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)

This chapter presents a re-understanding of the contents of our analog magnitude representations (e.g., approximate duration, distance, number). The approximate number system (ANS) is considered, which supports numerical representations that are widely described as fuzzy, noisy, and limited in their representational power. The contention is made that these characterizations are largely based on misunderstandings—that what has been called “noise” and “fuzziness” is actually an important epistemic signal of confidence in one’s estimate of the value. Rather than the ANS having (...) noisy or fuzzy numerical content, it is suggested that the ANS has exquisitely precise numerical content that is subject to epistemic limitations. Similar considerations will arise for other analog representations. The chapter discusses how this new understanding of ANS representations recasts the learnability problem for number and the conceptual changes that children must accomplish in the number domain. (shrink)

We concur with Cohen Kadosh & Walsh (CK&W) that representation of numbers in the parietal cortex is format dependent. In addition, we suggest that all formats do not automatically, and equally, access analog magnitude representation in the intraparietal sulcus (IPS). Understanding how development, learning, and context lead to differential access of analog magnitude representation is a key question for future research.

The aim of this paper is to show that attention to Kant's philosophy of mathematics sheds light on the doctrine that there are two stems of the cognitive capacity, which are distinct, but equally necessary for cognition. Specifically, I argue for the following four claims: The distinctive structure of outer sensible intuitions must be understood in terms of the concept of magnitude. The act of sensibly representing a magnitude involves a special act of spontaneity Kant ascribes to a (...) capacity he calls the productive imagination. Contrary to what is assumed by many commentators, it is not the case that the Two Stems Doctrine implies that a representation is either sensible or spontaneity-dependent, but not both. Outer sensible intuitions are both sensible and spontaneity-dependent – they are sensible because they exhibit the kind of structure Kant takes to be distinctive of outer sensible intuitions, and they depend on spontaneity because they are cases of sensibly representing a magnitude. (shrink)

Extensive Magnitudes: Metaphysics, Representation and Epistemology.

According to Clarke and Beck (C&B), the approximate number system (ANS) represents numbers. We argue that the ANS represents pure magnitudes. Considerations of explanatory economy favor the pure magnitudes hypothesis. The considerations C&B direct against the pure magnitudes hypothesis do not have force.

Being able to estimate quantity is important in everyday life and for success in the STEM disciplines. However, people have difficulty reasoning about magnitudes outside of human perception. This study examines patterns of estimation errors across temporal and spatial magnitudes at large scales. We evaluated the effectiveness of hierarchical alignment in improving estimations, and transfer across dimensions. The activity was successful in increasing accuracy for temporal and spatial magnitudes, and learning transferred to the estimation of numeric magnitudes associated with events (...) and objects. However, there were also a number of informative differences in performance on temporal, spatial, and numeric magnitude measures, suggesting that participants possess different categorical information for these scales. Educational implications are discussed. (shrink)

Representational momentum is a well-known phenomenon that occurs when a moving object vanishes suddenly and the memory of its final or vanishing position is displaced forward in the direction of its motion. Many studies have shown evidence of various perceptual and cognitive characteristics of RM in various daily aspects, sports, development, and aging. Here we examined the longitudinal developmental changes in the displacement magnitudes of RM among younger and older nursery school children for pointing and judging tasks. In our experiments, (...) the children were asked to point at by their finger and judge the spatial location of the vanishing point of a moving stimulus. Our results showed that the mean magnitudes of RM significantly decreased from 5- to 6-year-old children for the pointing and judging tasks, although the mean magnitude of RM was significantly greater in the 5-year-old children for the pointing task but not for the judging task. We further examined the developmental changes in RM for a wide range of ages based on data from the present study and our previous study. This ad hoc examination showed that the magnitude of RM was significantly greater in 5-year-old children than in adults for the pointing and judging tasks. Our findings suggest that the magnitude of RM was significantly greater in young children than in adults and significantly decreased in young children through adults for the pointing and judging tasks. (shrink)

The issue of abstractness raises two distinct questions. First, is there a format-independent magnitude representation? Second, does analogue magnitude really play a crucial role in the development of human mathematics? We suggest that neither developmental nor cultural studies support this notion. The field needs to redefine the properties of the core number representation as used in human arithmetic.

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.

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)

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).

