David Slepian’s paper “On bandwidth” is ostensibly about the so-called effective dimensionality theorem, which states (roughly) that the space of signals that are approximately1 bandlimited to the frequency interval [-W,W] and approximately time-limited to the time interval [-T/2,T/2] has finite dimension equal to the time-bandwidth product 2TW. This theorem has profound consequences in telecommunications engineering. However, even if you are not interested in the fine details of prolate spheroidal wave functions, you should read Slepian’s paper for a fascinating discussion on the philosophy of mathematical models of engineering systems.
The main idea is that the theory and practice of engineering takes place in two distinct worlds (or what Slepian calls Facet A and Facet B):
Facet A consists of observations on, and manipulations of, the “real world.” Do not ask me what this real world is: my thoughts become hopelessly muddled here. Let us assume that we all understand the term and agree on what it means. For the electrical engineer, this real world contains oscilloscopes and wires and voltmeters and coils and transistors and thousands of other tangible devices. These are fabricated, interconnected, energized, and studied with other real instruments. Numbers describing the state of this real world are derived from reading meters, thermometers, counters, and dial settings. They are recorded in notebooks as rational real numbers. (No other kind of number seems to be directly obtained in this real world.)
Facet B is something else again. It is a mathematical model and the means for operating with the model. It consists of papers and pencils and symbols and rules for manipulating the symbols. It also consists of the minds of the men and women who invent and interpret the rules and manipulate the symbols, for without the seeming consistency of their thinking processes there would be no single model to consider. When numerical values are given to some of the symbols, the rules prescribe numerical values for other symbols of the model.
Distinctions of this sort are rather common in philosophy of science and related fields. For example, the philosopher Brian Cantwell Smith, in his monumental treatise On the Origin of Objects, talks about the three realms of physics (the particular realm of forces, fields, spatiotemporal positions; the material realm of physicists, documents, experiments; and the universal realm of types, sets, numbers, laws, mathematics) and about the relations between these realms. Both Slepian and Smith are wary of carelessly mixing the entities from different facets or realms without keeping track of it. This is what Slepian says:
There are certain constructs in our models (such as the first few significant digits of some numerical variable) to which we attach physical significance. That is to say, we wish them to agree quantitatively with certain measurable quantities in a real-world experiment. Let us call these the principal quantities of Facet B. Other parts of our models have no direct meaningful counterparts in Facet A but are mathematical abstractions introduced into Facet B to make a tractable model. We call these secondary constructs or secondary quantities. One can, of course, consider and study any model that one chooses to. It is my contention, however, that a necessary and important condition for a model to be useful in science is that the principal quantities of the model be insensitive to small changes in the secondary quantities. Most of us would treat with great suspicion a model that predicts stable flight for an airplane if some parameter is irrational but predicts disaster if that parameter is a nearby rational number. Few of us would board a plane designed from such a model.
Slepian then goes on to discuss the notions of bandwidth, bandlimitedness, and time-limitedness through this lens, touching upon both the benefits and the pitfalls of such seemingly innocuous (and obviously indispensable) mathematical abstractions as real numbers and continuous functions. But the issue of (in)sensitivity of the principal quantities to small changes in the secondary quantities is extremely salient and connects to many other issues of importance to engineers, such as the design of measurement devices and procedures and the interpretation of their readings, robustness, reliability, and the use and misuse of probability and statistics.
For example, in his recent post about confidence intervals, Ben Recht takes a critical look at how we interpret them (spoiler: proper interpretation requires heaps of caveats and quantifiers). Such things as Kolmogorov’s axioms, probability measures, and even the good old normal distribution are Facet B entities; empirical observations and measurements, sample means and variances (registered as rational real numbers), and device tolerances are Facet A entities. The thoroughly pragmatic practice of statistics is intimately entwined with Facet A concerns; the logic and syntax of probability theory belong to Facet B. The tenuous semantic relation and the two-way mismatch between the two facets that Slepian talks about becomes even more troublesome as soon as we start talking about chance, uncertainty, and randomness. I will have more to say about the gap between probabilistic and statistical concepts in future posts.
Any exactly bandlimited signal must have infinite support in time domain, and any exactly time-limited signal must have infinite support in frequency domain. However, these are mathematical idealizations, and Slepian’s paper is precisely about how to properly deal with them.