Richard Haberman - Collected Quotes

"If our intuition about a problem does not correspond to what a mathematical formula predicts, then further investigations of the problem are necessary. Perhaps the intuition is incorrect, in which case the mathematical formulation and solution has aided in directly improving one's qualitative understanding. On the other hand, it may occur that the intuition is correct and consequently that either there was a mathematical error in the derivation of the formula or the model upon which the analysis is based may need improvement." (Richard Haberman, "Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow", 1998)

"In any problem we should compare as much as possible our intuition about what should happen with what the formula predicts. If the two agree, then we expect that our formula gives us the quantitative effects for the given problem - one of the major purposes for using mathematics." (Richard Haberman, "Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow", 1998)

"In mathematical models, usually the qualitative effects are at least partially understood. Quantitative results are often unknown. When quantitative results are known (perhaps due to precise experiments), then mathematical models are desirable in order to discover which mechanisms best account for the known data, i.e., which quantities are important and which can be ignored. In complex problems sometimes two or more effects interact. Although each by itself is qualitatively and quantitatively understood, their interaction may need mathematical analysis in order to be understood even qualitatively." (Richard Haberman, "Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow", 1998)

"One cannot underestimate the importance of good experiments in developing mathematical models. However, mathematical models are important in their own right, aside from an attempt to mimic nature. This occurs because the real world consists of many interacting processes. It may be impossible in an experiment to entirely eliminate certain undesirable effects. Furthermore one is never sure which effects may be negligible in nature. A mathematical model has an advantage in that we are able to consider only certain effects, the object being to see which effects account for given observations and which effects are immaterial." (Richard Haberman, "Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow", 1998)

 "Physical problems cannot be analyzed by mathematics alone. This should be the first fundamental principle of an applied mathematician (although apparently some mathematicians would frequently wish it were not so)." (Richard Haberman, "Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow", 1998)

"Science attempts to establish an understanding of all types of phenomena. Many different explanations can sometimes be given that agree qualitatively with experiments or observations. However, when theory and experiment quantitatively agree, then we can usually be more confident in the validity of the theory. In this manner mathematics becomes an integral part of the scientific method." (Richard Haberman, "Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow", 1998)