Learning the Natural Numbers as a Child

How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the initial segment of the natural numbers on the basis of the Fregean definitions, but do not learn the natural number structure as a whole on the basis of Hume's principle. Therefore, we need to account for some of the consistency of our number concepts with the Dedekind‐Peano axioms in other terms.     

Reasoning about Arbitrary Natural Numbers from a Carnapian Perspective

Journal of Philosophical Logic - Inspired by Kit Fine’s theory of arbitrary objects, we explore some ways in which the generic structure of the natural numbers can be presented. Following a...     

Finite Sets and Natural Numbers in Intuitionistic TT Without Extensionality

In this paper, we prove that Heyting's arithmetic can be interpreted in an intuitionistic version of Russell's Simple Theory of Types without extensionality.     

The Variety of Lattice-Ordered Monoids Generated by the Natural Numbers

We study the variety Var() of lattice-ordered monoids generated by the natural numbers. In particular, we show that it contains all 2-generated positively ordered lattice-ordered monoids satisfying appropriate distributive laws. Moreover, we establish that the cancellative totally ordered members of Var() are submonoids of ultrapowers of and can be embedded into ordered fields. In addition, the structure of ultrapowers relevant to the finitely generated case is analyzed. Finally, we provide a complete isomorphy invariant in the two-generated case.     

How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like , is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers (specifically, the radicands of radical expressions) as natural numbers. Strategy self‐reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task‐specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics.     

Taurek, Numbers and Probabilities

In his paper, “Should the Numbers Count?" John Taurek imagines that we are in a position such that we can either save a group of five people, or we can save one individual, David. We cannot save David and the five. This is because they each require a life-saving drug. However, David needs all of the drug if he is to survive, while the other five need only a fifth each. Typically, people have argued as if there was a choice to be made: either numbers matter, in which case we should save the greater number, or numbers don't matter, but rather there is moral value in giving each person an equal chance of survival, and therefore we should toss a coin. My claim is that we do not have to make a choice in this way. Rather, numbers do matter, but it doesn't follow that we should always save the greater number. And likewise, there is moral value in giving each person an equal chance of survival, but it doesn't follow that we should always toss a coin. In addition, I argue that a similar approach can be applied to situations in which we can save one person or another, but the chances of success are different.     

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.     

Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence

The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the mind is based on the linear number representation. This classical conception is rejected and a competitive hypothesis is formulated according to which the basic mature representational system of cognitive arithmetic is a structure composed of many numerical axes which possess a common constituent, namely, the numeral zero. Arithmetic of indexed numbers is just a formal tool for modelling the basic mature arithmetic competence. The third task is to develop a standpoint called temporal pluralism, which is motivated by neo-Kantian philosophy of arithmetic.     

Propositions,Numbers, And The Problem Of Arbitrary Identification

Those inclined to believe in the existence of propositions as traditionally conceived might seek to reduce them to some other type of entity. However, parsimonious propositionalists of this type are confronted with a choice of competing candidates – for example, sets of possible worlds, and various neo-Russellian and neo-Fregean constructions. It is argued that this choice is an arbitrary one, and that it closely resembles the type of problematic choice that, as Benacerraf pointed out, bedevils the attempt to reduce numbers to sets – should the number 2 be identified with the set Ø or with the set Ø, Ø? An “argument from arbitrary identification” is formulated with the conclusion that propositions (and perhaps numbers) cannot be reduced away. Various responses to this argument are considered, but ultimately rejected. The paper concludes that the argument is sound: propositions, at least, are sui generis entities.

     

The Intolerable Wrestle: Words,Numbers, and Meanings

Repertory grids are used by some as the basis, and by others as the vehicle, for the transmission of meanings. This article is based on the premise that meanings are of paramount importance to those working within the framework of personal construct theory and attempts to establish two general propositions: (1) that, despite a considerable measure of content freedom, repertory grids severely constrain respondents; and (2) that it is difficult for meanings to pass through the linguistic constrictions of the grid matrix. Some implications of the argument are subsequently discussed.     

Motor Imagery Shapes Abstract Concepts

The concepts of “good” and “bad” are associated with right and left space. Individuals tend to associate good things with the side of their dominant hand, where they experience greater motor fluency, and bad things with their nondominant side. This mapping has been shown to be flexible: Changing the relative fluency of the hands, or even observing a change in someone else's motor fluency, results in a reversal of the conceptual mapping, such that good things become associated with the side of the nondominant hand. Yet, based on prior studies, it is unclear whether space–valence associations were determined by the experience of fluent versus disfluent actions, or by the mere expectation of fluency. Here, we tested the role of expected fluency by removing motor execution and perceptual feedback altogether. Participants were asked to imagine themselves performing a psychomotor task with one of their hands impaired, after which their implicit space–valence mapping was measured. After imagining that their right hand was impaired, right‐handed participants showed the “good is left” association typical of left‐handers. Motor imagery can change people's implicit associations between space and emotional valence. Although asymmetric motor experience may be necessary to establish body‐specific associations between space and valence initially, neither motoric nor perceptual experience is needed to change these associations subsequently. The mere expectation of fluent versus disfluenct actions can drive fluency‐based effects on people's implicit spatialization of “good” and “bad.” These results suggest a reconsideration of the mechanisms and boundary conditions of fluency effects.     

Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege" s="" grundgesetze="" in="" object="" theory"="

In this paper, the author derives the Dedekind–Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege"s Grundgesetze. The proofs of the theorems reconstruct Frege"s derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege"s conception of numbers and logical objects.     

Leibniz on Infinite Numbers,Infinite Wholes,and Composite Substances

Leibniz claims that nature is actually infinite but rejects infinite number. Are his mathematical commitments out of step with his metaphysical ones? It is widely accepted that Leibniz has a viable response to this problem: there can be infinitely many created substances, but no infinite number of them. But there is a second problem that has not been satisfactorily resolved. It has been suggested that Leibniz's argument against the world soul relies on his rejection of infinite number, and, as such, Leibniz cannot assert that any body has a soul without also accepting infinite number, since any body has infinitely many parts. Previous attempts to address this concern have misunderstood the character of Leibniz's rejection of infinite number. I argue that Leibniz draws an important distinction between ‘wholes’ – collections of parts that can be thought of as a single thing – and ‘fictional wholes’ – collections of parts that cannot be thought of as a single thing, which allows us to make sense of his rejection of infinite number in a way that does not conflict either with his view that the world is actually infinite or that the bodies of substances have infinitely many parts.