Movatterモバイル変換

[0]ホーム
URL:

Wikipedia

Axiomatic system

(Redirected fromAxiomatic method)
This article has multiple issues. Please helpimprove it or discuss these issues on thetalk page.(Learn how and when to remove these messages)
This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(March 2013) (Learn how and when to remove this message)
This articleneeds attention from an expert in Mathematics. The specific problem is:Few citations despite the degree of detail.WikiProject Mathematics may be able to help recruit an expert.(January 2025)
(Learn how and when to remove this message)

Inmathematics andlogic, anaxiomatic system oraxiom system is a standard type of deductive logical structure, used also intheoretical computer science. It consists of a set of formal statements known asaxioms that are used for thelogical deduction of other statements. In mathematics these logical consequences of the axioms may be known aslemmas ortheorems. Amathematical theory is an expression used to refer to an axiomatic system and all its derived theorems.

A proof within an axiomatic system is a sequence of deductive steps that establishes a new statement as a consequence of the axioms. By itself, the system of axioms is, intentionally, a syntactic construct: when axioms are expressed innatural language, which is normal in books and technical papers, thenouns are intended asplaceholder words. The use of an axiomatic approach is a move away from informal reasoning, in which nouns may carry real-world semantic values, and towardsformal proof. In a fully formal setting, a logical system such aspredicate calculus must be used in the proofs. The contemporary application of formal axiomatic reasoning differs from traditional methods both in the exclusion of semantic considerations, and in the specification of the system of logic in use.

The axiomatic method in mathematics

edit

The reduction of a body of propositions to a particular collection of axioms underliesmathematical research. This dependence was very prominent, and contentious, in the mathematics of the first half of the twentieth century, a period to which some major landmarks of the axiomatic method belong. Theprobability axioms ofAndrey Kolmogorov, from 1933, are a salient example.[1] The approach was sometimes attacked at "formalism", because it cut away parts of the working intuitions of mathematicians, and those applying mathematics. In historical context, this alleged formalism is now discussed asdeductivism, still a widespread philosophical approach to mathematics.[2]

Timeline of axiomatic systems to 1900

edit
Further information:History of mathematics

Major axiomatic systems were developed in the nineteenth century. They includednon-Euclidean geometry,Georg Cantor's abstractset theory, and Hilbert's revisionist axioms forEuclidean geometry.

DateAuthorWorkComments
Fourth century BCE tothird century BCEEuclid ofAlexandriaThe ElementsKnown as the earliest extant axiomatic presentation ofEuclidean plane geometry, covering also parts ofnumber theory.[3]
published 1677Baruch SpinozaEthica, ordine geometrico demonstrataJust as forPrincipia philosophiae cartesianae of 1663, Spinoza in hisEthics claimed to be using the "geometric method" of Euclid. A modern view is "the contrast is glaring between the aspiration to prove points by way of deductive argument from self-evident axioms and the obvious source of those points from experience of life and at best some mix of theory and intuition."[4]
1829Nikolai LobachevskyО началах геометрии ("On the Origin of Geometry")Lobachevsky's paper is now recognised as the first publication on axiomaticplane geometry developed without theparallel axiom of Euclid, so founding the subject ofnon-Euclidean geometry.
1879Gottlob FregeBegriffschriftFrege published a formal system for the foundations of mathematics. In modern parlance, it was asecond-order logic,[5] withidentity relation. It was expressed in a linear notation forparse trees.
1882/3 to 1890sWalther von DyckAxioms for abstractgroup theoryVon Dyck is credited with the now-standard group theory axioms.[6] It is clear from von Dyck's introduction offree groups that he was working with the standard concept ofabstract group. It is not, however, evident whether the existence ofinverse elements was axiomatic: it would follow from the semantic assumption that groups werepermutation groups (permutations being invertible by definition) or geometric transformations with the same property. The discursive style of the period did not labour such points.James Pierpont, one of the American "postulate theorists", did have by 1896 a set of axioms for groups. It is of the modern type, though uniqueness of the identity element (for example) was not assumed.[7]
1888Richard Dedekindconstruction of the real numbersWhen Dedekind introduced his construction of real numbers byDedekind cuts, axioms for the reals were alreadymathematical folklore; a subset of those would, later, defineordered field. The further requirement was a theory ofmathematical limits[8]. For example, to capture the idea that thereal number line forms alinear continuum means dealing with the historicalZeno's paradoxes; and also clarifying the issue ofdecimal representations not being unique, so that0.999...=1, by subjecting it to amathematical proof. Dedekind's modelling of axioms of the reals put these matters on a firm footing. In practice, the theorems proved using Dedekind cuts that were fundamental results inreal analysis could also be proved for other constructions, for example usingCauchy sequences of rational numbers. In other words, they were verifiable axioms, an example being theArchimedean property.
1889Giuseppe PeanoArithmetices principia, nova methodo expositaAfter some earlier work of others, thePeano axioms provided an axiomatic basis for the arithmetical operations onnatural numbers, andmathematical induction, that gained wide acceptance.
1898Alfred North WhiteheadTreatise on Universal AlgebraWhitehead gave the first axiomatic system forBoolean algebra, as introduced byGeorge Boole in fundamental work on logic and probability.[9]
1899David HilbertGrundlagen der GeometriePresented what are now known asHilbert's axioms, a revised axiomatization ofsolid geometry.

