Logical Synthesis
The process of logical synthesis begins with some arbitrary but defined starting point.
- 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.
- 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).
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.
Read more about this topic: Synthetic Geometry
Famous quotes containing the words logical and/or synthesis:
“Philosophy aims at the logical clarification of thoughts. Philosophy is not a body of doctrine but an activity. A philosophical work consists essentially of elucidations.”
—Ludwig Wittgenstein (1889–1951)
“It is in this impossibility of attaining to a synthesis of the inner life and the outward that the inferiority of the biographer to the novelist lies. The biographer quite clearly sees Peel, say, seated on his bench while his opponents overwhelm him with perhaps undeserved censure. He sees him motionless, miserable, his head bent on his breast. He asks himself: “What is he thinking?” and he knows nothing.”
—Andre Maurois (1885–1967)