Delineating the Gap
“Between the idea
And the reality
Falls the Shadow.”
T.S. Eliot — The Hollow Men
The ancient Greek invention of hypothesis, together with the highly developed abstractions they developed, allowed mathematics to be extended far beyond simply dealing with numbers. But the Greeks had rooted their mathematics firmly in geometry, which they considered a critical part of the way the world worked. So they were always able to come back to something real for a comparison. They never lost sight of the gap between the real world and their abstractions.
The ancient Greeks were not able to develop tools to delve into the gap, but they were able to work around it. While Euclid’s Elements is purely abstraction, Eratosthenes’ measurement of the Earth was a highly practical application of basic geometry. Eratosthenes knew that his data were imperfect, and made allowances, recognizing that he was providing a rough estimate. However, that was better than no estimate, which was the situation prior to his exercise. In fact, he was very close to modern estimates, and it would be over a millennium before his result was improved upon.
For many centuries, the gap between reality and abstraction remained unexplored. There were many reasons for this, including a lack of need and a lack of quality tools for measuring. As mathematics was used by relatively few people, was mostly arithmetic, and was mostly applied to prosaic tasks, there was little need to explore the differences between abstract and real.
This changed when some gamblers with mathematical inclinations decided to explore how games of chance worked. In their exploration they found patterns in the behavior of apparently random numbers, which allowed them to undertake general predictions about the large-scale behavior of apparently random events, such as rolling dice and being dealt cards. Much of the initial work in this field occurred in the 17th century.
Some years later, this work was found to be applicable to the behavior of the differences between reality and abstractions. Applied more broadly, this work coalesced into the discipline of Statistics. Early work on error analysis by Gauss and others produced the Gaussian or Normal distribution, an important description of the behavior of certain natural phenomena.
Abstraction: The Downside
For many years, mathematics (as compared to arithmetic) was abstract and the domain of specialists, while arithmetic (primarily calculation) was practical and used by a far wider range of people. Arithmetic was one of the Three R’s (Reading, wRiting, aRithmetic) in practical education. However, mathematics has been taught to progressively wider audiences over the years.
This widespread teaching of mathematics led, in part, to a mistaken effort to take the abstraction of mathematics and apply it to a range of other fields. Some of these fields, such as the physical sciences, found including mathematics straightforward, but others, such as the social sciences and economics, found it more difficult to use mathematical abstraction.
Perhaps the worst aspect of the mathematical abstraction is that there is a widespread assumption among many people that mathematics is all about deterministic computations, and that this abstract representation is reality, not a model. This leads to an unreasonable belief that many problems have just one exact solution, and that once a solution is found, it must be the one and only solution.
Along with this misunderstanding is an assumption that there is no variation or error in measurements. This is totally false, as there is no such thing as an error-free measurement. Every time that an estimate of a physical quantity is made, it’s a measurement and is subject to error.
Our only means of knowing the world outside our own minds is through measurement of some sort. Every measurement is subject to error, and therefore so is our means of knowing anything about the world.
To summarize, mathematical abstraction is a very useful tool. It allows you to form a model of reality and manipulate that. But all models are simplifications. There is always a gap between the model and the reality.
The differences between an abstract model and reality can be described numerically if there have been measurements made. Examination of the differences commonly shows a pattern, once we have enough measurements. One the one level, we see a pattern of behavior: measurements fall into a certain range and distribution. On another level, it is impossible to predict the exact value of the next measurement, so the values measured are apparently random.
Because everything we know about reality, i.e., what is outside our minds, must be measured in some way, and every measurement has some randomness associated with it, our perception of reality includes randomness at its core. We cannot escape from this. As we go deeper into the idea of measurement, we keep finding randomness. As a subatomic level, the latest hypotheses indicate that the randomness is an essential part of the nature of space, time and matter.
We can conclude that there is no certainty, no perfection in measurement, and so all our knowledge is necessarily imperfect. We can try to keep the uncertainty as small as possible, which we call making the measurements precise, but we cannot make them exact.
To allow us to think through problems in a way that avoid the uncertainty, we can use the tool of abstraction. This is a very powerful thinking tool, but we are still working with a model of reality, not reality itself. As soon as we want to move from the abstract to the practical, we find the randomness is there waiting for us.
The word error is derived from the Latin errare, which means to wander. This wandering is a description of the randomness inherent in measurement. We use error to mean wandering from the true value, although the true value is something we can never know. So we can never know the exact size of the error. All we can do is look at the large-scale behavior of the errors and try to understand what is happening with the error when we make a measurement.
Once we have our basic measurements, complete with their built-in errors, we start to derive other information from those measurements. Undertaking a calculation using deterministic mathematics does not eliminate the errors: they propagate through all our subsequent calculations, and also the decision we make based on those calculations.
If we want to know how good the final values are, we need to discover how the errors affect the final result. This is a key part of the geospatial discipline.