AAnalog Computation and Representation
Corey J. Maleyforthcoming in
The British Journal for the Philosophy of Science
Abstract
Relative to digital computation, analog computation has been neglected in thephilosophical literature. To the extent that attention has been paid to analog computation, ithas been misunderstood. The received view—that analog computation has to do essentiallywith continuity—is simply wrong, as shown by careful attention to historical examples ofdiscontinuous, discrete analog computers. Instead of the received view, I develop an account ofanalog computation in terms of a particular type of analog representation that allows fordiscontinuity. This account thus characterizes all types of analog computation, whethercontinuous or discrete. Furthermore, the structure of this account can be generalized to othertypes of computation: analog computation essentially involves analog representation, whereasdigital computation essentially involves digital representation. Besides being a necessarycomponent of a complete philosophical understanding of computation in general,understanding analog computation is important for computational explanation incontemporary neuroscience and cognitive science. Those damn digital computers!Vannevar Bush, MIT a r X i v : . [ c s . G L ] D ec Introduction2 Analog Computers2.1 Mechanical analog computers2.2 Electronic analog computers2.3 Discontinuous analog elements3 What makes analog computation ‘analog’ and ‘computational’3.1 Analog as continuity3.2 Analog as covariation3.3 What makes it ‘analog’3.4 What makes it ‘computation’4 Questions and Objections4.1 Aren’t these just hybrid computers?4.2 Is this really even computation?4.3 The Lewis-Maley account is problematic5 Concluding thoughts
Like clocks and audio recordings, computation comes in both digital and analog varieties. Relativeto digital computation, analog computation has been neglected, and as a result, not wellunderstood. This is partially due to the fact that an account of what counts as analog in general hasproven controversial, particularly in contrast to what counts as digital. Fifty years have passed sinceGoodman ([1968]) began the discussion of the so-called analog/digital distinction in thephilosophical literature, and still we have not reached a consensus. In the few places where analogcomputation has specifically been mentioned, it usually goes something like this: ‘Analogcomputation is often contrasted with digital computation, but analog computation is a vague and2lippery concept. . . Roughly, abstract analog computers are systems that manipulate continuousvariables to solve certain systems of differential equations’ (Piccinini [2015], p. 123).One might think, like Piccinini, that all we need to know about analog computation is that it iscontinuous, rather than discrete. One might think that we need no more than this kind of roughcharacterization of analog computation because analog computation is no more than a historicalcuriosity, devoid of interest or relevance to contemporary philosophy. But these thoughts would bemisguided.The received view of analog computation—that it is essentially about continuity—is simplywrong, as shown by studying actual analog computers from the 20 𝑡 ℎ century. Providing an accountof analog computation is not as simple as the received view would have it. Further, while it is truethat the heyday of analog computers has come and gone (there are no companies that produceanalog computers anymore), there are two reasons why providing such an account is still important.First, if we want to understand computation simpliciter, we need a clear account of analogcomputation in order to see how it might fit into a more general account of computation thatincludes analog and digital (and perhaps other) types as species. Second, if we take seriously the ideathat cognitive science and neuroscience are in the business of explaining what minds and brains doin terms of the computations that they literally perform, then we should understand all types ofcomputation that might be applicable to such explanations.Here is the structure of what follows. In order to get a feel for how analog computation actuallyworks, I will first present a few examples of different types of mid-20 𝑡 ℎ century analog computers.After making clear why continuity alone does not suffice, I will then argue that a refined version ofanalog representation—the Lewis-Maley view—explains what is analog about analogcomputation. Finally, with this account of analog computation in hand, I will conclude with somegeneral considerations about the relevance of this account to contemporary issues in philosophy andthe cognitive sciences.Before beginning in earnest, a preemptory note is in order. Space prohibits a full account of This is a stronger claim than that the brain can be computationally simulated, which is true ofvirtually any scientific object of interest (Piccinini [2007]). This is how Adams ([2019]) labels the view originally developed by Lewis ([1971]) and refinedand defended by Maley ([2011]). 3nalog and digital computation. As just mentioned, the standard approach is to present an accountof both simultaneously. I think this has been a mistake, because, ‘analog’ and ‘digital’ are neitheropposites not jointly exhaustive. I can only offer some comments on digital computation andrepresentation in passing. It should simply be noted that a system of representation or computationthat is not analog on the account presented here is not necessarily digital.
If one wants to trace the history of digital computers, one excellent place to start is the work ofTuring ([1938]). To be sure, digital computing machines predate Turing’s work: examples includeBabbage’s difference engine and Pascal’s calculator (sometimes known as the Pascaline).Nevertheless, Turing’s work on computable numbers is usually taken to have initiated what we nowunderstand as computer science, and later, digital computer engineering. There is no such conceptual birthplace when it comes to analog computers. Analog devices havebeen used since antiquity: an early review essay on analog computing machines begins ‘The use ofinstruments of computation and analysis is as old as mathematics itself’ (Bush [1936], p. 649), andthe function of the two-millenium-old Antikythera mechanism has only recently been discovered(Efstathiou and Efstathiou [2018]). As for a contemporary theoretical centerpiece—something likethe ‘Turing’ of analog computation—common citations include (Shannon [1941]) and (Pour-El[1974]); but, as we will see later, even these are insufficient to account for analog computation in itsentirety. Rather than tracing a complete history of these instruments, we will begin with those devices firstdeveloped in the 1930s, now referred to as the first analog computers. Even still, beginning here is However, some, such as Corry ([2017]), have argued that Turing’s theoretical work did notinfluence the birth of digital computation nearly as much as is commonly thought. For instance, a recent monograph (Piccinini [2015]) on physical computation states that ‘[t]heclearest notion of analog computation is that of Pour-El (1974)’. Using the term ‘computer’ to describe these machines is, of course, anachronistic: this termreferred to a particular job, usually held by women, that involved performing mathematicalcomputations by hand. More accurate would be ‘analog computing machines.’ For brevity,however, I will use the anachronistic terminology in what follows.4omething of a challenge because of the lack of attention paid to these machines, particularly relativeto digital computers. One historian puts the point this way:In general, historians of computing have neglected analog computing, viewing itprimarily as an obsolete predecessor to digital. . . [W]e have not yet begun to understandthe history and significance of analog computing, especially the relationship betweenanalog and digital machines. (Mindell [2002], p. 10)A computer scientist makes a similar point:Because digital computers and computation have been so successful, they haveinfluenced how we think about both computers as machines and computation as aprocess—so much so, it is difficult today to reconstruct what analog computing was allabout. (Nyce [1996], p.3)To be sure, there have been some limited discussions of analog computation in the philosophicalliterature. For example, O’Brien and Opie ([2008], [2010]) use analog computation to illustrate therole of representation in cognitive science. Shagrir ([2010], [forthcoming]) discusses the use of whathe calls ‘analog-model’ computing, which he takes to involve the simulation or modeling of asystem. Isaac ([2018]) has recently argued that analog computation is compatible with embodiedapproaches to cognition. Papayannopoulos ([2020]) has recently defended the orthodox view thatanalog computers are simply computers with continuous values (a point I will return to below).These and related works are undoubtedly important steps toward the project of understandinganalog computation; nevertheless, we do not yet have a complete philosophical treatment of thesubject. Providing such a treatment is one goal of this essay.I will describe three somewhat simplified examples of analog computers: one mechanical, oneelectrical, and one electromechanical. These examples will illustrate some of the general principlesneeded to motivate later discussion of the right way to think about analog computation. Shagrir’s view has much in common with the view on offer here, although his focus is oncomputation more generally. The differences are important, but they will have to wait for anothertime. 5 .1 Mechanical analog computers
One of the most well-known mechanical analog computers is the differential analyzer, developed byVannevar Bush ([1931]). ‘Mechanical’ in this context means that the elements doing the computinguse physical movement, such as rotation and displacement, to perform their tasks. Bush’sdifferential analyzer uses an interconnected series of components to compute solutions tomathematical operations, including differential equations. One such component is the integrator,which represents the values of functions (changing over time) using the rotation of rods connectedto a disk and ball bearing assembly. A simplified version of this component is shown in Figure 1.First, we will examine how it works mechanically, then how it functions as a computer.Figure 1: Disk B rotates faster near the edge of disk A than near the center.
