I seem to have a preoccupation with the boundary between mathematics and what we can call the other—subjects that aren’t mathematics, but that interact with it in interesting ways.
I’ll try to be clearer about this boundary. To some extent, we would be hard-pressed to find a subject that doesn’t interact with mathematics in some capacity. Reporting survey results invokes statistics, even if only for descriptive purposes. The simple reporting of survey results is an obvious situation where mathematics “says something” about the other.
Example: Linguistic accommodation
For a less superficial situation, let’s consider a distinctly non-mathematical phenomenon. One of Mark Liberman’s recent posts over at Language Log raised questions about dialect features and linguistic accommodation in recent speeches by the presidential hopefuls in the US. On the surface, it wouldn’t seem that mathematics has much to say on these topics. We could, of course, adapt the corpus analysis techniques used in forensic linguistics and try to analyze the extent to which a particular speech by a candidate fits with the corpus of other speeches made by the same candidate, in which case inferential statistics would have a great deal to say about the subject.
There is also a more fundamental way in which mathematics plays a role in this discussion. If a speaker engages in linguistic accommodation, they need to consciously modify various features of their speech, adapting it to the prevailing characteristics of their target audience. For example, if someone from Louisiana were addressing people in Boston, they might adapt the phonetic aspects of their speech (e.g., adopting a non-rhotic pronunciation) and making different lexical selections (e.g., using soda rather than coke). If they were addressing people in Albany, they would likely avoid the non-rhotic pronunciation, but would still use soda rather than coke when asking for a drink. For those choices that correlate primarily with geography (rather than, say, occupation or income level), linguistic maps like the Generic Names for Soft Drinks that prompt the speaker from Louisiana to use soda as a linguistic accommodation can be useful. When we look at the map, the colors provide a clear division into geographic regions that use each term. The boundaries between these regions are called isoglosses, and identify where there is a change in a specific dialectic feature. These isoglosses are determined by the statistical distribution of particular linguistic phenomena, which will be estimated by sampling some of the speech of the residents in each area. The success of a speaker’s linguistic accommodation will depend on how well they respect such statistical distributions—substituting pop for coke in Boston isn’t particularly accommodating, nor is using a non-rhotic pronunciation in Albany.
This example suggests that mathematics both informs the act of linguistic accommodation (via the identification of isoglosses that allow selection of appropriate linguistic features) and permits an evaluation of that act (via a comparison between a particular speech and a given corpus of speeches). Both the informative and evaluative aspects allow mathematics to contribute to the subject of linguistic accommodation.
A more direct and structured interaction between mathematics and the other shows up in mathematical modeling. Rutherford Aris (Mathematical Modeling: A Chemical Engineer’s Perspective, p. 3) suggests that
A mathematical model is a representation, in mathematical terms, of certain aspects of a nonmathematical system. The arts and crafts of mathematical modeling are exhibited in the construction of models that not only are consistent in themselves and mirror the behavior of their prototype, but also serve some exterior purpose.
The mathematics involved in our previous example involve a representation only of the observed distributional properties of the nonmathematical system. A mathematical model goes further in that it represents some aspects of the actual system. The basic idea is familiar to anyone who has encountered so-called word problems (in the same way that anyone who has encountered “See spot. See spot run.” is familiar with the basic idea of reading).
Example: Mixing tank problem
A typical introductory example is a mixing tank problem, such as this one (from Sanchez, Allen, and Kyner, Differential Equations 2nd ed., p. 13):
A 1000-liter tank contains a mixture of water and chlorine. In order to reduce the concentration of chlorine in the tank, fresh water is pumped in at a rate of 6 liters per second. The fluid is well stirred and pumped out at a rate of 8 liters per second. If the initial concentration of chlorine is 0.02 grams per liter, find the amount of chlorine in the tank as a function of and the interval of validity of the mathematical model.
When we model this type of situation, we identify a particular nonmathematical system (the mixing tank), some observable properties of the system (flow rates, tank size, and initial concentration), and we try to find a mathematical relationship that holds among those properties (in this case, its a particular differential equation).
A critical element of any mathematical model is the identification of boundary conditions—essentially, observable properties that occur where the nonmathematical system meets the environment in which it is embedded. Aris makes the following observation (p. 13):
When Amundson taught the graduate course in mathematics for chemical engineering, he always insisted that “all boundary conditions arise from nature.” He meant, I think, that a lot of simplification and imagination goes into the model itself, but the boundary conditions have to mirror the links between the system and its environment very faithfully. Thus if we have no doubt that the feed goes get into the reactor, then we must have a condition that ensures this in the model. We probably do not wish to model the hydrodynamics of the entrance region, but the inlet must be an inlet.
The points of correspondence between the system to be modeled and the observable phenomena at the boundary of the system are fundamental. If the correspondence fails, then the mathematical model will fail to say anything about the (nonmathematical) system; the situation will be somewhat analogous to the speaker from Louisiana substituting pop for coke while they’re visiting Boston.
The relationship between mathematics and the other does not need to involve quantification. Contemporary mathematics includes the study of systematic relationships and patterns (as I’ve already discussed to some extent here).
Example: Data flow diagrams
A common tool in the software developer’s toolbox since the 1970’s is a data flow diagram (DFD). A DFD indicates the interaction between two types of objects of which a system is composed: data and processes. Each process can use certain data (its inputs), and produce other data (its outputs); in the DFD, inputs are represented by an arrow from the input data to the process that will use it, and outputs are represented by an arrow from the process to the data that it produces. From a mathematical standpoint, this is known as a directed graph.
In a structured analysis and design approach to software engineering, the system will be modeled not with a single DFD, but with several of them in a hierarchic structure. The top level often represents the entire system as a single process, and indicate what inputs it receives (and possibly who, or what organizational unit, provides these inputs) and what it produces (and possibly who, or what organizational unit, uses these products). At this level, the arrows on the diagram are completely analogous to the boundary conditions in mathematical models. The next level of DFD “explodes” the process, showing how it is to be decomposed into smaller processes that interact with one another in order to realize the system. Each of these smaller processes can then be exploded to show a finer level of detail. At some point, the process will be simple enough that the designer can simply specify what the code is to accomplish; this is the code that will be written by the programmers.
It is not uncommon to be in a position where we have an existing software system that we need to understand, but don’t have reliable documentation for. Unfortunately, the code does not provide us with this organizational and conceptual information about the system. What we can do is to identify the fine-grained processes, and their inputs and outputs. This provides one huge DFD (which is identified with a directed graph) with no hierarchic organization, which isn’t really what we want. We can impose a hierarchic organization on this by grouping some of the processes together into a new process, and ignoring the internal structure of the new process. In graph-theoretic terms, we are simply contracting the arcs in the directed graph.
This example, like the other two I discussed above, illustrates a way in which mathematics can interact with the other, this time without involving any notion of quantity.
Just as there is a boundary between a nonmathematical system and its environment, there is a boundary between mathematics and the other. Frequently, we use mathematics as a tool for describing, analyzing, and modeling the other. How successfully we do this depends, I think, on how faithfully the mathematics reflects the systematic properties of the other.
Copyright © 2008 Michael L. McCliment.