Abstract
The ‘received view’ of the analog–digital distinction holds that analog representations are continuous while digital representations are discrete. In this paper I first provide support for the received view by showing how it (1) emerges from the theory of computation, and (2) explains engineering practices. Second, I critically assess several recently offered alternatives, arguing that to the degree they are justified they demonstrate not that the received view is incorrect, but rather that distinct senses of the terms have become entrenched specific fields, perhaps most notably in the cognitive psychology of mental imagery.
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Notes
For instance, the expressions of lambda calculus are explicitly defined as consisting of a finite number of symbols plus an “enumerably infinite” number of variable names, of which only a finite number are used in any given formulation of a lambda-definable function (Church 1936). Similarly, Post’s (1936) definition of an effectively calculable system uses one symbol, a finite set of operations, and any given problem uses a finite number of contiguous squares, each of which may contain a single instance of the symbol.
I am not endorsing Turing’s argument; I’m only noting that the fact he makes it supports the view that Turing machines (and hence the foundations of computational theory) are committed to discrete representations.
A particularly interesting instance of the commitment to discrete representation occurs in Turing's discussion of how symbols are instantiated in the memory of a Turing machine, a paper tape. The tape is divided into squares, each of which can contain a single symbol token. Each square is continuous along each dimension, but is closed and bounded, and hence compact (Turing 1936, p. 249, footnote). Consequently, the space in which symbols are instantiated is such that that the set of possible symbols ‘reduces’ to a finite number of types in the sense that any arbitrary subset of points will be closer to one of these finite subsets than another. Despite their being continuous, Turing's mathematical medium for symbols—points in a plane – respects the requirement that Γ be (finite and) discrete.
As Sarpeshkar (1998) notes, in signal processing and electrical engineering the terms ‘analog’ and ‘digital’ are associated with the discreteness or continuity of the signal rather than the time dimension. Consequently, there are in addition to ‘full-blooded’ analog and digital systems, asynchronous digital systems (continuous time, discrete signal) and clocked analog systems (discrete time, continuous signal). For purposes of this paper I bracket these variations, focusing only on the ‘full-blooded’ versions of each type. However, the fact the standard interpretation facilitates these distinctions should not be overlooked.
These historical contingencies help explain why ‘discrete’ and ‘continuous’ tend to be used in mathematical contexts, while ‘digital’ and ‘analog’ are often used when talking about physical devices. So, for instance, there are textbooks on ‘discrete’ math, but not on ‘digital’ math; similarly, there are textbooks on ‘analog’ computers, but not ‘continuous’ computers. Discrete and continuous formats are the mathematical blueprints for what gets implemented by digital and analog devices, respectively.
For the purposes of this paper I ignore the fact that some proponents of continuity hold that physical states described using real-valued equations are not representational.
Katz’s version is: “Call whatever stuff from which representations are constructed the medium of representation. Call whatever structure is imposed on that medium the format of representation.” (2008, p. 404).
While neither Maley nor Katz mention domain formats, they implicitly acknowledge media format, e.g., in Maley’s definition of ‘continuous’ (2011, p. 123), and in Katz’s occasional reference to the physical medium (rather than the representational format) being continuous or discrete (e.g., Katz 2008, p. 404).
There are, of course, theoretical and practical interactions between MFs, RFs, and DFs. For instance, on the theoretical side, one cannot use a discrete RF (e.g., binary code) to exhaustively represent a continuous DF (e.g., real-valued solutions) because the cardinality of the latter is greater than that of the former. For the same reason, one cannot use a discrete MF (e.g., grains of sand) as a basis for a continuous RF, even if the MF were infinite. On the practical side, limitations on resources (e.g., finite memory or error introduced by noise) mean that no RF, regardless of its format, can fully represent an infinite (discrete or continuous) DF.
A popular engineering textbook makes this point as follows: “The discipline of discretization states that we choose to deal with discrete elements or ranges and ascribe a single value to each discrete element or range. Consequently, the discretization discipline requires us to ignore the distribution of values within a discrete element [and] this … requires that systems built on this principle operate within appropriate constraints so that the single-value assumptions hold.” (Argawal and Lang 2005, p. 4). That is, implementing a discrete RF requires mapping ranges of MF states to individual representational types, and this requires that “constraints” are imposed to guarantee that this design goal is satisfied.
This correspondence in structure between MF, RF, and DF tends to make the distinction between MF and RF ‘transparent’ in analog systems, as the continuous states of the MF appear to be directly related to the representational domain (a situation sometimes referred to as “native” representation (Argawal and Lang 2005, p. 44)). Transparency is not unique to analog systems; a similar situation occurs when a discrete MF is used to realize a discrete RF for purposes of representing a discrete domain, as in Maley's example of grains of sand being used to represent an integer quantity. The lesson is that care is required to avoid conflating MF and RF.
My interpretation of the received view also agrees with Lewis' second proffered counterexample to Goodman (it is analog) and with his example of a digital system (it is digital). These observations raise the issue of how to properly interpret Goodman's version of the distinction in comparison to my own, but this project is outside the scope of the present paper.
