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Markus Pantsar – “On Radical Enactivist Accounts of Arithmetical Cognition”

Two children selling fruit from a basket count the coins they just received.
Detail of “The Little Fruit Seller” (c. 1670-1675) Bartolomé Esteban Murillo

In this post, Markus Pantsar discusses the article he recently published in Ergo. The full-length version of his article can be found here.

Traditionally, cognitive science has held the view that the human mind works through, or is at least best explained by, mental repre­sentations and computations (e.g., Chomsky 1965/2015; Fodor 1975; Marr 1982; Newell 1980). Radical enactivist accounts of cognition challenge this paradigm. According to them, the most basic forms of cognition do not to involve mental representations or mental content; representations (and content) exist only in minds that have access to linguistic and sociocultural truth-telling practices (Hutto and Myin 2013, 2017).

As presented by Hutto and Myin, radical enactivism is a general approach to the philosophy of cognition. It is partly from this generality that it gets much of its force and appeal. However, a general theory of cognition ultimately needs to be tested on particular cognitive phenomena. In my paper, I set out to do just that with regard to arithmetical cognition. I am not a radical enactivist, but neither am I antagonistic to the approach. My aim is to provide a dispassionate analysis based on the progress that has been made in the empirical study and philosophy of numerical cognition.

Arithmetical cognition is especially suited to test radical enactivism (Zahidi 2021). This is not because arithmetic itself suggests the existence of non-linguistic representations. In fact, since Dedekind and Peano presented an axiomatization for arithmetic, it became clear that the entire arithmetic of natural numbers can be presented in a very simple language with only a handful of rules (i.e., the axioms) (Dedekind 1888; Peano 1889).

It is not arithmetic as a mathematical theory that presents challenges for radical enactivism; it is rather the development of arithmetic. This development happens on two levels. First, at the level of individuals, we have the ontogenetic development of arithmetical cognition. Second, at the level of populations and cultures, we have the phylogenetic and cultural-historical development of arithmetic. In my paper I focus on the ontogenetic level, because it is at that level that radical enactivism faces its most serious challenge.

It is commonly accepted that, in learning arithmetical knowledge and skills, children apply their innate, evolutionarily-acquired proto-arithmetical abilities (Pantsar 2014, 2019). These abilities – sometimes also called “quantical” (Núñez 2017) – are already present in human infants, and we share them with many non-human animals.

According to the most common view, there are two main proto-arithmetical abilities (Knops 2020). The first is subitizing: the ability to determine the amount of objects in our field of vision without counting. Subitizing enables detecting exact quantities, but it stops working after three or four objects. For larger collections, there is an estimating ability. This ability is not limited to small quantities, but it gets increasingly inaccurate as the size of the observed collection increases.

For the present topic, the literature on subitizing and estimating presents interesting questions. Following the work of Elizabeth Spelke (2000) and Susan Carey (2009), it is commonplace to associate each ability with a special core cognitive system (Hyde 2011). Subitizing is associated with the object tracking system (OTS), which allows for the parallel observation of objects in the subitizing range, up to three or four. Estimating is associated with the approximate number system (ANS), which is thought to be a numerosity-specific system.

The problem for the radical enactivist is that, under most interpretations, both the OTS and ANS are based on non-linguistic representations. The OTS is based on the observed objects occupying mental object files, one file for one object (Beck 2017; Carey 2009). For example, when I see three apples, three object files are occupied, and we can understand this as a representation of the number of the apples.

The ANS, on the other hand, is usually interpreted as representing quantities on a mental number line (Dehaene 2011). This line is likely to be logarithmic, given that the estimating ability becomes less accurate as the quantities become larger. Studies on anumerical cultures in the Amazon provide further evidence of this; members of those cultures tend to place quantities on a (physical) number line in a logarithmic manner (Dehaene et al. 2008; but see Núñez 2011).

Therefore, we have good empirical evidence in support of the idea that proto-arithmetical abilities are to be interpreted in terms of non-linguistic representations. Now the question is: can radical enactivism provide an alternative explanation for proto-arithmetical abilities without evoking representations?

This proves to be difficult, because it requires answering what is perhaps the most fundamental question in the field: namely, what exactly is a mental representation? Should visual memories, for example, be considered representations? For the radical enactivist they should not, but little evidence or argumentation has been provided to support this denial. In the present context, we must ask: could the OTS and the ANS work without using representations? Radical enactivism says so, but there is little solid evidence in support of this view.

Nonetheless, it should also be noted that the object files and the mental number line as explanations of the functioning of the OTS and the ANS, respectively, are currently nothing more than theoretical postulations : neither object files nor a mental number line have been located in the brain at the neuronal level, although fMRI studies give us good clues on where to look (Nieder 2016).