Situation at the beginning of the 20th century

edit

David Hilbert "was the first who explicitly adopted the axiomatic method as an investigative framework for the study of thefoundations of mathematics".[10] For Hilbert, a major foundational issue was the logical status ofCantor's set theory. In hislist of 23 unsolved problems in mathematics from 1900, Hilbert made thecontinuum hypothesis the first problem on the list.[11]

Hilbert's sixth problem asked for "axiomatization of all branches of science, in which mathematics plays an important part". He had in mind at least major areas inmathematical physics and probability.[12][13] Of the effect on science, Giorgio Israel has written:

Founded by mathematicianFelix Klein ... the Göttingen School, under the influence of David Hilbert, turned its efforts towards ... set theory, functional analysis, quantum mechanics and mathematical logic. It did so by taking on as its methodical principle the axiomatic method that was to revolutionise the science of [the twentieth century], from the theory of probabilities to theoretical physics.[14]

Israel comments also on cultural resistance, at least in France and Italy, to this "German model" and its international scope.[14] The initialInternational Congress of Mathematicians had heard the views ofHenri Poincaré from France on mathematical physics; Hilbert's list was a submission to the second Congress.[15] TheItalian school of algebraic geometry took a different attitude to axiomatic work in theory building and pedagogy.[16]

Timeline of axiomatic systems from 1901

edit

In the period to 1950, much ofpure mathematics received widely-accepted axiomatic foundations. Multiple systems coexisted inaxiomatic set theory. Mathematics began to be written in a tighter, less discursive if still informal style.

On the other hand, the approach associated with Hilbert of regarding the axiomatic method as fundamental came under criticism. Part ofL. E. J. Brouwer's critique of Hilbert's entire program resulted in an axiomatisation ofintuitionistic propositional logic byArend Heyting.[17] It allowedconstructivism in mathematics to be reconciled with "deductivism", by an exchange of logical calculus, under the title of theBrouwer–Heyting–Kolmogorov interpretation.

