- 1 - Dedekind's Contributions to the Foundations of Mathematics Erich H. Reck, November 2007 — Draft; please do not quote! Richard Dedekind (1831-1916) It is widely acknowledged that Dedekind was one of the greatest mathematicians of the nineteenth-century, as well as one of the most important contributions to number theory and algebra of all time. Any comprehensive history of mathematics will mention him for, among others: his invention of the theory of ideals and his investigation of the notions of algebraic number, field, module, lattice, etc. (see, e.g., Boyer & Merzbach 1991, Kolmogorov & Yushkevich 2001, Alten et al. 2003). Dedekind's more foundational work in mathematics is also widely known, at least in parts. Often acknowledged in that connection are: his analysis of the notion of continuity, his construction of the real numbers in terms of Dedekind-cuts, his formulation of the Dedekind-Peano axioms for the natural numbers, his proof of the categoricity of these axioms, and his contributions to the early development of set theory (Ferreirós 1999, Jahnke 2003). While many of Dedekind's contributions to mathematics and its foundations are thus common knowledge, they are seldom discussed together. In particular, his mathematical writings are often treated separately from his foundational ones. This entry provides a comprehensive survey of his contributions, in all of his major writings, with the goal of making their close relationships apparent. It will be argued that foundational concerns are at play throughout Dedekind's work, so that any attempt to distinguish sharply
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Dedekind's Contributions to the Foundations of Mathematics Erich H. Reck, November 2007 — Draft; please do not quote!
Richard Dedekind (1831-1916)
It is widely acknowledged that Dedekind was one of the greatest mathematicians of the
nineteenth-century, as well as one of the most important contributions to number theory
and algebra of all time. Any comprehensive history of mathematics will mention him for,
among others: his invention of the theory of ideals and his investigation of the notions of
Another way to bring to the fore the radical character of Dedekind's work is by returning
to a particular innovation it contains: his definition of the infinite. Note that what he does
here is to take what was widely seen as a paradoxical property (to be equinumerous with
E. Reck, November 2007 — Draft; please do not quote!
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a proper subset) as a defining characteristic (of infinite sets). And again, what he then
provides is an analysis of the finite (the natural numbers) in terms of the infinite (infinite
sets), also a rather bold idea. In order for innovations like these to be acceptable at all, it
was important that they not only open up novel realms for mathematics, but also lead to
further clarifications and results concerning older parts of mathematics—as they did in
this case (the clarification of the notion of continuity, of mathematical induction, various
results in algebraic number theory, in the theory of algebraic functions, etc.).
While Dedekind was a great innovator, he was, of course, not alone in leading a large
part of mathematics in a set-theoretic, infinitary, and structuralist direction. In fact, he
can be seen as part of a group of mathematicians—also including Dirichlet and Riemann,
among others—who promoted a more "conceptual" approach in the second half of the
nineteenth century (Ferreirós 1999). Of these, Dirichlet has sometimes been seen as the
leader, or as the "poet's poet", including being a big influence on Dedekind (Stein 1988).
As Hermann Minkowski, another major figure in this mathematical tradition, put it later
(in a reflection on Dirichlet's significance on the occasion of his 100th birthday): He
impressed on other mathematicians "to conquer the problems with a minimum amount of
blind calculation, a maximum of clear-seeing thought" (as quoted in Stein 1988).
Riemann also had a strong influence on Dedekind, especially in two respects: first, with
his explicit emphasis, primarily in his development of complex function theory, on
finding simple, characteristic, and "intrinsic" concepts on which proofs were to be based,
rather than relying on "extrinsic" properties connected with, say, particular formalisms or
symbolisms (Mehrtens 1979b, Laugwitz 1996, Tappenden 2006); second, with his
exploration of new "conceptual possibilities" (Stein 1988), including his systematic study
of non-Euclidean, or "Riemannian", geometries. Another exploration of such novel
possibilities can, finally, be found in the work of Dedekind's correspondent Cantor,
namely Cantor's pioneering investigation of transfinite ordinal and cardinal numbers.
