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688:. The focus is on the manipulation of these symbols according to specified rules, rather than on the objects themselves. One common understanding of formalism takes mathematics as not a body of propositions representing an abstract piece of reality but much more akin to a game, bringing with it no more ontological commitment of objects or properties than playing
881:
to functions and so these entities are not included among the objects. Some authors make use of Fregeâs notion of âobjectâ when discussing abstract objects. But though Fregeâs sense of âobjectâ is important, it is not the only way to use the term. Other philosophers include properties and relations
886:, properties and relations of higher type (e.g., properties of properties, and properties of relations) may be all be considered âobjectsâ. This latter use of âobjectâ is interchangeable with âentity.â It is this more broad interpretation that mathematicians mean when they use the term 'object'.
728:
asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its
406:
708:: A leading mathematician of the early 20th century, Hilbert is one of the most prominent advocates of formalism. He believed that mathematics is a system of formal rules and that its truth lies in the consistency of these rules rather than any connection to an abstract reality.
536:: A philosopher known for his work in the philosophy of science and nominalism. He argued against the existence of abstract objects, proposing instead that mathematical objects are merely a product of our linguistic and symbolic conventions.
626:(Basic Laws of Arithmetic), Frege attempted to show that arithmetic could be derived from logical axioms. He developed a formal system that aimed to express all of arithmetic in terms of logic. Fregeâs work laid the groundwork for much of
333:
to express these ideas. Moreover, it is hard to imagine how areas like quantum mechanics and general relativity could have developed without their assistance from mathematics, and therefore, one could argue that mathematics is
796:
suggests that mathematical objects are defined by their place within a structure or system. The nature of a number, for example, is not tied to any particular thing, but to its role within the system of
524:
or shorthand for describing relationships and structures within our language and theories. Under this view, mathematical objects don't have an existence beyond the symbols and concepts we use.
278:
580:, and truths can be derived from purely logical principles and definitions. Logicism faced challenges, particularly with the Russillian axioms, the Multiplicative axiom (now called the
696:. In this view, mathematics is about the consistency of formal systems rather than the discovery of pre-existing objects. Some philosphers consider logicism to be a type of formalism.
813:: A philosopher known for his work in the philosophy of mathematics, particularly his paper "What Numbers Could Not Be," which argues for a structuralist view of mathematical objects.
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716:: German mathematician and philosopher who, while not strictly a formalist, contributed to formalist ideas, particularly in his work on the foundations of mathematics.
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arguments to values, and is denoted by an incomplete expression, whereas an object is a âcompleteâ entity and can be denoted by a singular term. Frege reduced
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was enormously influential, the effort to reduce all of mathematics to logic was ultimately seen as incomplete. However, it did advance the development of
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which offer different perspectives on the matter, and many famous mathematicians and philosophers each have differing opinions on which is more correct.
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might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the
508:, Penrose has argued for a Platonic view of mathematics, suggesting that mathematical truths exist in a realm of abstract reality that we discover.
572:
objects forming the subject matter of those branches of mathematics are logical objects. In other words, mathematics is fundamentally a branch of
250:, and it is in this latter which mathematical objects usually lie. What constitutes an "object" is foundational to many areas of philosophy, from
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argue that we should believe the mathematical objects for which these theories depend actually exist, that is, we ought to have an
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1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
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608:. This meant that not all mathematical truths could be derived purely from a logical system, undermining the logicist program.
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who, though not a mathematician, laid the groundwork for
Platonism by positing the existence of an abstract realm of perfect
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848:, is a function in the general sense; here as in the association of any of the four colored shapes in X to its color in Y.
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741:, which is at odds with its classical interpretation. There are many forms of constructivism. These include the program of
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to these theories. It is because of this unreasonable effectiveness and indispensibility of mathematics that philosophers
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552:," which argues that mathematical statements are useful fictions that don't correspond to any actual abstract objects.
457:, so are statements about numbers and sets. Mathematicians discover these objects rather than invent them. (See also:
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489:: A 20th-century logician and mathematician, Gödel was a strong proponent of mathematical Platonism, and his work in
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Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools and instructional design
801:. In a sense, the thesis is that mathematical objects (if there are such objects) simply have no intrinsic nature.
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258:(the study of knowledge). In mathematics, objects are often seen as entities that exist independently of the
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821:: Another prominent philosopher who has developed and defended structuralism, especially in his book
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denies the independent existence of mathematical objects. Instead, it suggests that they are merely
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Bridges, Douglas; Palmgren, Erik; Ishihara, Hajime (2022), Zalta, Edward N.; Nodelman, Uri (eds.),
1951:. Cambridge introductions to philosophy (1. publ ed.). Cambridge: Cambridge University Press.
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are simultaneously homes to mathematical objects and mathematical objects in their own right. In
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to one another. Philosophers debate whether objects have an independent existence outside of
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Falguera, JosĂ© L.; MartĂnez-Vidal, Concha; Rosen, Gideon (2022), Zalta, Edward N. (ed.),
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Falguera, JosĂ© L.; MartĂnez-Vidal, Concha; Rosen, Gideon (2022), Zalta, Edward N. (ed.),
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365:(Premise 2) Mathematical entities are indispensable to our best scientific theories.
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among the abstract objects. And when the background context for discussing objects is
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Rettler, Bradley; Bailey, Andrew M. (2024), Zalta, Edward N.; Nodelman, Uri (eds.),
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is anything that has been (or could be) formally defined, and with which one may do
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about electrons and planets are true or false as these objects contain perfectly
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A Subject with No Object: Strategies for
Nominalistic Reconstrual of Mathematics
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Bell, John L.; Korté, Herbert (2024), Zalta, Edward N.; Nodelman, Uri (eds.),
1076:- Commonly used as an example of a continuous, nowhere-differentiable function
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relies largely on large and often vastly different areas of mathematics. From
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2567:. Undergraduate texts in mathematics. New York: Springer-Verlag. p. 30.