There are two disks: the large bottom disk A, which is a turntable, and disk B, in contact with,and perpendicular to, disk A. Disk A rotates at a constant speed, and disk B is connected to an inputlinkage that can be slid left and right (closer to and farther from the center of A), changing the pointwhere B contacts A. As A rotates, B rotates in the perpendicular direction. However, while the Simplifications include using two connected disks rather than a disk and ball bearing assembly,as well as omitting various supporting structures to allow for easier visualization.6peed of A is constant, the speed of B depends on exactly where B contacts A: disk B will rotate fasterwhen near the edge of A, slower near the center of A, and not at all when it is at the exact center of A.So how, and what, does this mechanism compute? As the name implies, this componentintegrates a function over time. The value of the function determines the position of B, whichmoves to the right (farther from the center) as the value increases, and left (close to the center) asthat value decreases.As an example, suppose we want to integrate part of the sine function shown in Figure 2. Thismeans we want to compute the area under the black curve (shown in gray). The bottom disk Aserves as the independent variable. Because the function we are integrating starts at zero, the positionof B would begin at the center of A (where, again, even when A is moving, B will not move inresponse). As our function begins to increase, the position of B will move to the right, further fromthe center. As it does so, B will start rotating, and rotate faster as it moves closer to the edge of A. Atthe apex of the function, B will be closest to the edge of A; after that, B will move back toward thecenter of A. By tracking the total number of rotations of B, we compute the area under the curve.Figure 2:
The sine function from zero to 𝜋 . In summary, the rotation of disk A represents an independent variable, such as time. Once the Strictly speaking, the independent variable could be something other than time, but time is themost common and easiest to explain, so that is what we will use.7omputation begins, A rotates at a constant speed, not influenced by other variables. The positionof B, relative to the center of A, serves as the input, or dependent variable. This position representsthe value of the function to be integrated. For different functions, this position will move back andforth in different ways: in the case of the sine function, it will decelerate as it moves toward thecenter, then accelerate away from it after reaching the maximum of the function. Again, the speedthat disk B rotates is determined by where it is relative to the center of A; thus, the running totalnumber of rotations of the disk B serves as the output: it represents the definite integral—the areaunder the curve—of the input up to that point.Many other devices were used to compute various mathematical functions in similar ways:displacement and rotation were used to represent quantities, which were then mechanically engagedin clever ways to deliver the requisite output. The particular example of the integrator is merely onecomponent, which could be connected to other components to compute more complex functions.The output of the integrator, for example, might serve as the input to another component (and evenmore integrators in the case of higher-order differential equations). Similarly, the input could comefrom the output of a separate component (in fact, the next section will demonstrate an examplewhere a number of elements are connected together in just such a way). Besides the integrator, therewere mechanical adders, multipliers, function generators, and many others (presented in, forexample, (Soroka [1954]), (Truitt and Rogers [1960]), and (Ashley [1963])).
Rather than using moving mechanical parts, electronic analog computers largely consisted ofconfigurable electronic circuitry. Thus, in these machines, quantities are represented by variouselectrical properties (such as voltage or resistance) rather than mechanical properties. Once again, wewill look at a simplified example in order to give us a feel for how these machines worked. First, let us consider a simple physical problem that we can set up an analog computer to solve. InFigure 3, we have a mass M connected to a wall via a spring. The problem we would like to solve is: Note that the disk can represent both positive and negative values of a function: in thisexample, if the disk were to move to the left of the center of A, then it would subtract the totalnumber of rotations, just as we would expect when integrating negative values of a function. This example is adapted from a textbook on analog computers (Peterson [1967], p. 29).8ow does the mass move as a force is applied to its side? In other words, what is its left-rightdisplacement (the variable 𝑥 ) when a right-moving force (the variable 𝑦 ) is applied to the left side ofM and released?Figure 3: A mechanical system consisting of a mass M connected to a wall via a spring (with spring-constant K), friction (B), with a force applied to its left side (y).
This problem is not trivial: we must take into account the friction between the mass and theground (given by B), and the ‘springiness’ of the spring (given by K). Still, we can use some basicphysics to characterize this system mathematically. Using the values M = 1 kg, B = 3 nt/m/sec, K = 16nt/m, and 𝑦 = -80 nt, the differential equation (and initial conditions) for this system is given by thefollowing: 𝑦 = 𝑥 (cid:48)(cid:48) + 𝑥 (cid:48) + 𝑥 ; 𝑦 = −
80 (2.1) 𝑥 (cid:48) ( ) = − . 𝑥 ( ) = − 𝑥 (cid:48)(cid:48) = 𝑥 (cid:48) + 𝑥 − 𝑦 (2.2)Using this equation and the initial conditions as a guide, we connect four types of electroniccomponents together, as shown in Figure 4: an adder (triangle with (cid:205) ), two integrators (triangles9ith ∫ ), an inverter (triangle with − ), and two potentiometers (circles with × ), which act asmultipliers. Figure 4: Electronic analog computer schematic.
