Such reasons [moreover] ill agree with the mathematical sci-
ences' much vaunted claims of exactitude. It should also be
observed that they tend to attribute an indefinite extent to
algebraic formulas when, in fact, these formulas hold under
certain conditions and for only certain values of the variables
involved. In determining these conditions and these values
and in settling in a precise manner the sense of the notation
and the symbols I use, I eliminate all uncertainty. ... It is
true that in order to remain faithful to these principles, I
sometimes find myself forced to depend on several proposi-
tions that perhaps seem a little hard on first encounter . . . .
But, those who will read them will find, I hope, that such
propositions, implying the pleasant necessity of endowing the
theorems with a greater degree of precision and restricting
statements which have become too broadly extended, will
actually benefit analysis and will also provide a number of
topics for research, which are surely not without importance.
I dedicate this book to the memory of my parents in gratitude for many
years of love and laughter.
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