DateAuthorWorkComments
1908 to 1922Ernst Zermelo andAbraham FraenkelZermelo-Fraenkel set theoryBuilding onZermelo set theory from 1908, the Zermelo-Fraenkel (ZF) theory provided an axiomatic basis for set theory with a clarified axiom system (adopting a restriction tofirst-order logic). With the addition of theaxiom of choice, the ZFC theory provided a working foundation for much of classical mathematics..[18]
1910Ernst SteinitzAlgebraische Theorie der KörperSteinitz, under the influence of the introduction byKurt Hensel of thep-adic numbers, gave an axiomatic theory of thefield concept in abstract algebra.[19]
1911 to 1913Alfred North Whitehead andBertrand RussellPrincipia Mathematica (3 vols.)A work devoted to the principle of axiomatic formalization of mathematics, that addressed theset theory paradoxes by an idiosyncratic version oftype theory (theramified theory of types).[20]
1913Hermann WeylDie Idee der Riemannschen Fläche[21]Weyl gave theRiemann surface concept ofcomplex analysis an axiomatic treatment, defining it as acomplex manifold of dimension one in terms ofneighbourhood systems.[22]
1914Felix HausdorffGrundzüge der MengenlehreThe book included axioms for what is now called aHausdorff topological space, building on Weyl's use of neighbourhoods.[22]
1915Maurice Fréchetabstract measures onmeasure spacesThe ideas ofLebesgue measure and associated integral, introduced firstly on thereal line andEuclidean spaces, were handled axiomatically onset systems.[23]
1920Stefan Banachcomplete normed vector spaceKnown now asBanach space, it is the classic setting forfunctional analysis; initially areal vector space was assumed.[24]
1921John Maynard KeynesA Treatise on ProbabilityKeynes's work subordinated probability to logic, under the influence ofPrincipia Mathematica. It gave an axiomatic treatment ofprobability interpretations.
1921Emmy NoetherIdealtheorie in Ringbereichen[25]Noether's paper introduced theascending chain condition onideals as an axiom incommutative rings, giving a subclass now calledNoetherian rings. It allowed a straightforward inductive proof ofHilbert's basis theorem. It is also considered the beginning of an "epoch" in abstract algebra.[26][27]
1923Norbert WienerWiener processWiener constructed a measure defining astochastic process model ofBrownian motion.[28]
1932Oswald Veblen andJ. H. C. WhiteheadThe Foundations of Differential Geometry (1932)The work gave the accepted axiomatic definition ofdifferential manifold,[29] apart from certain issues withseparation axioms.
1932John von NeumannMathematische Grundlagen der Quantenmechanik,Dirac–von Neumann axiomsContribution to themathematical formulation of quantum mechanics, dating back to a 1927 paper by von Neumann, proposing an axiomatisation of the founding works ofquantum mechanics, modelled formally on the notations ofPaul Dirac, using abstractHilbert space methods andunbounded operators.[30]
1933Andrey Kolmogorovprobability axiomsKolmogorv's work subordinated, in effect, mathematical probability tomeasure theory, while leaving its interpretation open. It built thereforeexpected values on theLebesgue integral.[31]
1945Samuel Eilenberg andNorman SteenrodEilenberg–Steenrod axiomsAn axiomatic system forhomology theory inalgebraic topology, it reflected developments since Noether advocated that homology classes be organised on abstract algebra principles.[26]
1945–1950Laurent Schwartztheory of distributionsUsing duality fortopological vector spaces oftest functions, Schwartz gave a unified axiomatic treatment of theDirac delta-function and a number of other formaloperator methods, and the geometric theory ofcurrents.[32]

Situation at mid-20th century

edit

Three prominent features of mathematics in 1950 were:

Axiomatics à la Bourbaki

edit

The aims of Bourbaki were for a treatment in the large of mathematics, which would be: (a) axiomatic, based down on a stripped-down logical foundation in set theory; (b) in the tradition of Hilbert and the Göttingen School, though excluding the needs of physics and computation; (c) a French reception of current developments. The initial work was carried out in a sharpyoung Turk reaction against theCours d'analyse mathématique, a standard text onclassical analysis from the beginning of the 20th century, byÉdouard Goursat, and in favour of the textModerne Algebra from the early 1930s onabstract algebra, byBartel Leendert van der Waerden.[33]

A pseudonymous paper from 1950, in fact the work ofJean Dieudonné, explained the attitude of Bourbaki to the axiomatic method.[34][35] The principal advantage of working axiomatically is asserted to lie in "elaboration" of mathematical "forms", orstructures; this takes precedence over the foundational work and the clarification ofinference. What Dieudonné wrote was of his time, as a departure from Hilbert's approaches, and not yet an arrival at structure in the sense implied by themorphisms ofcategory theory.[35]

Timeline of abstract varieties

edit

For the purposes of the exposition of his proof of theRiemann hypothesis for curves over finite fields, Weil made use of theJacobian of a curve, and some results ofintersection theory. Since he was working over afield of characteristic p rather than thecomplex numbers, carrying over the classic results required purely algebraic proofs. Further, he used a construction of the Jacobian as an "abstract variety": an intrinsic mathematical object, rather than aprojective algebraic variety found in acomplex projective space.

