Empirical discussions of mental representation appeal to a wide variety of representational kinds. Some of these kinds, such as the sentential representations underlying language use and the pictorial representations of visual imagery, are thoroughly familiar to philosophers. Others have received almost no philosophical attention at all. Included in this latter category are analogue magnitude representations, which enable a wide range of organisms to primitively represent spatial, temporal, numerical, and related magnitudes. This article aims to introduce analogue magnitude representations to (...) a philosophical audience by rehearsing empirical evidence for their existence and analysing their format, their content, and the computations they support. 1 Background1.1 Evidence of analogue magnitude representations1.2 Weber’s law1.3 Scepticism about analogue magnitude representations2 Format2.1 Carey’s analogy2.2 Neural realization2.3 Analogue representation2.4 Analogue magnitude representation components3 Content3.1 Do analogue magnitude representations have representational content?3.2 What do analogue magnitude representations represent?3.3 What content types do analogue magnitude representations have?4 Computations4.1 Arithmetic computation4.2 Practical deliberation5 Conclusion. (shrink)

In a recent paper by Casasanto and Pitt (2019), the authors addressed a debate regarding the role of order and magnitude in SNARC and SNARC‐like effects. Their position is that all these effects can be explained by order, while magnitude could only account for a subset of evidence. Although we agree that order can probably explain the majority of these effects, in this commentary we argue that magnitude is still relevant, since there is evidence that cannot be explained based on (...) ordinality alone. We argue that SNARC‐like effects can occur for magnitudes not clearly characterized by overlearned ordinality and that magnitude can prevail on order, when the two are pitted against each other. Finally, we propose that different interpretations of the role of order and magnitude depend on the interaction of stimulus properties and task demands. (shrink)

Quantities are naturally viewed as functions, whose arguments may be construed as situations, events, objects, etc. We explore the question of the range of these functions: should it be construed as the real numbers (or some subset thereof)? This is Carnap's view. It has attractive features, specifically, what Carnap views as ontological economy. Or should the range of a quantity be a set of magnitudes? This may have been Helmholtz's view, and it, too, has attractive features. It reveals the (...) close connection between measurement and natural law, it makes dimensional analysis intelligible, and explains the concern of scientists and engineers with units in equations. It leaves the philosophical problem of the relation between the structure of magnitudes and the structure of the reals. What explains it? And is it always the same? We will argue that on the whole, construing the values of quantities as magnitudes has some advantages, and that (as Helmholtz seems to suggest in "Numbering and Measuring from an Epistemological Viewpoint") the relation between magnitudes and real numbers can be based on foundational similarities of structure. (shrink)

ABSTRACT W.L. Craig has argued that the universe has a beginning because (1) the infinitude of the past entails the existence of actual infinite multitudes of past intervals of time, and (2) the existence of actual infinite multitudes is impossible. Puryear has rejected (1) and argued that what the infinitude of the past entails is only the existence of an actual infinite magnitude of past time. But this does not preclude the infinitude of the past, Puryear claims, because there can (...) be no justification for the claim that actual infinite magnitudes are impossible. I argue, against Puryear, that there can be such a justification. I claim, nevertheless, that, for reasons entirely different from Puryear’s, the finitude of the past cannot be established based either on the impossibility of actual infinite multitudes or on the impossibility of actual infinite magnitudes. My arguments in this paper draw on insights from al-Kindī and Avicenna. (shrink)

I am going to argue for a robust realism about magnitudes, as irreducible elements in our ontology. This realistic attitude, I will argue, gives a better metaphysics than the alternatives. It suggests some new options in the philosophy of science. It also provides the materials for a better account of the mind’s relation to the world, in particular its perceptual relations.

Analog representations come in different types. One distinction is between those representations that have parts that are themselves representations and those that do not (i.e., those for which the Parts Principle is true and those for which it is not). I offer a unified account of analog representation, showing what all types have in common. This account clarifies when the Parts Principle applies and when it does not, thereby illuminating why the Parts Principle is less interesting than one might have (...) thought. Understanding analog representation instead requires understanding the kinds of magnitudes used in a particular representation, and the kinds of variation possible within a representational scheme. (shrink)

After writing my response to the commentaries, I sat back and reflected on the fascination and frustration of work on this topic. There is the ancient fascination of trying to understand the nature of the sensory bridge linking us to the external world. Also, discussing the measurability of sensation brings to the surface concepts we use and take for granted when we are working in other areas of psychology; and it holds them before us for critical examination. The frustration lies (...) chiefly in the difficulty of formulating concepts clearly, in a way understood by all. At least when constructing the original Tower of Babel, workers hearing strange words knew that communication was not occurring. The psychophysical Tower of Babel sometimes has workers using the same words, but with different meanings. Much of my response is based on attempts to clarify concepts and clear up confusions. (shrink)

Cohen Kadosh & Walsh (CK&W) present convincing evidence indicating the existence of notation-specific numerical representations in parietal cortex. We suggest that the same conclusions can be drawn for a particular type of numerical representation: the representation of time. Notation-dependent representations need not be limited to number but may also be extended to other magnitude-related contents processed in parietal cortex (Walsh 2003).

