Numerical ordering ability mediates the relation between number-sense and arithmetic competence

What predicts human mathematical competence? While detailed models of number representation in the brain have been developed, it remains to be seen exactly how basic number representations link to higher math abilities. We propose that representation of ordinal associations between numerical symbols is one important factor that underpins this link. We show that individual variability in symbolic number-ordering ability strongly predicts performance on complex mental-arithmetic tasks even when controlling for several competing factors, including approximate number acuity. Crucially, symbolic number-ordering ability fully mediates the previously reported relation between approximate number acuity and more complex mathematical skills, suggesting that symbolic number-ordering may be a stepping stone from approximate number representation to mathematical competence. These results are important for understanding how evolution has interacted with culture to generate complex representations of abstract numerical relationships. Moreover, the finding that symbolic number-ordering ability links approximate number acuity and complex math skills carries implications for designing math-education curricula and identifying reliable markers of math performance during schooling.     

How can we detect when language emerged?

Views differ radically as to how deep the roots of language lie in human phylogeny, largely because prior to the development of writing systems, this striking human attribute has to be inferred from indirect proxies preserved in the material record. Here I argue that the most appropriate such archaeological proxies encode the modern human symbolic cognitive system from which language emerges. Throughout the 2.5 million years or more for which an archaeological record has existed, change has been both sporadic and rare—until symbolic objects and behaviors begin to appear, well within the tenure of our highly apomorphic species Homo sapiens. I propose that the biology underwriting our unusual cognitive and linguistic systems was acquired in the major developmental reorganization that gave rise to our anatomically distinctive species around 200,000 years ago in Africa. However, the material record indicates that this new potential lay fallow for around 100,000 years, following which it was released by what was necessarily a behavioral stimulus. By far the best candidate for that stimulus is the spontaneous invention of language, which is plausibly underwritten by a relatively simple mental algorithm, and could readily have spurred symbolic cognitive processes in a feedback process. None of this means that earlier hominid vocal communication systems were not complex, or that extinct hominid species were not highly intelligent. But it does emphasize the qualitative distinctiveness of both modern symbolic cognition and language.     

Persons, punishment, and free will skepticism

The purpose of this paper is to provide a justification of punishment which can be endorsed by free will skeptics, and which can also be defended against the “using persons as mere means” objection. Free will skeptics must reject retributivism, that is, the view that punishment is just because criminals deserve to suffer based on their actions. Retributivists often claim that theirs is the only justification on which punishment is constrained by desert, and suppose that non-retributive justifications must therefore endorse treating the people punished as mere means to social ends. Retributivists typically presuppose a monolithic conception of desert: they assume that action-based desert is the only kind of desert. But there are also personhood-based desert claims, that is, desert claims which depend not on facts about our actions, but instead on the more abstract fact that we are persons. Since personhood-based desert claims do not depend on facts about our actions, they do not depend on moral responsibility, so free will skeptics can appeal to them just as well as retributivists. What people deserve based on the mere fact of their personhood is to be treated as they would rationally consent to be treated if all they had in view was the mere fact of their personhood. We can work out the implications of this view for punishment by developing a “hypothetical consent” justification in which we select principles of punishment in the Rawlsian original position, so long as we are careful not to smuggle in the retributivist assumption that it is under our control whether we end up as criminals or as law-abiding citizens once we raise the veil of ignorance.     

CONSCIOUSNESS FOR THE TWENTY-FIRST CENTURY

Abstract. Human action and experience are the outcome of genes and memes. Not only are both of these represented in consciousness, but consciousness mediates their claims and thus governs our choices. Hence it is important how consciousness is ordered and where it is directed. Sorokin's typology of the sensate and the ideational ("spiritual"), and the dialectic between them, is relevant to this issue. In our period of history, the sensate factors of materialism and secularism need to be dialectically counterbalanced by the reinforcement of memes that value the spiritual intimations of the realm beyond the senses. As we approach the twenty-first century, the memes that will undergird our spirituality will be those that resacralize nature and emphasize our unity as humans with all of universal reality, in an idea of common "beinghood." Spiritual systems that accord with this trend in evolution will have to respect three conditions. They will (1) integrate the sensate and the ideational; (2) reflect the importance of the "flow" state of optimal experience, which matches ever-complexifying skills with comparable challenges; and (3) move the fulcrum of their worldview from the human being to the network of beings and its evolution.     

