This paper describes a novel perspective on the foundations of mathematics: how mathematics may be seen to be largely about “information compression via the matching and unification of patterns”. That is itself a novel approach to IC, couched in terms of nonmathematical primitives, as is necessary in any investigation of the foundations of mathematics. This new perspective on the foundations of mathematics reflects the facts that mathematics is almost exclusively the product of human brains, and has been developed, as an (...) aid to human thinking, mathematics is likely to be consonant with much evidence for the importance of IC in human learning, perception, and cognition. This perspective on the foundations of mathematics has grown out of a long-term programme of research developing the SP Theory of Intelligence and its realization in the SP Computer Model, a system in which a generalised version of ICMUP—the powerful concept of SP-multiple-alignment—plays a central role. This paper shows with an example how mathematics, without any special provision, may achieve compression of information. Then, it describes examples showing how variants of ICMUP may be seen in widely used structures and operations in mathematics. Examples are also given to show how several aspects of the mathematics-related disciplines of logic and computing may be understood as ICMUP. Also discussed is the intimate relation between IC and concepts of probability, with arguments that there are advantages in approaching AI, cognitive science, and concepts of probability via ICMUP. Also discussed is how the close relation between IC and concepts of probability relates to the established view that some parts of mathematics are intrinsically probabilistic, and how that latter view may be reconciled with the all-or-nothing, “exact,” forms of calculation or inference that are familiar in mathematics and logic. There are many potential benefits and applications of the mathematics-as-IC perspective. (shrink)

Cognitive Science, Volume 46, Issue 7, July 2022.

Debates concerning the types of representations that aid reading acquisition have often been influenced by the relationship between measures of early phonological awareness (the ability to process speech sounds) and later reading ability. Here, a complementary approach is explored, analyzing how the functional utility of different representational units, such as whole words, bodies (letters representing the vowel and final consonants of a syllable), and graphemes (letters representing a phoneme) may change as the number of words that can be read gradually (...) increases. Utility is measured by applying a Simplicity Principle to the problem of mapping from print to sound; that is, assuming that the “best” representational units for reading are those which allow the mapping from print to sounds to be encoded as efficiently as possible. Results indicate that when only a small number of words are read whole-word representations are most useful, whereas when many words can be read graphemic representations have the highest utility. (shrink)

ABSTRACTProbability judgements entail a conjunction fallacy if a conjunction is estimated to be more probable than one of its conjuncts. In the context of predication of alternative logical hypothesis, Bayesian logic provides a formalisation of pattern probabilities that renders a class of pattern-based CFs rational. BL predicts a complete system of other logical inclusion fallacies. A first test of this prediction is investigated here, using transparent tasks with clear set inclusions, varying in observed frequencies only. Experiment 1 uses data where (...) BL makes dominant predictions; Experiment 2's predictions were less clear, and we additionally investigated judgements about the second-most probable hypotheses. The results corroborated a pattern-probability account and cannot be easily explained by other theories of CFs. IFs were not limited to conjunctions, but rather occurred systematically for several logical connectives. Thus, pattern-based prob... (shrink)

The ongoing generation of expectations is fundamental to listeners’ experience of music, but research into types of statistical information that listeners extract from musical melodies has tended to emphasize transition probabilities and n‐grams, with limited consideration given to other types of statistical learning that may be relevant. Temporal associations between scale degrees represent a different type of information present in musical melodies that can be learned from musical corpora using expectation networks, a computationally simple method based on activation and decay. (...) Expectation networks infer the expectation of encountering one scale degree followed in the near (but not necessarily immediate) future by another given scale degree, with previous work suggesting that scale degree associations learned by expectation networks better predict listener ratings of pitch similarity than transition probabilities. The current work outlines how these learned scale degree associations can be combined to predict melodic continuations and tests the resulting predictions on a dataset of listener responses to a musical cloze task previously used to compare two other models of melodic expectation, a variable‐order Markov model (IDyOM) and Temperley's music‐theoretically motivated model. Under multinomial logistic regression, all three models explain significant unique variance in human melodic expectations, with coefficient estimates highest for expectation networks. These results suggest that generalized scale degree associations informed by both adjacent and nonadjacent relationships between melodic notes influence listeners’ melodic predictions above and beyond n‐gram context, highlighting the need to consider a broader range of statistical learning processes that may underlie listeners’ expectations for upcoming musical events. (shrink)

