Abstract
A skeptical worry known as ‘the indeterminacy of computation’ animates much recent philosophical reflection on the computational identity of physical systems. On the one hand, computational explanation seems to require that physical computing systems fall under a single, unique computational description at a time. On the other, if a physical system falls under any computational description, it seems to fall under many simultaneously. Absent some principled reason to take just one of these descriptions in particular as relevant for computational explanation, widespread failure of computational explanation would appear to follow. This paper advances a new solution to the indeterminacy of computation. Very roughly, I argue that the computational identity of a physical system is determinate relative to a contextually specified way of regarding that system computationally—known as a labelling scheme. When a system simultaneously implements multiple computations, it does so relative to different labelling schemes. But relative to a fixed labelling scheme, a physical system has a unique computational identity. I argue that this relativistic conception of computational identity vindicates computational explanation in the face of simultaneous implementation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
16 August 2022
A Correction to this paper has been published: https://doi.org/10.1007/s11229-022-03835-3
Notes
The term ‘indeterminacy of computation' was coined by Jack Copeland (see Fresco, Copeland and Wolf 2021, p. 12755) and is gaining currency; see also (Fresco and Milkowski 2021). The issue is also known as ‘the problem of simultaneous implementation' (Shagrir 2001, 2020; Dewhurst 2018b), and ‘the multiplicity of computations' (Piccinini 2008, 2015; Coelho Mollo, 2018; Lee 2021).
As we will see, string-theoretic and numerical computations have very different implementation conditions. But from a mathematical point of view, the differences between these two kinds of computation are less important. It is typical to start with string-theoretic computations and then define numerical computations by taking strings as representations of numbers (e.g., Davis, 1982). However, it is also possible to develop computability theory directly in terms of numerical computations (e.g., Davis et al., 1994, ch. 2–4).
I will move back and forth between talk of physical systems ‘satisfying computational descriptions’ and ‘implementing’ or ‘realizing’ computations, taking these to be roughly equivalent.
Not everyone will agree with this way of framing the debate. For instance, Shagrir (2001, p. 382) claims that computational identity involves semantic properties over and above implementation. But this is because Shagrir endorses a wholly non-semantic account of implementation. Rather than claiming that computational identity involves something in addition to implementation, it is more straightforward to see Shagrir as proposing a semantic account of implementation.
There is some dispute about whether this is a genuine alternative to the computation in (2) (Piccinini, 2020b, p. 153). There is also dispute about whether there is one kind of indeterminacy at play here, or two (Papayannopoulos, Fresco, and Shagrir, forthcoming). My own view is that there is just one; see Sects. 3.2 and 4.4.
Note that simultaneous implementation falls short of pancomputationalism, the claim that every physical system simultaneously implements every computation. See (Shagrir, 2001) for discussion.
Fresco et al. (2021) make a similar point in the context of natural computing systems. They suggest that it is evolutionarily beneficial if a given neural component can subserve multiple distinct neural processes.
Fresco (2021) defends a ‘long-arm’ anti-individualist view, in contrast to Piccinini’s ‘short-arm’ view. Although I focus on Piccinini’s version here, the objections raised below apply equally to both.
See (Rescorla, 2014) for other examples in this vein.
There are a few interesting instances of the ternary computers in the history of computer science. Perhaps the most notable are the Setun machines, developed in the 1950s and 60s. For discussion see (Brusentsov & Ramil Alvarez, 2011) and references therein.
Thanks to an anonymous referee for urging me to clarify this point.
(Cormen et al., 2001, p. 26). Crudely, the running time of an algorithm is Θ(f) for some function f if for sufficiently large inputs the running time is bounded above and below by f, up to some constant factor.
This view resembles more traditional causal views of computation (e.g. Chalmers, 1996a), but goes beyond them by endorsing the empirical claim that functionally characterized computations are realized by mechanisms.
At least to a very rough first approximation. In saying this I do not claim that all computational explanations identify routine, step-by-step processes. I merely claim that it is sufficient for a given explanation to constitute a computational explanation that it take this form.
But see (Papayannopoulos et al., forthcoming) for an alternative take on the situation.
At least, up to any indeterminacy in the properties cited by the specific scheme. Although this is itself a serious issue, I will not pursue it here.
To be clear, I do not claim that a system’s computational identity is determined by the total set of computations it implements (as in Milkowski, 2013). Rather, I claim that computational identity is determined only relative to a fixed labelling scheme. Relative to a fixed scheme, a system’s computational identity is determined by the computation implemented under that scheme.
