Nature’s drawing: problems and resolutions in the mathematization of motion

The mathematical nature of modern science is an outcome of a contingent historical process, whose most critical stages occurred in the seventeenth century. ‘The mathematization of nature’ (Koyré 1957, From the closed world to the infinite universe, 5) is commonly hailed as the great achievement of the ‘scientific revolution’, but for the agents affecting this development it was not a clear insight into the structure of the universe or into the proper way of studying it. Rather, it was a deliberate project of great intellectual promise, but fraught with excruciating technical challenges and unsettling epistemological conundrums. These required a radical change in the relations between mathematics, order and physical phenomena and the development of new practices of tracing and analyzing motion. This essay presents a series of discrete moments in this process. For mediaeval and Renaissance philosophers, mathematicians and painters, physical motion was the paradigm of change, hence of disorder, and ipso facto available to mathematical analysis only as idealized abstraction. Kepler and Galileo boldly reverted the traditional presumptions: for them, mathematical harmonies were embedded in creation; motion was the carrier of order; and the objects of mathematics were mathematical curves drawn by nature itself. Mathematics could thus be assigned an explanatory role in natural philosophy, capturing a new metaphysical entity: pure motion. Successive generations of natural philosophers from Descartes to Huygens and Hooke gradually relegated the need to legitimize the application of mathematics to natural phenomena and the blurring of natural and artificial this application relied on. Newton finally erased the distinction between nature’s and artificial mathematics altogether, equating all of geometry with mechanical practice.     

Epistemic injustice in mathematics

We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept to highlight a potential danger of intellectual enculturation.

     

Nicolas Bourbaki and the concept of mathematical structure

In the present article two possible meanings of the term mathematical structure are discussed: a formal and a nonformal one. It is claimed that contemporary mathematics is structural only in the nonformal sense of the term. Bourbaki's definition of structure is presented as one among several attempts to elucidate the meaning of that nonformal idea by developing a formal theory which allegedly accounts for it. It is shown that Bourbaki's concept of structure was, from a mathematical point of view, a superfluous undertaking. This is done by analyzing the role played by the concept, in the first place, within Bourbaki's own mathematical output. Likewise, the interaction between Bourbaki's work and the first stages of category theory is analyzed, on the basis of both published texts and personal documents.Several persons have read and criticized earlier drafts of this paper. I thank them all for forcing me to formulate my ideas more simply and convincingly. Special thanks are due to Professors Pierre Cartier (Bures-sur-Yvette), Giorgio Israel (Rome), and Andrée C. Ehresmann (Amiens) for stimulating and illuminating conversations.     

Structuralism as a philosophy of mathematical practice

This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure and the set. In the second case, from algebraic topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but that structures may not be all there is to mathematics. I wish to thank Colin McLarty as well as the anonymous referees for helpful comments on earlier versions of this paper.     

The comparability of scores from three mathematics test of the College Entrance Examination Board

Scores from three mathematics tests of the College Entrance Examination Board were examined in order to determine the effect on the scores of (1) choice of test, (2) amount of training in mathematics, and (3) recency of training in mathematics. Groups of candidates were paired in a number of comparisons and matched by means of a regression technique which is described. On the average, students of similar ability made comparable scores on the mathematical section of the Scholastic Aptitude Test and on the Comprehensive Mathematics Test. The scores of candidates who took the Intermediate Mathematics Test averaged substantially higher than those of comparable students who took either of the other two tests. A greater amount of mathematical training and more recent training were both found to be positively related to scores on the mathematical section of the Scholastic Aptitude Test and on the Intermediate Mathematics Test, but the effect of recency appeared to be less than one might expect.The author is indebted to Mrs. L. B. Plumlee of the Educational Testing Service for her extensive aid in carrying out this project.     