When the voltages—which are determined by the values of the equations—are ‘run’ on thiscomputer, the output of the system is given by 𝑥 . The output starts at a value of two volts (the initialvalue given in the equations), then changes over time as a function of itself (note that the output 𝑥 isfed back in as an input to the computer in Figure 4) and the other terms in the equation. After arapid change and a few oscillations, the output reaches a steady state, as shown in Figure 5. This iswhat happens with the physical system depicted in Figure 3: the mass rapidly goes back to—andthen overshoots—its equilibrium point, oscillates a bit around that point, and finally settles.This particular type of analog computer is known as an electronic differential analyzer (EDA).EDAs were by far the most common type of electronic analog computer, simply because differentialequations are so common in science and engineering (most people who are familiar with analogcomputers generally have in mind an EDA). We need not attend to the details of the individual In real electronic analog computers, the values of the variables of interest would have to bescaled so as to stay within the electrical limits of the analog computer. So, for example, if our systemrequired a value of 700, we might scale everything by .01 so that the computer would use seven volts,rather than 700 volts. In such a case, the graphical output would look the same, but the axes wouldneed to be linearly scaled to make sure the values are correct. This is identical to how slide rules canonly be used within a certain range, so problems have to be scaled up (or down) in order to fit in thatrange, then scaled back down (or up) once the calculations are finished.10igure 5:
Solution of the system given in Equation 2.1. components of the EDA, but we can see how the overall structure of this computer solves the givenequations. For example, consider the adder on the top left (indicated by the triangle with thesummation sign), which sums its three inputs. The first (top) input to that component is simply thevalue -80, which here is represented by -80 volts. Tracing the path of the other two inputs, we seethat the bottom one is the output from the first integrator (which is 𝑥 (cid:48) ), multiplied by -3. Themiddle input is the result of the output from the second integrator (which is 𝑥 ), multiplied by 16.This is exactly what Equation 2.1 specifies.The design and physical implementation of the electronic analog computer looks very differentthan the mechanical one illustrated earlier. The important point to note, however, is that the basicprinciple is the same. In each case, we have a physical quantity representing a variable: rotation anddisplacement in the mechanical case, and voltage in the electronic case. Both computers are designedso that they manipulate the physical quantities that represent values in ways that correspond to therequisite mathematical operation.Insofar as the relevant physical quantities are continuous, both the mechanical and the electronicanalog computers illustrated here fit perfectly well with the received view of analog computationmentioned at the beginning of this essay: analog computation is simply computation usingcontinuous elements. However, analog computers also used discontinuous elements, an importantbut little-known fact that causes serious trouble for the received view. So let us look at someexamples of these discontinuous analog computers.11 .3 Discontinuous analog elements While many analog computer elements are continuous, including the EDA mentioned above, notall are. One textbook introduces this point quite nicely:Ninety-nine and forty-four one-hundredths percent of the time, when an engineerspeaks of an analog computer he is referring to an electronic differential analyzer (EDA), but the EDA is just one type of analog computer, one specific application ofthe general principal of computation by analogy. So let’s see first of all what is meant by analogy and how we use analogs in computation—in the general sense, not just in theEDA. (Peterson [1967], p. 1, emphasis original)Let us follow this lead and look at that 0.56 percent.Many phenomena that one might want to study using analog computers include discontinuitiesof various sorts; as such, analog computers implement many different kinds of discontinuities. Forexample, some physical systems that involve spur gears or other mechanical parts have a certainamount of slack that cannot be eliminated, because the components cannot be be in perfect physicalcontact (if they were, they would be unable to move). In the case of gears, this means that one gearmight begin moving for a very short time before it contacts another, at which point the second gearwill move. Thus, the movement of the second gear is not a continuous function of the movement ofthe first. In fact, if several such gears are connected, the slack can become an important feature of thesystem to be studied. As such, it was necessary to include this kind of discontinuity in analogcomputers to study systems with such discontinuities.More generally, we may want to model any number of discontinuous functions, for any numberof reasons. Figure 6 illustrates a few of these as they are implemented in an electronic analogcomputer, adapted from (Cadman and Smith [1969], p. 31). Note that the behavior of the second For the moment, what is meant by ‘discontinuous’ is that the function in question is notsmooth, which in turn means that it has a number of points at which the function is notdifferentiable. Although this accords with a commonsense understanding of discontinuous, itwould still count as continuous in the strict mathematical sense, in which the function has gaps orjumps. Later we will see examples of functions that are discontinuous even in this strict sense.12ear just mentioned is captured by using the ‘zero limiting’ circuit: as a function of the first gear’smovement, the second gear is zero until a single point, at which it abruptly (not smoothly) begins toincrease.Figure 6:
Four different circuits implementing discontinuities (left column) and their respective graphs(right column).
A more complex example is also one of the more surprising, particularly to those who endorse thereceived view of the analog, and who are only familiar with the EDA as the paradigm example of13nalog computation. Recall the example of the spring-mass system and the accompanying EDAfrom section 2.2. We began with a physical problem, which we were then able to preciselycharacterize in mathematical terms. This allowed us to create a circuit based solely on thatmathematical characterization, which in turn allows us to compute the solution via an EDA. In thiscase, this was because we could analyze the problem using known physical principles: characterizingspring-mass systems in terms of differential equations is a well-known technique.Some problems, however, do not admit of this kind of mathematical characterization. Forexample, we may know what a particular function looks like, although we do not know how totranslate that into a set of equations that we can then use to construct an analog computer. Manyproblems are unlike the spring-mass system in that regard: there may be no first principles fromwhich one can derive a set of equations that describe the system. Although this might seem to renderanalog computers useless for such problems, they have components to handle cases exactly like these:Such behavior presents almost insurmountable obstacles to purely mathematicalinvestigation, but poses no particular difficulty to analog-computer investigation.Again, we are not solving equations, we are modeling systems. Thus if we can describethe input-output relationship. . . , all we need to do is provide an element on thecomputer which has the same relationship between its input and output voltages. Suchelements are known as arbitrary function generators. (Peterson [1967], p. 109, emphasisoriginal)Using an arbitrary function generator, an analog computer could be set up to construct a discrete,piecewise-linear approximation to any function, even one without a known mathematicalcharacterization. The piecewise-linear approximation generated by this component consists of aseries of straight line segments, with discontinuities where those segments meet. Figure 7 shows anexample of a continuous function plotted with such a piecewise approximation. Depending on theapplication and the particular function generator used, better approximations could be achieved byvarying the number of points and the distances between the points.If—as the received view would have it—analog computers essentially use only continuouselements, one would expect that an analog computer would have to use a continuous function toapproximate a discontinuous problem of interest. But note that the exact opposite is happening in14igure 7:
A continuous function (grey), approximated by a piecewise-linear function (black). the example just given. For some applications in which one wanted to study a continuous functionwhose mathematical characterization is unknown, the analog computer could use a discontinuousfunction to model that continuous function.In an earlier footnote, I mentioned that ‘discontinuous’ in the examples just given simply meansthat the functions in question are not smooth. Although analog computer users and engineersreferred to these function as discontinuous, mathematically speaking these kinds of function are stillcontinuous. The rough idea is that if we were to draw these functions on a piece of paper, we wouldnot need to lift our pencil off of the surface, even if there are sharp (rather than smooth) changes indirection. Mathematically discontinuous functions require gaps or jumps where we would need tolift our pencil off of the surface to draw them. As it turns out, analog computers implemented thesetypes of functions, too.An excellent example is a particular electromechanical component. Like purely electronic analogcomputers, electromechanical analog computers use electrical properties, such as voltage, torepresent variables of interest. However, in the electromechanical case, these variables aremanipulated by mechanical means rather than purely electronic means. The example here is astep-function generator. Step-functions are constant for a specified interval, then ‘jump’ to a More technically, these functions are not everywhere-differentiable.15ifferent constant for a different interval. Figure 8 shows a schematic of a simplified version of such acomponent, plus the step function it generates (adapted from (Korn and Korn [1952], p. 254)). Thecomponent has a rotating element that makes contact with separate wires, which in turn connect toa variable resistor at different points. The farther from the input, the lower the resistance, and thusthe greater the output. As this element rotates (counterclockwise in this example), it momentarilybreaks contact with one wire, then makes contact with the next wire, resulting in a discontinuousjump from one voltage to another. By adjusting the number of wires, where they contact theresistor, and the speed of the rotating element, one could implement different step functions withdifferent characteristics.Figure 8:
Switch-type step function generator (left) and its resulting step function (right).