There is at present a lively debate in cognitive psychology concerning the origin of natural number concepts. At the center of this debate is the system of mental magnitudes, an innately given cognitive mechanism that represents cardinality and that performs a variety of arithmetical operations. Most participants in the debate argue that this system cannot be the sole source of natural number concepts, because they take it to represent cardinality approximately while natural number concepts are precise. In this paper, I (...) argue that the claim that mental magnitudes represent cardinality approximately overlooks the distinction between a magnitude and the increments that compose to form that magnitude. While magnitudes do indeed represent cardinality approximately, they are composed of a precise number of increments. I argue further that learning the number words and the counting routine may allow one to mark in memory the number of increments that composed to form a magnitude, thereby creating a precise representation of cardinality. (shrink)

Peter Godfrey-Smith recently introduced the idea of representational ‘organization’. When a collection of representations form an organized family, similar representational vehicles carry similar contents. For example, where neural firing rate represents numerosity (an analogue magnitude representation), similar firing rates represent similar numbers of items. Organization has been elided with structural representation, but the two are in fact distinct. An under-appreciated merit of representational organization is the way it facilitates computational processing. Representations from different organized families can interact, (...) for example to perform addition. Their being organized allows them to implement a useful computation. Many of the cases where organization has seemed significant, but which fall short of structural representation, are cases where representational organization underpins a computationally useful processing structure. (shrink)

The ability to represent, discriminate, and perform arithmetic operations on discrete quantities (numerosities) has been documented in a variety of species of different taxonomic groups, both vertebrates and invertebrates. We do not know, however, to what extent similarity in behavioral data corresponds to basic similarity in underlying neural mechanisms. Here, we review evidence for magnitude representation, both discrete (countable) and continuous, following the sensory input path from primary sensory systems to associative pallial territories in the vertebrate brains. We (...) also speculate on possible underlying mechanisms in invertebrate brains and on the role played by modeling with artificial neural networks. This may provide a general overview on the nervous system involvement in approximating quantity in different animal species, and a general theoretical framework to future comparative studies on the neurobiology of number cognition. (shrink)

This paper shows how two models of precedential constraint can be broadened to include legal information represented through dimensions. I begin by describing a standard representation of legal cases based on boolean factors alone, and then reviewing two models of constraint developed within this standard setting. The first is the “result model”, supporting only a fortiori reasoning. The second is the “reason model”, supporting a richer notion of constraint, since it allows the reasons behind a court’s decisions to be (...) taken into account. I then show how the initial representation can be modified to incorporate dimensional information and how the result and reason models can be adapted to this new dimensional setting. As it turns out, these two models of constraint, which are distinct in the standard setting, coincide once they are transposed to the new dimensional setting, yielding exactly the same patterns of constraint. I therefore explore two ways of refining the reason model of constraint so that, even in the dimensional setting, it can still be separated from the result model. (shrink)

Rips et al. consider whether representations of individual objects or analog magnitudes are building blocks for the concept natural number. We argue for a third core capacity – the ability to bind representations of individuals into sets. However, even with this addition to the list of starting materials, we agree that a significant acquisition story is needed to capture natural number.

In this paper, representational structures of arithmetical thinking, encoded in human minds, are described. On the basis of empirical research, it is possible to distinguish four types of mental number lines: the shortest mental number line, summation mental number lines, point-place mental number lines and mental lines of exact numbers. These structures may be treated as generative mechanisms of forming arithmetical representations underlying our numerical acts of reference towards cardinalities, ordinals and magnitudes. In the paper, the theoretical framework for a (...) formal model of mental arithmetical representations is constructed. Many competitive conceptions of the mental system responsible for our arithmetical thinking may be unified within the presented framework. The paradigm underlying our research may be interpreted philosophically as a neo-Kantian approach to modeling the mind’s representational structures. (shrink)

People implicitly associate different emotions with different locations in left-right space. Which aspects of emotion do they spatialize, and why? Across many studies people spatialize emotional valence, mapping positive emotions onto their dominant side of space and negative emotions onto their non-dominant side, consistent with theories of metaphorical mental representation. Yet other results suggest a conflicting mapping of emotional intensity (a.k.a., emotional magnitude), according to which people associate more intense emotions with the right and less intense emotions with (...) the left — regardless of their valence; this pattern has been interpreted as support for a domain-general system for representing magnitudes. To resolve the apparent contradiction between these mappings, we first tested whether people implicitly map either valence or intensity onto left-right space, depending on which dimension of emotion they attend to (Experiments 1a, b). When asked to judge emotional valence, participants showed the predicted valence mapping. However, when asked to judge emotional intensity, participants showed no systematic intensity mapping. We then tested an alternative explanation of findings previously interpreted as evidence for an intensity mapping (Experiments 2a, b). These results suggest that previous findings may reflect a left-right mapping of spatial magnitude (i.e., the size of a salient feature of the stimuli) rather than emotion. People implicitly spatialize emotional valence, but, at present, there is no clear evidence for an implicit lateral mapping of emotional intensity. These findings support metaphor theory and challenge the proposal that mental magnitudes are represented by a domain-general metric that extends to the domain of emotion. (shrink)