A quasi-nonmetric method for multidimensional scaling VIA an extended euclidean model

An Extended Two-Way Euclidean Multidimensional Scaling (MDS) model which assumes both common and specific dimensions is described and contrasted with the standard (Two-Way) MDS model. In this Extended Two-Way Euclidean model then stimuli (or other objects) are assumed to be characterized by coordinates onR common dimensions. In addition each stimulus is assumed to have a dimension (or dimensions) specific to it alone. The overall distance between objecti and objectj then is defined as the square root of the ordinary squared Euclidean distance plus terms denoting the specificity of each object. The specificity,s _j , can be thought of as the sum of squares of coordinates on those dimensions specific to objecti, all of which have nonzero coordinatesonly for objecti. (In practice, we may think of there being just one such specific dimension for each object, as this situation is mathematically indistinguishable from the case in which there are more than one.)We further assume that _ij =F(d _ij ) +e _ij where _ij is the proximity value (e.g., similarity or dissimilarity) of objectsi andj,d _ij is the extended Euclidean distance defined above, whilee _ij is an error term assumed i.i.d.N(0, ²).F is assumed either a linear function (in the metric case) or a monotone spline of specified form (in the quasi-nonmetric case). A numerical procedure alternating a modified Newton-Raphson algorithm with an algorithm for fitting an optimal monotone spline (or linear function) is used to secure maximum likelihood estimates of the paramstatistics) can be used to test hypotheses about the number of common dimensions, and/or the existence of specific (in addition toR common) dimensions.This approach is illustrated with applications to both artificial data and real data on judged similarity of nations.     

Arithmetic,Mathematical Intuition,and Evidence

Abstract

This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Gödel: Platonism and Rationalism in Mathematics and Logic (Oxford: Oxford University Press, 2011). The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the genetic analysis of arithmetic knowledge that is called for in Husserl’s philosophy. This issues in an elaboration on a number of Gödel’s remarks about the meaning of his incompleteness theorems for the notion of evidence in mathematics.     

Feature cues in probability learning: Data base information and judgment

Judgments of probabilistic events are often based partly on some information about past similar events. This study investigates the impact of summarized historical data termed a feature cue on performance in a cue probability learning task. Judges (n = 64) made 150 predictions of a criterion variable (Y_e) from a single cue variable (X). The feature cue variable (Z) provided judges with the “average past criterion” for the cue value on trial i, i.e., the conditional mean . Availability of the feature cue was varied with an AB-BA transfer design. Results demonstrate that the presence of the feature cue greatly imporved prediction achievement and accuracy. Under certain conditions, consistency and cue weighting were also improved by the feature cue aid. Although the feature cue value itself was not used as a prediction, it served as an anchor, around which judgments were dispersed. Implications for decision making with data base information are discussed.     

Gender and Mathematical Problem Solving

The relationship between gender and mathematical problem-solving among high ability students depends on the attributes of the problem solving questions. This was evident in the present study of 12-year-olds. The children were from predominately White families. Eighty-three males and 76 females were tested in both the fall and the spring on the Fennema-Sherman Mathematics Attitudes Scales and on the Canadian Test of Basic Skills (CTBS). In the Spring, students were also tested on the GAUSS. Both the CTBS and the GAUSS measure mathematical problem solving. Among high ability students, there were gender differences on the problem-solving scale of the CTBS but not on the GAUSS, even though the GAUSS was independently rated as the more abstract and difficult of the two tests. The present study describes the implications of this for the question of the origin of gender differences in mathematics, and also looked at the relationship between attitudes toward mathematics and mathematical problem-solving performance.     

A contemporary sense of existentialism

Philosophical existentialism has sought to understand the nature of human existence and the possible meaning(s) that might be made thereof. For the noteworthy existentialist Jean-Paul Sartre, the meaning of life cannot be said to subsist somewhere beyond the province of individual human existence, since meaning is born of a fundamental freedom which inheres in human consciousness. From a more contemporary poststructuralist philosophical perspective, however, one might argue that Sartre’s individualist conception of existential meaning in Being and Nothingness remains fettered to an order of signification reliant upon a vestigial “metaphysics of presence”, where the presence of the signified has simply been displaced from the transcendental domain to immanent human subjectivity. This is potentially problematic insofar as such an order of meaning qua signification is destined to suspend meaning at a perpetually deferred distance; and concomitantly, human existential meaning remains interminably frustrated. However, using the contemporary philosophical insights of Jean-Luc Nancy, it can be argued that a contemporary (re) conceptualisation of existentialist thought might allow existentialism to liberate itself from a ceaselessly suspended signification of meaning, specifically by arguing for a means-to-meaning(s) always already manifest(ing) between human beings oriented towards the contemporary world as a shared space of sense.     

