词条 | Hilbert's axioms |
释义 |
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie[1][2][3][4] (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff. The axiomsHilbert's axiom system is constructed with six primitive notions: three primitive terms:[5]
and three primitive relations:[6]
Line segments, angles, and triangles may each be defined in terms of points and straight lines, using the relations of betweenness and containment.{{how|date=August 2018}} All points, straight lines, and planes in the following axioms are distinct unless otherwise stated. I. Incidence
II. Order
III. Congruence
IV. Parallels
V. Continuity
Hilbert's discarded axiomHilbert (1899) included a 21st axiom that read as follows: II.4. Any four points A, B, C, D of a line can always be labeled so that B shall lie between A and C and also between A and D, and, furthermore, that C shall lie between A and D and also between B and D.E.H. Moore and R.L. Moore independently proved that this axiom is redundant, and the former published this result in an article appearing in the Transactions of the American Mathematical Society in 1902.[10] Editions and translations of Grundlagen der GeometrieThe original monograph, based on his own lectures, was organized and written by Hilbert for a memorial address given in 1899. This was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902. This translation incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. In the Preface of this edition Hilbert wrote: "The present Seventh Edition of my book Foundations of Geometry brings considerable improvements and additions to the previous edition, partly from my subsequent lectures on this subject and partly from improvements made in the meantime by other writers. The main text of the book has been revised accordingly." New editions followed the 7th, but the main text was essentially not revised. The modifications in these editions occur in the appendices and in supplements. The changes in the text were large when compared to the original and a new English translation was commissioned by Open Court Publishers, who had published the Townsend translation. So, the 2nd English Edition was translated by Leo Unger from the 10th German edition in 1971. This translation incorporates several revisions and enlargements of the later German editions by Paul Bernays. The Unger translation differs from the Townsend translation with respect to the axioms in the following ways:
Axiom of completeness. To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.
The last two modifications are due to P. Bernays. Other changes of note are:
ApplicationThese axioms axiomatize Euclidean solid geometry. Removing five axioms mentioning "plane" in an essential way, namely I.4–8, and modifying III.4 and IV.1 to omit mention of planes, yields an axiomatization of Euclidean plane geometry. Hilbert's axioms, unlike Tarski's axioms, do not constitute a first-order theory because the axioms V.1–2 cannot be expressed in first-order logic. The value of Hilbert's Grundlagen was more methodological than substantive or pedagogical. Other major contributions to the axiomatics of geometry were those of Moritz Pasch, Mario Pieri, Oswald Veblen, Edward Vermilye Huntington, Gilbert Robinson, and Henry George Forder. The value of the Grundlagen is its pioneering approach to metamathematical questions, including the use of models to prove axioms independent; and the need to prove the consistency and completeness of an axiom system. Mathematics in the twentieth century evolved into a network of axiomatic formal systems. This was, in considerable part, influenced by the example Hilbert set in the Grundlagen. A 2003 effort (Meikle and Fleuriot) to formalize the Grundlagen with a computer, though, found that some of Hilbert's proofs appear to rely on diagrams and geometric intuition, and as such revealed some potential ambiguities and omissions in his definitions.[11] See also
Notes1. ^{{cite journal|author=Sommer, Julius|title=Review: Grundlagen der Geometrie, Teubner, 1899|journal=Bull. Amer. Math. Soc.|year=1900|volume=6|issue=7|pages=287–299|url=http://www.ams.org/journals/bull/1900-06-07/S0002-9904-1900-00719-1/S0002-9904-1900-00719-1.pdf|doi=10.1090/s0002-9904-1900-00719-1}} 2. ^{{cite journal|author=Poincaré, Henri|authorlink=Henri Poincaré|title=Poincaré's review of Hilbert's "Foundations of Geometry", translated by E. V. Huntington|journal=Bull. Amer. Math. Soc.|year=1903|volume=10|pages=1–23|url=http://www.ams.org/journals/bull/1903-10-01/S0002-9904-1903-01061-1/S0002-9904-1903-01061-1.pdf|doi=10.1090/S0002-9904-1903-01061-1}} 3. ^{{cite journal|author=Schweitzer, Arthur Richard|title=Review: Grundlagen der Geometrie, Third edition, Teubner, 1909|journal=Bull. Amer. Math. Soc.|year=1909|volume=15|issue=10|pages=510–511|url=http://www.ams.org/journals/bull/1909-15-10/S0002-9904-1909-01814-2/S0002-9904-1909-01814-2.pdf|doi=10.1090/s0002-9904-1909-01814-2}} 4. ^{{cite journal|author=Gronwall, T. H.|authorlink=Thomas Hakon Gronwall|title=Review: Grundlagen der Geometrie, Fourth edition, Teubner, 1913|journal=Bull. Amer. Math. Soc.|year=1919|volume=20|issue=6|pages=325–326|url=http://www.ams.org/journals/bull/1914-20-06/S0002-9904-1914-02492-9/S0002-9904-1914-02492-9.pdf|doi=10.1090/S0002-9904-1914-02492-9}} 5. ^These axioms and their numbering are taken from the Unger translation (into English) of the 10th edition of Grundlagen der Geometrie. 6. ^One could count this as six relations as specified below, but Hilbert did not do so. 7. ^In the Townsend edition this statement differs in that it also includes the existence of at least one point D between A and C, which became a theorem in a later edition. 8. ^The existence part ("there is at least one") is a theorem. 9. ^This is Hilbert's terminology. This statement is more familiarly known as Playfair's axiom. 10. ^{{citation|first=E.H.|last=Moore|title=On the projective axioms of geometry|journal=Transactions of the American Mathematical Society|year=1902|volume=3|pages=142–158|url=http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf|doi=10.2307/1986321}} 11. ^On page 334: "By formalizing the Grundlagen in Isabelle/Isar we showed that Hilbert's work glossed over subtle points of reasoning and relied heavily, in some cases, on diagrams which allowed implicit assumptions to be made. For this reason it can be argued that Hilbert interleaved his axioms with geometric intuition in order to prove many of his theorems." References
External links
2 : David Hilbert|Foundations of geometry |
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