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Reck, Erich; Schiemer, Georg (2023), Zalta, Edward N.; Nodelman, Uri (eds.),
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and was highly influential, though it encountered difficulties, most notably
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121:. Typically, a mathematical object can be a value that can be assigned to a
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is an argument for the existence of mathematical objects based on their
234:), or if their existence is dependent on mental constructs or language (
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2660:(Summer 2022 ed.), Metaphysics Research Lab, Stanford University
2523:(Spring 2023 ed.), Metaphysics Research Lab, Stanford University
2404:(Summer 2024 ed.), Metaphysics Research Lab, Stanford University
2352:(Spring 2024 ed.), Metaphysics Research Lab, Stanford University
2274:(Winter 2023 ed.), Metaphysics Research Lab, Stanford University
2132:(Summer 2024 ed.), Metaphysics Research Lab, Stanford University
2107:(Winter 2023 ed.), Metaphysics Research Lab, Stanford University
2082:(Winter 2016 ed.), Metaphysics Research Lab, Stanford University
2057:(Summer 2024 ed.), Metaphysics Research Lab, Stanford University
2032:(Winter 2023 ed.), Metaphysics Research Lab, Stanford University
2007:(Summer 2022 ed.), Metaphysics Research Lab, Stanford University
1930:(Summer 2024 ed.), Metaphysics Research Lab, Stanford University
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2496:"Structuralism, Mathematical | Internet Encyclopedia of Philosophy"
2429:(Fall 2022 ed.), Metaphysics Research Lab, Stanford University
2202:(Fall 2020 ed.), Metaphysics Research Lab, Stanford University
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620:: Frege is often regarded as the founder of logicism. In his work,
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126:
372:) We ought to have ontological commitment to mathematical entities
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Linnebo, Ăystein (2024), Zalta, Edward N.; Nodelman, Uri (eds.),
1440:
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to avoid the paradoxes that Fregeâs system encountered. Although
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227:
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102:
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2220:"Mathematical Nominalism | Internet Encyclopedia of Philosophy"
2150:"Platonism, Mathematical | Internet Encyclopedia of Philosophy"
1812:
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1004:
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215:
130:
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Tennant, Neil (2023), Zalta, Edward N.; Nodelman, Uri (eds.),
2099:
Horsten, Leon (2023), Zalta, Edward N.; Nodelman, Uri (eds.),
2049:
Colyvan, Mark (2024), Zalta, Edward N.; Nodelman, Uri (eds.),
2024:
Horsten, Leon (2023), Zalta, Edward N.; Nodelman, Uri (eds.),
865:. According to his view, a function is a kind of âincompleteâ
3686:
3032:
2877:
2291:
2051:"Indispensability Arguments in the Philosophy of Mathematics"
1831:
693:
573:
469:
652:. They attempted to derive all of mathematics from a set of
2651:
2344:
Weir, Alan (2024), Zalta, Edward N.; Nodelman, Uri (eds.),
1998:
1479:
1404:
761:, the constructive recursive mathematics of mathematicians
689:
2791:
Thinking about mathematics: The philosophy of mathematics
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782:
481:
or ideas, which influenced later thinkers in mathematics.
157:. Mathematical objects can be very complex; for example,
384:(or sometimes Predicativism) which states that the only
2420:
564:
asserts that all mathematical truths can be reduced to
2292:"Frege, Gottlob | Internet Encyclopedia of Philosophy"
646:, further developed logicism in their monumental work
16:
Anything with which mathematical reasoning is possible
1043:
2759:
Proof and Other
Dilemmas: Mathematics and Philosophy
425:
asserts that mathematical objects are seen as real,
350:
to them. The argument is described by the following
129:. Commonly encountered mathematical objects include
634:, which revealed inconsistencies in Fregeâs system.
2074:Paseau, Alexander (2016), Zalta, Edward N. (ed.),
2540:Philosophy of Mathematics: Structure and Ontology
823:Philosophy of Mathematics: Structure and Ontology
548:who has developed the form of nominalism called "
181:
4592:
2517:"Structuralism in the Philosophy of Mathematics"
2448:(1977a). "Aspects of Constructive Mathematics".
1946:
269:
246:, such as physical objects in the world, to the
2304:
176:
2194:Bueno, OtĂĄvio (2020), Zalta, Edward N. (ed.),
592:, which showed that any sufficiently powerful
93:hypercube is an example of mathematical object
2848:
2823:AMOF: The Amazing Mathematical Object Factory
2196:"Nominalism in the Philosophy of Mathematics"
2076:"Naturalism in the Philosophy of Mathematics"
1921:
376:This argument resonates with a philosophy in
2514:
2346:"Formalism in the Philosophy of Mathematics"
2126:"Platonism in the Philosophy of Mathematics"
777:. Constructivism also includes the study of
449:exist, so do numbers and sets. And just as
329:, it allows these areas to have an elegant
109:. In the usual language of mathematics, an
3040:
2855:
2841:
829:
169:are considered as mathematical objects in
2727:Metaphysical Myths, Mathematical Practice
2590:"Frege's Theory of Functions and Objects"
2395:
2317:. Cambridge University Press. p. 1.
1947:Carroll, John W.; Markosian, Ned (2010).
1139:
69:Learn how and when to remove this message
2732:Burgess, John, and Rosen, Gideon, 1997.
2587:
1893:List of two-dimensional geometric shapes
1842:
833:
404:
80:
32:This article includes a list of general
2658:The Stanford Encyclopedia of Philosophy
2521:The Stanford Encyclopedia of Philosophy
2427:The Stanford Encyclopedia of Philosophy
2402:The Stanford Encyclopedia of Philosophy
2350:The Stanford Encyclopedia of Philosophy
2272:The Stanford Encyclopedia of Philosophy
2265:
2200:The Stanford Encyclopedia of Philosophy
2130:The Stanford Encyclopedia of Philosophy
2123:
2105:The Stanford Encyclopedia of Philosophy
2098:
2080:The Stanford Encyclopedia of Philosophy
2055:The Stanford Encyclopedia of Philosophy
2048:
2030:The Stanford Encyclopedia of Philosophy
2023:
2005:The Stanford Encyclopedia of Philosophy
1928:The Stanford Encyclopedia of Philosophy
1120:
325:), not only does mathematics help with
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2862:
2762:. Mathematical Association of America.
2557:
2368:
2073:
890:List of mathematical objects by branch
464:Some some notable platonists include:
2836:
2310:
2238:
2193:
2168:
1777:
1549:
1322:
901:
804:Some notable structuralists include:
395:
2696:. Academic Press. pp. 205â228.
2633:Routledge Encyclopedia of Philosophy
2474:Foundations of Constructive Analysis
2343:
388:standards on existence are those of
18:
2805:Stanford Encyclopedia of Philosophy
2369:Simons, Peter (2009). "Formalism".
1485:
1092:- Commonly used as an example of a
1082:- Commonly used as an example of a
982:
684:treats objects as symbols within a
153:of other mathematical objects, and
125:, and therefore can be involved in
13:
2676:
1163:
729:non-existence and then deriving a
588:, and later with the discovery of
527:Some notable nominalists incluse:
429:that exist independently of human
38:it lacks sufficient corresponding
14:
4632:
2797:
2542:. Oxford University Press, 1997.
1940:
930:
720:
699:Some notable formalists include:
576:, and all mathematical concepts,
4574:
2682:"Chapter VII. Cohomotopy Groups"
2626:
1592:
1209:
788:
611:Some notable logicists include:
23:
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2142:
1523:
855:famously distinguished between
590:Gödelâs incompleteness theorems
214:, objects are often considered
2756:, and Simons, Roger A., 2011.
