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22-08-2018

Computers in Abstraction/Representation Theory

Author: Samuel C. Fletcher

Published in: Minds and Machines | Issue 3/2018

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Abstract

Recently, Horsman et al. (Proc R Soc Lond A 470:20140182, 2014) have proposed a new framework, Abstraction/Representation (AR) theory, for understanding and evaluating claims about unconventional or non-standard computation. Among its attractive features, the theory in particular implies a novel account of what is means to be a computer. After expounding on this account, I compare it with other accounts of concrete computation, finding that it does not quite fit in the standard categorization: while it is most similar to some semantic accounts, it is not itself a semantic account. Then I evaluate it according to the six desiderata for accounts of concrete computation proposed by Piccinini (Physical computation: a mechanistic account, Oxford University Press, Oxford, 2015). Finding that it does not clearly satisfy some of them, I propose a modification, which I call Agential AR theory, that does, yielding an account that could be a serious competitor to other leading account of concrete computation.

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Note the similarity between a representation and a “partial” functor between categories. Horsman (2015, p. 10) raises the possibility of giving a categorical interpretation of AR theory but does not pursue it. Similarly, I also urge its pursuit, but in future work. (See also footnote 5.)

Horsman et al. (2014) use the terminology “computational entity,” superseded already in Horsman (2015).

Horsman (2015, 2017) and Horsman et al. (2017b) essentially define a theory to be a set of representation relations, but this immediately leads to difficulties accounting for how theories can make claims about possible but not actual concrete situations: such situations are simply not in the domain of physical object states. But this problem can be avoided if one simply assumes that, whatever theories are, they provide representation relations for a wide range of domains, including possible but not actual physical states.

In fact, it’s not clear that all aspects of AR theory are compatible with the structuralist accounts of scientific representation given by van Fraassen (2008), who takes representations to entail the proposal or assertion of an hypothesis that there is an isomorphic embedding of the abstract model into the concrete target of the representation. For AR theory, a representation relation is a map from the concrete to the abstract: it’s just the wrong kind of relation to be an isomorphic embedding. Perhaps a structuralist account of representation for AR theory would make the representation relation a homomorphism, but I won’t pursue this question here.

One might describe this with an approximately commuting diagram—see footnote 1 for more on the connection with category theory. Also cf. Corless and Fillion (2014, p. 30), who attribute the idea of an approximately commuting diagram to describe representation in the context of numerical computing to Robidoux (2002, Chap. 6).

One could say in such cases that the function \(C_\mathcal {T} = \mathcal {R_T}(\mathbf {H})\) is the corresponding “abstract evolution” (Horsman 2015, p. 4), but this should be understood metaphorically: since abstract objects do not exist in time they cannot literally evolve.

See also Horsman (2017, p. 198).

They continue, “We almost never talk about ‘information’ or ‘knowledge’ or ‘meaning’ in using AR theory” but this seems to be an overstatement on at least one count: “information” is used in the context of discussing AR theory on several occasions in the same paper (Horsman et al. 2018, pp. 138, 142, 148) and elsewhere. Perhaps it is best to interpret these positive usages as information in the mathematical sense (Shannon and Weaver 1949), devoid of semantic content.

In the interests of facilitating reference throughout this section, I have separated the desiderata by line breaks in the following quote instead of listing them in-line.

A third sense is the so-called “value-free ideal” of having no social, moral, or political value impinge on scientific method or product.

Horsman et al. (2017b) also do so for chemotaxis in bacteria and DNA, but these cases less plausibly fall under the category of intuitively evaluable computations.

There is always some valid theory for any domain, namely, the one whose representation maps each physical system to the same abstract object—this is the constant representation. However, we must assume here some theory whose representation relation has a range with an infinite cardinality. Unless the physical domain \(\mathbf {P}\) is much simpler than it appears, this will be easily satisfied.

Another option would be to restrict the admissible representations or encodings directly, although to avoid being ad hoc this would move AR theory in the direction of a syntactic account of concrete computation. I won’t pursue that option here, since it is inimical to the motivations for AR theory for understanding unconventional computation, as discussed in Sect. 3.

I am writing as if there could be only one agential community—a group of objects considered as agents simpliciter—presumably the one in which we are included. There could be, however, as many distinct such communities as there are groups with different shared capabilities for abstraction and representation, observation and prediction, and theorizing and confirmation. Thus one can think about agential communities as a sort of epistemic community (van Fraassen 1980, pp. 18–19), of which there can be many and which can in principle change over time (van Fraassen 2005). But delineating the extent and multitude of these boundaries is not necessary for AAR theory: it is always applied relative to some such community, of which we can at least identify a prototypical or otherwise representative member.

This would follow from one view of cognitive science’s status as a science. Failing that, however, one could always conservatively restrict to humans.

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Title: Computers in Abstraction/Representation Theory
Author: Samuel C. Fletcher
Publication date: 22-08-2018
Publisher: Springer Netherlands
Published in: Minds and Machines / Issue 3/2018
Print ISSN: 0924-6495
Electronic ISSN: 1572-8641
DOI: https://doi.org/10.1007/s11023-018-9470-9

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