Modeling Syntax and Semantics

We are most concerned here with mathematical modeling. To construct a model of something as Harvey did, we construct a representation that then substitutes for the real object. Model theory, a branch of mathematics, plays a role in logic that is similar to the role arithmetic plays in the rest of mathematics. However, beyond the basic ideas, advanced model theory is its own specialty.

In applied mathematics, a model abstractly represents some material reality taken from the world. This idea differs from how logicians use the idea of model. To them a model of a theory is something concrete, as much as any mathematical object can be concrete, and the model is the object that ultimately satisfies all the axioms of a theory. Logicians use formulas or sequences of symbols to write the axioms. Rules always determine formation of these sequences; these rules are syntax. Model theory presupposes we interpret the symbols in the formulas in some defined way, so that our interpretation is what brings meaning to our formulas. Our meaning ultimately translates into true or false statements about our observations. This interpretive notion is semantics.

What's important is that modeling occurs more at the level of semantics and uses little syntax. In building models we try not to learn what a specific structure we superimpose upon nature is, but we want to learn what is true about our model of nature and how we can prove this trueness or validity, referring back to the whole universe of our ideas, even though some of these ideas may as yet be inaccessible. In this way, our modeling contrasts with theoretical computer scientists' study of algorithms. Algorithmic studies are mainly syntactic.

A model of a bee's circulation becomes the basis for visualizing and generalizing a geometrical process. Because the model is what gives shape or form to our thinking, we begin by classifying and comparing our information or observations. Only later might we employ the implication symbol to indicate the formalization of the system that ultimately would grow into an axiomitzation formulating the rules for proof. If our model is effective, the model is easier to test and analyze than would be studying an actual bee, because within the limits we place on nature that we define by our observations, our model can respond in the same way we would expect a real bee to behave.

To model something as complex as the bee's circulation requires combining visual and analytical thinking and frequently the use of metaphor. To simulate a shape in a model is to define a shape that will come to house a process. Where we were once satisfied with two-dimensional graphs as being adequate representations, we now demand topologically valid and analytically complete models having three dimensions that change with time. Analysis then reduces our measurements and simplifies our 'data' into manageable concepts.

Modeling is then a loose integration of mathematical methods taken together to describe a shape or a process, often in terms of an appropriate metaphor. Computer aided geometric design (CAGD), for example, applies the mathematics of curves and surfaces usually employing the parametric equations of differential geometry. We use computational geometry to design and analyze geometrical algorithms.

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