2460::10.1016/S0049-237X(08)71127-3
2450:Handbook of Mathematical Logic
2117:
2092:
2067:
2042:
2017:
1992:
1965:
1949:An introduction to metaphysics
1915:
1206:are also mathematical objects
242:). Objects can range from the
182:Nature of mathematical objects
1:
4535:History of mathematical logic
2729:. Cambridge University Press.
1903:
1665:
785:and the study of philosophy.
783:Constructive ZermeloâFraenkel
733:from that assumption. Such a
512:
275:Quine-Putnam indispensability
270:Quine-Putnam indispensability
4460:Primitive recursive function
2767:What is Mathematics, Really?
2476:. New York: Academic Press.
2314:What is Analytic Philosophy?
2239:Field, Hartry (2016-10-27).
1237:Theoretical computer science
1094:Non-analytic smooth function
676:
596:(like those used to express
400:
177:In philosophy of mathematics
7:
2828:Mathematical Object Exhibit
2793:. Oxford University Press.
2748:The Mathematical Experience
2245:. Oxford University Press.
2101:"Philosophy of Mathematics"
2026:"Philosophy of Mathematics"
1974:, and Rosen, Gideon, 1997.
1878:List of mathematical shapes
1861:
1707:
1315:
726:Mathematical constructivism
623:Grundgesetze der Arithmetik
556:
10:
4639:
3524:SchröderâBernstein theorem
3251:Monadic predicate calculus
2910:Foundations of mathematics
2588:Marshall, William (1953).
2423:"Constructive Mathematics"
2268:"Logicism and Neologicism"
894:
279:unreasonable effectiveness
4570:
4557:Philosophy of mathematics
4506:Automated theorem proving
4488:
4383:
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4108:
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3677:
3653:
3631:Von NeumannâBernaysâGödel
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2375:. Elsevier. p. 292.
2372:Philosophy of Mathematics
2169:Roibu, Tib (2023-07-11).
1356:- 2-dimensional polytope
779:constructive set theories
493:was a major influence on
222:and can stand in various
188:Philosophy of mathematics
2769:Oxford University Press.
2750:. Mariner Books: 156â62.
2734:A Subject with No Object
2594:The Philosophical Review
1851:Differentiable manifolds
1060:Differentiable functions
1038:Order-theoretic lattices
1034:group-theoretic lattices
546:contemporary philosopher
254:(the study of being) to
4207:Self-verifying theories
4028:Tarski's axiomatization
2979:Tarski's undefinability
2974:incompleteness theorems
2242:Science without Numbers
1980:Oxford University Press
1837:Alexander horned sphere
1391:3-dimensional polytope
1190:Natural transformations
1153:Heaviside step function
830:Objects versus mappings
194:" touches on topics of
53:more precise citations.
4606:Philosophical concepts
4581:Mathematics portal
4192:Proof of impossibility
3840:propositional variable
3150:Propositional calculus
1898:Mathematical structure
1141:Differential equations
1069:Pathological functions
849:
739:existential quantifier
735:proof by contradiction
682:Mathematical formalism
644:Alfred North Whitehead
642:: Russell, along with
506:mathematical physicist
459:Mathematical Platonism
419:
374:
348:ontological commitment
94:
4616:Mathematical concepts
4450:Kolmogorov complexity
4403:Computably enumerable
4303:Model complete theory
4095:Principia Mathematica
3155:Propositional formula
2984:BanachâTarski paradox
2765:Hersh, Reuben, 1997.
2736:. Oxford Univ. Press.
2446:Troelstra, Anne Sjerp
1844:Differential topology
1623:Arithmetic operations
1049:Mathematical Analysis
920:Algebraic expressions
837:
775:constructive analysis
662:Principia Mathematica
649:Principia Mathematica
408:
356:
303:differential geometry
84:
4601:Mathematical objects
4398:ChurchâTuring thesis
4385:Computability theory
3594:continuum hypothesis
3112:Square of opposition
2970:Gödel's completeness
2817:Mathematical Objects
2311:Glock, H.J. (2008).
1148:Dirac delta function
1122:Nonstandard analysis
1074:Weierstrass function
910:Algebraic operations
455:objective properties
412:The School of Athens
323:Mathematical biology
4552:Mathematical object
4443:P versus NP problem
4408:Computable function
4202:Reverse mathematics
4128:Logical consequence
4005:primitive recursive
4000:elementary function
3773:Free/bound variable
3626:TarskiâGrothendieck
3145:Logical connectives
3075:Logical equivalence
2925:Logical consequence
2783:. Lawrence Erlbaum.
2725:Azzouni, J., 1994.
2635:, London: Routledge
2171:"Sir Roger Penrose"
1692:Membership relation
1563:Logical connectives
1504:SierpiĆski triangle
1056:Continuous funtions
915:Algebraic functions
670:analytic philosophy
522:convenient fictions
378:applied mathematics
119:mathematical proofs
115:deductive reasoning
99:mathematical object
4350:Transfer principle
4313:Semantics of logic
4298:Categorical theory
4274:Non-standard model
3788:Logical connective
2915:Information theory
2864:Mathematical logic
2812:"âby Gideon Rosen.
