Synthetic geometry

Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.

Logical synthesis[edit]

The process of logical synthesis begins with some arbitrary but defined starting point. This starting point is defined in terms of primitive notions or primitives and axioms about these primitives:

• Primitives are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are things like points, lines and planes while the fundamental relationship is that of incidence – of one object meeting or joining with another. The terms themselves are generally undefined beyond basic terms. For example Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer glasses.^[1]
• Axioms are statements about these primitives, for example that any two points are together incident with just one line (i.e. that for any two points, there is just one line which passes through both of them). Axioms are true, but not provable. They are rather the building blocks of geometric constructions.

From a given set of axioms, synthesis proceeds as a carefully constructed logical argument. Where a significant result is proved rigorously, it becomes a theorem.

Any given set of axioms leads to a different logical system. In the case of geometry, each distinct set of axioms leads to a different geometry.

Properties of axiom sets[edit]

If the axiom set is not categorical (so that there is more than one model) one has the geometry/geometries debate to settle^[vague]. That's not a serious issue for a modern axiomatic mathematician, since the implication of axiom is now a starting point for theory rather than a self-evident plank in a platform based on intuition. Since the Erlangen program of Klein, the nature of any given geometry has been seen as the connection of symmetry and the content of propositions, rather than the style of development.

History[edit]

One of the early French analysts summarized synthetic geometry this way:

The Elements of Euclid are treated by the synthetic method. This author, after having posed the axioms, and formed the requisites, established the propositions which he proves successively being supported by that which preceded, proceeding always from the simple to compound, which is the essential character of synthesis.^[2]

The heyday of synthetic geometry can be considered to have been the 19th century, when analytic methods based on coordinates and calculus were ignored by some geometers such as Jakob Steiner, in favour of a purely synthetic development of projective geometry. For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory (with more models) than is found by starting with a vector space of dimension three. Projective geometry has in fact the simplest and most elegant synthetic expression of any geometry.

In his Erlangen program, Felix Klein played down the tension between synthetic and analytic methods:

On the Antithesis between the Synthetic and the Analytic Method in Modern Geometry:

The distinction between modern synthesis and modern analytic geometry must no longer be regarded as essential, inasmuch as both subject-matter and methods of reasoning have gradually taken a similar form in both. We choose therefore in the text as common designation of them both the term projective geometry. Although the synthetic method has more to do with space-perception and thereby imparts a rare charm to its first simple developments, the realm of space-perception is nevertheless not closed to the analytic method, and the formulae of analytic geometry can be looked upon as a precise and perspicuous statement of geometrical relations. On the other hand, the advantage to original research of a well formulated analysis should not be underestimated, - an advantage due to its moving, so to speak, in advance of the thought. But it should always be insisted that a mathematical subject is not to be considered exhausted until it has become intuitively evident, and the progress made by the aid of analysis is only a first, though a very important, step.^[3]

The close axiomatic study of Euclidean geometry led to the construction of the Lambert quadrilateral and the Saccheri quadrilateral. These structures introduced the field of non-Euclidean geometry where Euclid's parallel axiom is denied. Gauss, Bolyai and Lobachevski independently constructed hyperbolic geometry, where parallel lines have an angle of parallelism that depends on their separation. This study became widely accessible through the Poincaré disc model where motions are given by Möbius transformations.

Another example concerns inversive geometry as advanced by Ludwig Immanuel Magnus, which can be considered synthetic in spirit. The closely related operation of reciprocation expresses analysis of the plane.

Karl von Staudt showed that algebraic axioms, such as commutativity and associativity of addition and multiplication, were in fact consequences of incidence of lines in geometric configurations. David Hilbert showed^[4] that the Desargues configuration played a special role. Further work was done by Ruth Moufang and her students. The concepts have been one of the motivators of incidence geometry.

When parallel lines are taken as primary, synthesis produces affine geometry. Though Euclidean geometry is both affine and metric geometry, in general affine spaces may be missing a metric. The extra flexibility thus afforded makes affine geometry appropriate for the study of spacetime, as discussed in the history of affine geometry.

In 1955 Herbert Busemann and Paul J. Kelley sounded a nostalgic note for synthetic geometry:

Although reluctantly, geometers must admit that the beauty of synthetic geometry has lost its appeal for the new generation. The reasons are clear: not so long ago synthetic geometry was the only field in which the reasoning proceeded strictly from axioms, whereas this appeal — so fundamental to many mathematically interested people — is now made by many other fields.^[5]

For example, college studies now include linear algebra, topology, and graph theory where the subject is developed from first principles, and propositions are deduced by elementary proofs. In an abstract sense, these subjects are also synthetic geometry.

Today's student of geometry has axioms other than Euclid's available: see Hilbert's axioms and Tarski's axioms.

Computational synthetic geometry[edit]

In conjunction with computational geometry, a computational synthetic geometry has been founded, having close connection, for example, with matroid theory. Synthetic differential geometry is an application of topos theory to the foundations of differentiable manifold theory.

Notes and references[edit]

• Hilbert & Cohn-Vossen, Geometry and the imagination.
• Mlodinow, L; Euclid's window.