Was beweisbar ist, soll in der Wissenschaft nicht ohne Beweis geglaubt werden. gestellte Frage lautet: die Zahlen sind freie Schöpfungen des menschlichen. Zahlen. In the introduction to this paper he points out that the real . In addition the recent work by R. Dedekind Was sind und was sollen. In his penetrating monograph Dedekind returns to numbers. The published it under the title Was sind und was sollen die Zahlen?.
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die Zahlen? variously translated as What are numbers and what should they be? Much of Dedekind's terminology and notation as translated by Beman differs. Irrationale Zahlen”  () and “Was sind und was sollen die Zahlen?”  ( ) The invention of set-theory by Cantor and Dedekind. Febr. Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.
The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether. Thus Dedekind can be said to have been Kummer's most important disciple. In an article, Dedekind and Heinrich Martin Weber applied ideals to Riemann surfaces , giving an algebraic proof of the Riemann—Roch theorem. In , he published a short monograph titled Was sind und was sollen die Zahlen?
He also proposed an axiomatic foundation for the natural numbers, whose primitive notions were the number one and the successor function. The next year, Giuseppe Peano , citing Dedekind, formulated an equivalent but simpler set of axioms , now the standard ones.
Dedekind made other contributions to algebra. For instance, around , he wrote the first papers on modular lattices. In , while on holiday in Interlaken , Dedekind met Georg Cantor. Thus began an enduring relationship of mutual respect, and Dedekind became one of the very first mathematicians to admire Cantor's work concerning infinite sets, proving a valued ally in Cantor's disputes with Leopold Kronecker , who was philosophically opposed to Cantor's transfinite numbers.
A Source Book in Mathematical Logic, — Harvard Univ. Press: 98— Essays on the Theory of Numbers. Beman, W.
Contains English translations of Stetigkeit und irrationale Zahlen and Was sind und was sollen die Zahlen? He studied for a while with Peter Gustav Lejeune Dirichlet , and they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studied elliptic and abelian functions.
About this time, he became one of the first people to understand the importance of the notion of groups for algebra and arithmetic.
When the Collegium Carolinum was upgraded to a Technische Hochschule Institute of Technology in , Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in , but did occasional teaching and continued to publish.
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He never married, instead living with his sister Julia. He received honorary doctorates from the universities of Oslo , Zurich , and Braunschweig. Work[ edit ] Dedekind, before While teaching calculus for the first time at the Polytechnic school, Dedekind developed the notion now known as a Dedekind cut German : Schnitt , now a standard definition of the real numbers. The idea of a cut is that an irrational number divides the rational numbers into two classes sets , with all the numbers of one class greater being strictly greater than all the numbers of the other lesser class.
For example, the square root of 2 defines all the nonnegative numbers whose squares are less than 2 and the negative numbers into the lesser class, and the positive numbers whose squares are greater than 2 into the greater class.
Every location on the number line continuum contains either a rational or an irrational number. Thus there are no empty locations, gaps, or discontinuities. Dedekind's theorem  states that if there existed a one-to-one correspondence between two sets, then the two sets were "similar". He invoked similarity to give the first precise definition of an infinite set : a set is infinite when it is "similar to a proper part of itself," in modern terminology, is equinumerous to one of its proper subsets.
Dedekind edited the collected works of Lejeune Dirichlet , Gauss , and Riemann. Dedekind's study of Lejeune Dirichlet's work led him to his later study of algebraic number fields and ideals.
How one comes to invent a concept and how one choses to define it are often different things. The way in which a mathematician choses to define a concept is historically situated, and shaped by the disciplinary norms of legitimacy and rigour established by the scientific community.
Such a definition is chosen as the most accurate presentation, and is ultimately the one that should be remembered. It is the use made in his drafts on Dualgruppen.
However, it is possible to distinguish the main movements of the research, whose exposition will constitute the guiding thread of this paper. On the matter, one can see [Mehrtens, ], [Corry, , ] and [Schlimm, ]. The archival material on which this paper relies has not, as far as I am aware, been studied so far.
They contain a total of folios. Dedekind III 14 s is the only one that can be related to [Dedekind, ] and exclusively to it. It mostly contain documents laying the grounds of the paper, with a lot of fully written material, in particular next to last and last draftings of [Dedekind, ].
Since I am, here, mostly interested in the phases of research before writing a paper for publication, they are slightly out of the scope of my investigation and will not be used. I should add that this file is not devoid of the kind of preliminary researches that I will consider. However Dedekind uses a strange notation, which can be described as figurative numbers: three modules are denoted by the number 1, 2 and 3, and their product by 12, 23, 13, etc.
Dedekind X 9 after up to ; — Cod.