W.L. Craig has argued that the universe has a beginning because (1) the infinitude of the past entails the existence of actual infinite multitudes of past intervals of time, and (2) the existence of actual infinite multitudes is impossible. Puryear has rejected (1) and argued that what the infinitude of the past entails is only the existence of an actual infinite magnitude of past time. But this does not preclude the infinitude of the past, Puryear claims, because there can be (...) no justification for the claim that actual infinite magnitudes are impossible. I argue, against Puryear, that there can be such a justification. I claim, nevertheless, that, for reasons entirely different from Puryear’s, the finitude of the past cannot be established based either on the impossibility of actual infinite multitudes or on the impossibility of actual infinite magnitudes. My arguments in this paper draw on insights from al-Kindī and Avicenna. (shrink)

In their target article, Rips et al. have presented the view that there is no necessary dependency between natural numbers and internal magnitude. However, they do not give enough weight to neuroimaging and neuropsychological studies. We provide evidence demonstrating that the acquisition of natural numbers depends on magnitude representation and that natural numbers develop from a general magnitude mechanism in the parietal lobes.

Continuous magnitude estimation and continuous cross-modality matching with line length can efficiently track the momentary loudness of time-varying sounds in behavioural experiments. These methods are known to be prone to systematic biases but may be checked for consistency using their counterpart, magnitude production. Thus, in Experiment 1, we performed such an evaluation for time-varying sounds. Twenty participants produced continuous cross-modality matches to assess the momentary loudness of fourteen songs by continuously adjusting the length of a line. In Experiment 2, the (...) resulting temporal line length profile for each excerpt was played back like a video together with the given song and participants were asked to continuously adjust the volume to match the momentary line length. The recorded temporal line length profile, however, was manipulated for segments with durations between 7 to 12 s by eight factors between 0.5 and 2, corresponding to expected differences in adjusted level of −10, −6, −3, −1, 1, 3, 6, and 10 dB according to Stevens’s power law for loudness. The average adjustments 5 s after the onset of the change were −3.3, −2.4, −1.0, −0.2, 0.2, 1.4, 2.4, and 4.4 dB. Smaller adjustments than predicted by the power law are in line with magnitude-production results by Stevens and co-workers due to “regression effects.” Continuous cross-modality matches of line length turned out to be consistent with current loudness models, and by passing the consistency check with cross-modal productions, demonstrate that the method is suited to track the momentary loudness of time-varying sounds. (shrink)

Given a rich environment, how do we decide on what information to use? A view of a single entity (e.g., a group of birds) affords many distinct interpretations, including their number, average size, and spatial extent. An enduring challenge for cognition, therefore, is to focus resources on the most relevant evidence for any particular decision. In the present study, subjects completed three tasks—number discrimination, surface area discrimination, and convex hull discrimination—with the same stimulus set, where these three features were orthogonalized. (...) Therefore, only the relevant feature provided consistent evidence for decisions in each task. This allowed us to determine how well humans discriminate each feature dimension and what evidence they relied on to do so. We introduce a novel computational approach that fits both feature precision and feature use. We found that the most relevant feature for each decision is extracted and relied on, with minor contributions from competing features. These results suggest that multiple feature dimensions are separately represented for each attended ensemble of many items and that cognition is efficient at selecting the appropriate evidence for a decision. (shrink)

Do people represent space, time, number, and other conceptual domains using a generalized magnitude system (GMS)? To answer this question, numerous studies have used the spatial‐numerical association of response codes (SNARC) task and its variants. Yet, for a combination of reasons, SNARC‐like effects cannot provide evidence for a GMS, even in principle. Rather, these effects support a broader theory of how people use space metaphorically to scaffold their understanding of myriad non‐spatial domains, whether or not these domains exhibit variation in (...) magnitude. (shrink)

In recent work, Christopher Peacocke has argued for a kind of realism (or anti-reductionism) about magnitudes such as temperature and spatial distance. Peacocke’s argument is that magnitudes are an ineliminable commitment of scientific and everyday explanations (including high-level explanations), and that they are the natural candidates for semantic values of our ordinary magnitude talk, and for contents of our mental states. I critique these arguments, in particular focusing on whether the realist has a satisfactory account of how high-level (...) magnitude facts are grounded in lower-level facts. I argue that a less realist (i.e., more reductionist approach) is preferable, or at least viable. I also aim to substantially clarify what is at stake in the debate. (shrink)

Extensive Magnitudes: Metaphysics, Representation and Epistemology.

