Empirical evaluation of the axioms of multiplicativity, commutativity, and monotonicity in ratio production of area

Although direct scaling methods have been widely used in the behavioral sciences since the 1950s, theoretical approaches which could clarify the implicit assumptions inherent in Stevens' ratio scaling approach were developed only recently. Today, it is generally accepted that the axioms of commutativity and multiplicativity are fundamental to the subjects' ratio scaling behavior. Therefore, both axioms were evaluated in ratio production of area. Participants were required to adjust the area of a variable circle to prescribed ratio production factors. The results are in accordance with previous empirical findings: commutativity was satisfied, whereas multiplicativity failed to hold. Additionally, the validity of the monotonicity property was analyzed, which postulates that the subjects' adjustments in a ratio production experiment preserve the mathematical order of the ratio production factors. Monotonicity was satisfied empirically, which is consistent with all the current theories of ratio scaling.     

Empirical evaluation of axioms fundamental to Stevens's ratio-scaling approach: I. Loudness production

Stevens's direct scaling methods rest on the assumption that subjects are capable of reporting or producing ratios of sensation magnitudes. Only recently, however, did an axiomatization proposed by Narens (1996) specify necessary conditions for this assumption that may be put to an empirical test. In the present investigation, Narens's central axioms of commutativity and multiplicativity were evaluated by having subjects produce loudness ratios. It turned out that the adjustments were consistent with the commutativity condition; multiplicativity (the fact that consecutive doubling and tripling of loudness should be equivalent to making the starting intensity six times as loud), however, was violated in a significant number of cases. According to Narens's (1996) axiomatization, this outcome implies that although in principle a ratio scale of loudness exists, the numbers used by subjects to describe sensation ratios may not be taken at face value.     

Empirical evaluation of axioms fundamental to Stevens’s ratio-scaling approach: I. Loudness production

Stevens’s direct scaling methods rest on the assumption that subjects are capable of reporting or producing ratios of sensation magnitudes. Only recently, however, did an axiomatization proposed by Narens (1996) specify necessary conditions for this assumption that may be put to an empirical test. In the present investigation, Narens’s central axioms ofcommutativity andmultiplicativity were evaluated by having subjects produce loudness ratios. It turned out that the adjustments were consistent with the commutativity condition; multiplicativity (the fact that consecutive doubling and tripling of loudness should be equivalent to making the starting intensity six times as loud), however, was violated in a significant number of cases. According to Narens’s (1996) axiomatization, this outcome implies that although in principle a ratio scale of loudness exists, the numbers used by subjects to describe sensation ratios may not be taken at face value.     

Macaques’ (Macaca mulatta) use of numerical cues in maze trials

We tested the ability of number-trained rhesus monkeys to use Arabic numeral cues to discriminate between different series of maze trials and anticipate the final trial in each series. The monkeys prior experience with numerals also allowed us to investigate spontaneous transfer between series. A total of four monkeys were tested in two experiments. In both experiments, the monkeys were trained on a computerized task consisting of three reinforced maze trials followed by one nonreinforced trial. The goal of the maze was an Arabic numeral 3, which corresponded to the number of reinforced maze trials in the series. In experiment 1 (n=2), the monkeys were given probe trials of the numerals 2 and 4 and in experiment 2 (n=2), they were given probe trials of the numerals 2–8. The monkeys receiving the probe trials 2 and 4 showed some generalization to the new numerals and developed a pattern of performing more slowly on the nonreinforced trial than the reinforced trial before it for most series, indicating the use of the changing numeral cues to anticipate the nonreinforced trial. The monkeys receiving probe trials of the numerals 2–8 did not predict precisely when the nonreinforced trial would occur in each series, but they did incorporate the changing numerals into their strategy for performing the task. This study provides the first evidence that number-trained monkeys can use Arabic numerals to perform a task involving sequential presentations.     