Once again, we have a counterexample to the received view. These elements (and ones like them)were not uncommon in analog computers, yet they do not use continuously-varying elements.Instead, the variation is as discontinuous as could be.At this point, one might wonder why analog computer engineers would go to the trouble ofmaking these discontinuous components, given the well-known fact that these types of functionscan be approximated by continuous functions. For example, a simple step function that goes backand forth between two values can be approximated by a sum of sine and cosine functions. Whilethis may be good enough for some purposes, it is not good enough for others. In particular, thedifference between the approximation and the actual value of the step function one wants can be This particular example is of a non-shorting switch, or ‘break-before-make’ contact. Thisensures that the voltage truly jumps from one value to another. Other switches had overlappingcontacts, called shorting switches, or ‘make-before-break’ contact, which could be used if the truediscontinuity was not wanted. 16articularly bad at transition points for certain kinds of approximations, as illustrated in Figure 9.So, rather than trying to use continuous functions to approximate discontinuous functions, analogFigure 9: Step function (black) and a Fourier series continuous approximation (grey). computers were able to directly implement discontinuous functions. In a discussion of the use ofrelays and switches to incorporate a discontinuous voltage change (like the electromechanicalcomponent just illustrated), rather than using a continuous approximation (the diode limitedamplifier circuit, which would result in an approximation similar to that shown in Figure 9), theauthor of an analog computing monograph explains:For the problem being studied, it is not immediately obvious why the relay is needed.The voltage from the diode limited amplifier circuit can be made to closelyapproximate a delayed step function.. . . There are two reasons why this is not practical.The slope of the ‘step’ function out of this circuit is not exactly zero after the originaldiscontinuity. In the process of adding two of these step functions, a small error thatincreases with [time] would be applied to the integrator and cause an unwanted ‘drift’in the output of the integrator. (Ashley [1963], p. 201)The idea is simply that small approximation errors in different components can combine to formincreasingly large errors. Better to use the exact values of a discontinuous function to model a This is known as the Gibbs phenomenon. 17iscontinuous function, rather than a continuous approximation.These are just a few of the examples that demonstrate how analog computation was not onlyabout continuity. Discontinuous components are important for approximating continuousfunctions, as well as their use in directly implementing discontinuous functions of interest.Although this may seem surprising today, it was known during the heyday of analog computation.For example:Any mechanism which involves continuous variables is nowadays in danger of beingcalled “analog,” of course to distinguish it from “digital”. . . To make matters still moreconfounded, the common usage for computing structures, whereby only continuousmethods are called analog, is wrong, since it is clear that discrete or digital machinesmay also embody and constitute analogs of prototype phenomena. (Philbrick [1961],p.7)It is clear that the received view—that analog computation is essentially about continuity—issimply false. However, we still need to make clear what analog computation is essentially about. Inthe next section, I will discuss different accounts of the analog, and argue that only one particularaccount of analog representation makes sense of analog computation.
Nearly all accounts of the analog focus on analog representation, rather than analog computationmore specifically. Moreover, nearly all accounts of analog representation are made in contrast withdigital representation. This is unfortunate. The similarities and differences between analog anddigital representation—and analog and digital computation—are lost when ‘analog’ and ‘digital’ aretaken to be opposites, and jointly exhaustive of representational types. In other words, many haveassumed that once we know the right way to characterize analog representation, then digitalrepresentation comes for free (or vice versa): digital is whatever is not analog. While I take thisassumption to be false, I cannot engage in a full defense here, so I will limit the discussion to therelevant aspects of the positive accounts of what counts as analog. I will first review several accountsof analog representation, and then argue that only the so-called Lewis-Maley account is able to makesense of analog computation. What follows is captured quite well by Peacocke: ‘Analog18epresentation is representation of magnitudes, by magnitudes. Analog computation is theoperation on representing magnitudes to generate further representing magnitudes’ (Peacocke[2019], p.52), although how that slogan is elaborated here differs from Peacocke’s own elaboration.Again, I will not provide a detailed account of digital representation (much less digitalcomputation), although I will offer a few remarks to make clear the contrasts that other authorshave made, and outline the case for why ‘digital’ is neither synonymous with ‘discrete’, nor with‘non-analog’.Philosophical accounts of analog representation generally fall into one of two camps, which Beck([2018]) calls the ‘continuous’ and the ‘mirroring’ conceptions. I will use ‘covariation’ instead of‘mirroring’ (for reasons explained below), but the idea is roughly the same. In what follows, we willlook at each type of account, plus some specific instances of each.
