223:. However, if someone asks for the "true" identity statements for relating natural numbers to pure sets, then different set-theoretic methods yield contradictory identity statements when these elementarily equivalent sets are related together. This generates a set-theoretic falsehood. Consequently, Benacerraf inferred that this set-theoretic falsehood demonstrates it is impossible for there to be any Platonic method of reducing numbers to sets that reveals any abstract objects.
235:. The fundamental epistemological problem thus arises for the Platonist to offer a plausible account of how a mathematician with a limited, empirical mind is capable of accurately accessing mind-independent, world-independent, eternal truths. It was from these considerations, the ontological argument and the epistemological argument, that Benacerraf's anti-Platonic critiques motivated the development of structuralism in the philosophy of mathematics.
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structuralism, arguing that structural properties such as symmetry are instantiated in the physical world and are perceivable. In reply to the problem of uninstantiated structures that are too big to fit into the physical world, Franklin replies that other sciences can also deal with uninstantiated
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in ontology). Structures are held to exist inasmuch as some concrete system exemplifies them. This incurs the usual issues that some perfectly legitimate structures might accidentally happen not to exist, and that a finite physical world might not be "big" enough to accommodate some otherwise
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exist, and have structural features in common. If something is true of a structure, it will be true of all systems exemplifying the structure. However, it is merely instrumental to talk of structures being "held in common" between systems: they in fact have no independent
199:. By the mid-20th century, however, these anti-Platonist theories had a number of their own issues. This subsequently resulted in a resurgence of interest in Platonism. It was in this historic context that the motivations for structuralism developed. In 1965,
301:). Structures are held to have a real but abstract and immaterial existence. As such, it faces the standard epistemological problem, as noted by Benacerraf, of explaining the interaction between such abstract structures and flesh-and-blood mathematicians.
230:
test. Benacerraf contended that there does not exist an empirical or rational method for accessing abstract objects. If mathematical objects are not spatial or temporal, then
Benacerraf infers that such objects are not accessible through the
101:). The kind of existence that mathematical objects have would be dependent on that of the structures in which they are embedded; different sub-varieties of structuralism make different ontological claims in this regard.
210:
Firstly, Benacerraf argued that
Platonic approaches do not pass the ontological test. He developed an argument against the ontology of set-theoretic Platonism, which is now historically referred to as
51:
but are defined by their external relations in a system. For instance, structuralism holds that the number 1 is exhaustively defined by being the successor of 0 in the structure of the theory of
47:. Mathematical objects are exhaustively defined by their place in such structures. Consequently, structuralism maintains that mathematical objects do not possess any
367:
approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure. According to this view mathematical
55:. By generalization of this example, any natural number is defined by its respective place in that theory. Other examples of mathematical objects might include
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universals; for example the science of color can deal with a shade of blue that happens not to occur on any real object.
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published an article entitled "What
Numbers Could Not Be". Benacerraf concluded, on two principal arguments, that
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In the late 19th and early 20th century, a number of anti-Platonist programs gained in popularity. These included
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An
Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure
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144:. In the philosophy of mathematics, an abstract object is traditionally defined as an entity that:
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The historical motivation for the development of structuralism derives from a fundamental problem of
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divides structuralism into three major schools of thought. These schools are referred to as the
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times, philosophers have argued as to whether the ontology of mathematics contains
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denies the existence of any such abstract objects in the ontology of mathematics.
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Structuralism in the philosophy of mathematics is particularly associated with
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Resnik, Michael (1982). "Mathematics as a
Science of Patterns: Epistemology".
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legitimate structures. The
Aristotelian realism of James Franklin is also an
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Mathematics and Its
Applications: A Transcendental-Idealist Perspective
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mathematical objects or structures have (not, in other words, to their
601:(Winter 2017 ed.), Metaphysics Research Lab, Stanford University
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Secondly, Benacerraf argued that
Platonic approaches do not pass the
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view in that it holds that mathematical statements have an objective
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Platonism cannot succeed as a philosophical theory of mathematics.
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722:"Uninstantiated Properties and Semi-Platonist Aristotelianism"
513:(2nd ed.). Cambridge University Press. pp. 403–420.
