Skip to main content
Log in

Analog representations and their users

  • Published:
Synthese Aims and scope Submit manuscript


Characterizing different kinds of representation is of fundamental importance to cognitive science, and one traditional way of doing so is in terms of the analog–digital distinction. Indeed the distinction is often appealed to in ways both narrow and broad. In this paper I argue that the analog–digital distinction does not apply to representational schemes but only to representational systems, where a representational system is constituted by a representational scheme and its user, and that whether a representational system is analog or non-analog depends on facts about that user. This aspect of the distinction has gone unnoticed, and I argue that the failure to notice it can be an impediment to scientific progress.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. See e.g., Von Neumann (1958).

  2. See e.g., Churchland and Churchland (2000).

  3. See e.g., Pylyshyn (1984) and Tye (1991).

  4. See e.g., Wynn (1992) and Gallistel et al. (2006).

  5. If necessary it is also possible to talk of the tertiary user, and so on.

  6. In this case of course, the primary user and the secondary user do not create and interpret the same representations. The primary user creates and interprets representations in the machine language. The human user creates and interprets representations in the computer’s input–output language.

  7. The view grows out of the history of the development of computing technology. Thus, Newell (1983) writes, “When computers were first developed in the 1940s they were divided into two large families. Analog computers represented quantity by continuous physical variables, such as current or voltages...Digital computers represented quantities by discrete states...”(p. 195). See also Von Neumann (1958).

  8. The accounts offered by Haugeland (1998), Maley (2011), and Montemayor and Balci (2007) do not face this same problem of implying that there are no analog systems. Thus in those cases I will argue directly that the accounts as they are imply that whether a system is analog or not depends on facts about its user. Therefore the discussion of Goodman’s account is considerably longer than each of the others.

  9. Goodman does not draw the distinction between schemes and systems in the way I am. But it will be part of the burden of my argument that his account of the a/d distinction implicitly assumes the notion of a user. I’m attributing the distinction as I use it to him, so as to make that argument clearer.

  10. Goodman at times suggests continuity and not density, but the distinction will not be relevant to my argument.

  11. Goodman’s account of the a/d distinction grows out of an account of notational systems for art, which according to Goodman serve to “[mark] off performances that belong to the work from those that do not” (1968, p. 128). There are further requirements that Goodman identifies for notational systems.

  12. For present purposes, it will suffice that a technique for differentiating characters and inscriptions is theoretically possible if it is imaginable, even if it does not exist.

  13. Note that here and elsewhere Goodman assumes that space–time is infinitely divisible. I take it this in fact remains an open question, but it will not affect my argument, so I grant it for present purposes.

  14. The way Goodman states this worry (quoted above) in fact employs a universal quantifier in the left-most position. But such a formulation will imply the one in-text, with the existential quantifier. The latter makes clear that the problem is an alternation of the universal quantifier on \(\mu \) with the existential quantifiers on \({ K }\) and \({ K }^{\prime }\).

  15. More specifically, although (G) is true of the set of straight marks, (G) does not imply (NFD), and (NFD) is not true of the set. In other words, although (G) is true of it, the set of straight marks is finitely differentiated.

  16. In fact, I think the notion of possibility in practice is more in the spirit of Goodman’s account anyway. After all, his account of the a/d distinction grows out of an account of the distinction between notational and non-notational systems, where notational systems are those systems that allow for unique identification of artwork. His project is thus at root pragmatic, even though Haugeland accuses him of “betray[ing] a mathematician’s distaste for the nitty-gritty of practical devices” (1998, p. 80). It therefore seems at odds with his project that some of the criteria he sets for notational systems are wholly theoretical.

  17. Of course, Haugeland developed his account well before the mass availability of “digital” photography.

  18. This is not quite precise enough. If space–time is not infinitely divisible, then it is possible for a typical carpenter using typical tools to cut a board to exactly six feet, but it is not possible for her to know that she has done so.

  19. See for example Horak (2000).

  20. Maley holds that the a/d distinction applies only to representations of number. Though they do not argue for it, most authors disagree, assuming that representations with all manner of content may be either analog or digital. Maley gives historical reasons for his view, but I will not address the topic here.

  21. Again, on my view a representational scheme is neither analog nor non-analog, but may with some users constitute an analog system and with others constitute a non-analog system. But no system will be both analog and non-analog.

  22. Note that Montemayor and Balci use ‘analog’, ‘metric’, and ‘magnitude–based’ interchangeably.

  23. Indeed, Montemayor and Balci call this the “resemblance constraint”.

  24. See e.g., Lewis (1971), Blachowicz (1997), Frigerio et al. (2013), and Schonbein (2014). See Dretske (1981) for a non-standard account, and for a closely related discussion see Cummins (1996) and Cummins et al. (2001).

  25. See Carey and Spelke (1996) for a useful discussion of looking-time methods and results.

  26. See Xu and Spelke (2000).

  27. See McCrink and Wynn (2004).

  28. Other models posit a system in which representations compress logarithmically as the number represented grows. See for example, Dehaene et al. (2008).

