Hans Reichenbach and Euclid’s Axioms

In Euclid’s work, the geometrical achievements of the ancients reached their final form: geometry was established as a closed and complete system. The basis of the system was given by the geometrical axioms, from which all theorems were derived. The great practical significance of this construction consisted in the fact that it endowed geometry with a certainty never previously attained by any other science. The small number of axioms forming the foundation of the system were so self-evident that their truth was accepted without reservation. [ … ]

Euclid’s axiomatic construction was also important in another respect. The problem of demonstrability of a science was solved by Euclid in so far as he had reduced the science to a system of axioms. But now arose the epistemological question how to justify the truth of those first assumptions. If the certainty of the axioms was transferred to the derived theorems by means of the system of logical concatenations, the problem of the truth of this involved the construction was transferred, conversely, to the axioms. It is precisely the assertion of the truth of the axioms which epitomizes the problem of scientific knowledge, once the connection between axioms and theorems has been carried through.

Hans Reichenbach, The Philosophy of Space and Time, trans. Maria Reichenbach and John Freud (Dover Books: 1958) pp. 1-2


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