In the preceding discussion I interpret Maley as asserting his account of ‘digital’ should replace the standard interpretation. An anonymous reviewer suggests that I misinterpret Maley's proposal: Rather than offering an account of ‘digital’ intended to supplant the received view, he is doing nothing more than what I propose, namely, identifying a distinct sense as it appears in a particular context. In response, I believe my interpretation is justified. For example, Maley writes, “I claim that the term ‘digital’ should be reserved only for representations of this type [i.e., those that represent integers using a place-value scheme], rather than discrete representations more generally.” (p. 125). In other words, Maley asserts we should limit ‘digital’ to a subset of discrete systems instead of allowing the term to apply to all discrete systems as specified in the standard interpretation. Regarding ‘analog’ and ‘continuous’, Maley is not as straightforward, writing “I have presented a distinction between analog and continuous representation, and suggested that this distinction be adopted on the grounds that it provides a more useful way to classify representations” (p. 124). However, given his treatment of ‘digital’ and ‘discrete’, I assume his proposal regarding ‘analog’ and ‘continuous’ is also one of replacing the standard view.
The restriction that the relationship between P and Q be linear is relaxed to monotonicity in a footnote.
My proposed abbreviated version of Maley’s definition is intended to gloss over some complications for purposes of facilitating discussion. Most notably, R is said to be a ‘representation of a number Q’, which suggests that analog representations have restricted contents (like digitalE representations). For present purposes I assume that analog representations are not restricted to representing numbers – they could represent colors, propositions, objects, etc. The important point is that what is represented can be described quantitatively, which, taken in tandem with a similar description of the physical parameter P, results in a (monotonic) function from the latter to the former.
To be clear, whereas Maley considers specific contexts (binary digital engineering and mental rotation research) in order to uncover alternative senses, Katz does not explicitly adopt the same strategy; rather, he begins by noting the importance of the analog–digital distinction to numerical cognition literature but justifies his account by appealing to his thought experiment, not the literature. A reviewer suggests that this makes my subsequent consideration of the numerical cognition literature misplaced. Nonetheless, as I argue below, (1) Katz's account contradicts the received view, (2) his thought experiment does not represent the received view, and (3) his thought experiment is very similar to the ‘mental accumulator’ model of numerical reasoning as it appears in the literature he cites. Since his proposal departs from the received view, is not motivated by the received view, and is seemingly based on a model drawn from the numerical cognition literature, it is reasonable to consider whether his nonstandard alternative can be justified by appeal to that literature, even if Katz does not explicitly pursue that strategy. After all, the goal is to see if the nonstandard interpretation can be justified, and there is nowhere else to turn.
His justification for labeling C ‘digital’ is that, despite not being able to track individual water molecules, a human user “can readily distinguish representations of some number n from representations of other nearby numbers.” (p. 405) So, for example, John can't tell how many molecules there are in beaker #2, but he can tell that there are two (rather than one or three) increments of water in that beaker, and hence distinguish the overall representation of 302 from 402 and 202. Now, for this claim to hold of device C, it must also hold for each individual beaker in the array, and since device C is built from instances of device A, it follows that the user can readily distinguish nearby representations in device A. Since the only difference between A and D is their medium, the fact the user cannot discriminate individual molecules in device A should not be an impediment to the user distinguishing representations in D, either – after all, it's the same situation only rendered at a different physical scale. Yet, Katz unexpectedly asserts that the user cannot discriminate nearby representations in case D:
Because a large number of marbles are employed in each increment, the user will likely be unaware exactly how many marbles are employed in any given representation. Because of this, the user is likely to be unable to readily discern whether a representation is of some number n, or whether it is of some other relatively nearby number. (p. 405)
Substituting ‘molecules’ for ‘marbles’, it seems that the premise given in this passage holds for systems A and C, and so by parallel reasoning the user of C should not be reliable in discriminating nearby representations if they are not for system D; or, if they are reliable when using C, they should also be reliable in using D. In short, somewhere in this chain of inferences Katz is begging the question that the discriminatory capacities of the user are relevant to the analog–digital distinction.
Technically there are two possibilities: The first is that a full-blooded digital scheme is imposed on the underlying continuous MF, and the second is that a continuous RF is ‘notched’ (e.g., stable attractors are added) so that the system tends to settle into integer values. In this case an analog scheme is used to represent both reals and integers. In an effort to simplify the exposition, I've deliberately collapsed these two possibilities into one.
The authors also equate ‘discrete’ with ‘countable’, which is precisely how the received view conceives of discrete representation.
Katz also argues that his account is implied by Haugeland (1981, 1998), and hence draws support from that source. This seems to be based on a misinterpretation. As mentioned in "Making sense of implementation" section, Haugeland claims that analog devices are distinguished from digital devices in that the former use approximation procedures for reading and writing representational states while the latter use positive procedures. Approximation procedures are subject to noise – they are uncertain about the type of representation they read or write – and hence can only represent within a margin of error. In contrast, positive procedures reliably succeed in typing representational tokens. Katz interprets Haugeland as claiming that an analog device can be transformed into a digital one by making the read-write mechanisms more tolerant to noise (so that the error in typing token representational states is eliminated) and that a digital device can be transformed into an analog one by making the typing criteria more strict (so that error is introduced). This is only partially correct: The reason analog devices use approximation procedures is that they implement continuous representations, and continuous representations are always subject to noise ("Applying the standard interpretation"). This is why Haugeland notes that no amount of technological innovation can eliminate error in analog devices (p. 83). In contrast, digital devices use positive procedures because they must quantize the representational medium, and this process does not tolerate any error if the resulting states are going to be treated as representations. In short, Haugeland's account is properly viewed as an articulation of the engineering consequences of the received view.
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Acknowledgments
Thanks to Corey Maley and two anonymous reviewers for valuable comments on an earlier draft of this paper, and to Bill Schonbein for teaching me how to use a slide rule.
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Schonbein, W. Varieties of Analog and Digital Representation. Minds & Machines 24, 415–438 (2014). https://doi.org/10.1007/s11023-014-9342-x
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DOI: https://doi.org/10.1007/s11023-014-9342-x