Computer models of cognitive processes

Conclusion From the small sample of achievements that I have had time to mention, we can only conclude that automation is here to stay. Nor is there any doubt that more powerful automata will be built. A great many of the higher human abilities will be given to machines. The great rush to automation is sure to stimulate psychologists to learn more about the human symbolic processes being mimicked by the machines. And the computers, which are the ultimate cause of the feverish scramble toward automation, are providing both the framework for describing complex models of behavior and also the means for testing these models. With both the means and the motivation at hand, psychologists are sure to make rapid progress in understanding complex human behavior.Operated with support from the United States Army, Navy, and Air Force.     

The potential of the visual image at the interface

This article shows how the study of graphic interfacing has great potential for being an important applied domain for testing, evaluating, and implementing basic cognitive theories. It also describes various applications in which the inclusion of computer graphics can be beneficial while being inappropriate for others. Emphasis is placed on the need for future developments to be based on psychological and ergonomic principles in order that the merit of computer graphics is not based purely on cosmetic grounds. Concomitantly, it is argued that academic research needs also to continue refining our understanding of human performance and subsequently to direct these findings into a set of practical guidelines for designers. If this approach is taken, we may indeed find ourselves in an Information Age in which the visual image is seen to be playing a much more prominent role. In particular, the situation may arise in which tasks previously performed only through difficult and tedious programming language will become more accessible through easy-to-use and enjoyable graphic interfaced systems. The ultimate challenge, however, will be to see how the combination of visual and other forms of symbolic communication can be utilized at the interface, in such a way that a variety of computer systems for a diversity of applications can be designed that will be readily accessible to a greater number. Ours is the visual age. We are bombarded with images from morning till night. —Gombrich (1982)     

Representations of numerical and non‐numerical magnitude both contribute to mathematical competence in children

A growing body of evidence suggests that non‐symbolic representations of number, which humans share with nonhuman animals, are functionally related to uniquely human mathematical thought. Other research suggesting that numerical and non‐numerical magnitudes not only share analog format but also form part of a general magnitude system raises questions about whether the non‐symbolic basis of mathematical thinking is unique to numerical magnitude. Here we examined this issue in 5‐ and 6‐year‐old children using comparison tasks of non‐symbolic number arrays and cumulative area as well as standardized tests of math competence. One set of findings revealed that scores on both magnitude comparison tasks were modulated by ratio, consistent with shared analog format. Moreover, scores on these tasks were moderately correlated, suggesting overlap in the precision of numerical and non‐numerical magnitudes, as expected under a general magnitude system. Another set of findings revealed that the precision of both types of magnitude contributed shared and unique variance to the same math measures (e.g. calculation and geometry), after accounting for age and verbal competence. These findings argue against an exclusive role for non‐symbolic number in supporting early mathematical understanding. Moreover, they suggest that mathematical understanding may be rooted in a general system of magnitude representation that is not specific to numerical magnitude but that also encompasses non‐numerical magnitude.     

Archetypes Reconsidered as Emergent Outcomes of Cognitive Complexity and Evolved Motivational Systems

Advances in contemporary cognitive science suggest that our internal representational systems are powerfully shaped by interacting evolutionary, developmental, and neuro-computational processes. Although Jung’s archetypes of the collective unconscious are largely dismissed by modern psychological science, something very much like them emerges from the intersection of these perspectives. Functional analysis suggests that a variety of conserved systems—basic biological ones, like self-protection and mating, as well as more complex social ones, like cheater-detection—need to make use of more general representational systems (like face perception) to simulate and predict adaptive responses to recurring environmental problems. Furthermore, analogous to the capacity to develop language, these systems depend on specific input at critical developmental stages. Archetypes reflect the interaction of domain-specific challenges and domain-general simulations. They are dynamic patterns of perception, memory, and action, resonating with ancient motivational and emotional systems. They shed light on how the symbolic emerges from the subsymbolic. Archetypes are thus the natural consequence of our fundamental social goals playing out in three nested dynamics: the online representation of reality by mental simulation systems, the history of personal experiences that build a particular instantiation of these systems, and the evolutionary dynamics that selected the web of cognitive and affective capacities that all normally developing humans share. This modern elaboration of the idea of archetypes fuses disparate conceptual perspectives, provokes methodological reorientations, generates novel hypotheses, and will likely open whole new lines of integrative inquiry.     

The Explicit and Implicit Domains in Psychoanalytic Change

New research findings of the development and organization of the mind, brain, and behavior bolster the ongoing relational- or intersubjective-field paradigmatic revision of psychoanalytic theory. A multisystems view of learning, memory, and knowledge provide us with a more complex picture of information processing that has fundamental implications for a psychoanalytic theory of therapeutic action.