Examining the validity of numerical ratios in loudness fractionation

Direct psychophysical scaling procedures presuppose that observers are able to directly relate a numerical value to the sensation magnitude experienced. This assumption is based on fundamental conditions (specified by Luce, 2002), which were evaluated experimentally. The participants' task was to adjust the loudness of a 1-kHz tone so that it reached a certain prespecified fraction of the loudness of a reference tone. The results of the first experiment suggest that the listeners were indeed able to make adjustments on a ratio scale level. It was not possible, however, to interpret the nominal fractions used in the task as "true" scientific numbers. Thus, Stevens's (1956, 1975) fundamental assumption that an observer can directly assess the sensation magnitude a stimulus elicits did not hold. In the second experiment, the possibility of establishing a specific, strictly increasing transformation function that related the overt numerals to the latent mathematical numbers was investigated. The results indicate that this was not possible for the majority of the 7 participants.     

Structural complexity of motor patterns: A study on reaction times and movement times of handwritten letters

This study investigates the effects of psychomotor complexity on latencies for beginning to write single letters and numerals, and on times taken to complete the first strokes of letters and numerals. An experiment measured simple and choice reaction times for writing homogeneous graphemes (i.e., letters or numerals made up of similar strokes) and for writing heterogeneous graphemes (i.e., characters made up of dissimilar strokes). It was assumed that the motor programme for writing a letter is retrieved from long-term memory and briefly held, until it is used, in a short-term buffer store. The experiment examined the hypothesis that it is more difficult to read out homogeneous than heterogeneous stroke-structures from this store. For three out of four allographic grapheme pairs, homogeneous graphemes required longer initiation times or longer movement times for completion of the first stroke than did heterogeneous graphemes. These results are discussed in relation to recent findings on motor programming based on the use of reaction-time paradigms.     

Scaling relationships in indentation of power-law creep solids using self-similar indenters

We use dimensional analysis to derive scaling relationships for self-similar indenters indenting solids that exhibit power-law creep. We identify the parameter that represents the indentation strain rate. The scaling relationships are applied to several types of indentation creep experiment with constant displacement rate, constant loading rate or constant ratio of loading rate over load. The predictions compare favourably with experimental observations reported in the literature. Finally, a connection is found between creep and 'indentation-size effect' (i.e. changing hardness with indentation depth or load).     

Empirical evaluation of a model of global psychophysical judgments: IV. Forms for the weighting function

Understanding the psychological interpretation of numerals is of both practical and theoretical interest. In classical magnitude estimation, respondents match numerals to sensations and in magnitude production they select sensations that stand in a prescribed numerical ratio to a given standard. The present work focusses on evaluating several possible, and related, forms for the function W formulating the distortion of numerals. The main form, of which a power function is a special case, is the Prelec exponential/power representation. Behavioral equivalents to power and to Prelec functions are formulated, tested, and rejected. It is argued that either the mathematical form or the assumption W(1)=1 is wrong. Whereas, the axiomatic literature has focussed exclusively on the former inference, we explore the alternate that W(1)≠1. Behavioral axioms are formulated in each case and experimentally tested. We conclude that most respondents satisfy a general power function and that those who do not, satisfy the general Prelec function.     

Relations as transformations: Implications for analogical reasoning

We present two experiments assessing whether the size of a transformation instantiating a relation between two states of the world (e.g., shrinks) is a performance factor affecting analogical reasoning. The first experiment finds evidence of transformation size as a significant factor in adolescent analogical problem solving while the second experiment finds a similar effect on adult analogical reasoning using a markedly different analogical completion paradigm. The results are interpreted as providing evidence for the more general framework that cognitive representations of relations are best understood as mental transformations.     

