An empirically feasible approach to the epistemology of arithmetic

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.     

Individual differences in cognitive arithmetic

Unities in the processes involved in solving arithmetic problems of varying operations have been suggested by studies that have used both factor-analytic and information-processing methods. We designed the present study to investigate the convergence of mental processes assessed by paper-and-pencil measures defining the Numerical Facility factor and component processes for cognitive arithmetic identified by using chronometric techniques. A sample of 100 undergraduate students responded to 320 arithmetic problems in a true-false reaction-time (RT) verification paradigm and were administered a battery of ability measures spanning Numerical Facility, Perceptual Speed, and Spatial Relations factors. The 320 cognitive arithmetic problems comprised 80 problems of each of four types: simple addition, complex addition, simple multiplication, and complex multiplication. The information-processing results indicated that regression models that included a structural variable consistent with memory network retrieval of arithmetic facts were the best predictors of RT to each of the four types of arithmetic problems. The results also verified the effects of other elementary processes that are involved in the mental solving of arithmetic problems, including encoding of single digits and carrying to the next column for complex problems. The relation between process components and ability measures was examined by means of structural equation modeling. The final structural model revealed a strong direct relation between a factor subsuming efficiency of retrieval of arithmetic facts and of executing the carry operation and the traditional Numerical Facility factor. Furthermore, a moderate direct relation between a factor subsuming speed of encoding digits and decision and response times and the traditional Perceptual Speed factor was also found. No relation between structural variables representing cognitive arithmetic component processes and ability measures spanning the Spatial Relations factor was found. Results of the structural modeling support the conclusion that information retrieval from a network of arithmetic facts and execution of the carry operation are elementary component processes involved uniquely in the mental solving of arithmetic problems. Furthermore, individual differences in the speed of executing these two elementary component processes appear to underlie individual differences on ability measures that traditionally span the Numerical Facility factor.(ABSTRACT TRUNCATED AT 400 WORDS)     

Virtue epistemology and tacit cognitive processes in high-grade knowledge

Duncan Pritchard has recently argued that a certain brand of virtue epistemology, known as “virtue responsibilism”, cannot account for knowledge acquired through the use of tacit reasoning processes. I defend virtue responsiblism by showing that Pritchard's charge is founded on a mischaracterization of the view. Contra Pritchard, responsibilists do not demand that agents have complete access to the grounds for their beliefs in order to know. A closer examination of prominent accounts of virtue responsiblism, including Zagzebski's and Hookway's, reveals that the accessibility requirement is much weaker than Pritchard presumes. Zagzebski maintains that it is only intellectually virtuous motivations which drive the agent to adopt truth-conducive procedures and habits that must be accessible, rendering the agent responsible for her belief. Hookway writes that agents may display virtue not by actively monitoring or accessing each step of their deliberation, but by allowing deeply embedded intellectual traits to tacitly guide or shape their process of inquiry. Additional support for Hookway's claim comes from Dreyfus and Dreyfus's model of learning and mastery, which will be briefly discussed at the end. I conclude, therefore, that virtue responsibilists can accommodate knowledge acquired through nonreflectively accessible cognitive operations.     

Information acquisition strategies and the cognitive structure of arithmetic

In three experiments, we examined the sensitivity of information acquisition strategies to the underlying cognitive structure of arithmetic tasks. Previous work has shown that individuals solve arithmetic problems more quickly when they consider operators before operands. The operators establish the goal, and the operands are then assimilated into that goal. In the present experiments, participants performed arithmetic tasks in which they controlled the display of operators and operands, using separate keystrokes. The participants chose to see the operator before the operands in most conditions. This ordering was more likely when task constraints made it easier to achieve, when feedback emphasized execution time, and in a more complex multiple-step task. These results extend previous research on strategy selection and information acquisition, supporting the idea that coordination between the environment and ongoing cognition is an important feature of cognitive skill.     

The cognitive act and the first-person perspective: an epistemology for constructive type theory

The notion of cognitive act is of importance for an epistemology that is apt for constructive type theory, and for epistemology in general. Instead of taking knowledge attributions as the primary use of the verb ‘to know’ that needs to be given an account of, and understanding a first-person knowledge claim as a special case of knowledge attribution, the account of knowledge that is given here understands first-person knowledge claims as the primary use of the verb ‘to know’. This means that a cognitive act is an act that counts as cognitive from a first-person point of view. The method of linguistic phenomenology is used to explain or elucidate our epistemic notions. One of the advantages of the theory is that an answer can be given to some of the problems in modern epistemology, such as the Gettier problem.     

Deliberative epistemology: Towards an ubuntu-based epistemology that accounts for a priori knowledge and objective truth

The field of epistemology is concerned with the nature, generation and justification of knowledge as well as the rationality of belief; addressing what makes beliefs justified, what it means to say that one knows something and how one knows it. It has been characterised by many debates such as those between rationalism and empiricism or those between absolute truth and relativism. In light of the often bipolar framing of such epistemic debates, the scholarship on ubuntu contains insights that help advance the idea of a “deliberative” epistemology, which is well positioned to resolve many such tensions. With this idea, which speaks to like-minded approaches that are couched within frameworks such as conversational philosophy and complementary reflection, I add the dimension of discourse (i.e. how we think, speak and therefore do) and show that by applying an alternative discursive lens, an ubuntu-inspired epistemology need not only account for experience and subjectivity, as typically presumed, but can also embrace a priori knowledge and objective truth.     