If the implicit and explicit learning/memory systems are viewed as parallel processes, not easily translatable from one to the other, then new implicit relational experience carries considerably more power as compared to explicit/declarative processing in changing and establishing new implicit mental models. When these cognitive processing systems are viewed as more closely interconnected with a developmental emphasis on connecting them through language, then exploratory/interpretive work becomes more central. REM, dream, infant, and cognitive research evidence suggests that imagistic symbolic capacity exists at birth. Learning and remembering using imagistic symbolic processing could suggest a more easily translatable connection with later developed verbal symbolic processing and a closer, although varied, interconnection between implicit and explicit (symbolic) memory systems.

Explicit attitudes are more directly modifiable through an explicit/declarative focus. Implicit mental models, it is proposed, are variably modifiable through two different change processes: (1) accommodation or transformation of expectancies through new implicit procedural experience (not requiring explicit focus) and (2) diminished activation and increased capacity to deactivate implicit mental models through explicit/declarative processing and the establishment of contrasting implicit models through new implicit procedural experience.     

Category learning and multiple memory systems

Categorization is a vitally important skill that people use every day. Early theories of category learning assumed a single learning system, but recent evidence suggests that human category learning may depend on many of the major memory systems that have been hypothesized by memory researchers. As different memory systems flourish under different conditions, an understanding of how categorization uses available memory systems will improve our understanding of a basic human skill, lead to better insights into the cognitive changes that result from a variety of neurological disorders, and suggest improvements in training procedures for complex categorization tasks.     

The Value of Symbolic Computation

Standard generative linguistic theory, which uses discrete symbolic models of cognition, has some strengths and weaknesses. It is strong on providing a network of outposts that make scientific travel in the jungles of natural language feasible. It is weak in that it currently depends on the elaborate and unformalized use of intuition to develop critical supporting assumptions about each data point. In this regard, it is not in a position to characterize natural language systems in the lawful terms that ecological psychologists strive for. Connectionist learning models offer some help: They define lawful relations between linguistic environments and language systems. But our understanding of them is currently weak, especially when it comes to natural language syntax. Fortunately, symbolic linguistic analysis can help connectionism if the two meet via dynamical systems theory. I discuss a case in point: Insights from linguistic explorations of natural language syntax appear to have identified information structures that are particularly relevant to understanding ecologically appealing but analytically mysterious connectionist learning models.     

Symbolic play and language development

Symbolic play and language are known to be highly interrelated, but the developmental process involved in this relationship is not clear. Three hypothetical paths were postulated to explore how play and language drive each other: (1) direct paths, whereby initiation of basic forms in symbolic action or babbling, will be directly related to all later emerging language and motor outputs; (2) an indirect interactive path, whereby basic forms in symbolic action will be associated with more complex forms in symbolic play, as well as with babbling, and babbling mediates the relationship between symbolic play and speech; and (3) a dual path, whereby basic forms in symbolic play will be associated with basic forms of language, and complex forms of symbolic play will be associated with complex forms of language. We micro-coded 288 symbolic vignettes gathered during a yearlong prospective bi-weekly examination (N = 14; from 6 to 18 months of age). Results showed that the age of initiation of single-object symbolic play correlates strongly with the age of initiation of later-emerging symbolic and vocal outputs; its frequency at initiation is correlated with frequency at initiation of babbling, later-emerging speech, and multi-object play in initiation. Results support the notion that a single-object play relates to the development of other symbolic forms via a direct relationship and an indirect relationship, rather than a dual-path hypothesis.     

Measuring the approximate number system

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.     

Does Science Influence the Logic we Ought to Use: A Reflection on the Quantum Logic Controversy

In this article I argue that there is a sense in which logic is empirical, and hence open to influence from science. One of the roles of logic is the modelling and extending of natural language reasoning. It does so by providing a formal system which succeeds in modelling the structure of a paradigmatic set of our natural language inferences and which then permits us to extend this structure to novel cases with relative ease. In choosing the best system of those that succeed in this, we seek certain virtues of such structures such as simplicity and naturalness (which will be explained). Science can influence logic by bringing us, as in the case of quantum mechanics, to make natural language inferences about new kinds of systems and thereby extend the set of paradigmatic cases that our formal logic ought to model as simply and naturally as possible. This can alter which structures ought to be used to provide semantics for such models. I show why such a revolution could have led us to reject one logic for another through explaining why complex claims about quantum mechanical systems failed to lead us to reject classical logic for quantum logic.     