The recognition that human minds/brains are finite systems with limited resources for computation has led some researchers to advance theTractable Cognition thesis: Human cognitive capacities are constrained by computational tractability. This thesis, if true, serves cognitive psychology by constraining the space of computational‐level theories of cognition. To utilize this constraint, a precise and workable definition of “computational tractability” is needed. Following computer science tradition, many cognitive scientists and psychologists define computational tractability as polynomial‐time computability, leading to theP‐Cognition thesis. This article (...) explains how and why the P‐Cognition thesis may be overly restrictive, risking the exclusion of veridical computational‐level theories from scientific investigation. An argument is made to replace the P‐Cognition thesis by theFPT‐Cognition thesisas an alternative formalization of the Tractable Cognition thesis (here, FPT stands for fixed‐parameter tractable). Possible objections to the Tractable Cognition thesis, and its proposed formalization, are discussed, and existing misconceptions are clarified. (shrink)

Through cultural transmission, repeated learning by new individuals transforms cultural information, which tends to become increasingly compressible . Existing diffusion chain studies include in their design two processes that could be responsible for this tendency: learning and reproducing . This paper manipulates the presence of learning in a simple iterated drawing design experiment. We find that learning seems to be the causal factor behind the increase in compressibility observed in the transmitted information, while reproducing is a source of random heritable (...) innovations. Only a theory invoking these two aspects of cultural learning will be able to explain human culture's fundamental balance between stability and innovation. (shrink)

Smith gives an excellent overview on research in language evolution, in which he discusses several recent models of how linguistic systems and the cognitive capacities involved in language learning may have co‐evolved. He illustrates how combined pressures on language learning and communication/use produce compositionally structured languages. Once in place, a (culturally transmitted) communication system creates new selection pressures on the capacity for acquiring these systems.

Cognitive science depends on abstractions made from the complex reality of human behaviour. Cognitive scientists typically wish the abstractions in their theories to be universals, but seldom attend to the ontology of universals. Two sorts of universal, resulting from Galilean abstraction and materialist abstraction respectively, are available in the philosophical literature: the abstract universal—the one-over-many universal—is the universal conventionally employed by cognitive scientists; in contrast, a concrete universal is a material entity that can appear within the set of entities it (...) describes, of which it represents the essential, paradigmatic case. The potential role of concrete universals in cognitive science is discussed. (shrink)

The distribution of cultural variants in a population is shaped by both neutral evolutionary dynamics and by selection pressures. The temporal dynamics of social network connectivity, that is, the order in which individuals in a population interact with each other, has been largely unexplored. In this paper, we investigate how, in a fully connected social network, connectivity dynamics, alone and in interaction with different cognitive biases, affect the evolution of cultural variants. Using agent‐based computer simulations, we manipulate population connectivity dynamics (...) (early, mid, and late full‐population connectivity); content bias, or a preference for high‐quality variants; coordination bias, or whether agents tend to use self‐produced variants (egocentric bias), or to switch to variants observed in others (allocentric bias); and memory size, or the number of items that agents can store in their memory. We show that connectivity dynamics affect the time‐course of variant spread, with lower connectivity slowing down convergence of the population onto a single cultural variant. We also show that, compared to a neutral evolutionary model, content bias accelerates convergence and amplifies the effects of connectivity dynamics, while larger memory size and coordination bias, especially egocentric bias, slow down convergence. Furthermore, connectivity dynamics affect the frequency of high‐quality variants (adaptiveness), with late connectivity populations showing bursts of rapid change in adaptiveness followed by periods of relatively slower change, and early connectivity populations following a single‐peak evolutionary dynamic. We evaluate our simulations against existing data collected from previous experiments and show how our model reproduces the empirical patterns of convergence. (shrink)

This paper presents a computational model of the way humans inductively identify and aggregate concepts from the low-level stimuli they are exposed to. Based on the idea that humans tend to select the simplest structures, it implements a dynamic hierarchical chunking mechanism in which the decision whether to create a new chunk is based on an information-theoretic criterion, the Minimum Description Length (MDL) principle. We present theoretical justifications for this approach together with results of an experiment in which participants, exposed (...) to meaningless symbols, have been implicitly encouraged to create high-level concepts by grouping them. Results show that the designed model, called hereafter MDLChunker, makes precise quantitative predictions both on the kind of chunks created by the participants and also on the moment at which these creations occur. They suggest that the simplicity principle used to design MDLChunker is particularly efficient to model chunking mechanisms. The main interest of this model over existing ones is that it does not require any adjustable parameter. (shrink)