The dynamics of cognition: An ACT-R model of cognitive arithmetic

Zusammenfassung. Forschungsarbeiten zur Kognitiven Arithmetik untersuchen die mentale Repräsentation von Zahlen und arithmetischen Fakten sowie die kognitiven Prozesse die diese generieren, abrufen und manipulieren. Das Spannungsfeld zwischen der scheinbar einfachen formalen Struktur dieses Aufgabenbereichs und den Schwierigkeiten, die Kinder bei seiner Bewältigung haben, stellt einen einzigartigen Zugang zum Studium kognitiver Prozesse dar. Der vorliegende Beitrag präsentiert einen Erklärungsansatz der zentralen Befunde des Forschungsgebietes auf der Grundlage eines ACT-R Modells zur Lebenszeit-Simulation des Erwerbs arithmetischen Wissens. Die Anwendung der Bayesischen Lernmechanismen der ACT-R Architektur zeigen auf, wie sich diese Befunde auf die statistische Struktur des Aufgabengebiets zurückführen lassen. Aus den präzisen Vorhersagen der Simulation werden sowohl Hinweise zur Vermittlung arithmetischen Wissens abgeleitet als auch Erkenntnisse über die Architektur ACT-R selbst gewonnen. Im Rahmen einer formalen Analyse wird gezeigt, daß sich die vorgestellte Simulation als dynamisches System betrachten läßt, dessen Lernergebnis unmittelbar von Parametern der Architektur abhängt. Eine Untersuchung der Sensitivität der Parameter der Simulation belegt, daß die Werte, die zur besten Anpassung an die empirischen Daten führen, auch eine in einer optimalen Performanz resultieren. Die Implikationen dieses Ergebnis für die grundlegende Adaptivität menschlicher Kognition werden diskutiert. Abstract. Cognitive arithmetic studies the mental representation of numbers and arithmetic facts and the processes that create, access, and manipulate them. The contradiction between the apparent straightforwardness of its exact formal structure and the difficulties that every child faces in mastering it provides an important window into human cognition. An ACT-R model is proposed which accounts for the central results of the field through a single simulation of a lifetime of arithmetic learning. The use of the architecture's Bayesian learning mechanisms explains how these effects arise from the statistics of the task. Because of the precise predictions of the simulation, a number of lessons are derived concerning the teaching of arithmetic and the ACT-R architecture itself. A formal analysis establishes that the simulation can be viewed as a dynamical system whose ultimate learning outcome is fundamentally dependent upon some architectural parameters. Finally, an empirical study of the sensitivity of the simulation to its parameters determines that the values that yield the best fit to the data also provide optimal performance. The implications of these findings for the fundamental adaptivity of human cognition are discussed.     

An epistemology for practical knowledge

Anscombe thought that practical knowledge – a person’s knowledge of what she is intentionally doing – displays formal differences to ordinary empirical, or ‘speculative’, knowledge. I suggest these differences rest on the fact that practical knowledge involves intention analogously to how speculative knowledge involves belief. But this claim conflicts with the standard conception of knowledge, according to which knowledge is an inherently belief-involving phenomenon. Building on John Hyman’s account of knowledge as the ability to use a fact as a reason, I develop an alternative, two-tier, epistemology which allows that knowledge might really come in a belief-involving and an intention-involving form.     

Building blocks of self-control: increased tolerance for delay with bundled rewards

Impulsive choice can be defined as temporary preference for a smaller-sooner reward (SS) over a larger-later reward (LL). Hyperbolic discounting implies that impulsive choices will occur less when organisms choose between a series of SSs versus LLs all at once than when they choose between single SS versus LL pairs. Eight rats were exposed to two conditions of an intertemporal choice paradigm using sucrose solution as reward. In both conditions, the LL was 150 microl delayed by 3 s, while the SS was an immediate reward that ranged from 25-150 microl across sessions. Preference for the LL was greater when the chosen reward was automatically delivered three times in succession (bundled) than when it was chosen singly and delivered after each choice. For each of the 8 rats, the estimated SS amount that produced indifference was higher in the bundled condition than in the single condition. Because bundling in humans may be based on the perception that one's current choice is predictive of future choices, the data presented here may demonstrate an important building block of self-control.     

The epistemological status of a naturalized epistemology

Philosophically inclined psychologists and psychologically inclined philosophers often hold that the substantive discoveries of psychology can provide an empirical foundation for epistemology. In this paper it is argued that the ambition to found epistemology empirically faces certain unnoticed difficulties. Empirical theories concerned with knowledge‐gaining abilities have been historically associated with specific epistemological views such that the epistemology gives preferential support to the substantive theory, while the theory empirically supports the epistemology. Theories attribute to the subject just those epistemic abilities which associated epistemologies attribute to the scientist. The concept of epistemological significance is introduced as the significance a psychological theory can have for modifying the epistemological suppositions with which the theory was originally associated. Substantive psychological theories are strongly constrained by the epistemologies used in their development; the endorsement an epistemology receives from its associated theory should carry no weight. The alliance between psychology and epistemology is not progressive to the development of either field. Alternative sources of progress in epistemology and psychology are suggested.     