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that holds that mathematical theories describe structures of
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460:
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760:"Inferentialism and Structuralism: A Tale of Two Theories"
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of entity a mathematical object is, not to what kind of
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Benacerraf, Paul (1965). "What
Numbers Could Not Be".
180:—are such abstract objects. Contrarily, mathematical
500:
498:
219:, set-theoretic ways of relating natural numbers to
150:(2) exists independent of the empirical world; and
89:. However, its central claim only relates to what
868:Philosophy of Mathematics: Structure and Ontology
632:Zalta, Edward N.; Nodelman, Uri (February 2011).
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160:maintains that some set of mathematical elements—
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398:Aristotelian realist philosophy of mathematics
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918:Foundations of Structuralism research project
634:"A Logically Coherent Ante Rem Structuralism"
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511:Philosophy of Mathematics: Selected Readings
153:(3) has eternal, unchangeable properties.
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509:. In Putnam, H.W.; Benacerraf, P. (eds.).
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359:about structures in a way that parallels
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657:Thin Objects: An Abstractionist Account
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147:(1) exists independent of the mind;
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847:Mathematics as a Science of Patterns
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215:. Benacerraf noted that there are
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1193:List of category theory topics
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277:(particularly associated with
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681:da Silva, Jairo José (2017).
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323:(realism in truth value, but
293:) has a similar ontology to
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1188:Glossary of category theory
1062:Zermelo–Fraenkel set theory
1014:Mathematical constructivism
920:, University of Bristol, UK
877:10.1093/0195139305.001.0001
871:. Oxford University Press.
660:. Oxford University Press.
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10:
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1183:Mathematical structuralism
1120:Intuitionistic type theory
956:Foundations of Mathematics
903:Mathematical Structuralism
595:"Logicism and Neologicism"
388:Foundations of mathematics
233:causal theory of knowledge
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18:Mathematical structuralism
1229:Philosophy of mathematics
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1087:List of set theory topics
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865:Shapiro, Stewart (1997).
776:10.2143/LEA.244.0.3285352
687:. Springer. p. 265.
654:Linnebo, Øystein (2018).
505:Benacerraf, Paul (1983).
433:Philosophy of Mathematics
349:eliminative structuralism
269:("before the thing"), or
156:Traditional mathematical
41:philosophy of mathematics
844:Resnik, Michael (1997).
794:Franklin, James (2014).
720:Franklin, James (2015).
643:. University of Bristol.
620:abstractionist Platonism
618:Not to be confused with
347:("after the thing"), or
319:), is the equivalent of
29:Not to be confused with
1067:Constructive set theory
758:Nefdt, Ryan M. (2018).
551:Philosophia Mathematica
363:. Like nominalism, the
217:elementarily equivalent
1239:Abstract object theory
1219:History of mathematics
1168:Higher category theory
1072:Descriptive set theory
977:Mathematical induction
800:. Palgrave Macmillan.
593:Tennant, Neil (2017),
563:10.1093/philmat/4.2.81
383:Abstract object theory
271:abstract structuralism
1130:Univalent foundations
1115:Dependent type theory
1105:Axiom of reducibility
726:Review of Metaphysics
437:. Routledge. p.
429:Brown, James (2008).
393:Univalent foundations
311:("in the thing"), or
128:Historical motivation
1125:Homotopy type theory
1052:Axiomatic set theory
507:"Mathematical Truth"
469:Philosophical Review
321:Aristotelian realism
78:Structuralism is an
49:intrinsic properties
45:mathematical objects
850:. Clarendon Press.
313:modal structuralism
39:is a theory in the
1224:Mathematical logic
1110:Simple type theory
1057:Zermelo set theory
1004:Mathematical proof
964:Mathematical logic
764:Logique et Analyse
299:modal neo-logicism
67:, or elements and
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997:Natural deduction
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857:978-0-19-825014-2
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667:978-0-19-255896-1
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16:(Redirected from
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732:(1): 25–45.
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770:: 489–512.
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1234:Set theory
1208:Categories
1173:∞-groupoid
1034:Set theory
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416:References
404:Precursors
372:existence.
361:nominalism
297:(see also
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221:pure sets
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174:relations
170:functions
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