VOL. IV (XXI) 2009 - Departamentul de Filosofie si Stiinte ale ...

72 ON WITTGENSTEIN AND THE IMAGINARY NUMBER
Wittgenstein's additional references about “√–1” could be found also in
works like Remarks on the Foundations of Mathematics 3 (Bemerkungen über
die Grundlagen der Mathematik). In those comments or remarks, which don't
have a limit being only ascertaining or critical on this irrational number (which
is still used in various natural sciences such as phisics, statistics etc.), can
easily capture how Wittgenstein seems to approach this interesting number,
which is like an aporia to what could be named mystique (par excellence).
But before presenting this “lemmatic plurality” of the “√–1” posibilities
("plurality" which in part seems to be strange for the current understanding of
mathematics), I will try to present briefly, a sort of "historiography" of this “√–
1”. Therefore square root of a negative number is represented as
mathematical conventionalism by the letter "i". Thus the several centuries
research for the most effective methods to solve equations with an
increasing degree of difficulty, in Italy Bologna, dissatisfied Rafael Bombelli
(1526 - 1572), with the mathematical demonstrations from Ars magna
written by the great encyclopedic scholar, Hieronymus Cardanus (Gerolamo
Cardano, from 1501 to 1576), so he writes L'Algebra, his important work
written during two decades, in which it is shown:
...that we can not avoid the square roots of negative numbers.
But that requires a mental leap. After all, what is the square root
of –1 ? Obviously, no normal number (real) multiplied by itself
does not –1, given that even multiplying a negative number with
itself gives a positive result. However cubic equation solution
sometimes produces as intermediate step, a square root of a
negative number, even if the end result was a real number.
Cardano, who was intrigued by these "sophistic" numbers
concluded that they were "so subtle, that were unnecessary,
and they had to use in computing, said that one is "ignoring the
3 Ludwig Wittgenstein, Remarks on the Foundations of Mathematics, Third Edition,
Basil Blackwell Oxford, 1978, Reprinted 1989, 1994, 1998, 2001or Bemerkungen
über die Grundlagen der Mathematik, Werkausgabe Band 6, Herausgegeben von G.
E. M. Anscombe, Rush Rhees, G. H. von Wright, Suhrkamp; PART/Teil V, 5, pag.
261, 262;