From Searle’s Chinese room to the mathematics classroom: technical and cognitive mathematics

Employing Searle’s views, I begin by arguing that students of Mathematics behave similarly to machines that manage symbols using a set of rules. I then consider two types of Mathematics, which I call Cognitive Mathematics and Technical Mathematics respectively. The former type relates to concepts and meanings, logic and sense, whilst the latter relates to algorithms, heuristics, rules and application of various techniques. I claim that an upgrade in the school teaching of Cognitive Mathematics is necessary. The aim is to change the current mentality of the stakeholders so as to compensate for the undue value presently attached to Technical Mathematics, due to advances in technology and its applications, and thus render the two sides of Mathematics equal. Furthermore, I suggest a reorganization/systematization of School Mathematics into a cognitive network to facilitate students’ understanding of the subject. The final goal is the transition from mechanical execution of rules to better understanding and in-depth knowledge of Mathematics.

Hick’s law equivalent for reaction time to individual stimuli

Hick’s law, one of the few law-like relationships involving human performance, expresses choice reaction time as a linear function of the mutual information between the stimulus and response events. However, since this law was first proposed in 1952, its validity has been challenged by the fact that it only holds for the overall reaction time (RT) across all the stimuli, and does not hold for the reaction time (RT_i) for each individual stimulus. This paper introduces a new formulation in which RT_i is a linear function of (1) the mutual information between the event that stimulus i occurs and the set of all potential response events and (2) the overall mutual information for all stimuli and responses. Then Hick’s law for RT follows as the weighted mean of each side of the RT_i equation using the stimulus probabilities as the weights. The new RT_i equation incorporates the important speed–accuracy trade-off characteristic. When the performance is error-free, RT_i becomes a linear function of two entropies as measures of stimulus uncertainty or unexpectancy. Reanalysis of empirical data from a variety of sources provide support for the new law-like relationship.     

On Some Properties of Humanly Known and Humanly Knowable Mathematics

We argue that the set of humanly known mathematical truths (at any given moment in human history) is finite and so recursive. But if so, then given various fundamental results in mathematical logic and the theory of computation (such as Craig’s in J Symb Log 18(1): 30–32(1953) theorem), the set of humanly known mathematical truths is axiomatizable. Furthermore, given Godel’s (Monash Math Phys 38: 173–198, 1931) First Incompleteness Theorem, then (at any given moment in human history) humanly known mathematics must be either inconsistent or incomplete. Moreover, since humanly known mathematics is axiomatizable, it can be the output of a Turing machine. We then argue that any given mathematical claim that we could possibly know could be the output of a Turing machine, at least in principle. So the Lucas-Penrose (Lucas in Philosophy 36:112–127, 1961; Penrose, in The Emperor’s new mind. Oxford University Press, Oxford (1994)) argument cannot be sound.     

Monotone regression: Continuity and differentiability properties

Least-squares monotone regression has received considerable discussion and use. Consider the residual sum of squaresQ obtained from the least-squares monotone regression ofy _i onx _i . TreatingQ as a function of they _i , we prove that the gradient Q exists and is continuous everywhere, and is given by a simple formula. (We also discuss the gradient ofd=Q ^1/2.) These facts, which can be questioned (Louis Guttman, private communication), are important for the iterative numerical solution of models, such as some kinds of multidimensional scaling, in which monotone regression occurs as a subsidiary element, so that they _i and hence indirectlyQ are functions of other variables.     

Understanding Public Reaction to Energy Proposals: An Application of the Fishbein Model1

The Fishbein attitude model was applied to voter decision-making on an energy ballot proposal. Questionnaires were sent to a random sample of potential voters in Oregon's 1976 general election and dealt with the Nuclear Safeguards Initiative, a measure that would place restrictions on future nuclear power plants. Questionnaire items probed the attitude toward the act of voting “Yes” on the measure (A_act), perceived likelihood of various consequences of voting “Yes” (B_i), evaluations of these consequences (e_i), the subjective norm (SN), normative beliefs (NB_i), motivation to comply with several referents (Mc_i), and voting intention (VI). A follow-up interview determined the actual voting behavior (VB) of persons responding to the questionnaire. The following model predictions were tested and strongly supported by the data: (a) VB=VI; and (e) external variables have a nonsignificant relation to VB once VI is partialled out. It was concluded that the Fishbein model should be extremely useful in understanding public reaction to future energy proposals.     

