An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...) are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

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.

One approach to understanding how the human cognitive system stores and operates with quantifiers such as “some,” “many,” and “all” is to investigate their interaction with the cognitive mechanisms for estimating and comparing quantities from perceptual input (i.e., nonsymbolic quantities). While a potential link between quantifier processing and nonsymbolic quantity processing has been considered in the past, it has never been discussed extensively. Simultaneously, there is a long line of research within the field of numerical cognition on the relationship between (...) processing exact number symbols (such as “3” or “three”) and nonsymbolic quantity. This accumulated knowledge can potentially be harvested for research on quantifiers since quantifiers and number symbols are two different ways of referring to quantity information symbolically. The goal of the present review is to survey the research on the relationship between quantifiers and nonsymbolic quantity processing mechanisms and provide a set of research directions and specific questions for the investigation of quantifier processing. (shrink)

We evaluate here the performance of four models of cross-situational word learning: two global models, which extract and retain multiple referential alternatives from each word occurrence; and two local models, which extract just a single referent from each occurrence. One of these local models, dubbed Pursuit, uses an associative learning mechanism to estimate word-referent probability but pursues and tests the best referent-meaning at any given time. Pursuit is found to perform as well as global models under many conditions extracted from (...) naturalistic corpora of parent-child interactions, even though the model maintains far less information than global models. Moreover, Pursuit is found to best capture human experimental findings from several relevant cross-situational word-learning experiments, including those of Yu and Smith (), the paradigm example of a finding believed to support fully global cross-situational models. Implications and limitations of these results are discussed, most notably that the model characterizes only the earliest stages of word learning, when reliance on the co-occurring referent world is at its greatest. (shrink)

Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that arithmetical (...) knowledge developed in such a way cannot be totally conceptual in the sense relevant to the philosophy of arithmetic, but neither can arithmetic understood to be empirical. Rather, we need to develop a contextual a priori notion of arithmetical knowledge that preserves the special mathematical characteristics without ignoring the roots of arithmetical cognition. Such a contextual a priori theory is shown not to require any ontologically problematic assumptions, in addition to fitting well within a standard framework of general epistemology. (shrink)

This paper proposes and develops the feelings of goals hypothesis (FGH). It has two aims: first, to describe the evolutionary function of emotional feelings (EFs), and second, to describe the content and the format of EFs. According to FGH, the evolutionary function of EFs is to enable motoric flexibility. Specifically, EFs are a component of a psychological mechanism that permits differential motoric reactions to the same stimulus. Further, according to FGH, EF is a special type of mental representation with the (...) content of an action goal, and with a non-motoric, non-conceptual format. This paper thoroughly clarifies the assumptions underlying FGH and discusses its theoretical implications and empirical predictions. (shrink)

A science of prelinguistic infant cognition must take seriously the language-of-thought (LoT) hypothesis. I show how the LoT framework enables us to identify the representational and computational capacities of infant minds and the developmental factors that act on these capacities, and explain how Quilty-Dunn et al.'s take on LoT has important upshots for developmental theory-building.

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)

Conviction Narrative Theory (CNT) is a theory of choice underradical uncertainty– situations where outcomes cannot be enumerated and probabilities cannot be assigned. Whereas most theories of choice assume that people rely on (potentially biased) probabilistic judgments, such theories cannot account for adaptive decision-making when probabilities cannot be assigned. CNT proposes that people usenarratives– structured representations of causal, temporal, analogical, and valence relationships – rather than probabilities, as the currency of thought that unifies our sense-making and decision-making faculties. According to CNT, (...) narratives arise from the interplay between individual cognition and the social environment, with reasoners adopting a narrative that feels “right” to explain the available data; using that narrative to imagine plausible futures; and affectively evaluating those imagined futures to make a choice. Evidence from many areas of the cognitive, behavioral, and social sciences supports this basic model, including lab experiments, interview studies, and econometric analyses. We identify 12 propositions to explain how the mental representations (narratives) interact with four inter-related processes (explanation, simulation, affective evaluation, and communication), examining the theoretical and empirical basis for each. We conclude by discussing how CNT can provide a common vocabulary for researchers studying everyday choices across areas of the decision sciences. (shrink)

René Jagnow [2012] argues that David Rosenthal's theory of consciousness cannot account for certain experiences that involve colours so fine-grained that we do not and cannot have concepts of them. Jagnow claims that an appeal to comparative concepts such as being slightly darker than cannot help Rosenthal, since, in order to apply such concepts, we would already need to be conscious of two distinct fine-grained colours. The present paper contests this claim. It appeals to the Cornsweet illusion and some other (...) visual phenomena, in order to argue that our visual systems can and do represent brutely comparative matters. (shrink)