A Theory of Magnitude suggests that space, time, and quantities are processed through a generalized magnitude system. ATOM posits that task-irrelevant magnitudes interfere with the processing of task-relevant magnitudes as all the magnitudes are processed by a common system. Many behavioral and neuroimaging studies have found support in favor of a common magnitude processing system. However, it is largely unknown whether such cross-domain monotonic mapping arises from a change in the accuracy of the magnitude judgments or results (...) from changes in precision of the processing of magnitude. Therefore, in the present study, we examined whether large numerical magnitude affects temporal accuracy or temporal precision, or both. In other words, whether numerical magnitudes change our temporal experience or simply bias duration judgments. The temporal discrimination paradigm was used to present numerical magnitudes across varied durations. We estimated temporal accuracy and precision for each numerical magnitude. The results revealed that temporal accuracy for large numerical magnitude was significantly lower than that of small and identical magnitudes. This implies that the temporal duration was overestimated for large numerical magnitude compared to small and identical numerical magnitude, in line with ATOM’s prediction. However, no influence of numerical magnitude was observed on temporal precision. The findings of the present study suggest that task-irrelevant numerical magnitude selectively affects the accuracy of processing of duration but not duration discrimination itself. Further, we argue that numerical magnitude may not directly affect temporal processing but could influence via attentional mechanisms. (shrink)

This paper addresses a number of closely related questions concerning Kant's model of intentionality, and his conceptions of unity and of magnitude [Gröβe]. These questions are important because they shed light on three issues which are central to the Critical system, and which connect directly to the recent analytic literature on perception: the issues are conceptualism, the status of the imagination, and perceptual atomism. In Section 1, I provide a sketch of the exegetical and philosophical problems raised by Kant's views (...) on these issues. I then develop, in Section 2, a detailed analysis of Kant's theory of perception as elaborated in both the Critique of Pure Reason and the Critique of Judgment; I show how this analysis provides a preliminary framework for resolving the difficulties raised in Section 1. In Section 3, I extend my analysis of Kant's position by considering a specific test case: the Axioms of Intuition. I contend that one way to make sense of Kant's argument is by juxtaposing it with Russell's response to Bradley's regress; I focus in particular on the concept of ‘unity’. Finally, I offer, in Section 4, a philosophical assessment of the position attributed to Kant in Sections 2 and 3. I argue that, while Kant's account has significant strengths, a number of key areas remain underdeveloped; I suggest that the phenomenological tradition may be read as attempting to fill precisely those gaps. (shrink)

Theoria, Volume 88, Issue 3, Page 629-652, June 2022.

I reply to comments by David Miguel Gray and Grant Gillett concerning my paper, ‘The Generality Constraint and the Structure of Thought’.

Leibovich et al. challenge the prevailing view that non-symbolic number sense is innate, that detection of numerosity is distinct from detection of continuous magnitude. In the present commentary, the authors' viewpoint is discussed in light of the integrative theory of numerical development along with implications for understanding mathematics disabilities.

ABSTRACT This article examines Bergson’s critique of intensive magnitude in Time and Free Will. I demonstrate how his rejection of a different kind of quantity that is ordinal and does not allow measurement, and the underlying strict dualism of quantity and quality, is inconsistent with both the letter and the spirit of his later philosophy. I dismantle two main strategies for explaining these inconsistencies. Furthermore, I argue that Bergson’s simplistic conception of quantity in terms of homogeneous multiplicity, which is operative (...) in his rejection of an alternative conception of quantity, lacks justification in the face of the transformations that the concept of quantity underwent in the history of mathematics and physics. (shrink)

The present study examined whether a dissociation among formats for rational numbers can be obtained in tasks that require comparing a number to a non-symbolic quantity. In Experiment 1, college students saw a discrete or else continuous image followed by a rational number, and had to decide which was numerically larger. In Experiment 2, participants saw the same displays but had to make a judgment about the type of ratio represented by the number. The magnitude task was performed more quickly (...) using decimals, whereas the relation task was performed more accurately with fractions. The pattern observed for percentages was very similar to that for decimals. A dissociation between magnitude comparison and relational processing with rational numbers can be obtained when a symbolic number must be compared to a non-symbolic display. (shrink)

Accumulating evidence suggests that different magnitudes (e.g., number, size, and duration) are spatialized in the mind according to a common left–right metric, consistent with a generalized system for representing magnitude. A previous study conducted by two of us (Holmes & Lourenco, ) provided evidence that this metric extends to the processing of emotional magnitude, or the intensity of emotion expressed in faces. Recently, however, Pitt and Casasanto () showed that the earlier effects may have been driven by a left–right (...) mapping of mouth size rather than emotional magnitude, and they found no evidence for an emotional magnitude mapping when using words as stimuli. Here, we report two new experiments that further examine these conclusions. In Experiment 1, using face stimuli with mouths occluded, we replicate the original finding: Less emotional faces were associated with the left and more emotional faces with the right. However, we also find that people can reliably infer the sizes of the occluded mouths, and that these inferred mouth sizes can explain the observed left–right mapping. In Experiment 2, we show that comparative judgments of emotional words yield a left–right mapping of emotional magnitude not attributable to stimulus confounds. Based on these findings, we concur with Pitt and Casasanto that faces pose challenges for isolating the forces driving spatialization, but we suggest that emotional magnitude, when assessed using unconfounded stimuli in a sufficiently sensitive task, may indeed be spatialized as originally proposed. Suggestions for further research on the spatialization of emotional magnitude are discussed. (shrink)