Natural Number Concepts Research Articles

Where Our Number Concepts Come From

Where Our Number Concepts Come From

Proto-numerosities and concepts of number: Biologically plausible and culturally mediated top-down mathematical schemas

AbstractEarly quantitative skills cannot be directly extended to provide the richness, precision, and sophistication of the concept of natural number. These skills must interact with top-down mathematical schemas, which can be explained by bodily grounded everyday mechanisms for abstraction and imagination (e.g., conceptual metaphor, blending) that are both biologically plausible and culturally shaped (established beyond the child's mind).

The innate schema of natural numbers does not explain historical, cultural, and developmental differences

AbstractRips et al.'s proposition cannot account for the facts that (1) a historical look at the word number systems suggests that the concept of natural numbers has been progressively elaborated; (2) people from cultures without an elaborate counting system do not master the concept of natural numbers; (3) children take time to master natural numbers; and (4) the competing advantage of the postulated math schema in the natural selection process is not obvious.

Natural number concepts: No derivation without formalization

AbstractThe conceptual building blocks suggested by developmental psychologists may yet play a role in how the human learner arrives at an understanding of natural number. The proposal of Rips et al. faces a challenge, yet to be met, faced by all developmental proposals: to describe the logical space in which learners ever acquire new concepts.

Why cardinalities are the “natural” natural numbers

AbstractAccording to Rips et al., numerical cognition develops out of two independent sets of cognitive primitives – one that supports enumeration, and one that supports arithmetic and the concepts of natural numbers. I argue against this proposal because it incorrectly predicts that natural number concepts could develop without prior knowledge of enumeration.

Music training, engagement with sequence, and the development of the natural number concept in young learners

AbstractStudies by Gardiner and colleagues connecting musical pitch and arithmetic learning support Rips et al.'s proposal that natural number concepts are constructed on a base of innate abilities. Our evidence suggests that innate ability concerning sequence (“Basic Sequencing Capability” or BSC) is fundamental. Mathematical engagement relating number to BSC does not develop automatically, but, rather, should be encouraged through teaching.

Bridging the gap between intuitive and formal number concepts: An epidemiological perspective

AbstractThe failure of current bootstrapping accounts to explain the emergence of the concept of natural numbers does not entail that no link exists between intuitive and formal number concepts. The epidemiology of representations allows us to explain similarities between intuitive and formal number concepts without requiring that the latter are directly constructed from the former.

Do mental magnitudes form part of the foundation for natural number concepts? Don't count them out yet

AbstractThe current consensus among most researchers is that natural number is not built solely upon a foundation of mental magnitudes. On their way to the conclusion that magnitudes do not form any part of that foundation, Rips et al. pass rather quickly by theories suggesting that mental magnitudes might play some role. These theories deserve a closer look.

From numerical concepts to concepts of number

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas.

The origins of number: Getting developmental

AbstractRips et al. raise important questions about the relation between infant quantification and achievement of natural number concepts. However, they may be oversimplifying the interactions that characterize actual development in real time. Though they propose a worthwhile agenda for future research, its explanatory power will be limited if it does not address developmental issues with greater sensitivity.

Natural Numbers and Infinitesimals: A Discussion between Benno Kerry and Georg Cantor

During the first months of 1887, while completing the drafts of his Mitteilungen zur Lehre vom Transfiniten, Georg Cantor maintained a continuous correspondence with Benno Kerry. Their exchange essentially concerned two main topics in the philosophy of mathematics, namely, (a) the concept of natural number and (b) the infinitesimals. Cantor's and Kerry's positions turned out to be irreconcilable, mostly because of Kerry's irremediably psychologistic outlook, according to Cantor at least. In this study, I will examine and reconstruct the main points in the discussion around (a) and (b) and stress some interesting aspects of the philosophical and mathematical thought of Benno Kerry.

The generative basis of natural number concepts

Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality - that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies the arithmetic principle, supports exact equality and also enables computational compatibility with real- or rational-valued mental magnitudes.

The development of language and abstract concepts: The case of natural number.

What are the origins of abstract concepts such as "seven," and what role does language play in their development? These experiments probed the natural number words and concepts of 3-year-old children who can recite number words to ten but who can comprehend only one or two. Children correctly judged that a set labeled eight retains this label if it is unchanged, that it is not also four, and that eight is more than two. In contrast, children failed to judge that a set of 8 objects is better labeled by eight than by four, that eight is more than four, that eight continues to apply to a set whose members are rearranged, or that eight ceases to apply if the set is increased by 1, doubled, or halved. The latter errors contrast with children's correct application of words for the smallest numbers. These findings suggest that children interpret number words by relating them to 2 distinct preverbal systems that capture only limited numerical information. Children construct the system of abstract, natural number concepts from these foundations.