Although I will not argue for it here, this appears to be a general feature of applications of mathematics. To take Frege’s (1884) example, a single expanse of matter may simultaneously constitute one deck and fifty-two cards. There is no inconsistency in describing the matter simultaneously as one and as fifty-two, because these descriptions apply relative to the concepts DECK and CARD, respectively.
In this respect, computability theory and related branches of theoretical computer science play a familiar unifying role encountered elsewhere in science (Kitcher, 1981).
Thanks to an anonymous referee for pressing me on this point.
Scientific perspectivalism is itself a large, challenging topic, and I cannot hope to do justice to all the subtleties involved here. For more, see (Chakravartty, 2010), (Massimi, 2018), and (Massimi & McCoy, 2019), among many others. For detailed discussion of computational perspectivalism in particular see (Coelho Mollo, 2019).
At times, Dewhurst’s remarks suggest a more epistemic reading; see (Coelho Mollo, 2019) for an interpretation along these lines.
There is, of course, longstanding philosophical disagreement about the nature and status of mathematical objects. Rather than rehearse that debate here, I would instead emphasize that my view is compatible with a wide range of background views in the philosophy of mathematics. While some such views might lead to ontic perspectivalism, others arguably do not.
References
Blackmon, J. (2013). Searle’s Wall. Erkenntnis, 78(1), 109–117.
Brusentsov, N. P., & Alvarez, J. R. (2011). Ternary computers: The setun and the setun 70. In J. Impagliazzo & E. Proydakov (Eds.), Perspectives on Soviet and Russian Computing (Vol. 357, pp. 74–80). Berlin: Springer. https://doi.org/10.1007/978-3-642-22816-2_10
Chakravartty, A. (2010). Perspectivism, inconsistent models, and contrastive explanation. Studies in History and Philosophy of Science Part A, 41(4), 405–412. https://doi.org/10.1016/j.shpsa.2010.10.007
Chalmers, D. J. (1996a). Does a rock implement every finite-state automaton? Synthese, 108(3), 309–333.
Chalmers, D. J. (1996b). The conscious mind. Oxford University Press.
Coelho Mollo, D. (2018). Functional individuation, mechanistic implementation: The proper way of seeing the mechanistic view of concrete computation. Synthese, 195(8), 3477–3497.
Coelho Mollo, D. (2019). Against computational perspectivalism. The British Journal for the Philosophy of Science, axz036.
Copeland, B. J. (1996). What is computation? Synthese, 108(3), 335–359.
Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2001). Introduction to algorithms (2nd ed.). McGraw-Hill Higher Education.
Cutland, N. (1980). Computability, an introduction to recursive function theory. Cambridge University Press.
Davis, M. (1982). Computability & unsolvability. Dover.
Davis, M., Segal, R., & Weyuker, E. (1994). Computability, complexity and languages (2nd ed.). Academic Press.
Dewhurst, J. (2018a). Computing mechanisms without proper functions. Minds and Machines, 28(3), 569–588.
Dewhurst, J. (2018b). Individuation without representation. The British Journal for the Philosophy of Science, 69(1), 103–116.
Egan, F. (1995). Computation and content. The Philosophical Review, 104(2), 181. https://doi.org/10.2307/2185977
Frege, G. (1884). The foundations of arithmetic. Northwestern University Press.
Fresco, N. (2021). Long-arm functional individuation of computation. Synthese. https://doi.org/10.1007/s11229-021-03407-x
Fresco, N., Jack Copeland, B., & Wolf, M. J. (2021). The indeterminacy of computation. Synthese. https://doi.org/10.1007/s11229-021-03352-9
Fresco, N., & Milkowski, M. (2021). Mechanistic computational individuation without biting the bullet. British Journal for the Philosophy of Science, 72(2), 431–438.
Harbecke, J., & Shagrir, O. (2019). The role of the environment in computational explanations. European Journal for Philosophy of Science, 9(3), 37. https://doi.org/10.1007/s13194-019-0263-7
Hennessy, J. L., & Patterson, D. A. (2003). Computer architecture: A quantitative approach. Morgan Kaufmann Publishers.
Hitchcock, C. (2013). Contrastive explanation. In M. Blaauw (Ed.), Contrastivism in philosophy Routledge Studies in Contemporary Philosophy (Vol. 39, pp. 11–35). New York: Routledge.
Horowitz, A. (2007). Computation, external factors, and cognitive explanations. Philosophical Psychology, 20(1), 65–80. https://doi.org/10.1080/09515080601085856
Kitcher, P. (1981). Explanatory unification. Philosophy of Science, 48(4), 507–531.