But can the significance of Dedekind's methodology, or of such a "conceptual" approach
in general, be captured more succinctly and analyzed more deeply? One way to do so is
by highlighting the methodological values embodied in it: systematicity, generality,
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purity, etc. (Avigad 2006). Another is by pointing to the kind of reasoning involved,
namely "conceptual" reasoning or, in terminology later made prominent by Hilbert,
"axiomatic" reasoning (Sieg & Schlimm 2005). One may even talk about a novel style of
reasoning as having been introduced, where "style" is not to be understood in a
psychological or sociological, but in an epistemological sense. Finally, this new style
might be taken to have brought with it, not just new theorems and proofs, but a distinctive
kind of understanding of mathematical phenomena (Reck forthcoming).
Such attempts at analyzing the epistemological significance of Dedekind's innovations
further are only recent, initial forays, clearly in need of further elaboration. But one more
observation can be added already. The methodological structuralism that shapes all of
Dedekind's works is not independent of the logical and metaphysical views to be found in
his foundational writings. That is to say, if one adopts his general approach it is hardly
possible to hold on to metaphysical views about the nature of mathematical phenomena
common previously, especially to narrowly formalist, empiricist, and similar views—a
structuralist epistemology, along Dedekindian lines, calls for a structuralist metaphysics.
In fact, these should be seen as two sides of the same coin (Reck forthcoming, earlier also
Gray 1986). Dedekind seems to have been keenly aware of this fact, even if he never
spelt out his philosophical views in a direct, systematic way.
LITERATURE: Besides Mehrtens (1979b), Gray (1986), and Stein (1988), very recently
see Avigad (2006), McLarty (2006), Tappenden (2006), and Reck (forthcoming).
9. Concluding Remarks
We started out by sketching Dedekind's contributions to the foundations of mathematics
in his overtly foundational works, especially "Stetigkeit und Irrationale Zahlen" and "Was
sind und was sollen die Zahlen?" We then added a survey, and preliminary analysis, of
the methodological innovations in his more mainstream mathematical writings, from his
work in algebraic number theory to other areas. We also noted that the logical and
metaphysical position one can find in his foundational work is intimately connected with
his general methodological and epistemological perspective. In this sense it is misguided
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to separate his "foundational" works sharply from his "mathematical" works—he made
foundationally relevant contributions throughout. In fact, the case of Dedekind is a good
illustration of a more general lesson: Any strict distinction between "foundational" or
"philosophical" questions about mathematics, on the one hand, and "inner-mathematical"
questions, on the other, is problematic in the end, especially if one does not want to
impoverish both sides (compare again Tappenden 2006, Reck forthcoming).
If we look at Dedekind's contributions to the foundations of mathematics from such a
perspective, the sum total looks impressive indeed. He was not just one of the greatest
mathematicians of the nineteenth and twentieth centuries, but also one of the subtlest,
most insightful philosophers of mathematics. With his structuralist views about the
nature of mathematical entities and the way in which to investigate them, he (together
with Dirichlet, Riemann, and Cantor) was far ahead of his time. He was even, arguably,
ahead of much contemporary philosophy of mathematics, especially in terms of his
sensitivity to both sides. This is not to say that his position is without problems.
Dedekind himself was deeply troubled about Russell's Antinomy; and the twentieth
century produced additional surprises, such as Gödel's Incompleteness Theorems, which
are hard to accommodate for anyone. Moreover, the methodology of mathematics has
developed further since Dedekind's time, including attempts to reconcile, and integrate,
"conceptual" and "computational" thinking. Then again, is there a philosophical position
available today that addresses all corresponding problems and answers all relevant
questions? If not, updating a Dedekindian position may be a worthwhile project.
