The main purpose of this post is to summarise D&M's main arguments in favour of discrete representations so that I can refer to these in other posts. I make several comments about the quality of these arguments, but this is in no way meant to be a systematic response to their paper.
D&M spend the rest of the paper presenting seven arguments for why representations must be discrete rather than continuous. They provide seven defences for this position.
1) People discriminate between things in the environment.
“Individual items are selected from their environment by discriminating them from their surround” p. 106
The authors argue that continuous representations are unable to discriminate among states. They use a thermostat as an example. A thermostat has a continuous representation (a temperature-sensitive coil of metal) that acts differently in different temperatures (coiled tightly or loosely). Thermostats also have discrete representations. This is a liquid-filled vial attached to the coil that has electrical contacts at both ends. When the coil is tightly wound, the liquid in the vial reaches one of the contacts and the thermostat changes state (on or off depending on how the thing is set up). When the coil is loose, the liquid in the vial reaches the other electrical contact, and the thermostat switches to the opposite state. D&M argue that the discrete representation is doing all the important work because it discriminates between too hot and too cold. The continuous representation changes state, but it does not discriminate.
“Just because a system uses continuous representations doesn’t mean those representations (or any other continuous representations) are crucial to the functioning of the system.” p. 105
They connect this idea of discrimination with an organism’s ability to refer to individual entities.
2) People have to be able to refer to properties of objects in the world
They use a bunch of evidence from Markman & Gentner’s similarity research (see Markman & Gentner, 2000 for a review) to support this idea. Basically, people are good at listing commonalities and differences between objects. This suggests that people think about concepts in terms of sets of features, rather than as holistic entities. They argue that similarity models that use continuous representation are unable to describe concepts as sets of features, so these models must be inadequate for describing how we represent concepts.
To illustrate their point, D&M use geometric models of similarity (e.g., Shepard, 1962) as examples of continuous conceptual relations. The models represent concepts as points in multi-dimensional space and the similarity between concepts as the distance between points according to some metric. These models capture overall similarity, and do not decompose concepts into commonalities and differences. Thus, D&M conclude that continuous representations are incapable of capturing these properties.
This strikes me as unfair. These spatial models were constructed from pair-wise similarity ratings and were meant to capture relative similarity between concepts. As far as I know, no one using these models claimed that commonalities and differences didn’t exist, or were irrelevant to comparison. One of the main applications of spatial models was to capture animal cognition – for instance, stimulus generalisation in a learning experiment. I think D&M would be hard-pressed to argue that this phenomena would be better described using set-theoretic models (e.g., Tversky, 1977). Also, it is completely possible to model discrete factors in a spatial model. So, if someone built a model representing the features of a concept, this could accommodate both continuous (e.g., hue) and discrete (e.g., has wheels) properties.
D&M conclude that continuous representations cannot refer to properties
3) People combine representations
The authors argue that one way people combine representations is through concept formation. For instance, to acquire the concept of pony, I combine my representations of individual ponies I have encountered. This, according to D&M, is only possible with discrete representations.
“When continuous representations are combined, all that results is another continuous representation where all the original information is lost due to blurring.” p. 107
The authors make a comparison to the blending inheritance hypothesis that Darwin proposed to account for the transmission of traits across generations. Fleeming Jenkin compared this hypothesis to mixing a bunch of colours of paint together in a bucket, which blends all the colours together beyond recognition. D&M then go on to describe Mendel’s revelation that inheritance results from the transmission of discrete packets of information called genes. There is no blending going on. Traits that appear to blend, like flower colour (white + red = pink) are actually instances of incomplete dominance.