In the first Critique, Kant attempts to prove what we can call the "Principle of Intensive Magnitudes," according to which every possible object of experience will possess a determinate "degree" of reality. Curiously, Kant argues for this principle by inferring from a psychological premise about internal sensations (they have intensive magnitudes) to a metaphysical thesis about external objects (they also have intensive magnitudes). Most commentators dismiss the argument as a failure. In this article I give a reconstruction of Kant's argument (...) that attempts to rehabilitate the argument back into his broader transcendental theory of experience. I argue that we can make sense of the argument's central inference by appeal to Kant's theory of empirical intuition and by an analysis of the way in which Kant thinks sensory matter constitutes our most basic representations of objects. (shrink)

Legal Judgment Prediction (LJP) is an essential component of legal assistant systems, which aims to automatically predict judgment results from a given criminal fact description. As a vital subtask of LJP, researchers have paid little attention to the numerical LJP, i.e., the prediction of imprisonment and penalty. Existing methods ignore numerical information in the criminal facts, making their performances far from satisfactory. For instance, the amount of theft varies, as do the prison terms and penalties. The major challenge is how (...) the model can obtain the ability of numerical comparison and magnitude perception, e.g., 400 < 500 < 800, 500 is closer to 400 than to 800. To this end, we propose a judicial knowledge-enhanced magnitude-aware reasoning architecture, called NumLJP, for the numerical LJP task. Specifically, we first implement a contrastive learning-based judicial knowledge selector to distinguish confusing criminal cases efficiently. Unlike previous approaches that employ the law article as external knowledge, judicial knowledge is a quantitative guideline in real scenarios. It contains many numerals (called anchors) that can construct a reference frame. Then we design a masked numeral prediction task to help the model remember these anchors to acquire legal numerical commonsense from the selected judicial knowledge. We construct a scale-based numerical graph using the anchors and numerals in facts to perform magnitude-aware numerical reasoning. Finally, the representations of fact description, judicial knowledge, and numerals are fused to make decisions. We conduct extensive experiments on three real-world datasets and select several competitive baselines. The results demonstrate that the macro-F1 of NumLJP improves by at least 9.53% and 11.57% on the prediction of penalty and imprisonment, respectively. (shrink)

The Approximate Number System (ANS) is a system that allows us to distinguish between collections based on the number of items, though only if the ratio between numbers is high enough. One of the questions that has been raised is what the representations involved in this system represent. I point to two important constraints for any account: (a) it doesn’t involve numbers, and (b) it can account for the approximate nature of the ANS. Furthermore, I argue that representations of pure (...) magnitude with vehicles that have an imprecision in the value of the unit of measurement (further clarified through a formal model from measurement theory) fit both these requirements. (shrink)

Carey has argued that there is a system of core numerical cognition – the analog magnitude system – in which cardinal numbers are explicitly represented in iconic format. While the existence of this system is beyond doubt, this paper aims to show that its representations cannot have the combination of features attributed to them by Carey. According to the argument from abstractness, the representation of the cardinal number of a collection of individuals as such requires the representation (...) of individuals as such, and this in turn requires non-iconic format, from which it is concluded that the explicit representation of the cardinal number of some individuals requires non-iconic representational format. In support of the first premise, an account is given of what approximate cardinal numbers might be, and in support of the second, a direct argument is articulated and defended. Finally, in response to an objection, a second argument for the central thesis is provided. While the discussion is couched in the terms of Carey’s work, the considerations it adduces are perfectly general, and the conclusion should therefore be taken into consideration by all those aiming to characterize the AM system. (shrink)

Qualitative Reasoning (QR) is an area of research within Artificial Intelligence that automates reasoning and problem solving about the physical world. QR research aims to deal with representation and reasoning about continuous aspects of entities without the kind of precise quantitative information needed by conventional numerical analysis techniques. Order-of-magnitude Reasoning (OMR) is an approach in QR concerned with the analysis of physical systems in terms of relative magnitudes. In this paper we consider the logic OMR_N for order-of-magnitude (...) reasoning with the bidirectional negligibility relation. It is a multi-modal logic given by a Hilbert-style axiomatization that reflects properties and interactions of two basic accessibility relations (strict linear order and bidirectional negligibility). Although the logic was studied in many papers, nothing was known about its decidability. In the paper we prove decidability of OMR N by showing that the logic has the strong finite model property. (shrink)