Dedekind's Writings (Original Texts and English Translations) Dedekind, Richard (1854): "Über die Einführung neuer Funktionen in der Mathematik;
Habilitationsvortrag"; in Dedekind (1930-32), Vol. 3, pp. 428-438
_____ (1857): "Abriß einer Theorie der höheren Kongruenzen in bezug auf einen reellen Primzahl-Modulus"; reprinted in Dedekind (1930-32), Vol. 1, pp. 40-67
_____ (1872): "Stetigkeit und Irrationale Zahlen", Vieweg: Braunschweig; reprinted in Dedekind (1930-32), Vol. 3, pp. 315-334, and in Dedekind (1965), pp. 1-22; English trans., Dedekind (1901b)
_____ (1876a): "Bernhard Riemanns Lebenslauf", in Riemann (1876), pp. 539-558
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_____ (1876b): "Briefe an Lipschitz (1-2)", in Dedekind (1930-32), Vol. 3, pp. 468-479
_____ (1877): Sur la Théorie des Nombres Entiers Algébrique, Gauthier-Villars; reprinted in Dedekind (1930-32), Vol. 3, pp. 262-296; English trans. Dedekind (1996)
_____ (1882): "Theorie der algebraischen Funktionen einer Veränderlichen", with H. Weber; reprinted in Dedekind (1930-32), Vol. 1, pp. 238-350
_____ (1888a): "Was sind und was sollen die Zahlen?", Vieweg: Braunschweig; reprinted in Dedekind (1930-32), Vol. 3, pp. 335-91, and in Dedekind (1965), pp. III-XI and 1-47; English trans., (Dedekind 1901c) and (revised) Dedekind (1995)
_____ (1888b): "Brief an Weber", in Dedekind (1930-32), Vol. 3, pp. 488-490
_____ (1890): "Letter to Keferstein' reprinted, in English trans., in From Frege to Gödel, J. v. Heijenoort, ed., Harvard University Press: Cambridge, MA, 1967, pp. 98-103
_____ (1897): "Über Zerlegungen von Zahlen durch ihre größten gemeinsamen Teiler"; reprinted in Dedekind (1930-32), Vol. II, pp. 103-147.
_____ (1900): "Über die von drei Moduln erzeugte Dualgruppe", Mathematische Annalen 53, 371-403; reprinted in Dedekind (1930-32), Vol. II, pp. 236-271.
_____ (1901a): Essays on the Theory of Numbers, W.W. Beman, ed. and trans., Open Court Publishing Company: Chicago, 1901; reprinted by Dover: New York, 1963; English version of Dedekind (1965)
_____ (1901b): "Continuity and Irrational Numbers", in (Dedekind 1901a), pp. 1-27; English trans. of Dedekind (1872)
_____ (1901c): "The Nature and Meaning of Numbers", in (Dedekind 1901a), pp. 29-115; English trans. of Dedekind (1888)
_____ (1930-32): Gesammelte Mathematische Werke, Vols. 1-3, R. Fricke, E. Noether & Ö. Ore, eds., Vieweg: Braunschweig; reprinted—except the separately published Lejeune-Dirichlet (1893)—by Chelsea Publishing Company: New York, 1969
_____ (1964): Über die Theorie der ganzen algebraischen Zahlen, Nachdruck des elften Supplements mit einem Geleitwort von Prof. Dr. B. L. van der Waerden, Vieweg: Braunschweig
_____ (1965): Was sind und was sollen die Zahlen/Stetigkeit und Irrationale Zahlen. Studienausgabe, G. Asser, ed., Vieweg: Braunschweig; earlier English version, Dedekind (1901a)
_____ (1981): "Eine Vorlesung über Algebra", in Richard Dedekind 1831/1981. Eine Würdigung zu seinem 150. Geburtstag, W. Scharlau, ed., Vieweg: Braunschweig, pp. 59-100
_____ (1982): "Unveröffentlichte algebraische Arbeiten Richard Dedekinds aus seiner Göttinger Zeit, 1855-1858", Archive for History of Exact Sciences 27:4, 335-367
_____ (1985): Vorlesungen über Diffential- und Integralrechnung 1861/62, based on notes by H. Bechtold, M.-A. Knus & W. Scharlau, eds., Vieweg: Braunschweig
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_____ (1995): What Are Numbers and What Should They Be? H. Pogorzelski, W. Ryan & W. Snyder, eds. and trans., Research Institute for Mathematics: Orono, ME; revised English trans. of Dedekind (1888)