A generation later, with the publication of the textbookAlgebraic Geometry byRobin Hartshorne, "abstract variety" gained a standard definition withinscheme theory.[36]

DateAuthorWorkComments
1882Richard Dedekind andHeinrich Martin WeberTheorie der algebraischen Functionen einer VeränderlichenFor an irreduciblealgebraic curveC, defined over the complex numbers, and itsfunction fieldF, Dedekind and Weber considered a subringR such thatF was itsfield of quotients. The study ofideals inR recovered the points ofC, with a finite number of exceptions. The setting was adequate to prove theRiemann-Roch theorem.[37]
1910Ernst Steinitzalgebraic closureAny fieldK has an algebraic closure, a field that is essentially unique, consisting of all the roots of all the polynomials in one variable having coefficients inK.[38] The content of theFundamental Theorem of Algebra amounts to saying that the complex numbers are the algebraic closure of the real numbers. Algebraic geometry over any fieldK can be conceived of as studying the sets of solutions in its algebraic closure for systems of polynomials in any number of variables.
c.1911–1921Heinrich Kornblum (1890–1914),Emil Artinlocal zeta-functionsAfter Kornblum's dissertation on a polynomial ring analogue ofDirichlet's theorem on arithmetic progressions used the analogue of non-vanishing of anL-function, Artin's dissertationQuadratische Körper im Gebiete der höheren Kongruenzen onhyperelliptic curve]s over afinite field discussed the generating function now called the local zeta-function of a variety over a finite field.[39] As arational function, it had obvious poles; its zeroes became a research topic, as an analogue of theRiemann hypothesis.
1931Friedrich Karl Schmidtfunctional equation for local zeta-function of curvesWeil commented that both Schmidt's work, which applied the Riemann-Roch theorem to prove an analogue ofRiemann's functional equation, andHasse's theorem on elliptic curves, used a straightforward extension of the Dedekind–Weber foundations. It took thealgebraic closure of a finite field as field of constants.[40]
1932Wolfgang KrullAllgemeine Bewertungstheorie[41]Krull gave axioms for thevaluation concept. The set of valuations of thefunction field of an algebraic variety is related to thebirational geometry of the variety; only in the case of curves is the relationship to points of the variety straightforward. The terminology ofplaces, building on valuations, was used by the geometersOscar Zariski andShreeram Abhyankar.[42] Zariski stated that his work was influenced from the 1930s by the Dedekind–Weber paper.[37]
1941André Weilabstract varietiesWeil, at Princeton in spring 1941, in attempting complete foundations for his proof of theRiemann hypothesis for curves over finite fields, required some use of theJacobian variety over the algebraic closure. He later commented that the algebraists of the school of Emmy Noether were too close to the birational view of the Italian geometers: his need was not met by the birational approach to Jacobians viasymmetric products. He used a "piece" of the Jacobian, with its additive structure, as an "abstract" variety. He then found this idea had been implied byFrancesco Severi inTrattato di geometria algebrica: pt. 1. Geometria delle serie lineari (1926), pp. 283–4.[43]
1944Oscar ZariskiZariski's abstract Riemann surface (manifold)TheZariski topology, which foraffine space makes thealgebraic sets theclosed sets, arose around 1941, after a colloquium talk given by Zariski inPrinceton.[44] After some years in which it was mathematical folklore, Zariski published a related result, for valuations. For a fieldK and subringA, Zariski considered the set ofvaluation ring inK containingA, and having field of quotients equal toK. These subsets of all such valuation rings inK provided the base of open sets for a topology; and Zariski in geometric cases proved that the space of valuation rings thereby becamequasi-compact (i.e. not in generalHausdorff spaces but having theopen cover property ofcompact spaces).[45]
1942–1944André Weilcharts for abstract varietiesOn his own account, Weil was writing up his Ch. VII ofFoundations of Algebraic Geometry, published some years later, under some working assumptions. He adopted thecartographic method, as he called it, as applied by Weyl, Hausdorff, and Veblen and Whitehead; he made no use of the Zariski topology, not yet in print for varieties and associated with birational geometry. He definedintersection number only locally.[46]
c.1954Claude Chevalleyschémas (Mark I)Chevalley came to a foundational concept consisting of a set oflocal rings, such as the local rings associated with valuations.[47] He lectured on it in Japan, in 1954.[48] With the introduction ofsheaf theory, it could be considered aringed space. This definition was transitional.
c.1956Alexander Grothendieckscheme theoryA fresh start on axiomatic, abstract foundations for algebraic geometry was made with the definition of a scheme as a ringed space with each point having a neighbourhood of the form Spec(A), whereA is a commutative ring and Spec meansspectrum of a commutative ring, with points theprime ideals. Grothendieck was working on the theory, for Noetherian rings, in Chevalley's seminar, in 1956. The theory was developed in the book seriesÉléments de géométrie algébrique, co-authored by Grothendieck and Dieudonné, started in 1958.[49]