2654:"Abstract Objects"
2629:"Abstract objects"
2001:"Abstract Objects"
1779:Algebraic topology
1741:Topological spaces
1660:Successor function
1551:Mathematical logic
1324:Euclidian geometry
1084:nowhere continuous
1080:Dirichlet function
1064:Analytic functions
903:Elementary algebra
850:
838:In mathematics, a
666:mathematical logic
420:
409:Plato depicted in
396:Schools of thought
307:general relativity
264:schools of thought
190:, the concept of "
95:
4588:
4587:
4520:Abstract category
4323:Theories of truth
4133:Rule of inference
4123:Natural deduction
4104:
4103:
3649:
3648:
3354:Cartesian product
3259:
3258:
3165:Many-valued logic
3140:Boolean functions
3023:Russell's paradox
2998:diagonal argument
2895:First-order logic
2815:Wells, Charles. "
2574:978-0-387-90092-6
2324:978-0-521-87267-6
2252:978-0-19-877791-5
1958:978-0-521-82629-7
1873:Impossible object
1791:Cohomotopy groups
1786:Cohomology groups
1509:SierpiĆski carpet
1349:Regular polytopes
1158:Laplace transform
632:Russellâs paradox
600:) cannot be both
586:Axiom of Infinity
504:: A contemporary
475:Greek philosopher
427:abstract entities
299:quantum mechanics
287:branch of science
147:geometric objects
79:
78:
71:
4628:
4579:
4578:
4530:History of logic
4525:Category of sets
4418:Decision problem
4197:Ordinal analysis
4138:Sequent calculus
4036:Boolean algebras
3976:
3975:
3950:
3921:logical/constant
3675:
3674:
3661:
3584:ZermeloâFraenkel
3335:Set operations:
3270:
3269:
3207:
3038:
3037:
3018:LöwenheimâSkolem
2905:Formal semantics
2857:
2850:
2843:
2834:
2833:
2810:Abstract Objects
2713:
2712:
2710:
2708:
2695:
2686:
2674:
2668:
2667:
2666:
2665:
2649:
2643:
2642:
2641:
2640:
2624:
2618:
2617:
2585:
2579:
2578:
2564:Naive set theory
2555:
2549:
2537:
2531:
2530:
2529:
2528:
2512:
2506:
2505:
2503:
2502:
2492:
2486:
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2443:
2437:
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2418:
2412:
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2393:
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2308:
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2263:
2257:
2256:
2236:
2230:
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2227:
2226:
2216:
2210:
2209:
2208:
2207:
2191:
2185:
2184:
2182:
2181:
2175:Geometry Matters
2166:
2160:
2159:
2157:
2156:
2146:
2140:
2139:
2138:
2137:
2121:
2115:
2114:
2113:
2112:
2096:
2090:
2089:
2088:
2087:
2071:
2065:
2064:
2063:
2062:
2046:
2040:
2039:
2038:
2037:
2021:
2015:
2014:
2013:
2012:
1996:
1990:
1969:
1963:
1962:
1944:
1938:
1937:
1936:
1935:
1919:
1888:List of surfaces
1697:Cardinal numbers
1616:Rational numbers
1487:Fractal geometry
1233:Computer science
984:Abstract algebra
639:Bertrand Russell
495:modern platonism
439:physical objects
433:, often in some
283:natural sciences
103:abstract concept
91:four-dimensional
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
4638:
4637:
4631:
4630:
4629:
4627:
4626:
4625:
4611:Category theory
4591:
4590:
4589:
4584:
4573:
4566:
4511:Category theory
4501:Algebraic logic
4484:
4455:Lambda calculus
4393:Church encoding
4379:
4355:Truth predicate
4211:
4177:Complete theory
4100:
3969:
3965:
3961:
3956:
3948:
3668: and
3664:
3659:
3645:
3621:New Foundations
3589:axiom of choice
3572:
3534:Gödel numbering
3474: and
3466:
3370:
3255:
3205:
3186:
3135:Boolean algebra
3121:
3085:Equiconsistency
3050:Classical logic
3027:
3008:Halting problem
2996: and
2972: and
2960: and
2959:
2954:Theorems (
2949:
2866:
2861:
2800:
2787:Stewart Shapiro
2720:Further reading
2717:
2716:
2706:
2704:
2693:
2690:Homotopy theory
2684:
2675:
2671:
2663:
2661:
2650:
2646:
2638:
2636:
2625:
2621:
2606:10.2307/2182877
2586:
2582:
2575:
2559:Halmos, Paul R.
2556:
2552:
2538:
2534:
2526:
2524:
2513:
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2500:
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2222:
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2213:
2205:
2203:
2192:
2188:
2179:
2177:
2167:
2163:
2154:
2152:
2148:
2147:
2143:
2135:
2133:
2122:
2118:
2110:
2108:
2097:
2093:
2085:
2083:
2072:
2068:
2060:
2058:
2047:
2043:
2035:
2033:
2022:
2018:
2010:
2008:
1997:
1993:
1970:
1966:
1959:
1945:
1941:
1933:
1931:
1920:
1916:
1906:
1868:Abstract object
1864:
1847:
1801:Homotopy groups
1796:Homology groups
1782:
1712:
1702:Ordinal numbers
1670:
1606:Natural numbers
1597:
1554:
1528:
1490:
1395:Platonic solids
1327:
1320:
1244:Data structures
1240:
1214:
1168:
1165:Category theory
1144:
1125:
1115:Jacobian matrix
1052:
1024:Chain complexes
987:
935:
906:
899:
892:
832:
818:Stewart Shapiro
810:Paul Benacerraf
791:
723:
679:
582:Axiom of Choice
559:
515:
403:
398:
272:
184:
179:
151:transformations
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
4636:
4635:
4624:
4623:
4618:
4613:
4608:
4603:
4586:
4585:
4571:
4568:
4567:
4565:
4564:
4559:
4554:
4549:
4544:
4543:
4542:
4532:
4527:
4522:
4513:
4508:
4503:
4498:
4496:Abstract logic
4492:
4490:
4486:
4485:
4483:
4482:
4477:
4475:Turing machine
4472:
4467:
4462:
4457:
4452:
4447:
4446:
4445:
4440:
4435:
4430:
4425:
4415:
4413:Computable set
4410:
4405:
4400:
4395:
4389:
4387:
4381:
4380:
4378:
4377:
4372:
4367:
4362:
4357:
4352:
4347:
4342:
4341:
4340:
4335:
4330:
4320:
4315:
4310:
4308:Satisfiability
4305:
4300:
4295:
4294:
4293:
4283:
4282:
4281:
4271:
4270:
4269:
4264:
4259:
4254:
4249:
4239:
4238:
4237:
4232:
4225:Interpretation
4221:
4219:
4213:
4212:
4210:
4209:
4204:
4199:
4194:
4189:
4179:
4174:
4173:
4172:
4171:
4170:
4160:
4155:
4145:
4140:
4135:
4130:
4125:
4120:
4114:
4112:
4106:
4105:
4102:
4101:
4099:
4098:
4090:
4089:
4088:
4087:
4082:
4081:
4080:
4075:
4070:
4050:
4049:
4048:
4046:minimal axioms
4043:
4032:
4031:
4030:
4019:
4018:
4017:
4012:
4007:
4002:
3997:
3992:
3979:
