1. How science works (adapted from Coombs, 1983)

2. Empirical Systems T Segment of the real world Empirical generalization
Language development
Attitude measurement
Memory
Illusions T
Empirical generalization
Segment of the real world
Testing by
observation or experiment
To draw conclusion from the data about the existence of empirical regularities

3. Empirical Systems Can make inferences Bring novelty
Descriptions of observations T
Empirical generalization
Segment of the real world
Extrapolation beyond the bounds of the observation is unjustified
Have no explanatory power

4. Mathematical Systems M Axiom system Logical consequences
Set of abstract objects
Unproven assertions to avoid circularity of reasoning (e.g. element, belong to, set, point, and or) M
Logical consequences
Axiom system
Mathematical reasoning
(proof)
Consequences that are derived from the axiom system
(theorems)

5. Mathematical Systems (Model)
Totally deductive
Brings nothing new
Seeks logical consistency
Can explain M
Logical consequences
Axiom system

6. Empirical and Mathematical Systems
Empirical generalization
Segment of the real world A I
(interpretation)
(abstraction) M
Logical consequence
Axiom system

7. Model A I T M Segment of the real world Axiom system
Empirical generalization
Logical consequence

8. Theory A I T M Segment of the real world Axiom system
Empirical generalization
Logical consequence

9. The empirical world is rich, the mathematical world is powerful
The empirical world is rich, the mathematical world is powerful. The match is fruitful.
If AMI = T, then scientific knowledge has increased A I M
Logical consequence
Axiom system T
Empirical generalization
Segment of the real world

10. Explanations for simple phenomenon are lacking!
Invariance
Translation A
Rotation A
Size A A