Lee, J. (2021). Mechanisms, wide functions, and content: Towards a computational pluralism. British Journal for the Philosophy of Science, 72(1), 221–244.
Massimi, M. (2018). Four kinds of perspectival truth. Philosophy and Phenomenological Research, 96(2), 342–359.
Massimi, M., & McCoy, C. D. (Eds.). (2019). Understanding perspectivism: Scientific and methodological prospects. routledge studies in the philosophy of science 20. Taylor & Francis.
Matthews, R. J., & Dresner, E. (2017). Measurement and computational skepticism. Nous, 51(4), 832–854.
Milkowski, M. (2013). Explaining the computational mind. MIT Press.
Papayannopoulos, P., Fresco, N., & Shagrir, N. (forthcoming) On two different kinds of computational indeterminacy. The Monist.
Peacocke, C. (1999). Computation as involving content: A response to egan. Mind and Language, 14(2), 195–202.
Piccinini, G. (2020). Neurocognitive mechanisms: Explaining biological cognition. Oxford University Press. https://doi.org/10.1093/oso/9780198866282.001.0001
Piccinini, G. (2020a). Computation and information processing. In Neurocognitive mechanisms (pp. 128–55). Oxford University Press. https://doi.org/10.1093/oso/9780198866282.003.0007.
Piccinini, G. (2008). Computation without representation. Philosophical Studies, 137(2), 205–241.
Piccinini, G. (2015). Physical computation: A mechanistic account. Oxford University Press.
Potochnik, A. (2017). Idealization and the aims of science. The University of Chicago Press.
Rescorla, M. (2013). Against structuralist theories of computational implementation. The British Journal for the Philosophy of Science, 64(4), 681–707.
Rescorla, M. (2014). A theory of computational implementation. Synthese, 191(6), 1277–1307. https://doi.org/10.1007/s11229-013-0324-y
Rescorla, M. (2017). From Ockham to turing—and back again. In J. Floyd & A. Bokulich (Eds.), Philosophical explorations of the legacy of alan turing: turing 100 (pp. 279–304). Springer.
Ritchie, J. B., & Piccinini, G. (2019). Computational implementation. In M. Sprevak & M. Colombo (Eds.), Routledge handbook of the computational mind (pp. 192–204). Routledge.
Savage, J. E. (2008). Models of computation: Exploring the power of computing. http://cs.brown.edu/people/jsavage/book/.
Schiller, H. (2018). The swapping constraint. Minds and Machines, 28(3), 605–622.
Schweizer, P., & Piotr J. (2013). Abstract procedures and the physical world. In Proceedings of the AISB’13 Symposium on Computing and Philosophy (pp. 66–73).
Schweizer, P. (2019). Computation in physical systems: A normative mapping account. In M. V. d’Alfonso & D. Berkich (Eds.), On the cognitive, ethical, and scientific dimensions of artificial intelligence: Themes from IACAP 2016 (pp. 27–47). Springer.
Schweizer, P. (2019b). Triviality arguments reconsidered. Minds and Machines, 29(2), 287–308.
Shagrir, O. (2001). Content, computation and externalism. Mind, 110(438), 369–400.
Shagrir, O. (2020). In defense of the semantic view of computation. Synthese, 197, 4083–4108.
Sprevak, M. (2010). Computation, individuation, and the received view on representation. Studies in History and Philosophy of Science Part A, 41(3), 260–270.
Sprevak, M. (2019). Triviality arguments about computational implementation. In M. Sprevak & M. Colombo (Eds.), Routledge handbook of the computational mind (pp. 175–191). Routledge.
Turing, A. (1936). On computable numbers, with an application to the entscheidungsproblem. Proceedings of the London Mathematical Society, 42(1), 230–265.
Acknowledgements
Thanks to Soyeong An, Matteo Biachetti, Preston Lennon, Daniel Olson, Richard Samuels, Stewart Shapiro, Declan Smithies, Damon Stanley, and audiences in Bergamo and online at BSPS 2021 for comments and discussion. Special thanks to three anonymous referees, whose suggestions led to substantial improvements in the final paper. Any errors that remain are entirely my own.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author reports no conflicts of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The original online version of this article was revised: footnote 1 was updated.
Appendix
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Curtis-Trudel, A. The determinacy of computation. Synthese 200, 43 (2022). https://doi.org/10.1007/s11229-022-03568-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11229-022-03568-3