A down-to-earth admission of abstract objects can be based on detailed explanation of where the objectivity of mathematics comes from, and how a ‘thin’ notion of object emerges from objective mathematical discourse or practices. We offer a sketch of arguments concerning both points, as a basis for critical scrutiny of the idea that mathematical and social objects are essentially of the same kind—which is criticized. Some authors have proposed that mathematical entities are indeed institutional objects, a product of our collective (...) imposition of function onto reality (the phrase comes from Searle) and of surrogation or hypostasis. Yet there are significant disanalogies between the typical social objects and mathemata, on which basis I argue that one should make a clear distinction between both. The comparison of mathematical with social objects helps understanding how non-physical objects can figure prominently in our explanations of reality. Yet mathematical objects have a different kind of cognitive grounding, and the more elementary of them emerge under relatively very simple sociocultural conditions. The differences are also reflected in the wide scope of use of mathematical concepts, and the much higher degree of variation found among social objects. On the basis of all of these features, I defend the thesis that one can significantly distinguish degrees of objectivity, and I use the distinction to articulate a graded ontology where one can locate the different kinds of mathematical and social objects. (shrink)

This paper defends a conceptualistic version of structuralism as the most convincing way of elaborating a philosophical understanding of structuralism in line with the classical tradition. The argument begins with a revision of the tradition of “conceptual mathematics”, incarnated in key figures of the period 1850 to 1940 like Riemann, Dedekind, Hilbert or Noether, showing how it led to a structuralist methodology. Then the tension between the ‘presuppositionless’ approach of those authors, and the platonism of some recent versions of philosophical (...) structuralism, is presented. In order to resolve this tension, we argue for the idea of ‘logical objects’ as a form of minimalist realism, again in the tradition of classical authors including Peirce and Cassirer, and we introduce the basic tenets of conceptual structuralism. The remainder of the paper is devoted to an open discussion of the assumptions behind conceptual structuralism, and—most importantly—an argument to show how the objectivity of mathematics can be explained from the adopted standpoint. This includes the idea that advanced mathematics builds on hypothetical assumptions (Riemann, Peirce, and others), which is presented and discussed in some detail. Finally, the ensuing notion of objectivity is interpreted as a form of particularly robust intersubjectivity, and it is distinguished from fictional or social ontology. (shrink)

ABSTRACTWe discuss the thesis of universality of geometric notions and offer critical reflections on the concept of “natural geometry” employed by Spelke and others. Promoting interdisciplinary wor...

Most accounts of our knowledge of the successor axiom claim that this is based on the procedure of adding one. While they usually don’t claim to provide an account of how children actually acquire this knowledge, one may well think that this is how they get that knowledge. I argue that when we look at children’s responses in interviews, the time when they learn the successor axiom and the intermediate learning stages they find themselves in, that there is an empirically (...) viable alternative. I argue that they could also learn it on the basis of a method that has to do with the structure of the numeral system. Specifically, that they (1) use the syntactic structure of the numeral system and (2) attend to the leftmost digits, the one with the highest place-value. Children can learn that this is a reliable method of forming larger numbers by combining two elements. First, a grasp of the syntactic structure of the numeral system. That way they know that the leftmost digit receives the highest value. Second, an interpretation of numerals as designating cardinal values, so that they also realise that increasing or adding digits on the lefthand side of a numeral produces a larger number. There are thus two, currently equally well-supported, ways in which children might learn that there are infinitely many natural numbers. (shrink)

ABSTRACT How do non-experts single out numbers for reference? Linnebo has argued that they do so using a criterion of identity based on the ordinal properties of numerals. Neo-logicists, on the other hand, claim that cardinal properties are the basis of individuation, when they invoke Hume’s Principle. I discuss empirical data from cognitive science and linguistics to answer how non-experts individuate numbers better in practice. I use those findings to develop an alternative account that mixes ordinal and cardinal properties to (...) provide a detailed answer to the question: how do we in fact semantically individuate numbers? (shrink)

