# 2.7: Multiple Levels of Investigation and Explanation

- Page ID
- 35705

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Imagine bringing several different calculating devices into a class, with the goal of explaining how they work. How would you explain those devices? The topics that have been covered in the preceding pages indicate that several different approaches could—and likely should—be taken.

One approach would be to explain what was going on at a physical or implementational level. For instance, if one of the devices was an old electronic calculator, then you would feel comfortable in taking it apart to expose its internal workings. You would likely see an internal integrated circuit. You might explain how such circuits work by talking about the properties of semiconductors and how different layers of a silicon semiconductor can be doped with elements like arsenic or boron to manipulate conductivity (Reid, 2001) in order to create components like transistors and resistors.

Interestingly, the physical account of one calculator will not necessarily apply to another. Charles Babbage’s difference engine was an automatic calculator, but was built from a set of geared columns (Swade, 1993). Slide rules were the dominant method of calculation prior to the 1970s (Stoll, 2006) and involved aligning rulers that represented different number scales. The abacus is a set of moveable beads mounted on vertical bars and can be used by experts to perform arithmetic calculations extremely quickly (Kojima, 1954). The physical accounts of each of these three calculating devices would be quite different from the physical account of any electronic calculator.

A second approach to explaining a calculating device would be to describe its basic architecture, which might be similar for two different calculators that have obvious physical differences. For example, consider two different machines manufactured by Victor. One, the modern 908 pocket calculator, is a solar-powered device that is approximately 3" × 4" × ½" in size and uses a liquid crystal display. The other is the 1800 desk machine, which was introduced in 1971 with the much larger dimensions of 9" × 11" × 4½". One reason for the 1800’s larger size is the nature of its power supply and display: it plugged into a wall socket, and it had to be large enough to enclose two very large (inches-high!) capacitors and a transformer. It also used a gas discharge display panel instead of liquid crystals. In spite of such striking physical differences between the 1800 and the 908, the “brains” of each calculator are integrated circuits that apply arithmetic operations to numbers represented in binary format. As a result, it would not be surprising to find many similarities between the architectures of these two devices.

Of course, there can be radical differences between the architectures of different calculators. The difference engine did not use binary numbers, instead representing values in base 10 (Swade, 1993). Claude Shannon’s THROBACK computer’s input, output, and manipulation processes were all designed for quantities represented as Roman numerals (Pierce, 1993). Given that they were designed to work with different number systems, it would be surprising to find many architectural similarities between the architectures of THROBACK, the difference engine, and the Victor electronic machines.

A third approach to explaining various calculators would be to describe the procedures or algorithms that these devices use to accomplish their computations. For instance, what internal procedures are used by the various machines to manipulate numerical quantities? Algorithmic accounts could also describe more external elements, such as the activities that a user must engage in to instruct a machine to perform an operation of interest. Different electronic calculators may require different sequences of key presses to compute the same equation.

For example, my own experience with pocket calculators involves typing in an arithmetic expression by entering symbols in the same order in which they would be written down in a mathematical expression. For instance, to subtract 2 from 4, I would enter “4 – 2 =” and expect to see 2 on display as the result. However, when I tested to see if the Victor 1800 that I found in my lab still worked, I couldn’t type that equation in and get a proper response. This is because this 1971 machine was designed to be easily used by people who were more familiar with mechanical adding machines. To subtract 2 from 4, the following expression had to be entered: “4 + 2 –”. Apparently the “=” button is only used for multiplication and division on this machine!

More dramatic procedural differences become evident when comparing devices based on radically different architectures. A machine such as the Victor 1800 adds two numbers together by using its logic gates to combine two memory registers that represent digits in binary format. In contrast, Babbage’s difference engine represents numbers in decimal format, where each digit in a number is represented by a geared column. Calculations are carried out by setting up columns to represent the desired numbers, and then by turning a crank that rotates gears. The turning of the crank activates a set of levers and racks that raise and lower and rotate the numerical columns. Even the algorithm for processing columns proceeds in a counterintuitive fashion. During addition, the difference engine first adds the odd-numbered columns to the even-numbered columns, and then adds the even-numbered columns to the odd-numbered ones (Swade, 1993).

A fourth approach to explaining the different calculators would be to describe them in terms of the relation between their inputs and outputs. Consider two of our example calculating devices, the Victor 1800 and Babbage’s difference engine. We have already noted that they differ physically, architecturally, and procedurally. Given these differences, what would classify both of these machines as calculating devices? The answer is that they are both calculators in the sense that they generate the same input-output pairings. Indeed, all of the different devices that have been mentioned in the current section are considered to be calculators for this reason. In spite of the many-levelled differences between the abacus, electronic calculator, difference engine, THROBACK, and slide rule, at a very abstract level—the level concerned with input-output mappings—these devices are equivalent.

To summarize the discussion to this point, how might one explain calculating devices? There are at least four different approaches that could be taken, and each approach involves answering a different question about a device. What is its physical nature? What is its architecture? What procedures does it use to calculate? What input-output mapping does it compute?

Importantly, answering each question involves using very different vocabularies and methods. The next few pages explore the diversity of these vocabularies. This diversity, in turn, accounts for the interdisciplinary nature of cognitive science.