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)

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 (...) class='Hi'>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)

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)

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.

Extensive Magnitudes: Metaphysics, Representation and Epistemology.

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)

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)

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)

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)

For Kant, inquiry into nature requires seeking to explain all material wholes merely mechanically, in terms of their parts. There is no consensus on how he justifies this Principle of Mechanism. I argue that Kant seeks to derive this claim about part and wholes neither from his laws of mechanics, nor from the mere discursivity of our understanding (two standard options in the literature), but instead from a priori principles laid out in the first Critique, which govern parts, wholes, and (...) magnitudes. These principles are also fundamental to Kant’s account of mathematics. Therefore, Kant’s Principle of Mechanism and his philosophy of mathematics have common foundations. (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.

This paper examines the history of two related problems concerning earthquakes, and the way in which a theoretical advance was involved in their resolution. The first problem is the development of a physical, as opposed to empirical, scale for measuring the size of earthquakes. The second problem is that of understanding what happens at the source of an earthquake. There was a controversy about what the proper model for the seismic source mechanism is, which was finally resolved through advances in (...) the theory of elastic dislocations. These two problems are linked, because the development of a physically based magnitude scale requires an understanding of what goes on at the seismic source. I will show how the theoretical advances allowed seismologists to re-frame the questions they were trying to answer, so that the data they gathered could be brought to bear on the problem of seismic sources in new ways. (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)

The current consensus among most researchers is that natural number is not built solely upon a foundation of mental magnitudes. On their way to the conclusion that magnitudes do not form any part of that foundation, Rips et al. pass rather quickly by theories suggesting that mental magnitudes might play some role. These theories deserve a closer look.

For Kant, inquiry into nature properly requires seeking to explain all material wholes merely mechanically, in terms of their parts. There is no consensus on how he justifies this Principle of Mechanism. I argue that Kant seeks to derive this claim about part and wholes neither from his laws or mechanics, nor from the mere discursivity of our understanding (two standard options in the literature), but instead from a priori principles laid out in the first Critique, which govern parts, wholes, (...) and magnitudes. These principles are also fundamental to Kant’s account of mathematics. Therefore, Kant’s Principle of Mechanism and his philosophy of mathematics have common foundations. (shrink)

Although most deaf individuals could use sign language or sign/spoken language mix, hearing loss would still affect their language acquisition. Compensatory plasticity holds that the lack of auditory stimulation experienced by deaf individuals, such as congenital deafness, can be met by enhancements in visual cognition. And the studies of hearing individuals have showed that visual form perception is the cognitive mechanism that could explain the association between numerical magnitude processing and arithmetic computation. Therefore, we examined numerical magnitude processing (...) and its contribution to arithmetical ability in deaf adolescents, and explored the differences between the congenital and acquired deafness. 112 deaf adolescents and 58 hearing adolescents performed a series of cognitive and mathematical tests, and it was found there was no significant differences between the congenital group and the hearing group, but congenital group outperformed acquired group in numerical magnitude processing and arithmetic computation. It was also found there was a close association between numerical magnitude processing and arithmetic computation in all deaf adolescents, and after controlling for the demographic variables and general cognitive abilities, numerical magnitude processing could predict arithmetic computation in all deaf adolescents but not in congenital group. The role of numerical magnitude processing in deaf adolescents' mathematical performance should be paid attention in the training of arithmetical ability. (shrink)

The young Marx once remarked that political economy finds itself in an estranged form and is therefore in desperate need of a critical reconstruction of its object [Gegenstand]. He proposed a complete deconstruction of economic objectivity and its categories, hoping to recover the true species-life of man. In the article, we assert that contemporary economic theory remains confined by this estrangement, despite managing to ‘revolutionize’ itself out of the grip of classical political economy. The subjectivist-marginalist reliance on ‘measurable’ consumer preferences (...) not only solidified the discipline’s estrangement, but also wrested away any remaining basic principles of economics through neoclassical reconceptualization. A break with estrangement would require novel critical economic thinking that would do away with the discontinuity between classical and neoclassical economics. It would therefore need a rich enough framework to scrutinize its principal categories. We argue that Alain Badiou’s objective phenomenology possesses a complementary synchronic structure able to conform to basic economic tenets, allowing for a comparative and synthetic approach. This would then be the basis for a new model of economic theorizing. We conclude the article with Marx’s value form, seeing it as a possible central category of a newly proposed economic framework. (shrink)

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

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)

In an age characterized by impersonality and a fear of individuality this book is indeed unusual. It is personal, individualistic and idiosyncratic - a record of the scientific adventure of a single mind. Most scientific writing today is so depersonalized that it is impossible to recognize the man behind the work, even when one knows him. Costa de Beauregard's scientific career has focused on three domains - special relativity, statistics and irreversibility, and quantum mechanics. In Time, the Physical Magnitude (...) he has provided a personal vade mecum to those problems, concepts, and ideas with which he has been so long preoccupied. Some years ago we were struck by a simple and profound observa tion of Mendel Sachs, the gist of which follows. Relativity is based on very simple ideas but, because it requires highly complicated mathe matics, people find it difficult. Quantum mechanics, on the other hand, derives from very complicated principles but, since its mathematics is straightforward, people feel they understand it. In some ways they are like the bourgeois gentilhomme of Moliere in that they speak quantum mechanics without knowing what it is. Costa de Beauregard recognizes the complexity of quantum mechanics. A great virtue of the book is that he does not hide or shy away from the complexity. He exposes it fully while presenting his ideas in a non-dogmatic way. (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)