“The crucial property of genes that solved the problem of inheritance plaguing the theory of evolution was that genes are discrete. Each gene carries a fixed amount of information for constructing some protein. And, that information has to be discrete if it is to be combined, recovered, and used over and over again to do the job it is supposed to do. Further, the operations that are carried out on genes are discrete operations of the sort computational systems perform. For example, genes are copied and interpreted (to build proteins).” p. 109
I think this passage is interesting because it is contrary to the modern view of genetics. As far as I understand it, the whole idea of a gene as a discrete entity is falling apart. First, genes seem to be fairly distributed, rather than localised. Also, gene expression seems to be incredibly contingent and flexible, not the discrete behaviour D&M describe. One important characteristic human genetics is our hypervariability - instead of having only the normal diploid number of a particular gene, we might have 4 or 8 copies. The copy number sometimes changes the degree to which a trait is expressed. Furthermore, instead of modifying a particular gene, change in trait expression is often accomplished via some other regulator gene that simply changes the conditions under which the gene of interest is expressed. These properties of genetics are not consonant with D&Ms portrayal of genes as clearly discrete entities.
However, the larger issue here is D&Ms reliance on analogy as evidence for their position. This is problematic since support for their argument depends on selecting the appropriate analogy rather than testing the argument directly.
D&M go on to argue that the existence of mental combination proves that concepts are discrete because the original concepts are recoverable. For instance, we can recover wind and mill from windmill. Continuous entities (like fluids) are not (easily) recoverable following combination. Discrete entities (like solids) are separable after combination, so conceptual representations are discrete. Here we have more reliance on analogy, this time to the physical properties of objects.
4) Role / argument structures are important to cognition
“A representation is structured if and only if it is composed on parts; nothing that is homogeneous is structured and vice versa.” p. 110
Continuous representations aren’t homogeneous, but they have regions rather than parts. D&M think this is problematic because a region can’t be replaced by another region, but a part can be replaced by another part. Describing a continuous representation in terms of parts necessitates a discrete representation that supervenes on the continuous one (like their thermostat example). Again, I think this is unfair. Take a sine wave as an example of a continuous representation. Now focus on the regions of the wave defined by minimum amplitude. D&M would argue that these regions couldn’t be replaced by something else. However, we could add a different wave to our sine function and change the behaviour of the function at the regions of interest.. This seems to accomplish the same thing that D&M want to accomplish by substituting one part for another, but it does this via continuous representations.
5) People’s concepts are highly connected to one another
The connections between concepts create the functional content of representations. D&M argue that continuous concepts can’t have meaningful connections between them. We know people’s concepts are connected, therefore they must be discrete. D&M dismiss the notion of connections between continuous concepts out-of-hand, without seriously attempting to think of alternatives.
According to D&M, hierarchical categorisation also requires discrete representations. People can categorise the same object in multiple ways (Chihuahua, dog, animal). D&M argue that we couldn’t keep these different levels of abstraction distinct using continuous representation. Again, they don’t develop this idea at all, so it remains a very weak point.
6) We can abstract from our environment
Continuous representations are not abstract because they are tightly coupled with input.
“Not much is known about how concepts are constructed from the input we receive, indeed there is little agreement even on the best way to define concepts. There is general agreement, however, that concepts are abstractions.” p. 112
The evidence for abstraction is that we can treat discriminable items the as if they are the same (I can see the difference between these two ponies, but I know they’re both ponies, so I’ll treat them the same way). D&M suggest that abstraction necessitates losing information. For instance, there is some continuous representation of the environment coming in from the visual stream, but higher order representations extract discrete bits from this continuous flow. These discrete bits help us discriminate things in the environment (they give the example of partitioning sound waves into words). Abstractions get a lot of their content from functional roles, since they are not tightly coupled with the environment.
I think the premise that we can treat different things identically is suspect. I might refer to many different objects as coffee mugs, depending on their properties, but the way I interact with any one of those objects depends on its specific characteristics (e.g., size, distance from me, weight, etc).