Interactive effects of numerical surface form and operand parity in cognitive arithmetic

In Experiment 1, adults (n = 48) performed simple addition, multiplication, and parity (i.e., odd-even) comparisons on pairs of Arabic digits or English number words. For addition and comparison, but not multiplication, response time increased with the number of odd operands. For addition, but not comparison, this parity effect was greater for words than for digits. In Experiment 2, adults (n = 50) solved simple addition problems in digit and word format and reported their strategies (i.e., retrieval or procedures). Procedural strategies were used more for odd than even addends and much more for word than digit problems. The results indicate that problem encoding and answer retrieval processes for cognitive arithmetic are interactive rather than strictly additive stages.     

Vice epistemology has a responsibility problem

Vice epistemology is in the business of defining epistemic vice. One of the proposed requirements of epistemic vices is that they are reprehensible—blameworthy in a non‐voluntarist way. Our problem, as vice epistemologists, is giving an analysis of non‐voluntarist responsibility that will count just the right qualities, no more and no less, as epistemic vices. If our analysis of non‐voluntarist responsibility ends up being too narrow, then it risks excluding some qualities that we want to count as epistemic vices, such as implicit biases. Whereas, if it ends up being too broad, it risks including qualities that we do not want to count as epistemic vices, such as impaired vision. I recommend a three‐step program for vice epistemologists: 1. admit that we have a responsibility problem; 2. strive to define the responsibility problem; 3. work together with specialists in non‐voluntarist responsibility to solve the responsibility problem.     

Social epistemology as a rhetoric of inquiry

Fuller's program of social epistemology engages a rhetoric of inquiry that can be usefully compared and contrasted with other discursive theories of knowledge, such as that of Richard Rorty. Resisting the model of conversation, Fuller strikes an activist posture and lays the groundwork for normative knowledge policy, in which persuasion and credibility play key roles. The image of investigation is one that overtly rejects the storehouse conception of knowledge and invokes the metaphors of distributive economics. Productive questions arise as to how notions of creation and distribution might guide this rhetoric.     

Relationship of arithmetic problem solving and reflective-impulsive cognitive styles in third-grade students

Different individuals approach mathematical problems in a variety of ways, with these different approaches also reflected in over-all cognitive styles. This investigation had two purposes, first, to determine whether good and poor arithmetic problem solvers differ substantially in cognitive style, and second, to determine whether the students, after training in techniques of solving arithmetic problems, improve their performance with no significant change in cognitive style. A total of 98 third graders participated (mean age 8.1 yr.; 50 boys, 48 girls). The Matching Familiar Figure Test was used to classify the students by cognitive style as either Reflective or Impulsive. Students also were given training with different problem-solving exercises for different arithmetic problems. The training program in problem-solving strategies did not improve performance on arithmetic problems for Reflective students; however, Impulsive students' performance did improve after training.     

Towards a new epistemology of moral progress

Abstract: Awareness that moral beliefs and practices have changed across time threatens our confidence in our current moral beliefs: if past moral beliefs turned out to be wrong, how can we be sure ours aren't likewise mistaken? In this paper, I set up four desiderata for a successful theory of moral progress: it must allow us to judge that progress has occurred, avoid the image of increasing correspondence towards ahistorical truthmakers, allow for revision in belief, and yet not be disobligating. Rorty's pragmatist account of moral progress delivers on the first three, but at the cost of failing to meet the fourth: it drains moral beliefs of their categorical force. I then outline K.E. Løgstrup's understanding of the relation between the ‘ethical demand’ and changing, socially mediated norms. While Løgstrup does posit an unchanging ground of normativity ‐ the “ethical demand” to act for the sake of the other whose welfare is in our hands – he also thinks that changing social norms are an indispensable part of ethical life. I argue that Løgstrup's discussion of the ‘refraction‘ of the ethical demand through changing social norms provides resources for an account of moral progress that fulfils these four desiderata.     

Building a cognitive map by assembling multiple path integration systems

Path integration and cognitive mapping are two of the most important mechanisms for navigation. Path integration is a primitive navigation system which computes a homing vector based on an animal’s self-motion estimation, while cognitive map is an advanced spatial representation containing richer spatial information about the environment that is persistent and can be used to guide flexible navigation to multiple locations. Most theories of navigation conceptualize them as two distinctive, independent mechanisms, although the path integration system may provide useful information for the integration of cognitive maps. This paper demonstrates a fundamentally different scenario, where a cognitive map is constructed in three simple steps by assembling multiple path integrators and extending their basic features. The fact that a collection of path integration systems can be turned into a cognitive map suggests the possibility that cognitive maps may have evolved directly from the path integration system.