A Mathematician Reflects on the Useful and Reliable Illusion of Reality in Mathematics

Recent years have seen a growing acknowledgement within the mathematical community that mathematics is cognitively/socially constructed. Yet to anyone doing mathematics, it seems totally objective. The sensation in pursuing mathematical research is of discovering prior (eternal) truths about an external (abstract) world. Although the community can and does decide which topics to pursue and which axioms to adopt, neither an individual mathematician nor the entire community can choose whether a particular mathematical statement is true or false, based on the given axioms. Moreover, all the evidence suggests that all practitioners work with the same ontology. (My number 7 is exactly the same as yours.) How can we reconcile the notion that people construct mathematics, with this apparent choice-free, predetermined objectivity? I believe the answer is to be found by examining what mathematical thinking is (as a mental activity) and the way the human brain acquired the capacity for mathematical thinking.

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Association between basic numerical abilities and mathematics achievement

Various measures have been used to investigate number processing in children, including a number comparison or a number line estimation task. The present study aimed to examine whether and to which extent these different measures of number representation are related to performance on a curriculum‐based standardized mathematics achievement test in kindergarteners, first, second, and sixth graders. Children completed a number comparison task and a number line estimation task with a balanced set of symbolic (Arabic digits) and non‐symbolic (dot patterns) stimuli. Associations with mathematics achievement were observed for the symbolic measures. Although the association with number line estimation was consistent over grades, the association with number comparison was much stronger in kindergarten compared to the other grades. The current data indicate that a good knowledge of the numerical meaning of Arabic digits is important for children's mathematical development and that particularly the access to the numerical meaning of symbolic digits rather than the representation of number per se is important.     

Mathematics ability and related skills in preschoolers born very preterm

Children born very preterm (VPT) are at risk for academic, behavioral, and/or emotional problems. Mathematics is a particular weakness and better understanding of the relationship between preterm birth and early mathematics ability is needed, particularly as early as possible to aid in early intervention. Preschoolers born VPT (n = 58) and those born full term (FT; n = 29) were administered a large battery of measures within 6 months of beginning kindergarten. A multiple-mediation model was utilized to characterize the difference in skills underlying mathematics ability between groups. Children born VPT performed significantly worse than FT-born children on a measure of mathematics ability as well as full-scale IQ, verbal skills, visual–motor integration, phonological awareness, phonological working memory, motor skills, and executive functioning. Mathematics was significantly correlated with verbal skills, visual–motor integration, phonological processing, and motor skills across both groups. When entered into the mediation model, verbal skills, visual–motor integration, and phonological awareness were significant mediators of the group differences. This analysis provides insights into the pre-academic skills that are weak in preschoolers born VPT and their relationship to mathematics. It is important to identify children who will have difficulties as early as possible, particularly for VPT children who are at higher risk for academic difficulties. Therefore, this model may be used in evaluating VPT children for emerging difficulties as well as an indicator that if other weaknesses are found, an assessment of mathematics should be conducted.     

The developmental relations between spatial cognition and mathematics in primary school children

Spatial thinking is an important predictor of mathematics. However, existing data do not determine whether all spatial sub‐domains are equally important for mathematics outcomes nor whether mathematics–spatial associations vary through development. This study addresses these questions by exploring the developmental relations between mathematics and spatial skills in children aged 6–10 years (N = 155). We extend previous findings by assessing and comparing performance across Uttal et al.'s (2013), four spatial sub‐domains. Overall spatial skills explained 5%–14% of the variation across three mathematics performance measures (standardized mathematics skills, approximate number sense and number line estimation skills), beyond other known predictors of mathematics including vocabulary and gender. Spatial scaling (extrinsic‐static sub‐domain) was a significant predictor of all mathematics outcomes, across all ages, highlighting its importance for mathematics in middle childhood. Other spatial sub‐domains were differentially associated with mathematics in a task‐ and age‐dependent manner. Mental rotation (intrinsic‐dynamic skills) was a significant predictor of mathematics at 6 and 7 years only which suggests that at approximately 8 years of age there is a transition period regarding the spatial skills that are important for mathematics. Taken together, the results support the investigation of spatial training, particularly targeting spatial scaling, as a means of improving both spatial and mathematical thinking.