_____ (1996a): Theory of Algebraic Integers, John Stillwell, ed. and trans., Cambridge University Press; English trans. of Dedekind (1877)
_____ (1996b): "Julius Wilhelm Richard Dedekind (1831-1916)", in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Vol. 2, W.B. Ewald, ed., Oxford University Press, pp. 753-837; English translation of a variety of texts by Dedekind, partly in excerpt form
Lejeune-Dirichlet, P.G. (1893): Vorlesungen über Zahlentheorie, fourth edition, with supplements by Richard Dedekind, Richard Dedekind, ed.; Vieweg: Braunschweig; reprinted by Chelsea: New York, 1968
_____ (1999): Lectures on Number Theory, with supplements by Richard Dedekind, John Stillwell, ed. and trans., American Mathematical Society: Providence; English trans. of Lejeune-Dirichlet (1893)
Lipschitz, Rudolf (1986): Briefwechsel mit Cantor, Dedekind, Helmholtz, Kronecker, Weierstrass und anderen, W. Scharlau, ed., Vieweg: Braunschweig
Meschkowski, H. & Nilson, W., eds. (1991): Georg Cantor. Briefe, Springer: Berlin; especially "Die Periode des Briefwechsels mit Dedekind — die Entstehung der Mengenlehre", pp. 29-66
Noether, E. & Cavaillès, J., eds. (1937): Briefwechsel Cantor-Dedekind, Hermann, Paris
Riemann, Bernard (1876): Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass, H. Weber, ed., with assistance by R. Dedekind; 2nd modified ed. 1892, with a supplements added in1902, reprinted by Dover: New York, 1953
Further Bibliography (Secondary Literature in English, French, and German)
Alten, H.-W., et al., eds. (2003): 4000 Jahre Algebra: Geschichte, Kulturen, Menschen, Springer: Berlin; especially Chapter 9, "Algebra an der Wende zum 20. Jahrhundert", pp. 475-550
Avigad, Jeremy (2006): "Methodology and Metaphysics in the Development of Dedekind's Theory of Ideals", in The Architecture of Modern Mathematic, J. Ferreiros & J. Grey, eds., Oxford University Press, pp. 159-186
Awodey, S. & Reck, E. (2002): "Categoricity and Completeness, Part I: Nineteenth-Century Axiomatics to Twentieth-Century Metalogic", History and Philosophy of Logic 23, 1-30
Bell, E.T. (1937): Men of Mathematics; reprinted by Simon & Schuster: New York, 1965; especially Chapter 27, "Arithmetic the Second: Kummer (1810-1893), Dedekind (1831-1916)", pp. 510-525
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Belna, Jean-Pierre (1996): La Notion de Nombre chez Dedekind, Cantor, Frege: Théories, Conceptions, et Philosophie, Vrin: Paris
Biermann, Kurt R. (1971): "Dedekind", in Dictionary of Scientific Biography, Vol. 4, C.C. Gillespie, ed., pp. 1-5
Boyer, C. & Merzbach, U. (1991): A History of Mathematics, second ed. (revised), Wiley & Sons: New York; especially Chapter 25, "Analysis", pp. 553-574, and Chapter 26, "Algebra", pp. 575-598
Cooke, Roger (2005): "Richard Dedekind, Stetigkeit und Irrationale Zahlen (1872)", Chapter 43 of Landmark Writings in Western Mathematics, 1640-1940, I. Grattan-Guinness, ed., Elsevier, pp. 553-563
Corfield, David (2003): Towards a Philosophy of Real Mathematics, Cambridge University Press; especially Chapter 8, "Beyond the Methodology of Mathematical Research Programmes", pp. 175-203
Corry, Leo (2004): Modern Algebra and the Rise of Mathematical Structures, second edition (revised), Birkhäuser: Boston; especially Chapter 2: "Richard Dedekind: Numbers and Ideals", pp. 66-136
Demopoulos, W. & Clark, P. (2005): "The Logicism of Frege, Dedekind, and Russell", in The Oxford Handbook of Philosophy of Mathematics and Logic, S. Shapiro, ed., Oxford University Press, pp. 166-202
Dugac, Pierre (1976): Richard Dedekind et les Fondements des Mathématiques, Vrin: Paris
Edwards, Harold M. (1977): Fermat's Last Theorem. A Genetic Introduction to Algebraic Number Theory, Springer: New York
