Tarski and geometry

Szczerba, L. W.

doi:10.2307/2273904

Cited by 34 publications

(10 citation statements)

References 7 publications

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“…The first one and very influencial was the cooperation with Lesław W. Szczerba, the author of [42], in the early 80s of the previous century. Szczerba, who also submitted to the Mizar Mathematical Library twice, was at that time a head of the Institute of Mathematics in Białystok, Poland; the place the Mizar system was mainly developed (and that was also the affiliation of Trybulec).…”

Section: Mizaring Affine Geometrymentioning

confidence: 99%

Tarski's geometry modelled in Mizar computerized proof assistant

Grabowski¹

2016

Annals of Computer Science and Information Systems

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Abstract-In the paper, we discuss the formal approach to Tarski geometry axioms modelled with the help of the Mizar computerized proof assistant system. Although our basic development was inspired by Julien Narboux's Coq pseudo-code and is dated back to 2014, there are significant steps in the formalization of geometry done in the last decade of the previous century. Taking this into account, we will propose the reuse of existing results within this new framework (including Hilbert's axiomatic approach), with the ultimate future goal to encode the textbook Metamathematische Methoden in der Geometrie by Schwabhäuser, Szmielew and Tarski. We try however to go much further from the use of simple predicates in the direction of the use of structures with their inheritance, attributes as a tool of more human-friendly namespaces for axioms, and registrations of clusters to obtain more automation (with the possible use of external equational theorem provers like Otter/Prover9).

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Section: Mizaring Affine Geometrymentioning

confidence: 99%

Tarski's geometry modelled in Mizar computerized proof assistant

Grabowski¹

2016

Annals of Computer Science and Information Systems

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“…Szczerba reports an interesting observation about Tarski's attitude: "I remember Szmielew complaining jokingly that the more one worked with Tarski, the result tended to look less and less laborious. In fact Tarski would work over a mathematical presentation until it achieved an elegance and simplicity which disguised the difficulties hidden beneath the surface" [Szcz86,p. 910].…”

Section: Literaturementioning

confidence: 99%

Spheres, cubes and simple

Borgo

2013

LLP

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Abstract. In 1929 Tarski showed how to construct points in a region-based first-order logic for space representation. The resulting system, called the geometry of solids, is a cornerstone for region-based geometry and for the comparison of point-based and region-based geometries. We expand this study of the construction of points in region-based systems using different primitives, namely hyper-cubes and regular simplexes, and show that these primitives lead to equivalent systems in dimension n ≥ 2. The result is achieved by adopting a single set of definitions that works for both these classes of figures. The analysis of our logics shows that Tarski's choice to take sphere as the geometrical primitive might be intuitively justified but is not optimal from a technical viewpoint.

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“…The developm ent of familiar results of classical plane geometry in Tarski' s T has been carried out by several investigators (see Szczerba 1986 for references). A corresponding development of intuitionistic plane geometry has been begun (1999) by G abriela Stanica.…”

Section: Richard Vesleymentioning

confidence: 99%

“…The ® nal form ulations are included in Schwabhau$ ser et al (1983) and Tarski (1959), while the history of this development is reviewed in Szczerba (1986). The ® nal form ulations are included in Schwabhau$ ser et al (1983) and Tarski (1959), while the history of this development is reviewed in Szczerba (1986).…”

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confidence: 99%

Constructivity in Geometry

Vesley

1999

History and Philosophy of Logic

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W e review and contrast three ways to make up a forma l Euclidean geometry which one m ight call construc tive, in a computa tional sense. The starting poin t is the ® rst-orde r geom etry created by Tarski.Through much of his life Alfred Tarski worked oOEand on to achieve a neat ® rst order form ulation of Euclidean plane geom etry. The ® nal form ulations are included in Schwabhau$ ser et al. (1983) andTarski (1959), while the history of this development is reviewed in Szczerba (1986). As Tarski (1959) observed, the question whether this theory captures accurately the geom etry intended by Euclid is unrewarding. But it does capture a theory which deserves the m odern title of elem entary or ® rst order Euclidean plane geometry, as is supported by Tarski' s proof that the theory is complete.A notable feature of Tarski' s system T is its economy of language. Only one sort of variable is used, ranging over points. In addition to equality, only two prim itive predicates are introduced, B(a,b,c) for`point b is between points a and c ' and D (a,b,c,d) for`a is the same distance from b as c is from d ' . In this lim ited language, all the fam iliar results of Euclidean plane geom etry can be represented. A similarly spare axiom set allows proofs of these results, including theorems which assert the existence of points obtained by familiar Euclidean constructions, like that for ® nding the perpendicular bisector of a line segment, and also theorems which are less immediately correlated to Euclidean constructions, like the congruence theorem s for triangles.How much of this theory T is constructive ? This question did not concern Tarski, but it does suggest some interesting investigations. Evidently, any answers will depend on our ® xing sharply the meaning of the question. W e may see ourselves as continuing a centuries-old discussion concerning geometric constructions, dating from celebrated questions in which constructions are limited to those involving compass and straight edge, and generalizing now to broader views of what might be allowed in constructions.In this brief paper, three approaches to constructivity in Euclidean plane geometry are reviewed and put into context. The ® rst, urged by a referee of Lom bard and Vesley (1998), is proof-theoretic. The second has been developed by M oler and Suppes (1968) and Pam buccian (1989, 1992). The last derives from Brouwer and Heyting, with recent work by van Dalen (1990), von Plato (1994 and Lombard and Vesley (1998). There are other approaches possible and the last two of those mentioned actually subdivide into a number of related approaches.The ® rst view regards a formal geometry as constructive if there is an algorithm determining for any closed formula A whether or not A is a theorem , symbolically whether r-A or not r-A. As a consequence of com pleteness, Tarski' s formal geometry T has this property. Thus it is constructive in the sense that in principle no ingenuity is required in m aking proofs or refutations. Rather, there is a single master or universal algo...

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Tarski and geometry

Cited by 34 publications

References 7 publications

Tarski's geometry modelled in Mizar computerized proof assistant

Tarski's geometry modelled in Mizar computerized proof assistant

Spheres, cubes and simple

Constructivity in Geometry

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