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 logical English train a scholar as they train an engineer. Oxford is Greek factory, as Wilton mills weave carpet, and Sheffield grinds steel. They know the use of a tutor, as they know the use of a horse; and they draw the greatest amount of benefit from both. The reading men are kept by hard walking, hard riding, and measured eating and drinking, at the top of their condition, and two days before the examination, do not work but lounge, ride, or run, to be fresh on the college doomsday.”
—Ralph Waldo Emerson (1803–1882)
“It is not easy to construct by mere scientific synthesis a foolproof system which will lead our children in a desired direction and avoid an undesirable one. Obviously, good can come only from a continuing interplay between that which we, as students, are gradually learning and that which we believe in, as people.”
—Erik H. Erikson (20th century)