  29. Katz (2013).

  30. See Spelke (2003) for a description of the process.


  • Blachowicz, J. (1997). Analog representation beyond mental imagery. Journal of Philosophy, 94(2), 55–84.

    Article  Google Scholar 

  • Carey, S., & Spelke, E. (1996). Science and core knowledge. Philosophy of Science, 63, 515–533.

    Article  Google Scholar 

  • Churchland, P. M., & Churchland, P. S. (2000). Foreword. In von Neumann & J. New Haven (Eds.), The computer and the brain. New Haven: Yale University Press.

    Google Scholar 

  • Cummins, R. (1996). Systematicity. The Journal of Philosophy, 93(12), 591–614.

    Article  Google Scholar 

  • Cummins, R., Blackmon, J., Byrd, D., Poirier, P., Roth, M., & Schwarz, G. (2001). Systematicity and the cognition of structured domains. The Journal of Philosophy, 98(4), 167–185.

    Article  Google Scholar 

  • Dehaene, S., Izard, V., Spelke, E. S., & Pica, P. (2008). Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science, 320, 1217.

    Article  Google Scholar 

  • Dretske, F. (1981). Knowledge and the flow of information. Cambridge: MIT Press.

    Google Scholar 

  • Frigerio, A., Giordani, A., & Mari, L. (2013). On representing information: A characterization of the analog/digital distinction. Dialectica, 67(4), 455–483.

    Article  Google Scholar 

  • Gallistel, C. R., Gelman, R., & Cordes, S. (2006). The cultural and evolutionary history of the real numbers. In S. Levinson & P. Jaisson (Eds.), Culture and evolution. Cambridge: MIT Press.

    Google Scholar 

  • Goodman, N. (1968). Languages of art. Indianapolis: The Bobbs-Merrill Company Inc.

    Google Scholar 

  • Haugeland, J. (1998). Analog and analog. In J. Haugeland (Ed.), Having thought. Cambridge: Harvard University Press.

    Google Scholar 

  • Horak, R. (2000). Communications systems and networks (2nd ed.). Foster City, CA: M and T Books.

    Google Scholar 

  • Katz, M. (2013). Mental magnitudes and increments of mental magnitudes. The Review of Philosophy and Psychology, 4(4), 675–703.

    Article  Google Scholar 

  • Lewis, D. (1971). Analog and digital. Nous, 5(3), 321–327.

    Article  Google Scholar 

  • Maley, C. (2011). Analog and digital, continuous and discrete. Philosophical Studies, 155(1), 117–131.

    Article  Google Scholar 

  • Margolis, E., & Laurence, S. (2008). How to learn the natural numbers: Inductive inference and the acquisition of number concepts. Cognition, 106, 924–39.

    Article  Google Scholar 

  • McCrink, K., & Wynn, K. (2004). Large-number addition and subtraction by 9-month-old infants. Psychological Science, 15(11), 776–781.

    Article  Google Scholar 

  • Meck, W. H., & Church, R. M. (1983). A mode control mechanism of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes, 9(3), 320–34.

    Google Scholar 

  • Montemayor, C., & Balci, F. (2007). Compositionality in language and arithmetic. Journal of Theoretical and Philosophical Psychology, 27(1), 53–72.

    Article  Google Scholar 

  • Newell, A. (1983). Intellectual issues in the history of artificial intelligence. In F. Machlup & U. Mansfield (Eds.), The study of information: Interdisciplinary messages. New York: Wiley.

    Google Scholar 

  • Pylyshyn, Z. W. (1984). Computation and cognition. Cambridge: MIT Press.

    Google Scholar 

  • Schonbein, W. (2014). Varieties of analog and digital representation. Minds and Machines, 24(4), 415–438.

    Article  Google Scholar 

  • Spelke, E. S. (2003). What makes us smart? Core knowledge and natural language. In D. Gentner & S. Goldin-Meadow (Eds.), Language in mind: Advances in the investigation of language and thought. Cambridge: MIT Press.

    Google Scholar 

  • Tye, M. (1991). The imagery debate. Cambridge: MIT Press.

    Google Scholar 

  • Von Neumann, J. (1958). The computer and the brain (2nd ed.). New Haven: Yale University Press.

    Google Scholar 

  • Wynn, K. (1992). Evidence against Empiricist Accounts of the Origins of Numerical Knowledge. In A. I. Goldman (Ed.), Readings in Philosophy and Cognitive Science (pp. 209–27). Cambridge: MIT Press.

    Google Scholar 

  • Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month old infants. Cognition, 74, B1–B11.

    Article  Google Scholar 

Download references


I would like to thank Katherine Thomson-Jones, Joshua Smith, and three anonymous referees for comments on earlier drafts that have significantly improved the paper. I am especially grateful to Scott Weinstein, who first noticed the quantifier shift error in Goodman’s account of the a/d distinction.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Matthew Katz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Katz, M. Analog representations and their users. Synthese 193, 851–871 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: