Notes prepared by
March 13, 2001
What are numbers, and what is their meaning?:
Richard Dedekind (1831–1916)
1872 - Continuity and irrational numbers
1888 - What are numbers, and what is their meaning?
Let us recall that by 1850 the subject of analysis had been given a solid
footing in the real numbers — infinitesimals had given way to small positive
real numbers, the ε’s and δ’s. In 1858 Dedekind was in Zürich, lecturing
on the differential calculus for the first time. He was concerned about his
introduction of the real numbers, with crucial properties being dependent
upon the intuitive understanding of a geometrical line.1 In particular he was
not satisfied with his geometrical explanation of why it was that a monotone
increasing variable, which is bounded above, approaches a limit. By Novem-
ber of 1858 Dedekind had resolved the issue by showing how to obtain the
real numbers (along with their ordering and arithmetical operations) from
the rational numbers by means of cuts in the rationals — for then he could
prove the above mentioned least upper bound property from simple facts
about the rational numbers. Furthermore, he proved that applying cuts to
the reals gave no further extension.
These results were first published in 1872, in Stetigkeit und irrationale
Zahlen. In the introduction to this paper he points out that the real number
system can be developed from the natural numbers:
I see the whole of arithmetic as a necessary, or at least a natu-
ral, consequence of the simplest arithmetical act, of counting, and
counting is nothing other that the successive creation of the infi-
nite sequence of positive whole numbers in which each individual is
defined in terms of the preceding one.
In a single paragraph he simply states that, from the act of creating
successive whole numbers, one is led to the concept of addition, and then to
1Recall that in geometry some mathematicians had already taken efforts to eliminate
the dependence of the proofs on drawings.
multiplication. Then to have s