The authors seem to be suggesting that abstraction entails identifying shared properties among a class of items (e.g., that mugs hold liquid and have handles). There are two problems here. First, this doesn’t necessitate losing other information, as D&M argue. Second, categories based on necessary and sufficient features don’t work. Categories based on feature overlap don’t really work, either. These problems are apparent in their example of segmenting sounds into words. For instance, consider the pronunciation of “same eat” and “say meat.” These sound identical, even the spacing between each sound is identical (e.g., speakers don’t pause at word boundaries. You can convince yourself of this by listening to a foreign language. Notice how the sounds all seem to run together? That’s because they do.) So, the phonemes are insufficient for defining word boundaries.
Another illustration of the problems with their argument can be seen in atypical category members. We probably all agree that a two legged, hairless, dog that doesn’t bark is still a dog. However, it lacks most of the features that we would say define dogs. Any abstraction of the kind proposed by D&G would need to consist of discrete properties that properly characterise a class of objects. But our poor dog shows that such properties do not discriminate category members from non-members. You can try to refine the set of properties, but there will always be exceptions. Nonetheless, we usually cope with these exceptions pretty well, and there is substantial agreement between people about whether weird cases belong in or out of a given category.
7) There exist both nomic and nonnomic representations
Nomic representations are connected to the thing they represent by physical laws (the coil in the thermostat). Nonnomic representations are not connected to the thing they represent by physical laws (see Fodor, 1986). D&M argue that most concepts are nonnomic. Although, without knowing how to define concepts, or knowing how concepts are acquired, or knowing what the content of concepts actually is, it’s hard to know why they’re so sure about this.
The authors claim that continuous representations can’t be nonnomic. This seems to be the case by definition. Continuous representations are coupled with input from the environment. If these representations don’t for some reason vary in response to the relevant input, then they cease to represent anything. Nonnomic representations are necessarily not connected to the thing they represent, so continuous representation can’t be nonnomic. This seems like a trivial point to make, and not an argument against continuous representations at all. The burden should be on showing that our concepts are mostly nonnomic. Only then would this start to challenge the plausibility of continuous representation.
D&M point out that while input from the environment is sufficient to activate continuous representations, discrete representations are activated by both the environment and by their functional relations to other discrete representations. However, I don’t see why a continuous representation couldn’t be coupled with another continuous representation (which D&M dismiss out of hand as impossible) and therefore, be functionally related to other internal states.
Take home message
Representations must be discrete because: 1) Only discrete representations allow us to discriminate between things in the environment and 2) Discrete representations “allow a greater influence of functional role than do continuous representations” (p. 114).
This second point is a bit sneaky since they previously claim to have ruled out the possibility of functional connections between continuous representations. Why soften the message here?
The authors conclude that the necessity of discrete representations means that computationalism is the right way to characterise cognitive processes. Any finite system with discrete states can be described algorithmically. The mind is finite and the preceding arguments suggest that it has discrete states, so the mind can be described with algorithms. D&M conclude that because we can describe the mind computationally and because we’re pretty good at computation, we should use this framework until something better comes along.
So, that’s DC representation in a nutshell. I find this paper unsatisfying on many levels. My biggest complaint is that the authors do not devote any real energy to alternate possibilities. Instead, they either present their views as fact (e.g., people combine representations, p .107) or argue by analogy (e.g., the blending inheritance hypothesis, p. 108). This isn’t nearly rigorous enough to establish the authors’ claim that cognition requires discrete representations.
Dietrich, E. & Markman, A. B. (2003). Discrete thoughts: Why cognition must use discrete representations. Mind and Language, 18 (1), 95-119.
Fodor, J. (1986). The modularity of mind. In Z. Pylyshyn & W. Demopoulos, Meaning and cognitive structure: Issues in the computational theory of mind. Theoretical issues in cognitive science. (pp. 3-18). Westport, CT, US: Ablex Publishing. xi, 264 pp.
Markman, A. B. & Gentner, D. (2000). Structure mapping and the comparison process. American Journal of Psychology, 113 (4), 501-538.
Shepard, R. N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function. Part 1. Psychometrika, 27 (2), 125-140.
Tversky, A. (1977). Features of Similarity. Psychological Review, 84 (4), 327-352.