. Frege’s course exposed him to Bertrand Russell’s work on logic and philosophy, which gave a sense of direction to his studies; he accepted the effort to surpass traditional philosophy with logical innovations that inform the sciences.
Carnap worked on the doctoral thesis from 1919 to 1921. An earlier thesis of his defined an axiomatic theory of space and time; the physics department said it was too philosophical, and Bruno Bauch of the philosophy department said it was pure physics. Carnap then wrote another thesis in 1921, under Bauch’s supervision, on the theory of space in a more orthodox Kantian style.
Publication and Translation
Der Raum was published in Berlin by Reuther & Reichard in 1922. The catalog record describes the work as 87 pages, with diagrams, 24 cm, under the subject “Space and time,” and as no. 56 in the Kant-Studien series.
An English translation titled “Space: A Contribution to the Theory of Science” exists in a 2005 draft; the published version appeared in The Collected Works of Rudolf Carnap, Volume 1: Early Writings (2019), pp. 21–208, ISBN 978-0-19-874840-3. Oxford University Press is publishing a fourteen-volume set of Carnap’s collected works, organized by general editor Richard Creath. The first volume, Early Writings, appeared in 2019, and the seventh volume, Studies in Semantics, appeared in 2024.