3977:
3958:
3957:
3955:
3954:
3953:
3952:
3940:
3935:
3934:
3933:
3928:
3923:
3918:
3908:
3903:
3898:
3893:
3892:
3891:
3886:
3876:
3875:
3874:
3869:
3864:
3859:
3849:
3844:
3843:
3842:
3837:
3832:
3822:
3821:
3820:
3815:
3810:
3805:
3800:
3795:
3785:
3780:
3775:
3770:
3769:
3768:
3763:
3758:
3753:
3743:
3738:
3736:Formation rule
3733:
3728:
3727:
3726:
3721:
3711:
3710:
3709:
3699:
3694:
3689:
3684:
3678:
3672:
3655:Formal systems
3651:
3650:
3647:
3646:
3644:
3643:
3638:
3633:
3628:
3623:
3618:
3613:
3608:
3603:
3598:
3597:
3596:
3591:
3580:
3578:
3574:
3573:
3571:
3570:
3569:
3568:
3558:
3553:
3552:
3551:
3544:Large cardinal
3541:
3536:
3531:
3526:
3521:
3507:
3506:
3505:
3500:
3495:
3480:
3478:
3468:
3467:
3465:
3464:
3463:
3462:
3457:
3452:
3442:
3437:
3432:
3427:
3422:
3417:
3412:
3407:
3402:
3397:
3392:
3387:
3381:
3379:
3372:
3371:
3369:
3368:
3367:
3366:
3361:
3356:
3351:
3346:
3341:
3333:
3332:
3331:
3326:
3316:
3311:
3309:Extensionality
3306:
3304:Ordinal number
3301:
3291:
3286:
3285:
3284:
3273:
3267:
3261:
3260:
3257:
3256:
3254:
3253:
3248:
3243:
3238:
3233:
3228:
3223:
3222:
3221:
3211:
3210:
3209:
3196:
3194:
3188:
3187:
3185:
3184:
3183:
3182:
3177:
3172:
3162:
3157:
3152:
3147:
3142:
3137:
3131:
3129:
3123:
3122:
3120:
3119:
3114:
3109:
3104:
3099:
3094:
3089:
3088:
3087:
3077:
3072:
3067:
3062:
3057:
3052:
3046:
3044:
3035:
3029:
3028:
3026:
3025:
3020:
3015:
3010:
3005:
3000:
2988:Cantor's
2986:
2981:
2976:
2966:
2964:
2951:
2950:
2948:
2947:
2942:
2937:
2932:
2927:
2922:
2917:
2912:
2907:
2902:
2897:
2892:
2887:
2886:
2885:
2874:
2872:
2868:
2867:
2860:
2859:
2852:
2845:
2837:
2831:
2830:
2825:
2820:
2813:
2799:
2798:External links
2796:
2795:
2794:
2784:
2770:
2763:
2751:
2737:
2730:
2715:
2714:
2669:
2644:
2619:
2600:(3): 374â390.
2580:
2573:
2550:
2532:
2507:
2487:
2481:
2470:Bishop, Errett
2462:
2438:
2413:
2398:"Hermann Weyl"
2388:
2381:
2361:
2336:
2323:
2303:
2283:
2258:
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2211:
2186:
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2116:
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2016:
1991:
1964:
1957:
1939:
1913:
1912:
1905:
1902:
1901:
1900:
1895:
1890:
1885:
1883:List of shapes
1880:
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1689:
1680:
1678:Set partitions
1669:
1664:
1663:
1662:
1657:
1652:
1651:
1650:
1645:
1640:
1638:Multiplication
1635:
1630:
1620:
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1613:
1608:
1596:
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1590:
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1584:
1579:
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1548:
1547:
1546:
1527:
1522:
1521:
1520:
1515:
1506:
1501:
1496:
1494:Mandelbrot set
1489:
1484:
1483:
1482:
1473:
1455:
1454:
1453:
1448:
1443:
1438:
1431:Conic sections
1428:
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1423:
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1417:
1412:
1407:
1402:
1386:
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1383:
1382:
1377:
1372:
1367:
1360:Convex polygon
1342:
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1307:
1302:
1297:
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1287:
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1155:
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1143:
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1137:
1136:
1131:
1129:Infinitesimals
1124:
1119:
1118:
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1112:
1107:
1098:
1097:
1096:
1087:
1077:
1066:
1051:
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1040:
1031:
1026:
1021:
1016:
1007:
1002:
993:
986:
981:
980:
979:
977:Parallelepiped
974:
969:
964:
959:
954:
945:
934:
932:Linear algebra
929:
928:
927:
922:
917:
912:
905:
900:
898:
893:
891:
888:
831:
828:
827:
826:
814:
790:
787:
773:'s program of
722:
721:Constructivism
719:
718:
717:
709:
678:
675:
674:
673:
654:logical axioms
635:
566:logical truths
558:
555:
554:
553:
537:
533:Nelson Goodman
514:
511:
510:
509:
497:
482:
473:: The ancient
435:Platonic realm
417:Raphael Sanzio
402:
399:
397:
394:
295:Hilbert spaces
271:
268:
260:physical world
183:
180:
178:
175:
77:
76:
31:
29:
22:
15:
9:
6:
4:
3:
2:
4634:
4633:
4622:
4619:
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4612:
4609:
4607:
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4602:
4599:
4598:
4596:
4583:
4582:
4577:
4569:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4541:
4538:
4537:
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4533:
4531:
4528:
4526:
4523:
4521:
4517:
4514:
4512:
4509:
4507:
4504:
4502:
4499:
4497:
4494:
4493:
4491:
4487:
4481:
4478:
4476:
4473:
4471:
4470:Recursive set
4468:
4466:
4463:
4461:
4458:
4456:
4453:
4451:
4448:
4444:
4441:
4439:
4436:
4434:
4431:
4429:
4426:
4424:
4421:
4420:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4390:
4388:
4386:
4382:
4376:
4373:
4371:
4368:
4366:
4363:
4361:
4358:
4356:
4353:
4351:
4348:
4346:
4343:
4339:
4336:
4334:
4331:
4329:
4326:
4325:
4324:
4321:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4299:
4296:
4292:
4289:
4288:
4287:
4284:
4280:
4279:of arithmetic
4277:
4276:
4275:
4272:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4244:
4243:
4240:
4236:
4233:
4231:
4228:
4227:
4226:
4223:
4222:
4220:
4218:
4214:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4187:
4186:from ZFC
4183:
4180:
4178:
4175:
4169:
4166:
4165:
4164:
4161:
4159:
4156:
4154:
4151:
4150:
4149:
4146:
4144:
4141:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4115:
4113:
4111:
4107:
4097:
4096:
4092:
4091:
4086:
4085:non-Euclidean
4083:
4079:
4076:
4074:
4071:
4069:
4068:
4064:
4063:
4061:
4058:
4057:
4055:
4051:
4047:
4044:
4042:
4039:
4038:
4037:
4033:
4029:
4026:
