Mathematics as a field and modeling as a practice are allied, but have interests that sometimes conflict. This workshop will feature the many ways that mathematics and modeling can work together, and highlight their differences.
Carl Friedrich Gauss: “Mathematics is the queen of the sciences.”
Benjamin Peirce: “The science that draws necessary conclusions.”
Wikipedia: “Aristotle defined mathematics as 'the science of quantity,' and this definition prevailed until the 18th century. In the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. Today, there is no consensus on the definition of mathematics, even among professionals. Some emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. There is not even consensus on whether mathematics is an art or a science. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. Some say, 'Mathematics is what mathematicians do.'”
Albert Einstein: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality”.
George Box: “All models are wrong. Some models are useful.”
John Tukey: “An approximate answer to the right problem is worth a good deal more than an exact answer to an approximate problem.”
Richard Hamming: “Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.”
We will work in both directions. Putting settings first provides motivation and allows judgment and contextual knowledge to be taught. Putting math topics first provides a way to honor the logical dependencies between topics.