Hilbert’s Foundations of Geometry

Project Gutenberg has David Hilbert’s Foundations of Geometry available. It is a translation of Hilbert’s Grundlagen der Geometrie, which is famous as the first modern axiomization of Euclidean geometry. The difference between Hilbert’s approach and that of Euclid is that Hilbert fills in all of the fiddly little details required to meet modern standards of rigor.

The book is elementary, and (as translated by Townsend) is a pleasant read. Much of the book centers around constructing the field of real numbers in terms of the axiomized geometrical constructions. This in turn allows Hilbert to show that the set of axioms is complete. The topic leads naturally to one of the main themes of research in plane geometry in the early part of the last century, which is to consider different algebraic objects and how they can serve as coordinates for different notions of affine or projective planes. The reals can be replaced with an arbitrary division ring, for example. For a projective plane, the most general object is a planar ternary ring, with has a ternary operation that serves as a hybrid of addition and multiplication. Determining the projective planes with a finite number of points is still an open question.

2 thoughts on “Hilbert’s Foundations of Geometry

  1. Hilbert’s was the first widely-known modern axiomatization of geometry, but he was not first. That honour belongs to Italian mathematician Mario Pieri, who published in 1895 and 1897-98, before Hilbert. This was at a time when the leading geometers were Italian, and the leading Italian mathematicians geometers.

    @ARTICLE{pieri:geometry95,
    AUTHOR = “Mario Pieri”,
    TITLE = “Sui principi che reggiono la geometria di posizione”,
    JOURNAL = “Atti della Reale Accademia delle scienze di Torino”,
    YEAR = “1895″,
    volume = “30″,
    pages = “54–108″}

    @ARTICLE{pieri:geometry98,
    AUTHOR = “Mario Pieri”,
    TITLE = “I principii della geometria di posizione composti in sistema logico deduttivo”,
    JOURNAL = “Memorie della Reale Accademia delle Scienze di Torino 2″,
    YEAR = “1897–98″,
    volume = “48″,
    pages = “1–62″}

  2. Huh, I didn’t know that. It sounds like it wasn’t an obscure paper, either. Wikipedia suggests that it influenced Tarski’s axiomization of plane geometry.

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