4025:
4024:
4020:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3987:
3985:
3981:
3980:
3978:
3973:
3967:
3962:Example
3959:
3951:
3946:
3945:
3944:
3941:
3939:
3936:
3932:
3929:
3927:
3924:
3922:
3919:
3917:
3914:
3913:
3912:
3909:
3907:
3904:
3902:
3899:
3897:
3894:
3890:
3887:
3885:
3882:
3881:
3880:
3877:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3854:
3853:
3850:
3848:
3845:
3841:
3838:
3836:
3833:
3831:
3828:
3827:
3826:
3823:
3819:
3816:
3814:
3811:
3809:
3806:
3804:
3801:
3799:
3796:
3794:
3791:
3790:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3767:
3764:
3762:
3759:
3757:
3754:
3752:
3749:
3748:
3747:
3744:
3742:
3739:
3737:
3734:
3732:
3729:
3725:
3722:
3720:
3719:by definition
3717:
3716:
3715:
3712:
3708:
3705:
3704:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3679:
3676:
3673:
3671:
3667:
3662:
3656:
3652:
3642:
3639:
3637:
3634:
3632:
3629:
3627:
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3607:
3606:KripkeâPlatek
3604:
3602:
3599:
3595:
3592:
3590:
3587:
3586:
3585:
3582:
3581:
3579:
3575:
3567:
3564:
3563:
3562:
3559:
3557:
3554:
3550:
3547:
3546:
3545:
3542:
3540:
3537:
3535:
3532:
3530:
3527:
3525:
3522:
3519:
3515:
3511:
3508:
3504:
3501:
3499:
3496:
3494:
3491:
3490:
3489:
3485:
3482:
3481:
3479:
3477:
3473:
3469:
3461:
3458:
3456:
3453:
3451:
3450:constructible
3448:
3447:
3446:
3443:
3441:
3438:
3436:
3433:
3431:
3428:
3426:
3423:
3421:
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3382:
3380:
3378:
3373:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3340:
3337:
3336:
3334:
3330:
3327:
3325:
3322:
3321:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3296:
3292:
3290:
3287:
3283:
3280:
3279:
3278:
3275:
3274:
3271:
3268:
3266:
3262:
3252:
3249:
3247:
3244:
3242:
3239:
3237:
3234:
3232:
3229:
3227:
3224:
3220:
3217:
3216:
3215:
3212:
3208:
3203:
3202:
3201:
3198:
3197:
3195:
3193:
3189:
3181:
3178:
3176:
3173:
3171:
3168:
3167:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3148:
3146:
3143:
3141:
3138:
3136:
3133:
3132:
3130:
3128:
3127:Propositional
3124:
3118:
3115:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3086:
3083:
3082:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3055:Logical truth
3053:
3051:
3048:
3047:
3045:
3043:
3039:
3036:
3034:
3030:
3024:
3021:
3019:
3016:
3014:
3011:
3009:
3006:
3004:
3001:
2999:
2995:
2991:
2987:
2985:
2982:
2980:
2977:
2975:
2971:
2968:
2967:
2965:
2963:
2957:
2952:
2946:
2943:
2941:
2938:
2936:
2933:
2931:
2928:
2926:
2923:
2921:
2918:
2916:
2913:
2911:
2908:
2906:
2903:
2901:
2898:
2896:
2893:
2891:
2888:
2884:
2881:
2880:
2879:
2876:
2875:
2873:
2869:
2865:
2858:
2853:
2851:
2846:
2844:
2839:
2838:
2835:
2829:
2826:
2824:
2821:
2818:
2814:
2811:
2807:
2806:
2802:
2801:
2792:
2788:
2785:
2782:
2778:
2774:
2771:
2768:
2764:
2761:
2760:
2755:
2752:
2749:
2745:
2741:
2740:Davis, Philip
2738:
2735:
2731:
2728:
2724:
2723:
2722:
2721:
2703:
2699:
2692:
2691:
2683:
2679:
2673:
2659:
2655:
2648:
2634:
2630:
2623:
2615:
2611:
2607:
2603:
2599:
2595:
2591:
2584:
2576:
2570:
2566:
2565:
2560:
2554:
2548:
2547:0-19-513930-5
2545:
2541:
2536:
2522:
2518:
2511:
2497:
2491:
2484:
2482:4-87187-714-0
2479:
2475:
2471:
2466:
2459:
2455:
2451:
2447:
2442:
2428:
2424:
2417:
2403:
2399:
2392:
2384:
2382:9780080930589
2378:
2374:
2373:
2365:
2351:
2347:
2340:
2326:
2320:
2316:
2315:
2307:
2293:
2287:
2273:
2269:
2262:
2254:
2248:
2244:
2243:
2235:
2221:
2215:
2201:
2197:
2190:
2176:
2172:
2165:
2151:
2145:
2131:
2127:
2120:
2106:
2102:
2095:
2081:
2077:
2070:
2056:
2052:
2045:
2031:
2027:
2020:
2006:
2002:
1995:
1989:
1985:
1981:
1977:
1973:
1972:Burgess, John
1968:
1960:
1954:
1950:
1943:
1929:
1925:
1918:
1914:
1911:
1910:
1909:Cited sources
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1879:
1876:
1874:
1871:
1869:
1866:
1865:
1857:
1854:
1852:
1849:
1848:
1845:
1838:
1835:
1833:
1830:
1828:
1825:
1824:
1819:
1816:
1814:
1811:
1810:
1809:
1806:
1805:
1802:
1799:
1797:
1794:
1792:
1789:
1787:
1784:
1783:
1780:
1773:
1769:
1766:
1762:
1759:
1757:
1754:
1753:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1726:Neighborhoods
1724:
1722:
1719:
1717:
1714:
1713:
1710:
1703:
1700:
1698:
1695:
1693:
1690:
1688:
1684:
1681:
1679:
1675:
1672:
1671:
1668:
1661:
1658:
1656:
1653:
1649:
1646:
1644:
1641:
1639:
1636:
1634:
1631:
1629:
1626:
1625:
1624:
1621:
1617:
1614:
1612:
1609:
1607:
1604:
1603:
1602:
1599:
1598:
1595:
1594:Number theory
1588:
1585:
1583:
1580:
1578:
1575:
1573:
1569:
1566:
1564:
1561:
1559:
1556:
1555:
1552:
1545:
1541:
1537:
1533:
1530:
1529:
1526:
1519:
1516:
1514:
1513:Menger sponge
1510:
1507:
1505:
1502:
1500:
1497:
1495:
1492:
1491:
1488:
1481:
1477:
1474:
1471:
1467:
1463:
1459:
1456:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1434:
1433:
1432:
1429:
1421:
1418:
1416:
1413:
1411:
1408:
1406:
1403:
1401:
1398:
1397:
1396:
1393:
1392:
1390:
1387:
1381:
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1362:
1361:
1358:
1357:
1355:
1352:
1351:
1350:
1346:
1343:
1340:
1339:Line segments
1336:
1332:
1329:
1328:
1325:
1318:
1311:
1308:
1306:
1303:
1301:
1298:
1296:
1293:
1291:
1288:
1284:
1281:
1279:
1276:
1274:
1271:
1269:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1246:
1245:
1242:
1241:
1238:
1234:
1227:
1223:
1219:
1216:
1215:
1212:
1211:Combinatorics
1207:
1205:
1202:, proofs and
1201:
1197:
1191:
1188:
1185:
1183:
1180:
1178:
1175:
1173:
1170:
1169:
1166:
1159:
1156:
1154:
1151:
1149:
1146:
1145:
1142:
1135:
1132:
1130:
1127:
1126:
1123:
1116:
1113:
1111:
1108:
1106:
1102:
1099:
1095:
1091:
1090:Bump function
1088:
1085:
1081:
1078:
1075:
1072:
1071:
1070:
1067:
1065:
1061:
1057:
1054:
1053:
1050:
1046:
1039:
1035:
1032:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1014:Vector spaces