Each of our theories of mental representation provides some insight into how the mind works. However, these insights often seem incompatible, as the debates between symbolic, dynamical, emergentist, sub-symbolic, and grounded approaches to cognition attest. Mental representations—whatever they are—must share many features with each of our theories of representation, and yet there are few hypotheses about how a synthesis could be possible. Here, I develop a theory of the underpinnings of symbolic cognition that shows how sub-symbolic dynamics may give rise (...) to higher-level cognitive representations of structures, systems of knowledge, and algorithmic processes. This theory implements a version of conceptual role semantics by positing an internal universal representation language in which learners may create mental models to capture dynamics they observe in the world. The theory formalizes one account of how truly novel conceptual content may arise, allowing us to explain how even elementary logical and computational operations may be learned from a more primitive basis. I provide an implementation that learns to represent a variety of structures, including logic, number, kinship trees, regular languages, context-free languages, domains of theories like magnetism, dominance hierarchies, list structures, quantification, and computational primitives like repetition, reversal, and recursion. This account is based on simple discrete dynamical processes that could be implemented in a variety of different physical or biological systems. In particular, I describe how the required dynamics can be directly implemented in a connectionist framework. The resulting theory provides an “assembly language” for cognition, where high-level theories of symbolic computation can be implemented in simple dynamics that themselves could be encoded in biologically plausible systems. (shrink)

This study investigates the joint influences of three factors on the discovery of new word‐like units in a continuous artificial speech stream: the statistical structure of the ongoing input, the initial word‐likeness of parts of the speech flow, and the contextual information provided by the earlier emergence of other word‐like units. Results of an experiment conducted with adult participants show that these sources of information have strong and interactive influences on word discovery. The authors then examine the ability of different (...) models of word segmentation to account for these results. PARSER (Perruchet & Vinter, 1998) is compared to the view that word segmentation relies on the exploitation of transitional probabilities between successive syllables, and with the models based on the Minimum Description Length principle, such as INCDROP. The authors submit arguments suggesting that PARSER has the advantage of accounting for the whole pattern of data without ad‐hoc modifications, while relying exclusively on general‐purpose learning principles. This study strengthens the growing notion that nonspecific cognitive processes, mainly based on associative learning and memory principles, are able to account for a larger part of early language acquisition than previously assumed. (shrink)

Human cognition requires coping with a complex and uncertain world. This suggests that dealing with uncertainty may be the central challenge for human reasoning. In Bayesian Rationality we argue that probability theory, the calculus of uncertainty, is the right framework in which to understand everyday reasoning. We also argue that probability theory explains behavior, even on experimental tasks that have been designed to probe people's logical reasoning abilities. Most commentators agree on the centrality of uncertainty; some suggest that there is (...) a residual role for logic in understanding reasoning; and others put forward alternative formalisms for uncertain reasoning, or raise specific technical, methodological, or empirical challenges. In responding to these points, we aim to clarify the scope and limits of probability and logic in cognitive science; explore the meaning of the explanation of cognition; and re-evaluate the empirical case for Bayesian rationality. (shrink)

Much research has linked surprise to violation of expectations, but it has been less clear how one can be surprised when one has no particular expectation. This paper discusses a computational theory based on Algorithmic Information Theory, which can account for surprises in which one initially expects randomness but then notices a pattern in stimuli. The authors present evidence that a “randomness deficiency” heuristic leads to surprise in such cases.

Simplicity was recently described in the philosophy of science as “perhaps the most controversial theoretical virtue” (Schindler 2018). It has been also argued that contrary to the standard view, simplicity is not merely a pragmatic virtue but also an epistemic one. Virtue epistemologists are also interested in epistemic virtues, but simplicity is usually absent in their discussions. This paper adduces several contemporary approaches to simplicity showing that in philosophy and in psychology it can be considered either as a virtue or (...) as a vice. The paper then refers to the thought of Thomas Aquinas as an example of a premodern understanding of the virtue simplicity and suggests that his notion of simplicity as a virtue might be developed today within virtue epistemology into an interesting complement to this “perhaps the most controversial theoretical virtue” (Schindler 2018) known from contemporary philosophy. (shrink)