Stevens' power law and the problem of meaningfulness

Frequently, it is postulated that the results of a ratio production (resp., ratio estimation) experiment can be summarized by Stevens' power law psi=alphaphi(beta). In the present article, it is argued that the power law parameters depend, among other things, on the standard stimulus presented as a reference point, and the physical stimulus scale by which the physical intensities are measured. To formalize this idea, a new formulation of Stevens' power law is presented. We show that the exponent in Stevens' power law can only be interpreted in a meaningful way if the stimulus scale is a ratio scale. Furthermore, we present empirically testable axioms (termed invertibility and weak multiplicativity) which are both necessary and sufficient for the power law exponent to be invariant under changes of the standard stimulus. Finally, invertibility and weak multiplicativity are evaluated in a ratio production experiment. Ten participants were required to adjust the area of variable circles to prescribed ratio production factors. Both axioms are violated for all participants. The results cast doubts on the well-established practice of comparing power law exponents across different modalities.     

Applying Evidence-Centered Design for the Development of Game-Based Assessments in Physics Playground

Game-based assessment (GBA) is a specific use of educational games that employs game activities to elicit evidence for educationally valuable skills and knowledge. While this approach can provide individualized and diagnostic information about students, the design and development of assessment mechanics for a GBA is a nontrivial task. In this article, we describe the 10-step procedure that the design team of Physics Playground (formerly known as Newton's Playground) has established by adapting evidence-centered design to address unique challenges of GBA. The scaling method used for Physics Playground was Bayesian networks; thus this article describes specific actions taken for the iterative process of constructing and revising Bayesian networks in the context of the game Physics Playground.     

PARADOXICAL PROPOSITIONS

This paper concerns two paradoxes involving propositions. The first is Russell's paradox from Appendix B of The Principles of Mathematics, a version of which was later given by Myhill. The second is a paradox in the framework of possible worlds, given by Kaplan. This paper shows a number of things about these paradoxes. First, we will see that, though the Russell/Myhill paradox and the Kaplan paradox might appear somewhat different, they are really just variants of the same phenomenon. Though they do this in different ways, the core of each paradox is to use the notion of a proposition to construct a function, f, from the power set of some set into the set itself. Next we will see how this paradox fits into the Inclosure Schema. Finally, I will provide a model of the paradox in question, showing its results to be non‐trivial, though inconsistent.     

Digit identity influences numerical estimation in children and adults

Learning the meanings of Arabic numerals involves mapping the number symbols to mental representations of their corresponding, approximate numerical quantities. It is often assumed that performance on numerical tasks, such as number line estimation (NLE), is primarily driven by translating from a presented numeral to a mental representation of its overall magnitude. Part of this assumption is that the overall numerical magnitude of the presented numeral, not the specific digits that comprise it, is what matters for task performance. Here we ask whether the magnitudes of the presented target numerals drive symbolic number line performance, or whether specific digits influence estimates. If the former is true, estimates of numerals with very similar magnitudes but different hundreds digits (such as 399 and 402) should be placed in similar locations. However, if the latter is true, these placements will differ significantly. In two studies (N = 262), children aged 7–11 and adults completed 0–1000 NLE tasks with target values drawn from a set of paired numerals that fell on either side of “Hundreds” boundaries (e.g., 698 and 701) and “Fifties” boundaries (e.g., 749 and 752). Study 1 used an atypical speeded NLE task, while Study 2 used a standard non‐speeded NLE task. Under both speeded and non‐speeded conditions, specific hundreds digits in the target numerals exerted a strong influence on estimates, with large effect sizes at all ages, showing that the magnitudes of target numerals are not the primary influence shaping children's or adults’ placements. We discuss patterns of developmental change and individual difference revealed by planned and exploratory analyses.     

