Richard Dedekind: Completion

Richard Dedekind was German mathematician. Explore interesting quotes on completion.
Richard Dedekind: 26   quotes 1   like

“The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of all continuous domains.”

Richard Dedekind

p, 125
Stetigkeit und irrationale Zahlen (1872)

Question , Dependency

“Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication. While the performance of these two operations is always possible, that of the inverse operations, subtraction and division, proves to be limited. Whatever the immediate occasion may have been, whatever comparisons or analogies with experience, or intuition, may have led thereto; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new creative act; thus negative and fractional numbers have been created by the human mind; and in the system of all rational numbers there has been gained an instrument of infinitely greater perfection. This system, which I shall denote by R, possesses first of all a completeness and self-containedness which I have designated… as characteristic of a body of numbers [Zahlkőrper] and which consists in this, that the four fundamental operations are always performable with any two individuals in R, i. e., the result is always an individual of R, the single case of division by the number zero being excepted.”

Richard Dedekind

p, 125
Stetigkeit und irrationale Zahlen (1872)

Way , Motivation , Perfection

“Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this.”

Richard Dedekind

p, 125
Stetigkeit und irrationale Zahlen (1872)

Law , Question

“What advantage will be gained by even a purely abstract definition of real numbers 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.”

Richard Dedekind

p, 125
Stetigkeit und irrationale Zahlen (1872)