According to the continuous conception (the received view), the essential feature of analogrepresentation is that it is continuous in nature. What exactly ‘continuous’ means is not always madeprecise, but the basic idea is that analog representations vary smoothly, rather than in discrete steps.The first account of this kind in the philosophical literature is due to Goodman ([1968]). On thisview, analog representations are continuous, or dense, while digital representations aredifferentiated, or discrete. Haugeland ([1981]) draws from this account, and distinguishes betweenanalog and digital devices in terms of the reliability of the procedures that read and writerepresentation tokens. In short, digital devices read and write tokens that are completelydeterminate, with read/write procedures that are perfectly reliable. Analog devices, on the otherhand, read and write tokens that are not perfectly determinate, with read/write procedures that are There are still other types, such as (Frigerio et al. [2014]), that do not fit well into either camp;but these accounts are not relevant to the concerns noted here. Goodman speaks of representational schemes, rather than individual representations. The ideais that a single representation in isolation is neither continuous nor discrete, but representations canvary continuously or discretely in accordance with a scheme. While this is an importantconsideration, for brevity of exposition, I will simply refer to representations, rather thanrepresentational schemes. 19pproximate at best.Katz ([2016]), Schonbein ([2014]), and Papayannopoulos ([2020]) all offer elaborations anddefenses of these views. Katz clarifies a potential flaw in Haugeland’s view, specifying that whether agiven representation system is analog or digital is not a matter of objective facts about the system,but how the system is used (or supposed to be used). Katz makes clear that what counts as a user canbe very general, and need not be a human or other agent located outside of the system in question.The point is simply that we need to look carefully at the context in which representations are read,written, and otherwise used. Schonbein argues for the received view on historical grounds. On thisview, there is an entrenched engineering literature that treats analog as continuous, and digital asdiscrete. However, Schonbein also allows that there may be different varieties of analogrepresentation, such that different accounts might be better suited for different purposes.Papayannopoulos argues that a modified version of Goodman’s account is best suited forunderstanding analog computation, although he specifically discounts discrete components asbeing analog (although he does not use actual examples, he contends that two continuously-varyingwheels connected together would be analog, but wheels that move in discrete steps, constrained bythe teeth of gears, would be digital).Given the discontinuous examples from section 2.3, this family of accounts of the analog does notproperly characterize analog computation. There are, of course, differences among this family ofaccounts, and some of them disagree about hypothetical cases. However, what unites them is thethought that continuity is essential for any account of analog representation (or devices that useanalog representations, such as computers). Given that each implicitly accepts the idea that analogand digital are both opposites and jointly exhaustive, each account takes discontinuous cases like theone presented above to be non-analog, and thus digital.In summary, according to the continuous account, discontinuous analog computers are notreally analog. That is a shortcoming: although not as widely known as the continuous elements ofanalog computers, discontinuous elements were not uncommon. Fortunately, another account ofanalog representation is available, which does properly characterize analog computation as analog.20 .2 Analog as covariation
The second family of accounts of analog representation rejects the idea that this kind ofrepresentation is necessarily continuous. Instead, this family takes the essential feature to be somekind of mirroring, or covariation, between the representation and what is represented. What exactly‘covariation’ means is not always made precise (and differs somewhat between different accounts),but the basic idea is that analog representations are structurally isomorphic (or, in some cases,homomorphic) to what they represent.The first account of this kind in the philosophical literature is due to Lewis ([1971]). Lewissuggests—contra Goodman—that what is important about the voltage in an analog computer is notthat it is continuous, but that it covaries with what it represents. Specifically, as the quantity we wantto represent increases, so does (for example) the voltage, which is what is doing the representing.The value 34, for example, is represented by 34 volts; 34.8 is represented by 34.8 volts. Furthermore,this covariation would occur even if we had discrete, rather than continuous, variations of voltages:whether the voltage could only increase in increments of 1, volt, 0.1 volts, or continuously, thecovariation would still be present. According to Lewis, what does the representing in an analogrepresentation is some primitive or ‘almost primitive’ physical magnitude, such as voltage.Other examples of this kind of account include (Blachowicz [1997]), (Kulvicki [2015]), and(Peacocke [2019]). Blachowicz argues for what he calls the model interpretation: ‘the function ofanalog representation is to map or model what it represents’ (Blachowicz [1997], p. 83). In line withthe covariation account, Blachowicz argues that continuity is inessential to analog representation.However, Blachowicz is primarily concerned with examples of analog perception and thought,going beyond the kind of analog representation characterized here. Kulvicki agrees that importantexamples of analog representation need not be continuous, and that the preservation of structurebetween what is represented and what is doing the representing is the essential feature. Kulvicki isalso focused on more complex examples, including the ways in which these examples supportparticular kinds of psychological tasks or capacities. Peacocke’s view aligns almost perfectly with theLewis-Maley view to be developed below, although Peacocke’s overall purpose is to develop anaccount of the metaphysics that explains our perceptual capacities (Peacocke’s view of digitalrepresentation, however is at odds with the Lewis-Maley view).21aley ([2011]) builds on Lewis’s account by arguing that the notion of primitive or almostprimitive physical magnitude is too restrictive, and by offering a more precise characterization of thekind of covariation involved in analog representation. While in one sense Maley’s characterization isthe narrowest, it is also the most precise. However, we can improve upon Maley’s original view,resulting in the following characterization (with the formalism to be explained below):A representation 𝑅 of a quantity 𝑄 is analog (with resolution 𝑟 ) iff:1. there is some property 𝑃 (the representational property) of 𝑅 such that the physical quantityor amount of 𝑃 specifies 𝑄 ; and2. the quantity or amount of 𝑃 is a monotonic function 𝑓 of 𝑄 , and that function is ahomomorphism from 𝑄 to 𝑃 . Furthermore, let 𝑃 and 𝑃 be values of 𝑃 that representquantities 𝑄 and 𝑄 , respectively. If | 𝑃 − 𝑃 | ≥ 𝑟 , then (without loss of generality)stipulate that 𝑃 < 𝑃 (that is, let 𝑃 be the smaller of the two). In the case where 𝑓 ismonotonically increasing (non-decreasing), then 𝑄 < 𝑄 ; if 𝑓 is monotonically decreasing(non-increasing), then 𝑄 > 𝑄 . However, 𝑄 ≤ 𝑄 only implies 𝑃 ≤ 𝑃 for monotonicallyincreasing 𝑓 , or 𝑃 ≥ 𝑃 for monotonically decreasing 𝑓 .A few points about this characterization should be highlighted. First, this is not an account ortheory of representation in general; rather, this is a characterization of what makes somethingalready taken to be a representation a specifically analog representation. Second, and relatedly,‘specifies’ in the first clause refers to what it is about the representation that does the representing,and not about how the physical properties of the representation are caused to have the value thatthey do. Finally, specifying that the function is a monotonic homomorphism allows for maximalgenerality with respect to which functions will count as preserving an analog relationship. Ingeneral, this means that an increase in the property doing the representing necessarily means thatwhat is represented has increased, but an increase in what is to be represented does not necessarilymean that the property doing the representing increases (although it may).Consider a mercury thermometer. This counts as analog because an increase (or decrease) in aparticular property of the thermometer—the height of the mercury—represents an increase (ordecrease) in the ambient temperature. Another example is a vinyl record: these are analog because an22ncrease in the ‘frequency’ of the ridges within a groove represents an increase in the frequency ofthe sound represented (and similarly for the height of the ridges and the amplitude of the sound).Still another is an hourglass: the amount of substance (sand or liquid) in the bottom of the glassincreases as the amount of time has elapsed since the glass was turned over.Importantly, this characterization also counts these examples as analog if they happen to onlyvary in discrete steps. Consider the hourglass, for example. Many hourglasses use discrete particles,such