Constructivity in Geometry

Abstract

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...

Axiomatizing geometric constructions

“…The aim there is rather to show the advantages of P. Martin-Löf's intuitionistic type theory [62,63], in which the axiomatizations can be expressed, as pointed out in an earlier paper [60]. A short comparison between the constructive approach in the classical and intuitionistic setting can be found in [139].…”