To be sure, some monkey studies have detected the existence of number neurons: i.e., specific groups of neurons whose firing is connected to observing a particular (small) quantity of objects (Nieder 2016), and one could infer that such number neurons count as representations of quantities in the brain. But this inference is exactly the kind of inference that radical enactivists have warned us against. Radical enactivists agree that there is non-linguistic processing of information in the brain, but they deny that in such cases there is content, i.e., representations. In the words of Hutto and Myin, brains process non-linguistic information-as-covariance, but not information-as-content (Hutto and Myin 2013:67).

In conclusion, where do we stand? Is there a way forward in the debate on representations? I believe there is, but it would be spurious to claim that philosophers can find it on their own. Instead, we will need a better empirical understanding of the neuronal activity associated with the functioning of the OTS and the ANS. At the same time, it would also be misguided to expect empirical data alone to resolve the issue. We will not find groups of neurons that are unassailably non-linguistic representations, and philosophers will need to continue working with empirical researchers in an effort to gain more knowledge about the proto-arithmetical abilities.

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  • Beck, J. (2017). “Can Bootstrapping Explain Concept Learning?” Cognition 158:110–21.
  • Carey, S. (2009). The Origin of Concepts. Oxford: Oxford University Press.
  • Chomsky, Noam (2015). Aspects of the Theory of Syntax (50th anniversary ed.). MIT Press. (Original work published 1965)
  • Dedekind, Richard. (1888). Richard Dedekind: was sind und was sollen die Zahlen?: Stetigkeit und irrationale Zahlen. 1. Auflage. edited by S. Müller-Stach. Berlin [Heidelberg]: Springer Spektrum.
  • Dehaene, S., V. Izard, E. Spelke, and P. Pica. (2008). “Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures.” Science 320:1217–20.
  • Dehaene, Stanislas. (2011). The Number Sense: How the Mind Creates Mathematics, Revised and Updated Edition. Revised, Updated ed. edition. New York: Oxford University Press.
  • Fodor, J. (1975). The Language of Thought. New York: Harvard University Press.
  • Hutto, D. D., and E. Myin. (2013). Radicalizing Enactivism. Basic Minds without Content. Cambridge, MA: MIT Press.
  • Hutto, D. D., and E. Myin. (2017). Evolving enactivism. Basic minds meet content. Cambridge, MA: MIT Press.
  • Hyde, D. C. (2011). “Two Systems of Non-Symbolic Numerical Cognition.” Frontiers in Human Neuroscience 5:150.
  • Knops, A. (2020). Numerical Cognition. The Basics. New York: Routledge.
  • Marr, D. (1982). Vision: A Computational Investigation into the Human Representation and Processing of Visual Information. San Francisco: W.H. Freeman and Company.
  • Newell, A. (1980). “Physical symbol systems.” Cognitive Science 4(2):135–83.
  • Nieder, A. (2016). “The Neuronal Code for Number.” Nature Reviews Neuroscience 17(6):366.
  • Núñez, Rafael E. (2011). “No Innate Number Line in the Human Brain.” Journal of Cross-Cultural Psychology 42(4):651–68.
  • Núñez, Rafael E. (2017). “Is There Really an Evolved Capacity for Number?” Trends in Cognitive Science 21:409–24.
  • Pantsar, Markus. (2014). “An Empirically Feasible Approach to the Epistemology of Arithmetic.” Synthese 191(17):4201–29. doi: 10.1007/s11229-014-0526-y.
  • Pantsar, Markus. (2019). “The Enculturated Move from Proto-Arithmetic to Arithmetic.” Frontiers in Psychology 10:1454.
  • Peano, G. (1889). “The Principles of Arithmetic, Presented by a New Method.” Pp. 101–34 in Selected Works of Giuseppe Peano, edited by H. Kennedy. Toronto; Buffalo: University of Toronto Press.
  • Spelke, Elizabeth S. (2000). “Core Knowledge.” American Psychologist 55(11):1233–43. doi: 10.1037/0003-066X.55.11.1233.
  • Zahidi, K. (2021). “Radicalizing numerical cognition.” Synthese 198(Suppl 1):529–45.

About the author

Markus Pantsar is a guest professor at the RWTH University in Aachen. He has the title of docent at University of Helsinki. Pantsar’s main research fields are philosophy of mathematics and artificial intelligence. His upcoming book “Numerical Cognition and the Epistemology of Arithmetic” (Cambridge University Press) will present a detailed, empirically-informed philosophical account of arithmetical knowledge.

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Henry Clarke – “Mental Filing Systems: A User’s Guide”

The painting depicts a person looking out from an interior through a sash window, which echoes the theme of the article: a thinker's view of the world being made up of compartmentalized bodies of information.
“Tall Windows” (1913) Wilhelm Hammershøi

In this post, Henry Clarke discusses his article recently published in Ergo. The full-length version of Henry’s article can be found here.