In this essay, I explore how cognitive science could illuminate the concept of beauty. Two results from the extensive literature on aesthetics guide my discussion. As the term “beauty” is overextended in general usage, I choose as my starting point the notion of “perfect form.” Aesthetic theorists are in reasonable agreement about the criteria for perfect form. What do these criteria imply for mental representations that are experienced as beautiful? Complexity theory can be used to specify constraints on mental representations (...) abstractly formulated as vectors in a high-dimensional space. A central feature of the proposed model is that perfect form depends both on features of the objects or events perceived and on the nature of the encoding strategies or model of the observer. A simple example illustrates the proposed calculations. A number of interesting implications that arise as a consequence of reformulating beauty in this way are noted. (shrink)

The study aims to help characterize the sort of structures about which people can acquire unconscious knowledge. It is already well established that people can implicitly learn n-grams and also repetition patterns. We explore the acquisition of unconscious structural knowledge of symmetry. Chinese Tang poetry uses a specific sort of mirror symmetry, an inversion rule with respect to the tones of characters in successive lines of verse. We show, using artificial poetry to control both n-gram structure and repetition patterns, that (...) people can implicitly learn to discriminate inversions from non-inversions, presenting a challenge to existing models of implicit learning. (shrink)

Natural language is full of patterns that appear to fit with general linguistic rules but are ungrammatical. There has been much debate over how children acquire these “linguistic restrictions,” and whether innate language knowledge is needed. Recently, it has been shown that restrictions in language can be learned asymptotically via probabilistic inference using the minimum description length (MDL) principle. Here, we extend the MDL approach to give a simple and practical methodology for estimating how much linguistic data are required to (...) learn a particular linguistic restriction. Our method provides a new research tool, allowing arguments about natural language learnability to be made explicit and quantified for the first time. We apply this method to a range of classic puzzles in language acquisition. We find some linguistic rules appear easily statistically learnable from language experience only, whereas others appear to require additional learning mechanisms (e.g., additional cues or innate constraints). (shrink)

Identifying patterns in the world requires noticing not only unusual occurrences, but also unusual absences. We examined how people learn from absences, manipulating the extent to which an absence is expected. People can make two types of inferences from the absence of an event: either the event is possible but has not yet occurred, or the event never occurs. A rational analysis using Bayesian inference predicts that inferences from absent data should depend on how much the absence is expected to (...) occur, with less probable absences being more salient. We tested this prediction in two experiments in which we elicited people's judgments about patterns in the data as a function of absence salience. We found that people were able to decide that absences either were mere coincidences or were indicative of a significant pattern in the data in a manner that was consistent with predictions of a simple Bayesian model. (shrink)

Children learn their native language by exposure to their linguistic and communicative environment, but apparently without requiring that their mistakes be corrected. Such learning from “positive evidence” has been viewed as raising “logical” problems for language acquisition. In particular, without correction, how is the child to recover from conjecturing an over-general grammar, which will be consistent with any sentence that the child hears? There have been many proposals concerning how this “logical problem” can be dissolved. In this study, we review (...) recent formal results showing that the learner has sufficient data to learn successfully from positive evidence, if it favors the simplest encoding of the linguistic input. Results include the learnability of linguistic prediction, grammaticality judgments, language production, and form-meaning mappings. The simplicity approach can also be “scaled down” to analyze the learnability of specific linguistic constructions, and it is amenable to empirical testing as a framework for describing human language acquisition. (shrink)

Many of the problems studied in cognitive science are inductive problems, requiring people to evaluate hypotheses in the light of data. The key to solving these problems successfully is having the right inductive biases—assumptions about the world that make it possible to choose between hypotheses that are equally consistent with the observed data. This article explores a novel experimental method for identifying the biases that guide human inductive inferences. The idea behind this method is simple: This article uses the responses (...) produced by a participant on one trial to generate the stimuli that either they or another participant will see on the next. A formal analysis of this “iterated learning” procedure, based on the assumption that the learners are Bayesian agents, predicts that it should reveal the inductive biases of these learners, as expressed in a prior probability distribution over hypotheses. This article presents a series of experiments using stimuli based on a well-studied set of category structures, demonstrating that iterated learning can be used to reveal the inductive biases of human learners. (shrink)

