A Theory of Mathematical Knowledge: Can Rules Account for Creative Behavior?

1971 ◽  
Vol 2 (3) ◽  
pp. 183-196
Author(s):  
Joseph M. Scandura
Keyword(s):  
Mathematical Knowledge ◽  
Semantic Knowledge ◽  
Higher Order ◽  
Main Argument ◽  
Creative Behavior ◽  
Finite Set ◽  
Rule Sets ◽  
Mathematical Behavior ◽  
Mathematical Systems

This paper describes some elements of an emerging theory of mathematical knowledge. It is proposed that the mathematical behavior any given (idealized) S is potentially capable of at any given stage of learning can be accounted for precisely in terms of a finite set of rules. In particular, it is hypothesized that rule characterizations of this sort can account for creative, as well as more routine, behavior. The main argument centers on the notion of allowing rules to operate on other rules (i.e., to act as higher order rules) and thus allowing rule sets to “grow.” (Newly generated rules are assumed to be available from then on.) Attention is first centered on the behaviors involved in knowing mathematical systems (semantic knowledge). Then, the same ideas are extended to logical interrelationships between properties of mathematical systems, e.g., proving theorems.

The Constructivist Researcher as Teacher and Model Builder

1983 ◽  
Vol 14 (2) ◽  
pp. 83-94 ◽  
Author(s):  
Paul Cobb ◽  
Leslie P. Steffe
Keyword(s):  
Mathematical Knowledge ◽  
Extended Period ◽  
Teaching Experiment ◽  
Constructivist Teaching ◽  
Individual Interviews ◽  
Model Builder ◽  
Mathematical Behavior ◽  
Constructivist Teaching Experiment

The constructivist teaching experiment is used in formulating explanations of children's mathematical behavior. Essentially, a teaching experiment consists of a series of teaching episodes and individual interviews that covers an extended period of time—anywhere from 6 weeks to 2 years. The explanations we formulate consist of models—constellations of theoretical constructs--that represent our understanding of children's mathematical realities. However, the models must be distinguished from what might go on in children's heads. They are formulated in the context of intensive interactions with children. Our emphasis on the researcher as teacher stems from our view that children's construction of mathematical knowledge is greatly influenced by the experience they gain through interaction with their teacher. Although some of the researchers might not teach, all must act as model builders to ensure that the models reflect the teacher's understanding of the children.

scholarly journals Introducing: Second-order permutation

2019 ◽  
Author(s):  
Bassey Godwin
Keyword(s):  
Background Knowledge ◽  
Higher Order ◽  
Second Order ◽  
Single Line ◽  
Constraint Set ◽  
Finite Set

In this study we answer questions that have to do with finding out the total number of ways of arranging a finite set of symbols or objects directly under a single line constraint set of finite symbols such that common symbols between the two sets do not repeat on the vertical positions. We go further to study the outcomes when both sets consist of the same symbols and when they consist of different symbols. We identify this form of permutation as 'second-order permutation' and show that it has a corresponding unique factorial which plays a prominent role in most of the results obtained. This has been discovered by examining many practical problems which give rise to the emergence of second-order permutation. Upon examination of these problems, it becomes clear that a directly higher order of permutation exist. Hence this study aims at equipping mathematics scholars, educators and researchers with the necessary background knowledge and framework for incorporating second-order permutation into the field of combinatorial mathematics.

HIERARCHY OF CERTAIN TYPES OF DNA SPLICING SYSTEMS

2012 ◽  
Vol 09 ◽  
pp. 271-277
Author(s):  
YUHANI YUSOF ◽  
NOR HANIZA SARMIN ◽  
T. ELIZABETH GOODE ◽  
MAZRI MAHMUD ◽  
FONG WAN HENG
Keyword(s):  
Formal Language ◽  
Deoxyribonucleic Acid ◽  
Restriction Enzymes ◽  
Finite Alphabet ◽  
Theoretic Analysis ◽  
Original Idea ◽  
Finite Set ◽  
Rule Sets ◽  
Biological Interpretation ◽  
Splicing Systems

A Head splicing system (H-system)consists of a finite set of strings (words) written over a finite alphabet, along with a finite set of rules that acts on the strings by iterated cutting and pasting to create a splicing language. Any interpretation that is aligned with Tom Head's original idea is one in which the strings represent double-stranded deoxyribonucleic acid (dsDNA) and the rules represent the cutting and pasting action of restriction enzymes and ligase, respectively. A new way of writing the rule sets is adopted so as to make the biological interpretation transparent. This approach is used in a formal language- theoretic analysis of the hierarchy of certain classes of splicing systems, namely simple, semi-simple and semi-null splicing systems. The relations between such systems and their associated languages are given as theorems, corollaries and counterexamples.