Axiomatic QFT

edit

Plausible axioms for QFT, theWightman axioms, were introduced byArthur Wightman. The need for non-trivial examples for these axioms led toconstructive quantum field theory, launched by work ofArthur Jaffe andOscar Lanford, in doctoral dissertations supervised by Wightman in the mid-1960s.[50]

Axiomatization

edit

Inmathematics,axiomatization is the process of taking a body of knowledge and working backwards towards its axioms. It is the formulation of a system of statements (i.e.axioms) that relate a number of primitive terms — in order that aconsistent body ofpropositions may be deriveddeductively from these statements. Thereafter, theproof of any proposition should be, in principle, traceable back to these axioms.

Axiomatization typically involves choices, and once a theory is axiomatic, it may be possible to change the set of axioms without affecting the mathematical results implied.

Axioms and postulates

edit

InAncient Greek logic, a contrast between axioms andpostulates was recognised ("postulate" being, however, an English term taken frommedieval Latin). It reflected, without being applied consistently, axioms as speaking aboutprimitive notions in a way that should becommon ground; and postulates as "requests" or "demands", for the purposes ofargument.Aristotle's view wasminimalist about postulates.[51]

From the time of Boole's work in the 1840s, in thealgebra of logic tradition, logic itself was developed from "postulates" alone. The minimalist view was taken, by the end of the 19th century, to imply research on independence of axioms.Mathematical elegance was also a consideration.[52]Friedrich Schur criticised the lack of independence of Hilbert's axioms for geometry given inGrundlagen der Geometrie.[53]

Timeline of postulational analysis

edit

Postulational analysis, according toSusan Stebbing, is what is used "in the construction of adeductive system".[54] It is a term applied to the correcting or adjusting of axiomatic systems. Axioms may be added to, or removed from, the system; they may be strengthened or weakened. It is also possible to change the logical calculus used for deduction.

DateAuthorWorkComments
Fourth century BCE tothird century BCEEuclid ofAlexandriaThe ElementsThe Greek term used by Euclid was αἰτήματα (aitēmata).[51] Its standard English translation is "postulate".[55]
1882Moritz PaschPasch's axiomPasch introduced an axiom ofplane geometry not proved by Euclid, but used by him tacitly.[56] It was not a consequence of Euclid's axioms, i.e. wasindependent of Euclid's system.
2024Terence TaoEquational Theories Project[57]A project to have a complete calibration of theories inequational logic for amagma, where the binary operation is used at most four times. Apartial order on the theories makesT≤U whenT implies all the theorems implied byU. The purpose of the project was to determine all the cases of ≤, so that an accurateHasse diagram of the partial order can be drawn.Proof assistant software was used in some cases. The project was completed in April 2025.[58]

Properties

edit

Four important properties of an axiom system are consistency, relative consistency, completeness and independence. An axiomatic system is said to beconsistent if it lackscontradiction. That is, it is impossible to derive both a statement and its negation from the system's axioms.[59]Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explosion).Relative consistency comes into play when we can not prove the consistency of an axiom system. However, in some cases we can show that an axiom system A is consistent if anotheraxiom set B is consistent.[59]

In an axiomatic system, an axiom is calledindependent if it cannot be proven or disproven from other axioms in the system. A system is called independent if each of its underlying axioms is independent.[59] Unlike consistency, in many cases independence is not a necessary requirement for a functioning axiomatic system — though it is usually sought after to minimize the number of axioms in the system.