Dedekind X 10 mids ; — Cod. Dedekind X late s to early s ; — Cod. Dedekind XI-1 to , and a couple of folios from the late s ; — Cod. Dedekind XI-2 late s. It is difficult to pinpoint exactly which file is related to which paper. Indeed, their content consists of researches and computations on modules, groups, the duality and various prop- erties of the operations, Modulgruppen, Dualgruppen, which can be found in both papers.
In addition, we have no certainty that Dedekind himself put all the folios together in each file. Many of the files appear to be very fragmentary. Despite the seemingly thematic organization of the Nachlass, it can be doubtful that all the folios put together do indeed form a consistent thematic unit on the same researches.
Dedekind X contains manuscripts from the s and seems to be the file with the most material related precisely to [Dedekind, ].
The dating of these files is difficult, as Dedekind himself very rarely gives such indi- cations on his drafts. Indeed, he often wrote on the back of advertisements, invoices, administrative papers, letters, or drafts of his own published papers. However, a discrepancy of up to ten years can be observed, for example on manuscripts containing both a reference to a published work and the date of an invoice, which prompts caution in dating.
Other contentual aspects, such as the notations used, can help finding an order for the manuscripts and files, but again without a high degree of precision.
Besides, the establishment of the archive, as described above, adds to the difficulty: while it is certainly possible to date with a relative precision some of the folios, it cannot be applied to those surrounding them.
The broad range of appli- cation of the concept is, for Dedekind, a testimony of its usefulness and fruitfulness. It satisfies the laws of a partial order relation. The Modulgesetz cannot be derived from the fundamental laws. It should be considered as extending them. Its name comes from the fact that Dualgruppen formed by modules e.
Finally, let me mention two notions which were initially introduced in module theory, and are used in the theory of Dualgruppen: the number of congruence classes in a module m according to a module n and denoted by m, n , and the notion of chain Kette of modules.
His Nachlass gives us insights into these twenty years. They appear to be explorations of the properties of modules and their operations through computation. By this exploration, Dedekind is experimenting, so to speak, and observing the laws of module theory. It is the introduction of a new notation that allowed Dedekind not only to go from mimicking arithmetic to actually computing with modules, but also to highlight the duality of the operations.
Some of them can be identified as being part of a larger investigation. Other stand on their own. An example of such computations is the following, in which Dedekind works with finitely generated modules with a basis composed of two numbers. On the pages 3 to 8 of Cod. While not extremely complicated, the computations which resemble an identification of indeterminate coefficients are lengthy and require a lot of ingenuity and familiarity with the subject.
There, Dedekind works with arbitrary modules rather than finite modules or examples. A paradigmatic example, for this kind of research, are the operations between three modules a, b, c. Duality comes forth as a motivation to study in more detail the operations between modules. As the research goes on, the mere interest in the duality of operations is replaced by the developments of module theory and later Dualgruppen.
Yet, duality appears regularly as a sort of guiding light in the exploration of new aspects of module theory. Duality is, thus, omnipresent in the computations themselves. Dedekind X , p. For Dedekind, V. The results in IV. This numbering is also used in the published papers.
By doing so, they facilitate the manipulation of the said operations, as the duality can also be used to shorten some of the computations or proofs. The presentation of the operations in two columns given p.
Such lists are part of a larger toolbox developed by Dedekind to accompany his computations. Interestingly Dedekind does not draw a lot of different tables, but actually draws several times the same table with different notations, different organisations.
These tables, which suggest that even more computations happened in non-preserved drafts, appear to be both an intermediary result, and a tool for the research. The fact that he made so many different versions suggests that he was looking for the best version, an optimized table to display the results. One of the most striking examples of this is a table given in Cod. More- over, the complete table suggests that Dedekind was looking for a pattern, looking maybe for the clean symmetry which can be seen in the finished tables see the table of GCD and LCM for three modules, published in [Dedekind, , ].
As such, the tables appear to also be a device, for Dedekind, in conducting computa- tions. They help understanding the computations, the patterns, the duality that appears in the computations and which can be difficult to visualize or understand without a graphical device such as this one.Hence, his starting point for a critical examination of the foundation of analysis is that concepts can be clearly and unambiguously defined, yet that this does not mean that they can be regarded as being intrinsically determined.
Dedekind’s Contributions to the Foundations of Mathematics
I am grateful to the members of the project for their useful remarks. Further reflection on Dedekind's procedure and similar ones leads to a new question, however: What exactly is involved in it if it is thought through fully, i.
Veblen, etc. Oxford Uni. Concluding Remarks We started out by considering Dedekind's contributions to the foundations of mathematics in his overtly foundational writings: the two booklets, Stetigkeit und irrationale Zahlen and Was sind und was sollen die Zahlen?
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