Numerical magnitude information is assumed to be spatially represented in the form of a mental number line defined with respect to a body-centred, egocentric frame of reference. In this context, spatial language skills such as mastery of verbal descriptions of spatial position have been proposed to be relevant for grasping spatial relations between numerical magnitudes on the mental number line. We examined 4- to 5-year-old’s spatial language skills in tasks that allow responses in egocentric and allocentric frames of reference, (...) as well as their relative understanding of numerical magnitude. In addition, we evaluated influences of children’s absolute understanding of numerical magnitude assessed by their number word comprehension and of their knowledge on numerical sequences. Results indicated that when considering responses that corresponded to the egocentric perspective, children’s spatial language was associated significantly with their relative numerical magnitude understanding, even after controlling for covariates, such as children’s SES, mental rotation skills, and also absolute magnitude understanding or knowledge on numerical sequences. This suggests that the use of egocentric reference frames in spatial language may facilitate spatial representation of numbers along a mental number line and thus seem important for preschoolers’ relative understanding of numerical magnitude. (shrink)

Recent research in numerical cognition has begun to systematically detail the ability of humans and nonhuman animals to perceive the magnitudes of nonsymbolic ratios. These relationally defined analogs to rational numbers offer new potential insights into the nature of human numerical processing. However, research into their similarities with and connections to symbolic numbers remains in its infancy. The current research aims to further explore these similarities by investigating whether the magnitudes of nonsymbolic ratios are associated with space just as symbolic (...) numbers are. In two experiments, we found that responses were faster on the left for smaller nonsymbolic ratio magnitudes and faster on the right for larger nonsymbolic ratio magnitudes. These results further elucidate the nature of nonsymbolic ratio processing, extending the literature of spatial–numerical associations to nonsymbolic relative magnitudes. We discuss potential implications of these findings for theories of human magnitude processing in general and how this general processing relates to numerical processing. (shrink)

We argue that number principles may be learnable instead of innate, by suggesting that children acquire probabilistically true number concepts rather than algorithms. We also suggest that non-propositional representational formats (e.g., mental models) may implicitly provide information that supports the induction of numerical principles. Given probabilistically true number concepts, the problem of the acquisition of mathematical principles is eliminated.

Whether negative numbers have a fixed spatial–numerical association of response codes effect, and whether the spatial representation of negative numbers is associated with negative numbers’ absolute or signed values remains controversial. In this study, through three experiments, the coding level of the magnitude and the spatial-direction is manipulated. In the first experiment, participants are required to code the magnitude and spatial-direction explicitly by using a magnitude classification task. In the second experiment, participants are forced to code (...) the magnitude implicitly as well as to code the spatial-direction explicitly by utilizing a cuing task. In the third experiment, participants are obliged to code the magnitude explicitly as well as to code the spatial-direction implicitly by adopting a magnitude and arrow-direction classification tasks with Go/No-Go responses. The results show that the absolute value of negative numbers associates with space when the magnitude of negative numbers is explicitly coded, no matter employing the explicit or implicit spatial-direction; the signed value of negative numbers associates with space under the condition of implicit magnitude as well as explicit spatial-direction. In conclusion, the current study indicates that the SNARC effect of negative numbers is variable in different conditions, and the type of SNARC effect about negative numbers is modulated by the joint coding level of the magnitude and spatial-direction. (shrink)

A dual-code model of number processing needs to take into account the difference between a number symbol and its meaning. The transition of automatic non-abstract number representations into intentional abstract representations could be conceptualized as a translation of perceptual asemantic representations of numerals into semantic representations of the associated magnitude information. The controversy about the nature of number representations should be thus related to theories on embodied grounding of symbols.

To investigate mechanisms of rational representation, I consider construction of an ordered continuum of psychophysical scale of magnitude of sensation; counting mechanism leading to an approximate numerosity scale for integers; and conjoint measurement structure pitting the denominator against the numerator in tradeoff positions. Number sense of resulting rationals is neither intuitive nor expedient in their manipulation.

Leibovich et al. propose that non-symbolic numerosity abilities develop from the processing of more basic, continuous magnitudes such as size, area, and density. Here I review similar arguments arising in the visual perception field and further propose that the evolvement of discrete representations from continuous stimulus properties may be a fundamental characteristic of cognitive development.

