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Using the prover ANDP to simplify orthogonality

Dafa Li
2003 Annals of Pure and Applied Logic  
Recently, von Plato used di erent concepts to axiomatize the geometry: he used 14 axioms to describe the axiomatization for apartness geometry.  ...  In the 1920s, Heyting attempted at axiomatizing constructive geometry.  ...  The experiments whose results are reported in this paper were executed at RRL of department of computer science of SUNY Albany.  ... 
doi:10.1016/s0168-0072(03)00051-4 fatcat:aoch76uzlnh7xjhrwqspst5vpa

Simplifying the axiomatization for the order affine geometry [article]

Dafa Li
2022 arXiv   pre-print
Based on an ordering with directed lines and using constructions instead of existential axioms, von Plato proposed a constructive axiomatization of ordered affine geometry.  ...  There are 22 axioms for the ordered affine geometry, of which the axiom I.7 is about the convergence of three lines (ignoring their directions).  ...  By using theorem prover ANDP [9, 10] , we found a natural deduction of Halting problem [11] , and simplified von Plato's axiomatization of constructive apartness geometry and orthogonality [12, 13]  ... 
arXiv:2205.05162v1 fatcat:buqt2ocsnncvpg6od56ffyfcby


2000 Annals of Pure and Applied Logic  
von Plato's axiomatization of Constructive Apartness Geometry (1}2) 1} 26 Pynko, A.P., Erratum to &&De"nitional equivalence and algebraizability of generalized logical systems'' Annals of Pure and Applied  ...  , M., see Jensen, R.B. (1}2) 101}138 Kanovei, V., Linearization of de"nable order relations (1}2) 69}100 Kohlenbach, U., Things that can and things that cannot be done in PRA (3) 223}245 Li, D., Simplifying  ... 
doi:10.1016/s0168-0072(99)00051-2 fatcat:bbi7kmnhavfkjlzrwae7lktmyu

Page 8375 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
Dafa (PRC-TSI; Beijing); Jia, Peifa (PRC-TSI-IT; Beijing); Li, Xinxin (1-WYNS-C; Detroit, MI) Simplifying von Plato’s axiomatization of constructive apartness geometry.  ...  By replacing three pairs of axioms with three axioms, as well as simplifying one of von Plato’s axioms for apartness geometry, the authors obtain a new, equivalent axiom system with 11 axioms.  ... 

A Decision Procedure for Geometry in Coq [chapter]

Julien Narboux
2004 Lecture Notes in Computer Science  
The idea of the method is to express the goal to be proved using three geometric quantities and eliminate points in the reverse order of their construction thanks to some elimination lemmas.  ...  We present in this paper the development of a decision procedure for affine plane geometry in the Coq proof assistant.  ...  I want to thank Hugo Herbelin for his help during the elaboration of this work.  ... 
doi:10.1007/978-3-540-30142-4_17 fatcat:6ho42opbsjhg3cnybu33255ot4

The axiomatic melting pot. Teaching probability theory in Prague during the 1930's [article]

Stepanka Bilova, Laurent Mazliak, Pavel Sisma
2006 arXiv   pre-print
In this paper, we are interested in the teaching of probability theory in Prague and Czechoslovakia, in particular during the 1930's.  ...  We focus specially on a textbook, published in Prague by Karel Rychlik in 1938, which uses Kolmogorov's axiomatization, a very exceptional fact before World War II.  ...  Let us just mention Von Plato's book [74] whose subject is precisely to describe this new emergence of probability and renewal of the theory after the creations of the 17-18th centuries and the Laplacian  ... 
arXiv:math/0607217v1 fatcat:lp3oa4ekyvftjfnlcoreo6ho2i

Mathematical Interpretation of Plato's Third Man Argument [article]

George Chailos
2014 arXiv   pre-print
Thus, the notion of convergence is consistent with Plato's apprehension of mathematical concepts, and in particular these of "density" of magnitudes and the complete continuum in the sense that they include  ...  I also claim and prove that within this framework the logic of the TMA is consistent with that of the Third Bed Argument (TBA) as presented in Plato's "Republic".  ...  His construction is in accordance with Von Neumann set-theoretic one 19 .  ... 
arXiv:1406.1973v1 fatcat:2l2fdvdsjjdzvaqivtwbsrvlvu