Giving the boot to the bootstrap: How not to learn the natural numbers

According to one theory about how children learn the concept of natural numbers, they first determine that “one”, “two”, and “three” denote the size of sets containing the relevant number of items. They then make the following inductive inference (the Bootstrap): The next number word in the counting series denotes the size of the sets you get by adding one more object to the sets denoted by the previous number word. For example, if “three” refers to the size of sets containing three items, then “four” (the next word after “three”) must refer to the size of sets containing three plus one items. We argue, however, that the Bootstrap cannot pick out the natural number sequence from other nonequivalent sequences and thus cannot convey to children the concept of the natural numbers. This is not just a result of the usual difficulties with induction but is specific to the Bootstrap. In order to work properly, the Bootstrap must somehow restrict the concept of “next number” in a way that conforms to the structure of the natural numbers. But with these restrictions, the Bootstrap is unnecessary.

PAOLO MANCOSU, KLAUS FROVIN JORGENSEN AND STIG ANDUR PEDERSEN, eds. Visualization, Explanation and Reasoning Styles in Mathematics. Synthese Library, Vol. 327. Dordrecht: Springer, 2005. Pp. x + 300. ISBN 1-4020-3334-6 (cloth), 1-4020-3335-4 (e-book).

What is philosophy of mathematics and what is it about? The most popular answer, I suppose, to this question would be that philosophers should provide a justification for our presently most cherished mathematical theories and for the most important tool to develop such theories, namely logico-mathematical proof. In fact, it does cover a large part of the activity of philosophers that think about mathematics. Discussions about the merits and faults of classical logic versus one or other ‘deviant’ logics as the logical basis for mathematical theories, ranging from intuitionist over modal logic to paraconsistent logic, typically belong to this area, as do debates about the natural-number concept, its ‘nature’, its properties, especially its uniqueness, and so on. No doubt sociologists of knowledge could explain why (the community of) philosophers of mathematics came to select these particular problems and deal with them the way they do. What it does imply,...

Appropriation des proprietes ordinales du nombre par l'eleve du cours preparatoire

The aim of this research is the clarification of the process whereby the learner, in the first year of compulsory education (Cours Preparatoire) constructs and internalizes the concept of natural number. The research is based on a close and rigorous analysis of learners' behaviour when confronted with problem situations designed according to a priori established priorities. This analysis is based on one-to-one interviews, a method which enables the dynamics of the processes involved for each learner, and thus the evolution of his cognitive system with respect to the task in hand, to be apprehended.

A Further Test of the Ordinal Theory of Number Development

Abstract Two questions about recently reported findings that the natural number concept derives from a prior understanding of ordination rather than a prior understanding of cardination were examined: (a)Do the findings hold when Piaget's criterion of natural number competence (number conservation) is substituted for the mathematician's criterion (addition and subtraction of integers)? (b)What accounts for the later emergence of cardination relative to ordination and natural number? Concerning a, it was found that the sequence of emergence of number concepts in five- to seven-year-olds is ordination → natural number → cardination when number conservation is the natural number criterion. Concerning b, it was found that roughly half the subjects did not possess a concept which is both logically and empirically necessary for cardination: viz., the concept of number-as-class. The concept of number-as-class—as measured by a card sorting task—also was found to emerge after ordination and concurrently with natur...

Logical reflection and formalism

A “foundational crisis” occurred already in Greek mathematics, brought about by the Pythagorean discovery of incommensurable quantities. It was Eudoxos who provided new foundations, and since then Greek mathematics has been unshakeable. If one reads modern mathematical textbooks, one is normally told that something very similar occurred in modern mathematics. The calculus invented in the seventeenth century had to go through a crisis caused by the use of divergent series. One is told that by the achievements of the nineteenth century from Cauchy to Cantor this crisis has definitely been overcome. It is well known, but it is nevertheless very often not taken seriously into account, that this is an illusion. The so-called ε-δ-definitions of the limit concepts are an admirable achievement, but they are only one step towards the goal of a final foundation of analysis. The nineteenth century solution of the problem of foundations consists of recognizing, in addition to the concept of natural number as the basis of arithmetic, another basic concept for analysis, namely the concept of set. By the inventors of set theory it was strongly held that these sets are self-evident to our intuition; but very soon the belief in their self-evidence was destroyed by the set-theoretic paradoxes. After that, about 1908, the period of axiomatic set theory began. In analogy to geometry there was put forward an uninterpreted system of axioms, a formal system. This, of course, is quite possible. A formal system contains strings of marks; and a special class of these strings, the class of the so-called “theorems”, is inductively defined.

The Definition of Number

It is surely a very remarkable thing that despite the range, power and success of modern mathematics, the concept of natural number, on which the whole edifice rests, is still something of a mystery.Towards the end of the last century the great German mathematician and philosopher Gottlob Frege, dissenting strongly from the view current in his time—and perhaps still today—that number is either a physical or a mental entity, gave the first purely logical definition of a natural number. His work aroused so little interest at the time that his definition remained practically unknown until it was rediscovered by Bertrand Russell 20 years later. The definition, though a little strange at first sight, is quite simple and is based upon a notion which is of first importance in many branches of mathematics, the notion of a (1, 1) correspondence.