1011:
1008:
1006:
1003:
1001:
997:
994:
992:
989:
988:
985:
978:
975:
973:
970:
968:
965:
963:
960:
958:
955:
953:
949:
946:
944:
940:
937:
936:
933:
926:
923:
921:
918:
916:
913:
911:
908:
907:
904:
897:
887:
885:
880:
876:
872:
868:
864:
860:
859:
854:
847:
846:
842:
836:
824:
820:
819:
815:
812:
811:
807:
806:
805:
802:
800:
795:
794:Structuralism
789:Structuralism
786:
784:
780:
776:
772:
768:
764:
760:
756:
752:
748:
744:
740:
736:
732:
731:contradiction
727:
715:
714:
710:
707:
706:
705:David Hilbert
702:
701:
700:
697:
695:
691:
687:
686:formal system
683:
671:
667:
663:
659:
655:
651:
650:
645:
641:
640:
636:
633:
629:
625:
624:
619:
618:
617:Gottlob Frege
614:
613:
612:
609:
607:
603:
599:
595:
594:formal system
591:
587:
583:
579:
575:
571:
567:
563:
551:
547:
543:
542:
538:
535:
534:
530:
529:
528:
525:
523:
519:
507:
503:
502:
501:Roger Penrose
498:
496:
492:
488:
487:
483:
480:
476:
472:
471:
467:
466:
465:
462:
460:
456:
452:
448:
444:
440:
436:
432:
428:
424:
418:
414:
413:
407:
393:
391:
387:
386:authoritative
383:
379:
373:
371:
366:
363:
361:
355:
353:
349:
345:
344:Hilary Putnam
341:
340:Willard Quine
337:
336:indispensable
332:
328:
324:
320:
319:combinatorics
316:
312:
308:
304:
300:
296:
292:
288:
284:
280:
276:
267:
265:
261:
257:
253:
249:
245:
241:
237:
233:
229:
228:human thought
225:
221:
218:that possess
217:
213:
209:
205:
201:
197:
193:
189:
174:
172:
168:
164:
160:
156:
152:
148:
144:
140:
136:
132:
128:
124:
120:
116:
112:
108:
104:
100:
92:
88:
83:
73:
70:
62:
52:
48:
42:
41:
35:
30:
21:
20:
4572:
4551:
4370:Ultraproduct
4217:Model theory
4182:Independence
4118:Formal proof
4110:Proof theory
4093:
4066:
4023:real numbers
3995:second-order
3906:Substitution
3783:Metalanguage
3724:conservative
3697:Axiom schema
3641:Constructive
3611:MorseâKelley
3577:Set theories
3556:Aleph number
3549:inaccessible
3455:Grothendieck
3339:intersection
3226:Higher-order
3214:Second-order
3160:Truth tables
3117:Venn diagram
2900:Formal proof
2803:
2790:
2780:
2776:
2766:
2757:
2754:Gold, Bonnie
2747:
2744:Reuben Hersh
2733:
2726:
2719:
2718:
2705:. Retrieved
2689:
2678:Hu, Sze-Tsen
2672:
2662:, retrieved
2657:
2647:
2637:, retrieved
2632:
2622:
2597:
2593:
2583:
2563:
2553:
2539:
2535:
2525:, retrieved
2520:
2510:
2499:. Retrieved
2490:
2473:
2465:
2456:: 973â1052.
2453:
2449:
2441:
2431:, retrieved
2426:
2416:
2406:, retrieved
2401:
2391:
2371:
2364:
2354:, retrieved
2349:
2339:
2328:. Retrieved
2313:
2306:
2295:. Retrieved
2286:
2276:, retrieved
2271:
2261:
2241:
2234:
2223:. Retrieved
2214:
2204:, retrieved
2199:
2189:
2178:. Retrieved
2174:
2164:
2153:. Retrieved
2144:
2134:, retrieved
2129:
2119:
2109:, retrieved
2104:
2094:
2084:, retrieved
2079:
2069:
2059:, retrieved
2054:
2044:
2034:, retrieved
2029:
2019:
2009:, retrieved
2004:
1994:
1975:
1967:
1948:
1942:
1932:, retrieved
1927:
1917:
1908:
1907:
1818:Trefoil knot
1772:Klein bottle
1768:Möbius strip
1746:Uniformities
1568:Propositions
1558:Truth values
1525:Graph theory
1470:Hyperboloids
1415:Dodecahedron
1254:Linked lists
1226:combinations
1222:derangements
1218:permutations
1200:proof theory
1194:
862:
856:
851:
844:
840:
822:
816:
808:
803:
792:
743:intuitionism
724:
713:Hermann Weyl
711:
703:
698:
680:
661:
647:
637:
628:modern logic
621:
615:
610:
569:
560:
550:fictionalism
541:Hartry Field
539:
531:
526:
516:
499:
491:model theory
484:
468:
463:
421:
410:
375:
367:
364:
357:
335:
315:chaos thoery
273:
256:epistemology
185:
171:proof theory
110:
98:
96:
65:
56:
37:
4480:Type theory
4428:undecidable
4360:Truth value
4247:equivalence
3926:non-logical
3539:Enumeration
3529:Isomorphism
3476:cardinality
3460:Von Neumann
3425:Ultrafilter
3390:Uncountable
3324:equivalence
3241:Quantifiers
3231:Fixed-point
3200:First-order
3080:Consistency
3065:Proposition
3042:Traditional
3013:Lindström's
3003:Compactness
2945:Type theory
2890:Cardinality
2627:Hale, Bob,
1856:Hairy balls
1716:Closed sets
1633:Subtraction
1466:Paraboloids
1420:Icosahedron
1400:Tetrahedron
1259:Hash tables
972:Linear maps
962:Linear span
925:Polynomials
884:type theory
745:founded by
658:type theory
327:predictions
212:metaphysics
165:, and even
143:expressions
107:mathematics
105:arising in
51:introducing
4595:Categories
4291:elementary
3984:arithmetic
3852:Quantifier
3830:functional
3702:Expression
3420:Transitive
3364:identities
3349:complement
3282:hereditary
3265:Set theory
2707:August 28,
2664:2024-08-28
2639:2024-08-28
2527:2024-08-28
2501:2024-08-28
2433:2024-08-28
2408:2024-08-28
2356:2024-08-28
2330:2023-08-28
2297:2024-08-29
2278:2024-08-27
2225:2024-08-28
2206:2024-08-27
2180:2024-08-27
2155:2024-08-28
2136:2024-08-27
2111:2024-08-28
2086:2024-08-28
2061:2024-08-28
2036:2024-08-29
2011:2024-08-28
1988:0198236158
1934:2024-08-28
1904:References
1667:Set theory
1587:Signatures
1572:Predicates
1518:Cantor set
1499:Julia sets
1462:Ellipsoids
1451:Hyperbolas
1410:Octahedron
1196:Categories
1172:Categories
1101:Derivative
1029:Operations
875:properties
799:arithmetic
656:, using a
606:consistent
598:arithmetic
584:) and his
518:Nominalism
513:Nominalism
486:Kurt Gödel
451:statements
437:. Just as
382:Naturalism
370:Conclusion
313:'s use of
240:nominalism
220:properties
202:, and the
34:references
4621:Platonism
4562:Supertask
4465:Recursion
4423:decidable
4257:saturated
4235:of models
4158:deductive
4153:axiomatic
4073:Hilbert's
4060:Euclidean
4041:canonical
3964:axiomatic
3896:Signature
3825:Predicate
3714:Extension
3636:Ackermann
3561:Operation
3440:Universal
3430:Recursive
3405:Singleton
3400:Inhabited
3385:Countable
3375:Types of
3359:power set
3329:partition
3246:Predicate
3192:Predicate
3107:Syllogism
3097:Soundness
3070:Inference
3060:Tautology
2962:paradoxes
2773:Sfard, A.