Language, Concepts, and Abstract Thought

2017 ◽  
Author(s):  
Martin V. Butz ◽  
Esther F. Kutter
Keyword(s):  
Language Production ◽  
Common Ground ◽  
Complex Form ◽  
Semantic Knowledge ◽  
Language Abilities ◽  
Finite Set ◽  
Universal Communication ◽  
Further Development ◽  
The Brain

Language is probably the most complex form of universal communication. A finite set of words enables us to express a mere infinite number of thoughts and ideas, which we set together by obeying grammatical rules and compositional, semantic knowledge. This chapter addresses how human language abilities have evolved and how they develop. A short introduction to linguistics covers the most important conceptualized aspects, including language production, phonology, syntax, semantics, and pragmatics. The brain considers these linguistic aspects seemingly in parallel when producing and comprehending sentences. The brain develops some dedicated language modules, which strongly interact with other modules. Evolution appears to have recruited prelinguistic developmental neural structures and modified them into maximally language-suitable structures. Moreover, evolution has most likely evolved language to further facilitate social cooperation and coordination, including the further development of theories of the minds of others. Language develops in a human child building on prelinguistic concepts, which are based on motor control-oriented structures detailed in the previous chapter. A final look at actual linguistic communication emphasizes that an imaginary common ground and individual private grounds unfold between speaker and listener, characterizing what is actually commonly and privately communicated and understood.

scholarly journals Combining concepts across categorical domains: a linking role of the precuneus

2021 ◽  
pp. 1-50
Author(s):  
Giuseppe Rabini ◽  
Silvia Ubaldi ◽  
Scott L. Fairhall
Keyword(s):  
Semantic Category ◽  
Semantic Knowledge ◽  
Higher Order ◽  
Conceptual Combination ◽  
Human Thought ◽  
Semantic Domain ◽  
Fmri Study ◽  
Single Category ◽  
Semantic Domains

Abstract The human capacity for semantic knowledge entails not only the representation of single concepts but the capacity to combine these concepts into the increasingly complex ideas that underlie human thought. This process involves not only the combination of concepts from within the same semantic category but frequently the conceptual combination across semantic domains. In this fMRI study (N=24) we investigate the cortical mechanisms underlying our ability to combine concepts across different semantic domains. Using five different semantic domains (People, Places, Food, Objects and Animals), we present sentences depicting concepts drawn from a single semantic domain as well as sentences that combine concepts from two of these domains. Contrasting single-category and combinedcategory sentences reveals that the precuneus is more active when concepts from different domains have to be combined. At the same time, we observe that distributed category selectivity representations persist when higher-order meaning involves the combination of categories and that this category-selective response is captured by the combination of the single categories composing the sentence. Collectively, these results suggest that the precuneus plays a role in the combination of concepts across different semantic domains, potentially functioning to link together category-selective representations distributed across the cortex.

scholarly journals Cortical circuits for mathematical knowledge: evidence for a major subdivision within the brain's semantic networks

2018 ◽  
Vol 373 (1740) ◽  
pp. 20160515 ◽  
Author(s):  
Marie Amalric ◽  
Stanislas Dehaene
Keyword(s):  
Mathematical Knowledge ◽  
Developmental Disorders ◽  
Brain Activity ◽  
Semantic Networks ◽  
Brain Regions ◽  
Semantic Knowledge ◽  
Simple Fact ◽  
High Level ◽  
Language Areas ◽  
Fact Retrieval

Is mathematical language similar to natural language? Are language areas used by mathematicians when they do mathematics? And does the brain comprise a generic semantic system that stores mathematical knowledge alongside knowledge of history, geography or famous people? Here, we refute those views by reviewing three functional MRI studies of the representation and manipulation of high-level mathematical knowledge in professional mathematicians. The results reveal that brain activity during professional mathematical reflection spares perisylvian language-related brain regions as well as temporal lobe areas classically involved in general semantic knowledge. Instead, mathematical reflection recycles bilateral intraparietal and ventral temporal regions involved in elementary number sense. Even simple fact retrieval, such as remembering that ‘the sine function is periodical’ or that ‘London buses are red’, activates dissociated areas for math versus non-math knowledge. Together with other fMRI and recent intracranial studies, our results indicated a major separation between two brain networks for mathematical and non-mathematical semantics, which goes a long way to explain a variety of facts in neuroimaging, neuropsychology and developmental disorders. This article is part of a discussion meeting issue ‘The origins of numerical abilities’.

scholarly journals Generalized finite difference schemes with higher order Whitney forms

2021 ◽  
Author(s):  
Lauri Kettunen ◽  
Jonni Lohi ◽  
Jukka Räbinä ◽  
Sanna Mönkölä ◽  
Tuomo Rossi
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Solution Process ◽  
Higher Order ◽  
Finite Difference Schemes ◽  
Local Character ◽  
Whitney Forms ◽  
The Sixties ◽  
Finite Set ◽  
Yee’S Scheme

Finite difference kind of schemes are popular in approximating wave propagation problems in finite dimensional spaces. While Yee's original paper on the finite difference method is already from the sixties, mathematically there still remains questions which are not yet satisfactorily covered. In this paper, we address two issues of this kind. Firstly, in the literature Yee's scheme is constructed separately for each particular type of wave problem. Here, we explicitly generalize the Yee scheme to a class of wave problems that covers at large physics field theories. For this we introduce Yee's scheme for all problems of a class characterised on a Minkowski manifold by i) a pair of first order partial differential equations and by ii) a constitutive relation that couple the differential equations with a Hodge relation. In addition, we introduce a strategy to systematically exploit higher order Whitney elements in Yee-like approaches. This makes higher order interpolation possible both in time and space. For this, we show that Yee-like schemes preserve the local character of the Hodge relation, which is to say, the constitutive laws become imposed on a finite set of points instead of on all ordinary points of space. As a result, the usage of higher order Whitney forms does not compel to change the actual solution process at all. This is demonstrated with a simple example.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

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