Automatic quantity processing in 5-year olds and adults

In this study adults performed numerical and physical size judgments on a symbolic (Arabic numerals) and non-symbolic (groups of dots) size congruity task. The outcomes would reveal whether a size congruity effect (SCE) can be obtained irrespective of notation. Subsequently, 5-year-old children performed a physical size judgment on both tasks. The outcomes will give a better insight in the ability of 5-year-olds to automatically process symbolic and non-symbolic numerosities. Adult performance on the symbolic and non-symbolic size congruity tasks revealed a SCE for numerical and physical size judgments, indicating that the non-symbolic size congruity task is a valid indicator for automatic processing of non-symbolic numerosities. Physical size judgments on both tasks by children revealed a SCE only for non-symbolic notation, indicating that the lack of a symbolic SCE is not related to the mathematical or cognitive abilities required for the task but instead to an immature association between the number symbol and its meaning.     

The Mentality of Apes Revisited

Although early comparative psychology was seriously marred by claims of our species'supremacy, the residual backlash against these archaic evolutionary views is still being felt, even though our understanding of evolutionary biology is now sufficiently advanced to grapple with possible cognitive specializations that our species does not share with closely related species. The overzealous efforts to dismantle arguments of human uniqueness have only served to show that most comparative psychologists working with apes have yet to set aside the antiquated evolutionary "ladder." Instead, they have only attempted to pull chimpanzees up to the ladder's highest imaginary rung — or perhaps, to pull humans down to an equally imaginary rung at the height of the apes. A true comparative science of animal minds, however, will recognize the complex diversity of the animal kingdom, and will thus view Homo sapiens as one more species with a unique set of adaptive skills crying out to be identified and understood.     

Cognitive Evaluation of Machine Learning Agents

Advances in applying statistical Machine Learning (ML) led to several claims of human-level or near-human performance in tasks such as image classification & speech recognition. Such claims are unscientific primarily for two reasons, (1) They incorrectly enforce the notion that task-specific performance can be treated as manifestation of General Intelligence and (2) They are not verifiable as currently there is no set benchmark for measuring human-like cognition in a machine learning agent. Moreover, ML agent’s performance is influenced by knowledge ingested in it by its human designers. Therefore, agent’s performance may not necessarily reflect its true cognition. In this paper, we propose a framework that draws parallels from human cognition to measure machine’s cognition. Human cognitive learning is quite well studied in developmental psychology with frameworks and metrics in place to measure actual learning. To either believe or refute the claims of human-level performance of machine learning agent, we need scientific methodology to measure its cognition. Our framework formalizes incremental implementation of human-like cognitive processes in ML agents with an implicit goal to measure it. The framework offers guiding principles for measuring, (1) Task-specific machine cognition and (2) General machine cognition that spans across tasks. The framework also provides guidelines for building domain-specific task taxonomies to cognitively profile tasks. We demonstrate application of the framework with a case study where two ML agents that perform Vision and NLP tasks are cognitively evaluated.     

Knowledge on the line: Manipulating beliefs about the magnitudes of symbolic numbers affects the linearity of line estimation tasks

It has been suggested that differences in performance on number-line estimation tasks are indicative of fundamental differences in people’s underlying representations of numerical magnitude. However, we were able to induce logarithmic-looking performance in adults for magnitude ranges over which they can typically perform linearly by manipulating their familiarity with the symbolic number formats that we used for the stimuli. This serves as an existence proof that individuals’ performances on number-line estimation tasks do not necessarily reflect the functional form of their underlying numerical magnitude representations. Rather, performance differences may result from symbolic difficulties (i.e., number-to-symbol mappings), independently of the underlying functional form. We demonstrated that number-line estimates that are well fit by logarithmic functions need not be produced by logarithmic functions. These findings led us to question the validity of considering logarithmic-looking performance on number-line estimation tasks as being indicative that magnitudes are being represented logarithmically, particularly when symbolic understanding is in question.     

Models and mechanisms in psychological explanation

Mechanistic explanation has an impressive track record of advancing our understanding of complex, hierarchically organized physical systems, particularly biological and neural systems. But not every complex system can be understood mechanistically. Psychological capacities are often understood by providing cognitive models of the systems that underlie them. I argue that these models, while superficially similar to mechanistic models, in fact have a substantially more complex relation to the real underlying system. They are typically constructed using a range of techniques for abstracting the functional properties of the system, which may not coincide with its mechanistic organization. I describe these techniques and show that despite being non-mechanistic, these cognitive models can satisfy the normative constraints on good explanations.     

Measuring the approximate number system

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.