as sand, whereas others use liquids (which are presumably continuous). On the accountoffered here, it is the fact that the amount of substance is doing the representing that makes thehourglass analog, rather than whether that substance is continuous or discrete. Similarly, this characterization counts analog clocks correctly, whether they vary continuously ordiscretely. Consider the second hand of an analog clock. On the continuous conception, an analogclock with a hand that sweeps continuously would be analog, whereas one that ticks would not. Buton the account here, it is the fact that an increase in the angle of the second hand represents anincrease in time that matters, regardless of whether that increase happens continuously or discretely.This also illustrates the relevance of the homomorphism constraint (where an isomorphism wouldbe too strong). Assume that time is continuous, and consider its representation by adiscretely-ticking second hand. If the second hand has moved, we know that time has passed. On theother hand, small amounts of time—smaller than the resolution term 𝑟 , which in this case would beone second—can pass without the second hand moving.The resolution term 𝑟 also captures the fact that in a discrete analog representation, there mayalso be small amounts of ‘jitter’ or noise. When the second hand ticks to a new location, it mightoscillate very slightly. However, we do not want these oscillations to count as differences in what isbeing represented. Thus, if the difference in magnitude between two positions is not greater than 𝑟 ,then those magnitudes do not represent different quantities.Finally, some analog representations may represent in a kind of inverse way. Perhaps instead of a I take it to be a virtue of this account, relative to continuity-based accounts, that we can setaside issues about whether anything is really continuous (properties like voltage, liquids, spacetimeitself) in order to determine whether anything is really analog. Such questions simply do not matterfor the present account. 23hermometer with a substance whose height increases with temperature, one might have one wherethe height decreases as temperature increases. Also, some analog representations might be seen asincreasing in one way, but decreasing in another. As the second hand of a clock increases in anglefrom 30 to 36 degrees (with respect to the usual ‘12’ at the top), the reflex angle is decreasing from 330to 324 degrees (a necessary consequence of angular rotation). Roughly speaking, the relations stillhold between what is represented and what is doing the represented, except in reverse. However, theinclusion of both monotonically increasing and decreasing functions in the second clause capturessuch cases.For present purposes, I will adopt the improved Lewis-Maley account, simply because it is bestsuited for characterizing the kind of analog representation relevant to understanding analogcomputation. As mentioned above, this account is relatively limited. However, it is worthmentioning that this account may well be generalizable to coincide with other mirroring accounts,as well as what have been called structural representations (Ramsey [2007]), (Shea [2014]),(Nirshberg and Shapiro [2020]). The Lewis-Maley account as presented covers what we can callone-dimensional representations: one property of a representation represents some quantity (forexample, the height of liquid in a thermometer represents the temperature at a single point in space),and it does so by monotonically covarying with that quantity (that is, as the temperature literallyincreases/decreases over time, the height of the liquid literally increases/decreases over time). Infuture work, however, this account might be extendable to multiple dimensions to cover analog orstructural representations mentioned in other accounts, such as photographs and scale models. Forthe present essay, however, we only need a precise account of the one-dimensional representationsused by analog computers.
Different accounts of analog representation have been based on particular examples, oftenhypothetical. As mentioned above, where analog computation has been mentioned in thisliterature, it is usually taken as nothing more than computation using continuous elements, andthus used to support the view that to be analog is to be continuous. This is simply incorrect, as ourhistorical counterexamples have shown. Nevertheless, I endorse the underlying assumption that (all24lse being equal) we ought to prefer an account of the analog that applies to b analog representationand analog computation.We saw in section 2.3 that analog computation is not simply about continuity. Discontinuousanalog components were not uncommon in analog computers, and worked alongside continuouselements. This rules out accounts that take analog representation to involve continuity essentially.At the same time, this rules in covariation-based accounts, particularly the Lewis-Maley version. Letus see why.Consider again the example of the electromechanical step-function generator depicted in Figure8. Like purely electronic analog computer elements, this device uses voltage to represent a quantityof interest; unlike some other elements, it can only represent a finite set of values that varydiscontinuously. The way it represents a quantity is via covariation between that quantity and thevoltage: four volts represents a quantity of four; two volts represents a quantity of two. This isprecisely the Lewis-Maley account of analog representation. At the same time, the Lewis-Maleyaccount also covers continuous elements, such as the ones described in section 2.1 and section 2.2.The monotonic covariation required by that account holds whether the representation variescontinuously or discretely.Now, in order to avoid confusion, it is important to note what the Lewis-Maley account ofanalog representation does and does not require of analog representation. The idea of monotoniccovariation might lead one to think that the example from Figure 8 rules out this account; after all,the step function shown is absolutely not monotonic! However, this observation is beside the point.What is monotonic is the relationship between what is doing the representing (the voltage) andwhat it represents (the values of the function). Consider the first line segment (at the top left) of thisfunction, let us say it is at 9.6 volts. That represents the number 9.6. Next, it steps down to 7.5 volts.The voltage, which is what does the representing, has literally decreased. But so has what is beingrepresented: 7.5 volts represents the number 7.5, and 7.5 is less than 9.6. Similarly for when it stepsdown again to 2.1 volts: both the voltage and what the voltage represents have literally decreased. In Recall once again that this is an exegetical simplification: depending on the particulars of theproblem and the machine, the particular values represented by the voltages within a machine mayhave been uniformly scaled. 25he next jump, up to 5.9 volts, both the voltage and the value being represented increase.Let us further clarify this account of analog representation by contrasting it with a case of digitalrepresentation in a digital computer. Like electronic and electromechanical analog computers,digital computers also use voltage to represent numbers. How digital computers do so is quitedifferent. In short, analog computers represent numbers by representing their quantity, while digitalcomputers represent numbers by representing their names. Specifically, digital computers use abase-2, or binary, representation, which means that a number such as nine is represented as 1001.The way to interpret this sequence is as follows: starting from the rightmost place, there is a 1 in theones place, a 0 in the twos place, a 0 in the fours place, and a 1 in the eights place; thus, add one toeight, which is nine. In a digital computer, the 0s are typically represented in a circuit element byzero volts, while the 1s are represented by 5 volts. So to represent the number nine, a sequence of atleast four circuit elements would be required. From left to right, the first would be at five volts, thesecond at zero volts, the third at zero volts, and the last at five volts.Now imagine increasing the value that we want to represent from nine to ten. In binary, this is1010. This would result in the following change: the first circuit element would remain at five volts,the second element would remain at zero volts, the third would change from zero to five, and the lastwould change from five to zero. Note the difference between the changes in voltage here and thechanges in voltage in the electronic analog computer: as the value being represented in the digitalcomputer increases, some voltages increase, some decrease, and some stay the same. The way theychange reflects a change in the elements of the digital representation of the number: the 1s and 0shave to change in a systematic way to represent the digits of the larger number. However, in theanalog computer, the voltage does not represent the elements of a representation (the name, or partsof the name) of the number; rather, the voltage represents the magnitude of the number.To put this last point in still a different way, observe that the sequence of symbols 1010 is notlarger than the sequence 1001, even though what the sequence 1010 represents is larger than what thesequence 1001 represents. This is true even when we look at how the individual 1s and 0s arephysically implemented, in this case as voltage levels. But for the analog case, what