_____ (1980): "The Genesis of Ideal Theory", Archive for History of Exact Sciences 23, 321-378
_____ (1983): "Dedekind's Invention of Ideals", Bulletin of the London Mathematical Society 15, 8-17
Ewald, William B. (1996): "Julius Wilhelm Richard Dedekind", in From Kant to Hilbert. A Source Book in the Foundations of Mathematics, Vol. 2, Oxford University Press, pp. 753-754
Ferreirós, José (1993): "On the Relation between Georg Cantor and Richard Dedekind", Historia Mathematica 20, 343-363
_____ (1999): Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics, Birkhäuser: Boston; especially: Chapter III, "Dedekind and the Set-theoretic Approach to Algebra", pp. 81-116; Chapter IV, "The Real Number System", pp. 117-144; and Chapter VII, "Sets and Maps as a Foundation for Mathematics", pp. 215-255
_____ (2005): "Richard Dedekind (1888) and Giuseppe Peano (1989), Booklets on the Foundations of Arithmetic", Chapter 47 of Landmark Writings in Western Mathematics, 1640-1940, I. Grattan-Guinness, ed., Elsevier, pp. 613-626
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_____ (2007): "The Early Development of Set Theory", Stanford Encyclopedia of Philosophy (Spring 2007 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/entries/settheory-early//>
Gardies, Jean-Louis (1984): "Eudoxe et Dedekind", Revue d'histoire des sciences 37, 111-125
Geyer, Wulf-Dieter (1981): "Die Theorie der algebraischen Funktionen einer Veränderlichen nach Dedekind und Weber", in Scharlau (1981a), pp. 109-133
Gillies, D.A. (1982): Frege, Dedekind, and Peano on the Foundations of Arithmetic, Assen
Goldman, Jay R. (1998): The Queen of Mathematics: A Historically Motivated Guide to Number Theory, A.K. Peters: Wellesley, MA
Gray, Jeremy (1986): "The Nineteenth-Century Revolution in Mathematical Ontology", in Revolutions in Mathematics, D. Gillies, ed., Oxford University Press; second edition 1992, pp. 226-248
_____ (2000): Linear Differential Equations and Group Theory, from Riemann to Poincaré, second edition, Birkhäuser: Boston; especially Chapter IV, "Modular Equations", pp. 101-140
Haubrich, Ralf (1992): Zur Entstehung der algebraischen Zahlentheorie Richard Dedekinds, Ph.D. dissertation, University of Göttingen
Jahnke, Hans Niels, ed. (2003): A History of Analysis, American Mathematical Society; especially Chapter 1, "Antiquity", pp. 1-40, and Chapter 10: "The End of the Science of Quantity: Foundations of Analysis, 1860-1910", pp. 291-324
Jourdain, P.E.B. (1916): "Richard Dedekind (1833-1916)", The Monist 26, 415-427; reprinted in Modern Logic 3, 1993, 207-214
Kitcher, Philip (1986): "Frege, Dedekind, and the Philosophy of Mathematics", in Frege Synthesized, L. Haaparanta & J. Hintikka, eds., Reidel: Dordrecht, pp. 299-343
Kolmogorov, A.N. & Yushkevich, A.P. (2001): Mathematics of the 19th Century. Vol. I: Mathematical Logic, Algebra, Number Theory, Probability Theory, 2nd revised edition, Birkhäuser: Boston; especially Chapter 2, Section 3, "The Theory of Algebraic Numbers and the Beginnings of Commutative Algebra", pp. 86-135
Laugwitz, Detlef (1996): Bernhard Riemann 1826-1866. Wendepunkte in der Auffassung der Mathematik, Birkhäuser: Basel; English trans., Bernhard Riemann 1826-1866. Turning Points in the Conception of Mathematics, Birkhäuser: Boston, 1999
McCarty, David (1995): "The Mysteries of Richard Dedekind", in From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics, J. Hintikka, ed., Springer: New York, pp. 53-96
McLarty, Colin (2006): "Emmy Noether's 'Set Theoretic' Topology: From Dedekind to the Rise of Functors", in The Architecture of Modern Mathematics, J. Ferreiros & J. Grey, eds., Oxford University Press, pp. 187-208
