On representational content and format in core numerical cognition

Ball, Brian

doi:10.1080/09515089.2016.1263988

Cited by 8 publications

(8 citation statements)

References 12 publications

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“…He writes, "I conjecture that the early Greeks articulated and formalized basic animal and childhood capacities when they theorized about magnitudes and ratios in a way that is unspecific as to whether the magnitudes are numbers or continuous quantities" (Burge 2010, p.483). 4 4 Buijsman (2021) endorses Burge's suggestion, and supplements it with an account of indeterminate vehicles to explain the ANS's imprecision. Buijsman (2021, p. 310) acknowledges that readers might wonder why he says that the ANS represents pure magnitudes rather than natural numbers, and replies that natural numbers "cannot be indeterminate" because, "There are no alternative choices for '1' as the unit value of the natural numbers which are equally good, whereas there are alternative choices for '1 cm' which are equally good, namely 1 inch, 1 meter, and so on."…”

Section: Number Vs Numerositymentioning

confidence: 99%

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The number sense represents (rational) numbers

Clarke¹,

Beck²

2021

Behav Brain Sci

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On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique—the arguments from congruency, confounds, and imprecision—and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g. 7), but also non-natural rational numbers (e.g. 3.5). It does not represent irrational numbers (e.g. √2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research.

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Section: Number Vs Numerositymentioning

confidence: 99%

“…), precise numbers with a confidence estimation attached (Halberda 2016), numerical intervals (5-9, 1.25-1.75, etc.) (Ball 2017), or probability distributions over numerical intervals. While interesting and important, these questions are orthogonal to whether the ANS represents natural, real, or rational numbers.…”

Section: What Kind(s) Of Number?mentioning

confidence: 99%

The number sense represents (rational) numbers

Clarke¹,

Beck²

2021

Behav Brain Sci

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“…For everything that has been said here, participants could be representing precise quantities somewhat inaccurately (e.g., sometimes representing there to be precisely 5 dots when presented with 6) or representing these precise quantities imprecisely (e.g., representing there to be 6ish dots when presented with 6 – Lyons 2021). Alternatively, they could involve subjects representing some numerical range (e.g., 5–7 dots – Ball 2017) or (on my preferred view) a numerical range with some probability distribution or confidence level attached (Halberda 2016). All these possibilities can accommodate the abovementioned results provided that (e.g.,) noise in the extraction of number suffices to explain the ANS's characteristic imprecision and conformity to Weber's Law (Clarke & Beck 2021b).…”

Section: A Philosopher's Guide To Approximate Number Representationmentioning

confidence: 99%

Compositionality and constituent structure in the analogue mind

Clarke

2023

Philosophical Perspectives

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I argue that analogue mental representations possess a canonical decomposition into privileged constituents from which they compose. I motivate this suggestion, and rebut arguments to the contrary, through reflection on the approximate number system, whose representations are widely expected to have an analogue format. I then argue that arguments for the compositionality and constituent structure of these analogue representations generalize to other analogue mental representations posited in the human mind, such as those in early vision and visual imagery.

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“…Lyons takes the ANS's imprecision to imply that it represents “approximate number” (e.g., 13ish), suggests that this is at odds with our proposal, and claims that this is something which cannot be “easily squared” with our suggestion that the ANS represents rational numbers – a conjecture which attributed “greater precision [to the ANS]… when what was needed was less.” But we suggested that the ANS might represent “numerical intervals (5–9, 1.25–1.75, etc.) (Ball, 2017), or probability distributions over numerical intervals.” Either option would involve the ANS referencing numbers, and be compatible with the representation of rational numbers. If Lyons has something else in mind by “approximate number” and “13ish,” it's not clear to us what it is.…”

Section: What Kind(s) Of Number?mentioning

confidence: 99%

Numbers, numerosities, and new directions

Clarke¹,

Beck²

2021

Behav Brain Sci

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In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system (ANS) represents numbers or numerosities, and why the ANS represents rational (but not irrational) numbers.

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On representational content and format in core numerical cognition

Cited by 8 publications

References 12 publications

The number sense represents (rational) numbers

The number sense represents (rational) numbers

Compositionality and constituent structure in the analogue mind

Numbers, numerosities, and new directions

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