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On the relationship between plane and solid geometry. (English) Zbl 1242.51001

Questions regarding the relation between plane and solid geometry, in particular the legitimacy of using spatial concepts to prove theorems with planimetric statements came up to a certain extent in Euclid’s Elements, Archytas and Pappus, then in the 17th century in the works of Luca Valerio and Evangelista Torricelli, a century that also witnessed the discovery of the central item on the authors’ agenda, Desargues’ theorem on homological triangles. During the 19th century, the question whether solid geometry and plane geometry are to be treated in conjunction or separately, spawned the “debate on fusionism”, one side (the “fusionists”, defended by Gergonne, Bellavitis, Cremona, de Paolis, Bassani, Lazzeri, de Amicis) arguing for either “the parallel development of plane and solid geometry to point out analogies between the two areas” or “the use of stereometric considerations in the constructions, respectively proofs, of geometrical problems, respectively theorems”), the other (the anti-fusionists, such as Francesco Palatini, the only one whose position is described in some detail by the authors), who argued that “one should try to prove every proposition of elementary geometry without appeal to any theory on which it does not necessarily depend,” foreshadowing the later concern, championed by Hilbert, on preserving “the purity of method, i.e. to prove theorems if possible using means that are suggested by the content of the theorem”. It is this issue of purity that concerns the greatest part of the paper. Based on Desargues’ theorem, the positions of Klein (1873), Peano (1894), and Hilbert (1899) are closely scrutinized, to determine who first realized the independence of Desargues’ theorem (Klein is influenced by a result of Beltrami and thus sees geometry only in the Riemannian geometric context, Peano states the independence of Desargues theorem from the other incidence axioms, hinting that it fails on surfaces of non-constant curvature, without actually providing the details for such a surface (a gap filled by the authors with an ellipsoidal example due to Patrick Popescu-Pampu)). The remainder of the paper is devoted to a presentation of Hilbert’s axiomatic thinking, to his thoughts on Reinheit der Methode, and to a refutation of M. Hallett’s view in [P. Mancosu (ed.), The philosophy of mathematical practice, Oxford: Oxford University Press (2008; Zbl 1163.03001)], that Hilbert thought a spatial proof of Desargues’ theorem to be pure, given that the latter has “spatial content”. In the authors’ view, it has “planar content”, and only the “informal” content of a statement can be taken into account in ascribing purity to a proof, considerations based on the “formal” content, advocated by Hallett, being untenable.

MSC:

51-03 History of geometry
01A55 History of mathematics in the 19th century
00A30 Philosophy of mathematics
01A05 General histories, source books

Citations:

Zbl 1163.03001
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References:

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