E. Reck, November 2007 — Draft; please do not quote!
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Mehrtens, Herbert (1979a): Entstehung der Verbandstheorie, Gerstenberg: Hildesheim; especially Chapter 2, "Die 'Dualgruppe' von Richard Dedekind", pp. 66-126
_____ (1979b): "Das Skelett der modernen Algebra. Zur Bildung mathematischer Begriffe bei Richard Dedekind", in Disciplinae Novae, C.J. Sciba, ed., Vanderhoeck & Ruprecht: Göttingen, pp. 25-43
_____ (1982): "Richard Dedekind – Der Mensch und die Zahlen", Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft 33, 19-33
Potter, Michael (2000): Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap, Oxford University Press; especially Chapter 3, "Dedekind", pp. 81-104
Reck, Erich (2003a): "Dedekind's Structuralism: An Interpretation and Partial Defense", Synthese 137:3, 369-419
_____ (forthcoming): "Dedekind, Structural Reasoning, and Mathematical Understanding", in Perspectives on Mathematical Practices, Vol. 2, J.-P. van Bendegem & B. van Kerkhove, eds., Springer: Dordrecht
Reck, E. & Price, M. (2000): "Structures and Structuralism in Contemporary Philosophy of Mathematics", Synthese 125:3, 341-383
Reed, David (1995): Figures of Thought, Routledge: London; especially Chapter 4, "Number Theory in the Nineteenth Century", pp. 76-116
Scharlau, Winfried, ed. (1981a): Richard Dedekind, 1831/1981: Eine Würdigung zu seinen 150. Geburtstag, Vieweg: Braunschweig
_____ (1981b): "Erläuterungen zu Dedekinds Manuskript über Algebra", in Scharlau (1981a), pp. 101-108
Schlimm, Dirk (2000): Richard Dedekind: Axiomatic Foundations of Mathematics, M.A. thesis, Carnegie Mellon University, Pittsburgh
Sieg, W. & Schlimm, D. (2005): "Dedekind's Analysis of Number: System and Axioms", Synthese 147, 121-170
Sinaceur, M.A. (1974): "L'infini et les nombres – Commentaires de R. Dedekind à 'Zahlen' – La correspondence avec Keferstein", Revue d'histoire des sciences 27, 251-278
_____ (1979): "La méthode mathématique de Dedekind", Revue d'histoire des sciences 32, 107-142
Stein, Howard (1988): "Logos, Logic, and Logistiké: Some Philosophical Remarks on Nineteenth-Century Transformations of Mathematics", in History and Philosophy of Modern Mathematics. Minnesota Studies in the Philosophy of Science, Vol. XI, W. Aspray & P. Kitcher, eds., University of Minnesota Press, pp. 238-259
_____ (1990): "Eudoxos and Dedekind: On the Ancient Greek Theory of Ratios and its Relation to Modern Mathematics", Synthese 84, 163-211
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_____ (1998): "Dedekind, Julius Wilhelm Richard", in Routledge Encyclopedia of Philosophy, Vol. 2, Routledge: London/New York, pp. 840-842
Stillwell, John (1996): "Translator's Introduction", in Richard Dedekind: Theory of Algebraic Numbers, J. Stillwell, ed., Cambridge University Press, pp. 1-50
Tait, W.W. (1996): "Frege versus Cantor and Dedekind: On the Concept of Number", in Frege: Importance and Legacy, M. Schirn, ed., de Gruyter: Berlin, pp. 70-113
Tappenden, Jamie (2006): "The Riemannian Background to Frege's Philosophy", in The Architecture of Modern Mathematic, J. Ferreiros & J. Grey, eds., Oxford University Press, pp. 97-132
Wussing, Hans (1984): The Genesis of the Abstract Group Concept, MIT Press; reprinted by Dover: New York, 2007
Other Internet Resources (Online Information and Texts) Texts by Dedekind:
• www.ru.nl/w-en-s/gmfw/bronnen/dedekind1.html (facsimile of "Stetigkeit und Irrationale Zahlen")
• www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html (facsimile of "Was sind und was sollen die Zahlen?")