An axiomatic system is calledcomplete if for every statement, either itself or its negation is derivable from the system's axioms, i.e. every statement can be proven true or false by using the axioms.[59][60] However, note that in some cases it may beundecidable if a statement can be proven or not.

Axioms and models

edit

Amodel for an axiomatic system is aformal structure, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system. If an axiom system has a model, the axioms are said to have beensatisfied.[61] The existence of a model which satisfies an axiom system, proves theconsistency of the system.[62]

Models can also be used to show the independence of an axiom in the system. By constructing a model for a subsystem (without a specific axiom) shows that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem.[61]

Two models are said to beisomorphic if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship.[63] An axiomatic system for which every model is isomorphic to another is calledcategorical or categorial. However, this term should not be confused with the topic ofcategory theory. The property of categoriality (categoricity) ensures the completeness of a system, however the converse is not true: Completeness does not ensure the categoriality (categoricity) of a system, since two models can differ in properties that cannot be expressed by thesemantics of the system.

Incompleteness

edit

If the formal system is notcomplete not every proof can be traced back to the axioms of the system it belongs. For example, a number-theoretic statement might be expressible in the language of arithmetic (i.e. the language of the Peano axioms) and a proof might be given that appeals totopology orcomplex analysis. It might not be immediately clear whether another proof can be found that derives itself solely from the Peano axioms.

See also

edit
Wikiquote has quotations related toAxiomatic system.