2746:, 1999 .
2680:(1971) .
2614:0031-8108
1751:Manifolds
1736:Open sets
1687:Relations
1683:Functions
1655:Fractions
1648:Exponents
1476:Cylinders
1446:Parabolas
1389:Polyhedra
1375:Pentagons
1365:Triangles
1345:Polytopes
1310:Languages
1305:Variables
1182:Morphisms
879:relations
858:functions
677:Formalism
443:electrons
423:Platonism
401:Platonism
352:syllogism
224:relations
196:existence
139:functions
87:tesseract
59:June 2009
4547:Logicism
4540:timeline
4516:Concrete
4375:Validity
4345:T-schema
4338:Kripke's
4333:Tarski's
4328:semantic
4318:Strength
4267:submodel
4262:spectrum
4230:function
4078:Tarski's
4067:Elements
4054:geometry
4010:Robinson
3931:variable
3916:function
3889:spectrum
3879:Sentence
3835:variable
3778:Language
3731:Relation
3692:Automata
3682:Alphabet
3666:language
3520:-jection
3498:codomain
3484:Function
3445:Universe
3415:Infinite
3319:Relation
3102:Validity
3092:Argument
2990:theorem,
2789:, 2000.
2702:59-11526
2561:(1974).
2472:(1967).
1924:"Object"
1862:See also
1709:Topology
1643:Division
1628:Addition
1611:Integers
1577:Formulas
1441:Ellipses
1380:Hexagons
1354:Polygons
1317:Geometry
1295:Integers
1204:theorems
1177:Functors
1110:Integral
1105:Gradient
1086:function
1045:Calculus
1019:Algebras
948:Matrices
781:such as
751:finitism
602:complete
578:theorems
562:Logicism
557:Logicism
331:language
291:physics'
285:. Every
252:ontology
248:abstract
244:concrete
236:idealism
216:entities
200:identity
167:theories
159:theorems
127:formulas
123:variable
4489:Related
4286:Diagram
4184: (
4163:Hilbert
4148:Systems
4143:Theorem
4021:of the
3966:systems
3746:Formula
3741:Grammar
3657: (
3601:General
3314:Forcing
3299:Element
3219:Monadic
2994:paradox
2935:Theorem
2871:General
1832:Toruses
1756:Atlases
1721:Filters
1601:Numbers
1458:Spheres
1436:Circles
1370:Squares
1290:Objects
1186:objects
1134:Fluxion
1005:monoids
1000:Modules
952:Tensors
943:Vectors
939:Scalars
896:Algebra
863:objects
845:mapping
759:Bernays
755:Hilbert
747:Brouwer
568:, and a
447:planets
431:thought
390:science
380:called
360:Premise
311:Biology
293:use of
281:in the
232:realism
208:reality
192:objects
131:numbers
47:improve
4252:finite
4015:Skolem
3968:
3943:Theory
3911:Symbol
3901:String
3884:atomic
3761:ground
3756:closed
3751:atomic
3707:ground
3670:syntax
3566:binary
3493:domain
3410:Finite
3175:finite
3033:Logics
2992:
2940:Theory
2777:et al.
2700:
2612:
2571:
2379:
2321:
2249:
1986:
1955:
1813:Unknot
1761:Charts
1685:, and
1532:Graphs
1331:Points
1300:Floats
1278:Graphs
1268:Queues
1264:Stacks
1249:Arrays
1010:Fields
991:Groups
957:Kernel
867:entity
771:Bishop
769:, and
767:Markov
763:Shanin
749:, the
204:nature
163:proofs
155:spaces
111:object
101:is an
36:, but
4242:Model
3990:Peano
3847:Proof
3687:Arity
3616:Naive
3503:image
3435:Fuzzy
3395:Empty
3344:union
3289:Class
2930:Model
2920:Lemma
2878:Axiom
2694:(PDF)
2685:(PDF)
1827:Holes
1808:Knots
1582:Terms
1544:Edges
1540:Nodes
1536:Trees
1480:Cones
1335:Lines
1283:Trees
996:Rings
967:Bases
869:that
853:Frege
694:chess
574:logic
479:forms
470:Plato
441:like
321:(see
210:. In
4365:Type
4168:list
3972:list
3949:list
3938:Term
3872:rank
3766:open
3660:list
3472:Maps
3377:sets
3236:Free
3206:list
2956:list
2883:list
2742:and
2709:2024
2698:LCCN
2610:ISSN
2569:ISBN
2544:ISBN
2478:ISBN
2377:ISBN
2319:ISBN
2247:ISBN
1984:ISBN
1953:ISBN
1731:Nets
1674:Sets
1570:and
1405:Cube
1273:Heap
1266:and
1235:and
1047:and
877:and
871:maps
861:and
765:and
757:and
690:ludo
668:and
604:and
544:: A
445:and
342:and
317:and
301:and
238:and
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4052:of
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3277:Set
2808:: "
2602:doi
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692:or
415:by
309:to
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