- NUMBER SYSTEMS 205
can define arithmetic operations on such classes so that the resulting system has
all the properties we want the real numbers to have, especially the essential one
for calculus: continuity. Dedekind claimed that in this way he was able to prove
rigorously for the first time that y/2</3 = ÷/ä.^14
The practical-minded reader who is content to use approximations will probably
be getting somewhat impatient with the discussion at this point and asking if it
was really necessary to go to so much trouble to satisfy a pedantic desire for rigor.
Such a reader will be in good company. Many prominent mathematicians of the
time asked precisely that question. One of them was Rudolf Lipschitz (1832-1903).
Lipschitz didn't see what the fuss was about, and he objected to Dedekind's claims
of originality (Scharlau, 1986, p. 58). In 1876 he wrote to Dedekind:
I do not deny the validity of your definition, but I am nevertheless
of the opinion that it differs only in form, not in substance, from
what was done by the ancients. I can only say that I consider
the definition given by Euclid... to be just as satisfactory as your
definition. For that reason, I wish you would drop the claim that
such propositions as y/2\/3 = \/6 have never been proved. I think
the French readers especially will share my conviction that Euclid's
book provided necessary and sufficient grounds for proving these
things.Dedekind refused to back down. He replied (Scharlau, 1986, pp. 64-65):I have never imagined that my concept of the irrational numbers
has any particular merit; otherwise I should not have kept it to my-
self for nearly fourteen years. Quite the reverse, I have always been
convinced that any well-educated mathematician who seriously set
himself the task of developing this subject rigorously would be
bound to succeed... Do you really believe that such a proof can
be found in any book? I have searched through a large collec-
tion of works from many countries on this point, and what does
one find? Nothing but the crudest circular reasoning, to the effect
that */a\/b = \/ab because (\/ay/b) = (y/a) (\/b) = ab; not
the slightest explanation of how to multiply two irrational num-
bers. The proposition (mn)^2 = m^2 n^2 , which is proved for rational
numbers, is used unthinkingly for irrational numbers. Is it not
scandalous that the teaching of mathematics in schools is regarded
as a particularly good means to develop the power of reasoning,
while no other discipline (for example, grammar) would tolerate
such gross offenses against logic for a minute? If one is to proceed
scientifically, or cannot do so for lack of time, one should at least
honestly tell the pupil to believe a proposition on the word of the
teacher, which the students are willing to do anyway. That is bet-
ter than destroying the pure, noble instinct for correct proofs by
giving spurious ones.(^14) In his paper (1992) David Fowler (1937-2004) investigated a number of approaches to the
arithmetization of the real numbers and showed how the specific equation \/5\/3 = VG could have
been proved geometrically, and also how difficult this proof would have been using many other
natural approaches.