For many, if not all, of the objects we can think about, we have a reasonably rich conception of what they are like. One of the basic representational functions of the mind is to draw together different bits of information to make up these conceptions. An image that naturally suggests itself is of the mind as a kind of filing system, with folders dedicated to different objects, and descriptions placed in each to be called upon when needed.

The mental filing system idea is philosophically useful in that it provides a framework for understanding how a thinker can treat her beliefs as being about one and the same thing. Doing so results in her being prepared to draw certain implications from the content of a conception without having to decide whether that identity holds. For example, someone with a mental file that contains the descriptions tall, leafy, has brown bark can immediately infer that something is tall, leafy, with brown bark. The inference presupposes that being tall, leafy, etc. are all true of one and the same object. This sort of presupposition is a fundamental part of thought, and of a thinker’s perspective on her thought. Having such a perspective sustains the sort of rationally structured view of the world that we can develop and make use of, and mental files have seemed to many like a good way of capturing this.

But does the mental filing system idea actually tell us something about how the mind works? Or is it just a dispensable metaphor? Rachel Goodman and Aidan Gray (2022) have argued (taking François Recanati’s theory of mental files as their working example), that mental filing – taking one’s beliefs immediately to be about the same thing – doesn’t require taking the file image too seriously. The source of their skepticism comes from an analysis of what it means to say that we ‘can’ treat our beliefs as being about the same thing without having to figure out the relevant identity. They argue that this is a matter of rational permissibility, and if that’s so, then mental files aren’t needed. 

Why would that be? Inferences are permissible because of the contents of the beliefs they involve. If reference isn’t enough to account for permissibility, as it appears not to be, then there must be some other feature that is. Following Kit Fine (2007) Goodman and Gray call this feature coordination. Coordination, like other representational features, can be attributed because it helps to make sense of what and how a thinker thinks what they do. But if we have this representational feature, which can be attributed to make sense of treating beliefs as being about the same thing, then files add nothing. You can have mental filing without mental files. The appeal of this result is that it seems to give a more refined picture of what is actually involved in rationally structured thinking, without unnecessary metaphors.

That is their argument, in outline. Does it work? The main problem is that it overlooks the psychology of mental filing. What calls for explanation is not just the permissibility of inferences that presuppose identity, but also the fact that thinkers are prepared to make them. Mental files can be brought in as entities whose function is to account for when a thinker is prepared to draw these inferences. The causal basis of that function might be described in other terms that tell us how the function is carried out. But as a hypothesis – that at some functional level there is something that brings together different bits of information and so provides the causal basis for a thinker being disposed to presuppose identity – mental files do the job nicely.

This causal-functional view of files shows that there is a notion of a mental file that does some work. This undermines Goodman and Grey’s argument because it renders coordination, the extra representational feature, redundant. Suppose we have the identity-presupposing dispositions in place, because of the presence of a mental file. Then the question is, do we need to add a representational feature (coordination) to make them permissible? It seems not. If a thinker is disposed to make the inferences, and nothing indicates that there is something faulty going on, then the inferences are in good standing. The results of the inferences will be (at least potentially) relied upon by the thinker in pursuing her plans and projects – things that matter to her. This means that, were there to be something to indicate that the conception in question was somehow incorrectly formed, then the thinker should be motivated to check the inferences she is disposed to draw. If she is rational, then she will do this, and so were there to be a problem, the inferences would not be made. Having the dispositions and monitoring their viability is enough for permissibility because it makes them manifestly reliable. So Goodman and Grey’s conclusion ought to be inverted: coordination doesn’t add anything that would otherwise be missing from the account that files provide. 

There is other work for files to do as well. Goodman and Grey suggest that the basis for coordination is a thinker reliably gathering information together that does concern one and the same object. But we need files for this to happen. Interpreting new information when finding out more about what objects are like calls upon the conceptions we already have. The content of a mental file will tell us how to locate the new information we get from various sources: in order to recognize something we’ve already encountered, or to determine (for example) that a new person we meet really is someone new and not an old acquaintance, we need to use information integrated by mental files.

Forming a picture of the world means gathering together these smaller-scale pictures of objects, and to do that, we need the kind of structure that mental files provide. They are not just a metaphor. But how exactly they work remains to be uncovered.

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  • Kit Fine (2007). Semantic Relationism. Blackwell.
  • Rachel Goodman and Aidan Grey (2022). “Mental Filing”. Noûs, 56(1), 204–226.

About the author

Henry Clarke is a Senior Project Editor in the Humanities at Oxford University Press. He received his PhD from UCL in 2016. His research focuses on the philosophy of mind.