Cohen Kadosh & Walsh (CK&W) neglect the solid empirical evidence for a convergence of notation-specific representations onto a shared representation of numerical magnitude. Subliminal priming reveals cross-notation and cross-modality effects, contrary to CK&W's prediction that automatic activation is modality and notation-specific. Notation effects may, however, emerge in the precision, speed, automaticity, and means by which the central magnitude representation is accessed.

Repetition priming of object identification refers to the phenomenon whereby experience with an object induces systematic changes in subsequent processing of that same object. This data-driven form of priming is distinct from conceptually-driven priming. To date, considerable controversy exists about whether data-driven priming reflects facilitation in perceptual processing or mediation by preexisting object representations. The present study concerned priming of recognizing familiar and unfamiliar faces and how this priming is influenced by face inversion, which interferes with perceptual face processing. Perceptual (...) and representation-based loci conjointly contributed to priming; the perceptual locus was operative similarly for familiar and unfamiliar faces, whereas the representation-based locus was only invoked for familiar faces and resulted in a response-time reduction triple the magnitude of that from the perceptual locus. The results constrain theoretical accounts of data-driven priming by indicating that improved identification can result from the combination of perceptual and representation-based facilitation. (shrink)

Arguments against naturalistic style accounts of representations in humans and other animals would be obviated if scepticism concerning their conclusion could be justified. One such justification consists in showing, in detail, that the concept of representation has a purchase among 'non-linguistic' animals. Thereby the existence of natural or 'intrinsic' intentionality is secured. Four levels of explanation can be distinguished in the study of animal behavior and capacity rely on attributions of representations to animals (to what N. Humphrey calls 'nature's (...) psychologists'). Such explanations cite factors which do not covary directly with physical magnitudes in the stimuli. Additional animal studies are analysed, examining (1) the claim that not all representations are 'linguistic' (the Two Representations Hypothesis), (2) categorization in animals, (3) 'self-representation' in animals, (4) animal deception, and (5) comparative and weighting representations in animal learning and behavior. (shrink)

Arguments against naturalistic style accounts of representations in humans and other animals would be obviated if scepticism concerning their conclusion could be justified. One such justification consists in showing, in detail, that the concept of representation has a purchase among 'non-linguistic' animals. Thereby the existence of natural or 'intrinsic' intentionality is secured. Four levels of explanation can be distinguished in the study of animal behavior and capacity rely on attributions of representations to animals (to what N. Humphrey calls 'nature's (...) psychologists'). Such explanations cite factors which do not covary directly with physical magnitudes in the stimuli. Additional animal studies are analysed, examining (1) the claim that not all representations are 'linguistic' (the Two Representations Hypothesis), (2) categorization in animals, (3) 'self-representation' in animals, (4) animal deception, and (5) comparative and weighting representations in animal learning and behavior. (shrink)

The theory put forward by Cohen Kadosh & Walsh (CK&W) proposing that semantic representations of numerical magnitude in the parietal cortex are format-specific, does not specify how these representations might be constructed over the course of learning and development. The developmental predictions of the non-abstract theory are discussed and the need for a developmental perspective on the abstract versus non-abstract question highlighted.

The extent to which visual processing can proceed in the visual hierarchy without awareness determines the magnitude of perceptual delay. Increasing data demonstrate that primary visual cortex (V1) is involved in consciousness, constraining the magnitude of visual delay. This makes it possible that visual delay is actually within the optimal lengths to allow sufficient computation; thus it might be unnecessary to compensate for visual delay.

We investigated whether a pre-change representation is inhibited or weakened under correct change detection. Two arrays of six objects were rapidly presented for change detection in three experiments. After detection, the perceptual identification of degraded stimuli was tested in Experiments 1 and 2. The weakening of a pre-change representation was not observed under correct detection. The repetition priming effect was observed for a pre-change object and the magnitude was equivalent to the effect for a post-change object. Under (...) change blindness, repetition priming for a pre-change representation was observed when detection did not require report of location in Experiment 1 and was not observed when location was required to be reported in Experiment 2. The results of Experiment 3 showed that a pre-change representation was recognized at a higher rate under correct detection than under change blindness, reflecting a stronger rather than a weaker pre-change representation in the former context. (shrink)

We agree that the default numerical representation is best accessed by probing automatic processing. The locus of this representation is apparently at the horizontal intraparietal sulcus (HIPS), the convergence zone of magnitude information. The parietal lobes are the right place to look for non-abstract representation of magnitude, yet the proof for that is still to be found.