Continuity and Irrational Numbers / Dedekind

My attention was first directed toward the considerations which form the

subject of this pamphlet in the autumn of 1858. As professor in the Polytechnic

School in Zurich I found myself for the first time obliged to lecture upon the

elements of the differential calculus and felt more keenly than ever before the

lack of a really scientific foundation for arithmetic. In discussing the notion of

the approach of a variable magnitude to a fixed limiting value, and especially

in proving the theorem that every magnitude which grows continually, but not

beyond all limits, must certainly approach a limiting value, I had recourse to

geometric evidences. Even now such resort to geometric intuition in a first pre-

sentation of the differential calculus, I regard as exceedingly useful, from the

didactic standpoint, and indeed indispensable, if one do es not wish to lose to o

much time.

But that this form of introduction into the differential calculus

can make no claim to being scientific, no one will deny. For myself this feel-

ing of dissatisfaction was so overpowering that I made the fixed resolve to keep

meditating on the question till I should find a purely arithmetic and perfectly

rigorous foundation for the principles of infinitesimal analysis. The statement is

so frequently made that the differential calculus deals with continuous magni-

tude, and yet an explanation of this continuity is nowhere given; even the most

rigorous expositions of the differential calculus do not base their pro ofs upon

continuity but, with more or less consciousness of the fact, they either appeal

to geometric notions or those suggested by geometry, or depend upon theorems

which are never established in a purely arithmetic manner. Among these, for ex-

ample, belongs the above-mentioned theorem, and a more careful investigation

convinced me that this theorem, or any one equivalent to it, can be regarded in

some way as a su cient basis for infinitesimal analysis. It then only remained to

discover its true origin in the elements of arithmetic and thus at the same time

to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858,

and a few days afterward I communicated the results of my meditations to my

dear friend Dur‘ ege with whom I had a long and lively discussion. Later I ex-

plained these views of a scientific basis of arithmetic to a few of my pupils, and

here in Braunschweig read a paper upon the subject before the scientific club

of professors, but I could not make up my mind to its publication, because, in

the first place, the presentation did not seem altogether simple, and further, the

theory itself had little promise. Nevertheless I had already half determined to

select this theme as subject for this o ccasion, when a few days ago, March 14,

by the kindness of the author, the paper Die Elemente der Funktionenlehre by

E. Heine (Crelle’s Journal, Vol. 74) came into my hands and confirmed me in

my decision. In the main I fully agree with the substance of this memoir, and

indeed I could hardly do otherwise, but I will frankly acknowledge that my own

presentation seems to me to be simpler in form and to bring out the vital point

more clearly. While writing this preface (March 20, 1872), I am just in receipt

of the interesting paper Ueber die Ausdehnung eines Satzes aus der Theorie der

trigonometrischen Reihen, by G. Cantor (Math. Annalen, Vol. 5), for which I

owe the ingenious author my hearty thanks. As I find on a hasty perusal, the

axiom given in Section II. of that paper, aside from the form of presentation,

agrees with what I designate in Section III. as the essence of continuity.

But what advantage will be gained by even a purely abstract definition of real num-bers of

a higher type, I am as yet unable to see, conceiving as I do of the domain of real numbers as complete in itself.