Logic is the study of well-formed arguments. Logic is an analytical tool. It is concerned with the validity of inferences and whether conclusions flow in sequential steps from their premises. On the abstract plain of pure logic and pure mathematics, along with an austere beauty, we find certain truth that refers only to itself. Traditionally, philosophers have recognized that pure mathematics and deductive logic are a special category of knowing. This is category is “analytical knowledge.”
Analytical knowledge is an idealistic realm based on definite proof and absolute certainty. Analytical knowledge is contrasted with “synthetic knowledge” which refers to non-analytical categories of knowledge claims that refer to the real world. Applied mathematics, the hard sciences, the humanities and the arts all fall into this catch-all grouping.
It is easy to see why a logical syllogism and, say, the solution of a quadratic equation fall into the analytical categor and it is easy to see how a knowledge claim in economics or pschology or a protest song fall into the synthetic category; but consider the following:
PV = nRT
2x2 - 8x- 18 = 0
The two equations appear similar. Both have an equals sign and both consist of algebraic symbols. The ideal-gas equation is used in chemistry to describe real world relationships between temperature, volume, pressure and amount of gases. The equation does not take into account the size of the gas molecules and interactions between them. Predictions can be made about the behavior of gases from the ideal-gas equation that can be verified in the laboratory. But, like all equations used in technology and the sciences, it remains a simplification despite its effectiveness and utility. It is odd and counterintuitive that—from a meta-knowledge perspective—the ideal-gas law has more in common with a historical claim or poetic metaphor than it does with the purely abstract quadratic equation. Of course, the quadratic equation would immediately move into synthetic category if it was being applied to a real life situation.
Are traditional word problems in math text books that refer to tanks of water filling up or men digging trenches anaytical or synthetic? Was designing the Golden Gate Bridge in San Francisco in the anaytical or the synthetic realm? Is theoretical physics synthetic knowledge?
Is the analytical—synthetic demarcation useful? Is it a gross oversimplification? Are there any “grey areas” in the realm of logic and pure mathematics where absolute certainty breaks down or cannot be assumed?