References

edit
  1. ^Gauch, Hugh G. (2012).Scientific Method in Brief. Cambridge University Press. p. 136.ISBN 978-1-107-01962-1.
  2. ^Paseau, Alexander; Pregel, Fabian (Aug 25, 2023)."Deductivism in the Philosophy of Mathematics". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
  3. ^Lehman, Eric; Meyer, Albert R; Leighton, F Tom.Mathematics for Computer Science(PDF). Retrieved2 May 2023.
  4. ^Wong, David (Sep 5, 2024)."Comparative Philosophy: Chinese and Western". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
  5. ^Zalta, Edward N. (Aug 5, 2023)."Frege's Theorem and Foundations for Arithmetic". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
  6. ^Kline, Morris (1 March 1990).Mathematical Thought From Ancient to Modern Times, Volume 3. Oxford University Press. p. 1141.ISBN 978-0-19-977048-9.
  7. ^Zitarelli, David E.; Dumbaugh, Della; Kennedy, Stephen F. (28 July 2022).A History of Mathematics in the United States and Canada: Volume 2: 1900–1941. American Mathematical Society. p. 72.ISBN 978-1-4704-6730-2.
  8. ^Narens, Louis (11 September 2014).Theories of Meaningfulness. Psychology Press. p. 8.ISBN 978-0-415-65456-2.
  9. ^Padmanabhan, Ranganathan; Rudeanu, Sergiu (2008).Axioms for Lattices and Boolean Algebras. World Scientific. p. 73.ISBN 978-981-283-454-6.
  10. ^Baldwin, Thomas (27 November 2003).The Cambridge History of Philosophy 1870-1945. Cambridge University Press. p. 142.ISBN 978-0-521-59104-1.
  11. ^Gray, Jeremy (2000).The Hilbert Challenge. Oxford University Press. p. 7.ISBN 978-0-19-850651-5.
  12. ^Atiyah, Michael Francis (2014).Collected Works. Vol. VII. Clarendon Press. p. 70.ISBN 978-0-19-968926-2.
  13. ^Corry, L. (1 November 2004).David Hilbert and the Axiomatization of Physics (1898–1918): From Grundlagen der Geometrie to Grundlagen der Physik. Springer Science & Business Media. p. 101.ISBN 978-1-4020-2777-2.
  14. ^abEmmer, Michele, ed. (2 December 2003).Mathematics and Culture I. Springer Science & Business Media. p. 10.ISBN 978-3-540-01770-7.
  15. ^Arnold, V. I. (14 July 2015).Experimental Mathematics. American Mathematical Soc. pp. 4–5.ISBN 978-0-8218-9416-3.
  16. ^Centina, Andrea Del; Gimigliano, Alessandro (3 March 2025).From Here to Infinity: Tracing the Origin and Development of Projective Geometry. Springer Nature. p. 1739.ISBN 978-3-031-72585-2.
  17. ^Baldwin, Thomas (27 November 2003).The Cambridge History of Philosophy 1870-1945. Cambridge University Press. p. 589.ISBN 978-0-521-59104-1.
  18. ^Weisstein, Eric W."Zermelo-Fraenkel Axioms".mathworld.wolfram.com. Retrieved2019-10-31.
  19. ^Corry, Leo (6 December 2012).Modern Algebra and the Rise of Mathematical Structures. Birkhäuser. p. 195.ISBN 978-3-0348-7917-0.
  20. ^Ewald, William (Nov 17, 2018). "The Emergence of First-Order Logic". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
  21. ^Weyl, Hermann (1913).Die Idee der Riemannschen Fläche. Leipzig: B.G. Teubner.
  22. ^abWillard, Stephen (12 July 2012).General Topology. Courier Corporation. p. 298.ISBN 978-0-486-13178-8.
  23. ^Heyde, C. C.; Seneta, E. (9 August 2001).Statisticians of the Centuries. Springer Science & Business Media. p. 331.ISBN 978-0-387-95283-3.
  24. ^Giles, J. R. (13 March 2000).Introduction to the Analysis of Normed Linear Spaces. Cambridge University Press. p. 259.ISBN 978-0-521-65375-6.
  25. ^Noether, Emmy (1 March 1921)."Idealtheorie in Ringbereichen".Mathematische Annalen (in German).83 (1–2):24–66.doi:10.1007/BF01464225.
  26. ^abNewman, Harvey B.; Ypsilantis, Thomas (6 December 2012).History of Original Ideas and Basic Discoveries in Particle Physics. 955: Springer Science & Business Media.ISBN 978-1-4613-1147-8.{{cite book}}: CS1 maint: location (link)
  27. ^Kleiner, Israel (2 October 2007).A History of Abstract Algebra. Springer Science & Business Media. p. 94.ISBN 978-0-8176-4684-4.
  28. ^Prokhorov, Yurij V.; Shiryaev, Albert N. (14 March 2013).Probability Theory III: Stochastic Calculus. Springer Science & Business Media. p. 7.ISBN 978-3-662-03640-2.
  29. ^O'Connor, John J.;Robertson, Edmund F.,"John Henry Constantine Whitehead",MacTutor History of Mathematics Archive,University of St Andrews
  30. ^Grattan-Guinness, Ivor (11 February 2005).Landmark Writings in Western Mathematics 1640-1940. Elsevier. pp. 194–195.ISBN 978-0-08-045744-4.
  31. ^Vilenkin, N. Ya (29 June 2013).In Search of Infinity. Springer Science & Business Media. p. 107.ISBN 978-1-4612-0837-2.