represents the This is simply the standard way that we represent numbers, although we typically use base-10,or decimal, rather than binary. This is described in more detail in (Maley [2011])26umber ten (ten volts, possibly scaled by some constant 𝑐 ) is physically larger than what representsnine (nine volts, again possibly scaled by the same 𝑐 ). Both the physical representation and what isrepresented increase. This is true for other cases as well, such as the height of the liquid in thethermometer, or the angle of the hand in the watch.Let us now begin putting everything together. What makes analog computers analog is that theyuse analog representations, understood according to the Lewis-Maley account just described (andwhat makes analog computers computers will be discussed next). This characterization covers thewell-known continuous examples of analog computation such as the EDA, as well as thenot-so-well-known discontinuous examples discussed above. Furthermore, the Lewis-Maleyaccount is not an ad hoc characterization of the analog, custom-built for the purpose of makingsense of analog computation. Rather, it is a principled account that also makes sense of examples ofanalog representation outside of computation (as argued in (Maley [2011])).Importantly, requiring the involvement of representations excludes many non-computationalsystems from the account (and rightly so). For example, virtually any electrical system with a resistorcan be interpreted as multiplying voltages. But because voltages in general are not representations ofanything, it is not the case that virtually every electrical system is performing analog computations.The point applies to the mechanical and electromechanical components. If the above is the right way to understand what is analog about analog computation, what is theright way to understand what is computational about analog computation? The short answer is thatanalog computation is the mechanistic manipulation of analog representations. We have seen whatanalog representation is about, so let us look at what mechanistic manipulation is about.First, let us look at what is required of an account of computation simpliciter. A completeaccount of computation should (among other things) make sense of what makes digitalcomputation and analog computation similar enough such that both are genuine species ofcomputation, yet different enough to count as separate species. Assuming the orthodox view thatcomputation requires representation, the answer to what makes something analog computation iswhat was just mentioned: analog computation is computation that uses analog representation.27imilarly, digital computation is computation that uses digital representation. When we takeseriously the fact that analog representation can be discrete, then we have an account that is bothprincipled and does justice to extant examples of computation, such as the ones we examined above.What remains is the question of what kinds of manipulations count. The essential idea—whichwill have to be further developed in future work—is that computation is the mechanisticmanipulation of representations, where ‘mechanistic’ is understood in the sense developed byPiccinini ([2015]). Unlike Piccinini’s account, however, the positive account I have in mind requiresthe manipulation of representations. Importantly, the mechanisms doing the manipulating must besensitive only to the physical properties of the representations that are responsible for representing.Given the Lewis-Maley account of analog representation, we can say that a mechanism mustmanipulate analog representations qua analog representations in order for them to count as analogcomputational mechanisms. Finally, the manipulations of those representations are such that theresult is itself an analog representation.For example, in the electronic and electromechanical analog computers discussed above, it is thevoltage in the circuits elements does the representing. To count as computational, the mechanism(or mechanisms) that manipulate the representations must do so by manipulating their voltages.This is in contrast to mechanisms that manipulate some other property, such as the temperature, ofthe circuit elements (as a cooling fan might do). Or in the case of mechanical analog computers, themechanism must manipulate the position, speed, or angle (as the case may be) of the relevantcomponent. Changing the temperature of disk A in Figure 1 does not count as a manipulationrelevant to computation, because temperature is not the property that does the representing. Butchanging the displacement and rotation does count, because those are the properties that do therepresenting.Requiring the involvement of a mechanism separates analog computation from other systemsthat use analog representations, but where the representations are manipulated by other means. For example, the liquid height of a mercury thermometer is an analog representation of Of course, in some other computers, temperature may well be the property that does therepresenting. The discussion here is focused on devices that compute, rather than the human activity ofcomputation. 28emperature. But the thermometer does not have a mechanism that manipulates the level of liquid;rather, the height changes via a natural process of liquid expansion and contraction due totemperature change. Thus, a mercury thermometer does not count as an analog computer.Now, before concluding, it is worth pointing out that there will be borderline cases where it isunclear whether a computation is being performed or not. For example, is the playing of a record ona standard record player an instance of analog computation? The vinyl record is clearly an analogrepresentation. We also have a mechanism: the turntable spins at a particular speed, and the needle‘reads’ the properties of the record that are relevant to its being an analog representation. Similarly,placing a weight on a spring scale may twist a dial, indicating its mass. If these count ascomputations, then they are very simple computations, and it is not clear how simple a mechanisticmanipulation needs to be in order to not count as a computation. However, this question is notunique to the present account: it is a question for every account of computation, which we will nottry to answer here. Before concluding, in this section I will respond to what seem to me obvious questions andobjections.
First, some accounts of digital representation would take issue with the idea that components likethe discontinuous step-function generator mentioned in section 2.3 are analog. For example,Haugeland counts this kind of device as digital, simply because it is discrete: in responding to anearly identical example presented by Lewis ([1971]), Haugeland states ‘I think it’s clearly digital -just as digital as a stack of silver dollars, even when the croupier “counts” them by height’(Haugeland [1981], 218). So why not say that, when an otherwise-analog (continuous) computeruses a component like this, it is simply a hybrid analog-digital computer? For example, consider a digital kitchen timer that can only count down from an input setting.It is not clear whether that counts as a computation or not, and if it does, it is also a very simplecomputation. 29his kind of response does not do justice to actual hybrid analog-digital computers. To see whyrequires another brief digression into digital computation. For simplicity, I will limit the discussionto electronic digital computers.Recall the discussion in the previous section about how digital representation works. Discreteelements are used to represent the digits of numbers, and the digits of numbers represent thenumber in the usual place-based way. However, when we attend to the step-function generator, it isclear that although it is discrete and discontinuous, it is not digital in the way described above.Thus, it is a mistake to call it so, as Haugeland does. A digital device requires digital representation,and a hybrid analog-digital computer has components that convert analog representations to digitalones (or vice versa).Figure 10:
An analog-digital converter. The physical inputs (voltages) are shown on the left. Whatthose voltages represent is shown on the right. For the analog input, seven volts represents the numberseven (although the seven volts may be linearly scaled, per footnote 11); for the digital output, zero voltsrepresents the numeral 0, and five volts represents the numeral 1. These numerals, in turn, represent thenumber seven when the numerals are interpreted as the digits of a binary representation.
In short, what makes hybrid analog-digital computers hybrid is not that they use some discreteelements and some continuous ones. Rather, they are hybrid because they use analog componentsand digital components, where, again, ‘digital’ is understood in the sense discussed above. Forexample, the interface from an analog to a digital component would include a converter that takes asingle voltage level (for example, seven volts) as input and produces as output a series of separatevoltages, which themselves represent the binary representation of seven, or 0111 (as illustrated inFigure 10). The conversion from a digital to an analog representation would do the opposite (Kornand Korn [1964]), (Hyndman [1970]). The step-function generator does not operate via a digitalrepresentation at all, and requires no conversion to operate with other analog components (forexample, the output of the step function generator could be used as an input to some part of the30DA in Figure 4).