In recent years, scientists have increasingly taken to investigate the predictive nature of cognition. We argue that prediction relies on abstraction, and thus theories of predictive cognition need an explicit theory of abstract representation. We propose such a theory of the abstract representational capacities that allow humans to transcend the “here-and-now.” Consistent with the predictive cognition literature, we suggest that the representational substrates of the mind are built as ahierarchy, ranging from the concrete to the abstract; however, we argue that (...) there are qualitative differences between elements along this hierarchy, generating meaningful, often unacknowledged,diversity.Echoing views from philosophy, we suggest that the representational hierarchy can be parsed into:modality-specificrepresentations, instantiated on perceptual similarity;multimodalrepresentations, instantiated primarily on the discovery of spatiotemporal contiguity; andcategoricalrepresentations, instantiated primarily on social interaction. These elements serve as the building blocks ofcomplex structures discussed in cognitive psychology (e.g., episodes, scripts) and are the inputs for mental representations that behave like functions, typically discussed in linguistics (i.e.,predicators). We support our argument for representational diversity by explaining how the elements in our ontology are all required to account for humans’ predictive cognition (e.g., in subserving logic-based prediction; in optimizing the trade-off between accurate and detailed predictions) and by examining how the neuroscientific evidence coheres with our account. In doing so, we provide a testable model of the neural bases of conceptual cognition and highlight several important implications to research on self-projection, reinforcement learning, and predictive-processing models of psychopathology. (shrink)

Human cognition and behavior are dominated by symbol use. This paper examines the social learning strategies that give rise to symbolic communication. Experiment 1 contrasts an individual-level account, based on observational learning and cognitive bias, with an inter-individual account, based on social coordinative learning. Participants played a referential communication game in which they tried to communicate a range of recurring meanings to a partner by drawing, but without using their conventional language. Individual-level learning, via observation and cognitive bias, was sufficient (...) to produce signs that became increasingly effective, efficient, and shared over games. However, breaking a referential precedent eliminated these benefits. The most effective, most efficient, and most shared signs arose when participants could directly interact with their partner, indicating that social coordinative learning is important to the creation of shared symbols. Experiment 2 investigated the contribution of two distinct aspects of social interaction: behavior alignment and concurrent partner feedback. Each played a complementary role in the creation of shared symbols: Behavior alignment primarily drove communication effectiveness, and partner feedback primarily drove the efficiency of the evolved signs. In conclusion, inter-individual social coordinative learning is important to the evolution of effective, efficient, and shared symbols. (shrink)

Is formal simplicity a guide to learning in humans, as simplicity is said to be a guide to the acceptability of theories in science? Does simplicity determine the difficulty of various learning tasks? I argue that, similarly to how scientists sometimes preferred complex theories when this facilitated calculations, results from perception, learning and reasoning suggest that formal complexity is generally unrelated to what is easy to learn and process by humans, and depends on assumptions about available representational and processing primitives. (...) “Simpler” hypotheses are preferred only when they are also easier to process. Historically, “simpler”, easier‐to‐process, scientific theories might also be preferred if they are transmitted preferentially. Empirically viable complexity measures should build on the representational and processing primitives of actual learners, even if explanations of their behaviour become formally more complex. (shrink)

Although the mapping between form and meaning is often regarded as arbitrary, there are in fact well-known constraints on words which are the result of functional pressures associated with language use and its acquisition. In particular, languages have been shown to encode meaning distinctions in their sound properties, which may be important for language learning. Here, we investigate the relationship between semantic distance and phonological distance in the large-scale structure of the lexicon. We show evidence in 100 languages from a (...) diverse array of language families that more semantically similar word pairs are also more phonologically similar. This suggests that there is an important statistical trend for lexicons to have semantically similar words be phonologically similar as well, possibly for functional reasons associated with language learning. (shrink)

Working memory has been shown to be strongly related to fluid intelligence; however, our goal is to shed further light on the process of information compression in working memory as a determining factor of fluid intelligence. Our main hypothesis was that compression in working memory is an excellent indicator for studying the relationship between working-memory capacity and fluid intelligence because both depend on the optimization of storage capacity. Compressibility of memoranda was estimated using an algorithmic complexity metric. The results showed (...) that compressibility can be used to predict working-memory performance and that fluid intelligence is well predicted by the ability to compress information. We conclude that the ability to compress information in working memory is the reason why both manipulation and retention of information are linked to intelligence. This result offers a new concept of intelligence based on the idea that compression and intelligence are equivalent problems. (shrink)