Mathematical cognition has become an interesting case study for wider theories of cognition. Menary :1–20, 2015) argues that arithmetical cognition not only shows that internalist theories of cognition are wrong, but that it also shows that the Hypothesis of Extended Cognition is right. I examine this argument in more detail, to see if arithmetical cognition can support such conclusions. Specifically, I look at how the use of numerals extends our arithmetical abilities from quantity-related innate systems to systems that can deal (...) with exact numbers of arbitrary size. I then argue that the system underlying our grasp of small numbers is an unhelpful case study for Menary; it doesn’t support an argument for externalism over internalism. The system for large numbers, on the other hand, clearly displays important interactions between public numeral systems and our cognitive processes. I argue that the large number system supports an argument against internalist theories of arithmetical cognition, but that one cannot conclude that the Hypothesis of Extended Cognition is correct. In other words, the large number case doesn’t decide between the Hypothesis of Extended Cognition and the Hypothesis of EMbedded Cognition. (shrink)

How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the (...) initial segment of the natural numbers on the basis of the Fregean definitions, but do not learn the natural number structure as a whole on the basis of Hume's principle. Therefore, we need to account for some of the consistency of our number concepts with the Dedekind-Peano axioms in other terms. (shrink)

Can concepts be acquired by testing hypotheses about these concepts? Fodor famously argued that this is not possible. Testing the correct hypothesis would require already possessing the concept. I argue that this does not generally hold for mathematical concepts. I discuss specific, empirically motivated, hypotheses for number concepts that can be tested without needing to possess the relevant number concepts. I also argue that one can test hypotheses about the identity conditions of other mathematical concepts, and then fix the application (...) conditions based on those hypotheses—under the assumption that the neo‐logicist view on abstraction principles is correct. (shrink)

If there is content that we reason on, cognitive content, it is in the head and accessible to reasoning mechanisms. This paper discusses the phenomenal theory of cognitive content, according to which cognitive contents are the contents of phenomenal consciousness. I begin by distinguishing cognitive content from the closely associated notion of narrow content. I then argue, drawing on prior work, that the phenomenal theory can plausibly account for the cognitive contents of many relatively simple mental states. My main focus (...) in this paper is the question whether the phenomenal theory can account for the apparently abstract and complex cognitive contents of "high-level" thoughts. It might seem dubious that the theory can account for such cognitive contents, because cognitive phenomenology is too scarce and too thin. However, I argue there are in fact few abstract or complex cognitive contents, so the phenomenal theory's predictions are correct. This position explains some apparently irrational behavior that is otherwise hard to explain, and it makes sense of the central role of inner speech and other forms of consciousness in reasoning. (shrink)

One sometimes believes a proposition without grasping it. For example, a complete achromat might believe that ripe tomatoes are red without grasping this proposition. My aim in this paper is to shed light on the difference between merely believing a proposition and grasping it. I focus on two possible theories of grasping: the inferential theory, which explains grasping in terms of inferential role, and the phenomenal theory, which explains grasping in terms of phenomenal consciousness. I argue that the phenomenal theory (...) is more plausible than the inferential theory. (shrink)

Susan Carey's account of Quinean bootstrapping has been heavily criticized. While it purports to explain how important new concepts are learned, many commentators complain that it is unclear just what bootstrapping is supposed to be or how it is supposed to work. Others allege that bootstrapping falls prey to the circularity challenge: it cannot explain how new concepts are learned without presupposing that learners already have those very concepts. Drawing on discussions of concept learning from the philosophical literature, this article (...) develops a detailed interpretation of bootstrapping that can answer the circularity challenge. The key to this interpretation is the recognition of computational constraints, both internal and external to the mind, which can endow empty symbols with new conceptual roles and thus new contents. (shrink)

Kripke’s version of Wittgenstein’s rule-following paradox has been influential. My concern is with how it—and Wittgenstein’s views more generally—have been perceived as undercutting the individualistic picture of mathematical practice: the view that individuals— Robinson Crusoes —can, entirely independently of a community, engage in cogent mathematics, and indeed have “private languages.” What has been denied is that phrases like “correctly counting” can be applied to such individuals because these normative notions can only be applied cogently in a context involving community standards. (...) I attempt to show that this shocking corollary doesn’t follow even if Kripke’s Wittgensteinian objections to dispositional approaches to rule-following are largely right. My reason for claiming this is that there is another solution to the rule-following paradox, one that doesn’t favor community standards over individual ones. Furthermore, it doesn’t replace truth conditions with assertability conditions; and this latter maneuver is essential to Kripke’s sceptical solution favoring the community over the individual. (shrink)

External representations play a crucial role in learning. At the same time, cognitive load theory suggests that the possibility of learning depends on limited resources of the working memory and on cognitive load imposed by instructional design and representation tools. Both these observations motivate a critical look at Computer-Supported Argument Visualization tools that are supposed to facilitate learning. This paper uses cognitive load theory to compare the cognitive efficacy of RationaleTM 2 and AGORA.