Finding one's meaning: a test of the relation between quantifiers and integers in language development

We explored children’s early interpretation of numerals and linguistic number marking, in order to test the hypothesis (e.g., Carey (2004). Bootstrapping and the origin of concepts. Daedalus, 59-68) that children’s initial distinction between one and other numerals (i.e., two, three, etc.) is bootstrapped from a prior distinction between singular and plural nouns. Previous studies have presented evidence that in languages without singular-plural morphology, like Japanese and Chinese, children acquire the meaning of the word one later than in singular-plural languages like English and Russian. In two experiments, we sought to corroborate this relation between grammatical number and integer acquisition within English. We found a significant correlation between children’s comprehension of numerals and a large set of natural language quantifiers and determiners, even when controlling for effects due to age. However, we also found that 2-year-old children, who are just acquiring singular-plural morphology and the word one, fail to assign an exact interpretation to singular noun phrases (e.g., a banana), despite interpreting one as exact. For example, in a Truth-Value Judgment task, most children judged that a banana was consistent with a set of two objects, despite rejecting sets of two for the numeral one. Also, children who gave exactly one object for singular nouns did not have a better comprehension of numerals relative to children who did not give exactly one. Thus, we conclude that the correlation between quantifier comprehension and numeral comprehension in children of this age is not attributable to the singular-plural distinction facilitating the acquisition of the word one. We argue that quantifiers play a more general role in highlighting the semantic function of numerals, and that children distinguish between numerals and other quantifiers from the beginning, assigning exact interpretations only to numerals.     

Gleason's Theorem Has a Constructive Proof

Gleason's theorem for R ³ says that if f is a nonnegative function on the unit sphere with the property that f(x) + f(y) + f(z) is a fixed constant for each triple x,y,z of mutually orthogonal unit vectors, then f is a quadratic form. We examine the issues raised by discussions in this journal regarding the possibility of a constructive proof of Gleason"s theorem in light of the recent publication of such a proof.     

Intrinsicf0 differences in spoken and sung vowels and their perception by listeners

We explore how listeners perceive distinct pieces of phonetic information that are conveyed in parallel by the fundamental frequency (f0) contour of spoken and sung vowels. In a first experiment, we measured differences inf0 of /i/ and /a/ vowels spoken and sung by unselected undergraduate participants. Differences in “intrinsicf0” (withf0 of /i/ higher than of /a/) were present in spoken and sung vowels; however, differences in sung vowels were smaller than those in spoken vowels. Four experiments tested a hypothesis that listeners would not hear the intrinsicf0 differences as differences in pitch on the vowel, because they provide information, instead, for production of a closed or open vowel. The experiments provide clear evidence of “parsing” of intrinsicf0 from thef0 that contributes to perceived vowel pitch. However, only some conditions led to an estimate of the magnitude of parsing that closely matched the magnitude of produced intrinsicf0 differences.     

Does the cue help? Children's understanding of multiplicative concepts in different problem contexts

Background: Understanding arithmetical principles is a key part of a conceptual understanding of mathematics. However, very little attention has been paid to children's understanding of multiplicative, as compared to additive, principles. Aims: This study investigated(a) children's ability to use commutative and distributive cues to solve multiplication problems, (b) whether their ability to use these cues depends on the problem context, and(c) whether separate mechanisms might underlie children's understanding of commutativity and distributivity. Sample: Twenty‐seven 9‐year‐olds (Year 5) and thirty‐two 10‐year‐olds (Year 6). Methods: Forty‐eight multiplication problems (with a multiple‐choice response format) were presented to children. There were four types of problem: Commutative, Distributive, Combined commutative‐distributive(all preceded by a cue) and No cue problems. Each type of problem was presented in three different contexts: Isomorphism of measures, Area, and Cartesian product. Results: Children demonstrated a good understanding of commutativity but a very poor understanding of distributivity. A common mistake in the distributive problems was to select the number that was one more, or one less, than the answer in the cue. Children's understanding of distributivity (butnot commutativity) seemed to depend on the problem context. Factor analysis suggested that separate factors underlie the ability to solve commutative and distributive problems. Conclusions: Nine‐ and 10‐year‐olds understand commutativity, but are unable to use the distributive principle in multiplication. Their errors suggest that they may confuse some of the principles of multiplication with those of addition. When children do begin to understand the principle of distributivity, they most easily apply it in the context of Isomorphism of measures multiplication problems. The implications for mathematical education are discussed.     