  32. ^Lützen, J. (6 December 2012).The Prehistory of the Theory of Distributions. Springer Science & Business Media. pp. 2–3.ISBN 978-1-4613-9472-3.
  33. ^Mashaal, Maurice (2006).Bourbaki. American Mathematical Soc. p. 47.ISBN 978-0-8218-3967-6.
  34. ^Bourbaki, Nicholas (1950)."The Architecture of Mathematics".The American Mathematical Monthly.57 (4):221–232.doi:10.2307/2305937.ISSN 0002-9890.JSTOR 2305937.
  35. ^abFerreira, Fernando; Kahle, Reinhard; Sommaruga, Giovanni (13 October 2022).Axiomatic Thinking I. Springer Nature. p. 86.ISBN 978-3-030-77657-2.
  36. ^Hartshorne, Robin (29 June 2013).Algebraic Geometry. Springer Science & Business Media. p. 105.ISBN 978-1-4757-3849-0.
  37. ^abCorry, Leo (6 December 2012).Modern Algebra and the Rise of Mathematical Structures. Birkhäuser. p. 119 and note 128.ISBN 978-3-0348-7917-0.
  38. ^Chambert-Loir, Antoine (21 December 2007).A Field Guide to Algebra. Springer Science & Business Media. p. 36.ISBN 978-0-387-26955-9.
  39. ^Goldstein, Catherine; Schappacher, Norbert; Schwermer, Joachim, eds. (3 February 2007).The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae. Springer Science & Business Media. pp. 176–177.ISBN 978-3-540-34720-0.
  40. ^Weil, André (1979).Collected Papers (in French). Vol. I. Springer. pp. 540–541.ISBN 978-3-540-90330-7.
  41. ^Krull, Wolfgang (1999).Collected Papers (in German). Walter de Gruyter. p. 580.ISBN 978-3-11-012771-3.
  42. ^Lang, Serge (20 March 2019).Introduction to Algebraic Geometry. Courier Dover Publications. p. 19.ISBN 978-0-486-83422-1.
  43. ^Weil, André (1979).Collected Papers (in French). Vol. I. Springer. pp. 552–553.ISBN 978-3-540-90330-7.
  44. ^James, Ioan (2002).Remarkable Mathematicians: From Euler to Von Neumann. Cambridge University Press. p. 405.ISBN 978-0-521-52094-2.
  45. ^Fontana, Marco; Frisch, Sophie; Glaz, Sarah (15 July 2014).Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions. Springer. p. 154.ISBN 978-1-4939-0925-4.
  46. ^Weil, André (1979).Collected Papers (in French). Vol. I. Springer. p. 556.ISBN 978-3-540-90330-7.
  47. ^Krömer, Ralph (25 June 2007).Tool and Object: A History and Philosophy of Category Theory. Springer Science & Business Media. p. 164.ISBN 978-3-7643-7524-9.
  48. ^Fantechi, Barbara, ed. (2005).Fundamental Algebraic Geometry: Grothendieck's FGA Explained. American Mathematical Soc. p. 248.ISBN 978-0-8218-4245-4.
  49. ^Fantechi, Barbara, ed. (2005).Fundamental Algebraic Geometry: Grothendieck's FGA Explained. American Mathematical Soc. p. 249.ISBN 978-0-8218-4245-4.
  50. ^Gesztesy, Fritz (2007).Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday. American Mathematical Soc. pp. 71–72.ISBN 978-0-8218-4248-5.
  51. ^abSmith, David E. (1 June 1958).History of Mathematics. Courier Corporation. p. 280 and note 5.ISBN 978-0-486-20430-7.{{cite book}}:ISBN / Date incompatibility (help)
  52. ^Burris, Stanley; Legris, Javier (Feb 12, 2021). "The Algebra of Logic Tradition". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
  53. ^Gray, Jeremy (2000).The Hilbert Challenge. Oxford University Press. p. 104.ISBN 978-0-19-850651-5.
  54. ^Beaney, Michael; Chapman, Siobhan (May 7, 2025). "Susan Stebbing". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
  55. ^Torretti, Roberto (Oct 20, 2016). "Nineteenth Century Geometry". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
  56. ^Hvidsten, Michael (8 December 2016).Exploring Geometry. CRC Press. p. 66.ISBN 978-1-4987-6082-9.
  57. ^"Equational Theories Project".teorth.github.io. Retrieved25 September 2025.
  58. ^Cepelewicz, Jordana (30 April 2025)."Mathematical Beauty, Truth and Proof in the Age of AI".Quanta Magazine.
  59. ^abcdA. G. Howson A Handbook of Terms Used in Algebra and Analysis, Cambridge UP, ISBN 0521084342 1972 pp 6
  60. ^Weisstein, Eric W."Complete Axiomatic Theory".mathworld.wolfram.com. Retrieved2019-10-31.
  61. ^abC. C. Chang and H. J. Keisler "Model Theory" Elsevier 1990, pp 1-7
  62. ^C. C. Chang and H. J. Keisler "Model Theory" Elsevier 1990, pp 1-7, Theorem 1.2.11
  63. ^Hodges, Wilfrid; Scanlon, Thomas (2018),"First-order Model Theory", in Zalta, Edward N. (ed.),The Stanford Encyclopedia of Philosophy (Winter 2018 ed.), Metaphysics Research Lab, Stanford University, retrieved2019-10-31

Further reading

edit
[8]ページ先頭
©2009-2025 Movatter.jp