Another objection might go something like this. Analog ‘computers’ have very little to do withobvious paradigms of computers, such as digital computers and Turing Machines. After all, theentire branch of mathematics known as the theory of computation is concern almost exclusivelywith Turing Machines and other abstract automata, none of which operate on analogrepresentations. Thus, we should not take very seriously the idea that what these analog machinesdo is a genuine form of computation in the first place.This objection simply ignores the history of computation as it has actually been practiced.Analog computation was the dominant type of computation for several decades before digitalcomputation became efficient and cost-effective enough to replace it. Analog and digital computerswere used to solve similar problems in science and engineering, and although they work quitedifferently, they were seen as two types of the same kind of machine. Numerous textbooks andresearch monographs were published with ‘analog computation’ or ‘analog computers’ in the title.Furthermore, the hybrid computing machines just mentioned were not considered to be partially acomputer (the digital part), and partially something else (the analog part): they were seen as a singlecomputing machine that operated using two different computational paradigms.Perhaps there is some positive argument that only digital computation should really count ascomputation. But that argument will have to explain why so many scientists and engineers werewrong to call certain machines computers in the first place. Without a very strong argument to thecontrary, we should consider analog computers to be exactly what scientists and engineers tookthem to be: genuine computers.
Yet another objection might focus on the Lewis-Maley account of analog representation adoptedhere. One problem with this account is that some examples of representation count as both analogand digital. An example is unary notation, where the number four is represented as a series of fourstrokes, or four 1s: 1111. We have numerals in places, thus it is digital (for further elaboration of what31s meant by ‘digital’, see §4 of (Maley [2011])). But we also have a monotonic covariation between aproperty of the representation (number of strokes) and what is represented. So this seems to be botha digital and a discrete analog representation. Another example seems even worse: digitalrepresentations implemented in contemporary computers. Here, each numeral is represented by avoltage, where the amount of voltage (zero volts or five volts) monotonically increases with what isbeing represented (a 0 or 1). Again, the Lewis-Maley account seems to classify this as both analogand digital. The received view of the analog—equating ‘analog’ with ‘continuous’—does not havethese faults.While it is true that the first example is both analog and digital, this is a degenerate sense of digital.Unary, or base-1 notation, can only represent integers, unlike notation in any larger base. To see why,consider what the digits of a base-10 and a base-2 representation mean.314 = ( × ) + ( × ) + ( × ) = + + = ( × ) + ( × ) + ( × ) + ( × ) = + + + = 𝑏 and digits 𝑑 𝑛 . . . 𝑑 𝑑 𝑑 , the digital representation is: 𝑑 𝑛 ...𝑑 𝑑 𝑑 = ( 𝑑 𝑛 × 𝑏 𝑛 ) + ... + ( 𝑑 × 𝑏 ) + ( 𝑑 × 𝑏 ) + ( 𝑑 × 𝑏 ) Back to the unary notation:1111 = ( × ) + ( × ) + ( × ) + ( × ) = + + + . = ( × ) + ( × ) + ( × − ) = + + . . = ( × ) + ( × ) + ( × − ) + ( × − ) = + + + Concluding thoughts
Let us take stock of what we have done. We have seen different examples of analog computers usingmechanical, electronic, and electromechanical mechanisms. We saw that the received view—thatanalog computation is essentially continuous—fails: some analog computers use discrete,non-continuous elements. We then surveyed some accounts of analog representation, and saw thatthe the Lewis-Maley account is uniquely able to capture what is analog about analog computation.Next, we briefly looked at how this account of analog computation fits in with more generalaccounts of computation. Finally, we looked at how to answer some anticipated questions andaddress some possible objections.The upshot is that we have a principled account of analog computation that does justice toanalog computers implemented in different physical media, using both continuous and discreterepresentations.So after all that trouble, why care about analog computation? First, it has simply been neglected:the philosophical attention paid to computation has been almost exclusively aimed at digitalcomputation. As it turns out, analog computation is both interesting in its own right—as well asimportantly different from digital computation—in ways that have not been appreciated. Anycomplete philosophical treatment of computation simpliciter will have to attend to analogcomputation alongside digital computation.Second, understanding analog computation offers richer opportunities for the kinds ofcomputation that could play a role in the explanation of the mind and brain. Computationalismabout the mind—the view that the mind is literally a computer—has been a major philosophicalview for several decades (Piccinini [2009]). Computationalism about the brain—the view that thebrain is literally a computer—is taken seriously in the sciences of the mind, particularlyneuroscience (Shagrir [2006]). By couching analog computation in mechanistic terms, this accountis applicable to natural (and not just artificial) computational processes like those that may be foundin the mind/brain. After all, it seems that some natural processes are mechanistic ((Machamer et al. [2000]), (Glennan [2002]), (Woodward [2002])). Thus, natural analog computation is astraightforward matter. Some work, such as Maley ([2018]), has already argued that some neuralprocesses may well be analog—but not digital—computations. Furthermore, many psychologists34ave already appealed to the analog nature of certain mental processes (seminal examples includeShepard and Metzler ([1971]) and Kosslyn ([1994])), but without a clear and precise notion of howanalog representation and computation might go together. The account here may help provide justsuch an account. We would be foolish to limit our view of the kind of computation available inthese discussions to digital computation alone, given that there is an entirely separate,well-established second kind of computation that has received so little attention.Finally, this account paves the way for investigations into the possibility of still other types ofcomputation. As sketched above, one natural way to build a general account of computationcouches computation in terms of the mechanistic manipulation of representations. Computationsare then typed by the kinds of representation involved (which in turn constrains the kind ofmanipulations possible for the relevant mechanisms). By rejecting the thesis that ‘analog’ and‘digital’ are jointly-exhaustive opposites, we allow for the possibility of computation that is neitherdigital nor analog. Specifically, if there are representational types that are neither analog nor digital,and we can specify how such representations are mechanistically manipulated, we can come up withprincipled accounts of new kinds of computation. This is, of course, largely speculative, butscientists are actively exploring a menagerie of seemingly-exotic computations, including quantum,optical, molecular, membrane, and even physarum (slime mold) computation. It may well be thatsome of these kinds of computation—if indeed they are species of computation at all—are bestcharacterized as something other than analog or digital computation. We should be in a position tocharacterize computation more generally, and one first step is a full understanding of whatcomputation could be if it is not digital. I hope to have taken that first step.
Funding
This material is based upon work supported by the National Science Foundation under Grant No.1754974. Any opinions, findings, and conclusions or recommendations expressed in this material arethose of the author and do not necessarily reflect the views of the National Science Foundation.35 cknowledgements
I would like to thank the following lovely people for their helpful comments and suggestions: ZedAdams, Cameron Buckner, Zoe Drayson, Daniel Estrada, Gualtiero Piccinini, Daniel Schneider,and especially Sarah Robins; the Philosophy Departments at the University of Kansas, theUniversity of Wisconsin, and Western University; audiences at the 2019 SLAPSA, 2018 PSA, and2018 SSPP conferences; and the MCMP–Western Ontario Workshop on Computation in ScientificTheory and Practice. Finally, the referees for this article provided valuable comments andsuggestions for improvement.
Corey J. MaleyUniversity of Kansas3075 Wescoe HallLawrence, KS [email protected]
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