A Half-Century of Mathematics

Hermann Weyl
1951 The American mathematical monthly  
In the domain of elementary axiomatic geometry one strange discovery, that of von Neumann’s pointless “continuous geometries” stands out, because it is intimately inter- related with quantum mechanics,  ...  The axiomatic approach has often revealed inner relations between, and has made for unification of methods within, domains that apparently lie far apart.  ... 
doi:10.1080/00029890.1951.11999734 fatcat:tzhuvhvwynegnghcouhf4fp4d4

The Logic of Brouwer and Heyting [chapter]

Joan Rand Moschovakis
2009 Handbook of the History of Logic  
axiomatic extensions, each necessarily contained in classical logic (which is axiomatically complete).  ...  The logic of Brouwer and Heyting is effective. The conclusion of an intuitionistic derivation holds with the same degree of constructivity as the premises.  ...  Mark van Atten, Garyfallia Vafeiadou, Richard Vesley, Richard Zach and Yiannis Moschovakis generously read and commented on an earlier version of the manuscript.  ... 
doi:10.1016/s1874-5857(09)70007-x fatcat:q7ha5fv2w5hd3elbjvstdtbmcy

Page 1249 of Mathematical Reviews Vol. 32, Issue Index [page]

Mathematical Reviews  
(English summary) 2000d:51019 Li, Dafa (with Jia, Peifa; Li, Xinxin) Simplifying von Plato’s axiomatization of construc- tive apartness geometry.  ...  On non-spread representations of projective planes of order q’ in PG(2i,q). (English summary) 2000a:51008 von Plato, Jan A constructive theory of ordered affine geometry.  ... 

Page 74 of Mathematical Reviews Vol. 32, Issue Index [page]

Mathematical Reviews  
., (2000h:03006) Leblanc, Hugues see Roeper, Peter, 2000k:03030 Li, Dafa (with Jia, Peifa; Li, Xinxin) Simplifying von Plato’s axiomatization of construc- tive apartness geometry.  ...  Uber einen Satz von Krasner. II. [On a theorem of Krasner.  ... 

The Nature of Physical Knowledge

L. W. Friedrich, R. Bruce Lindsay
1961 Physics today  
Even though the elementary bodies are constructed of mathematical figures, they retain much of the mutability characteristic of the primordial source. 12 How modern Plato's assessment of the mutability  ...  The use of the axiomatic method amounts to this: one starts with postulates-these postulates are part of the axiomatic system.  ...  To the man who pursues his studies in the proper way all geo metrical constructions, all systems of numbers, all duly constituted melodic progressions, the single ordered scheme of all celestial revolutions  ... 
doi:10.1063/1.3057656 fatcat:w72z4itk3rdilnb7oywx5estzi

The Rhetoric of Failure: A Hyper-Dialog About Method in Economics and How to Get Things Going

Egmont Kakarot-Handtke
2012 Social Science Research Network  
of methodology.  ...  This paper gives first an overview of the methodological questions at issue. The frame of reference includes J. S. Mill, Jevons, Popper, Keynes, and Lawson.  ...  and at the same time its failure to construct any sort of axiomatic theoretical framework led to its marginalization.  ... 
doi:10.2139/ssrn.2047177 fatcat:c6h7v46uxbamtkqhy26xlnygte

Quantum groups and noncommutative geometry

Shahn Majid
2000 Journal of Mathematical Physics  
generalisation of geometry itself powerful enough to make sense in the quantum domain.  ...  Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in  ...  of Cambridge, England, where some of the work was completed.  ... 
doi:10.1063/1.533331 fatcat:iyelepep4jddvij5vykvnq6xla

The problem of the invariance of dimension in the growth of modern topology, part I

Dale M. Johnson
1979 Archive for History of Exact Sciences  
Several mathematicians argued that geometry ought to be constructed as a pure autonomous science and not on the basis of numbermanifolds. This is the axiomatic-deductive trend.  ...  KLEIN pointed the way to a solution avoiding the vicious circle, through VON STAUDT'S quadrilateral construction. By means of this construction we can avoid the usual metric.  ...  Dictionary of Scientific Biography. Ed. C.C. GILLISPIE. New York. Vol. X, p. 24.  ... 
doi:10.1007/bf00327627 fatcat:yxt26tgq2zeqbdac7tizcbeabe
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