Construction of new Objects; Views of R. Dedekind,
H. Hankel, R. Stolz.

DEDEKIND gives the name section to a division of the rational
number system into two classes such that any number in the first
class is smaller than any number in the second; and he shows
that every rational number generates a section, or properly
speaking two sections, but that there are sections not generated
by any rational number. He then goes on to say (§ 4, p. 14): ^A

'Now whenever we are presented with a section, (A1, A2)
not generated by any rational number, we construct a new,
irrational number a, which we regard as completely defined by
this section; we shall say that the number a corresponds to this
section, or generates this section.'

It is in this construction that the heart of the matter lies. We
must first notice that this procedure is quite different from
what is done in formalist arithmetic--the introduction of a new
sort of figures and special rules for manipulating them. There
the difficulty is how to tell whether these new rules may turn
out to conflict with those laid down previously and how to
straighten out such a conflict. Here we are concerned with the
question whether construction is possible at all; whether, if it is
possible, it is unrestrictedly possible; or whether certain laws must
be observed when we are constructing. In the last case it would
first have to be proved that the construction was justified in
accordance with these laws, before we might perform the act of
creation. These inquiries are her completely lacking, and thus
there is lacking the main thing--what the proofs carried out by
means of irrational numbers depend upon for their cogency.

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