The Legacy of Neoplatonism in F. W. J. Schelling's Thought

F.W.J. Schelling, one of the essential thinkers in the development of German Idealism, formed his own thought not only in a critical dialogue with Kant's and Fichte's transcendentalism and Hegel's earlier conception of thinking, but also in an intensive discussion with Plato and Aristotle. Over and above that, Neoplatonism – especially Plotinus, Proclus and the Christian Dionysius the Areopagite – played a decisive role in Schelling's reception and transformation of ancient philosophy. Selecting the manifold aspects which could be reflected on in this field, I want to make plausible as a transcendental analogy to Plotinus' concept of self-knowledge Schelling's requirement for a raising-up and transformation of the finite 'I' into the form of the Absolute, whose central features converge with the goal of the Plotinian self – transformation of thought into a timeless self-thinking and its ground. A main part of this paper discusses Schelling's and Plotinus' concept of nature as a dynamic process constituted by an immanent 'creating theoria'. Furthermore we find in Schelling's theory of the Absolute as the 'utterly One' a union of Plotinus' notion of a pure One beyond Being with that of the reflexive self-presence of nous, so that this Absolute can be understood as an All-Unity which grounds and embraces all actuality – because it is in itself the most unifying self-affirmation or self-mediation. What follows is a reflection on the anagogical function of art, especially from the viewpoint of Plotinus' non-Platonic rehabilitation of art as an imitation of nature. The last perspectives focus on Schelling's concept of matter and emanation – as different from and at the same time coherent with that of Plotinus – and on Schelling's theory of an absolute self – willing will in connection with Plotinus' Enneads VI.8, 'On free will and the will of the One' as a causa sui.     

Differential Method of Characterizing Gait Strategies From Step Lengths and Frequencies: Strategy of Velocity Modulation

The differential method consists of the analysis of the variation of gait parameters length, frequency, and velocity with respect to their mean values, respectively, ΔL = L — L_m , Δf = f — f_m , and Δv = v – v_m , where L_m , f_m , and v_m represent the mean values of those parameters. Assuming that the strategy of modulation of velocity implies that L and f are functions of v and that statistical analyses of ratios ΔL/Δv and Δf/Δv have established that there is a very significant linear correlation, close to 1, between those ratios, the mathematical procedure allows one to determine the equation of step length, L = a · f + b · v + K, where a and b are the slope and the intercept of the linear regression and K is close to L_m . The equation was experimentally tested on 140 gait sequences performed by 6 participants and for gait velocities ranging from 0.6 to 2.2 m/s and was found to be very representative of all individual values. The differential method provides another way of using the derivative of velocity, v = L·f, to characterize the strategy of velocity modulation, which then permits one to determine the linear equation of velocity, v = f · L_m + L · f_m — L_m · f_m , and to show that the respective parts played by each parameter in the progression velocity are approximately equal. The author establishes the uniqueness of the different linear adjustments and discusses the differential method's own modes of use, that is, interindividually or globally.     

Inference and exact numerical representation in early language development

How do children as young as 2 years of age know that numerals, like one, have exact interpretations, while quantifiers and words like a do not? Previous studies have argued that only numerals have exact lexical meanings. Children could not use scalar implicature to strengthen numeral meanings, it is argued, since they fail to do so for quantifiers [Papafragou, A., & Musolino, J. (2003). Scalar implicatures: Experiments at the semantics–pragmatics interface. Cognition, 86, 253–282]. Against this view, we present evidence that children’s early interpretation of numerals does rely on scalar implicature, and argue that differences between numerals and quantifiers are due to differences in the availability of the respective scales of which they are members. Evidence from previous studies establishes that (1) children can make scalar inferences when interpreting numerals, (2) children initially assign weak, non-exact interpretations to numerals when first acquiring their meanings, and (3) children can strengthen quantifier interpretations when scalar alternatives are made explicitly available.