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:
“The contention that a standing army and navy is the best security of peace is about as logical as the claim that the most peaceful citizen is he who goes about heavily armed. The experience of every-day life fully proves that the armed individual is invariably anxious to try his strength. The same is historically true of governments. Really peaceful countries do not waste life and energy in war preparations, with the result that peace is maintained.”
—Emma Goldman (1869–1940)
“In order to begin an analysis, there must already be a synthesis present in the mind.”
—Johan Huizinga (1872–1945)