Set representations required for the acquisition of the “natural number” concept

2008 ◽  
Vol 31 (6) ◽  
pp. 655-656 ◽  
Author(s):  
Justin Halberda ◽  
Lisa Feigenson
Keyword(s):  
Natural Number ◽  
Building Blocks ◽  
Number Concept ◽  
Set Representations ◽  
Individual Objects

AbstractRips et al. consider whether representations of individual objects or analog magnitudes are building blocks for the concept natural number. We argue for a third core capacity – the ability to bind representations of individuals into sets. However, even with this addition to the list of starting materials, we agree that a significant acquisition story is needed to capture natural number.

Arithmetic, philosophical issues in

2018 ◽  
Author(s):  
Michael Potter
Keyword(s):  
Natural Number ◽  
Decision Procedure ◽  
Mathematical Induction ◽  
Universal Quantifier ◽  
Number Concept ◽  
Special Character ◽  
The Status ◽  
Proof By Induction ◽  
Philosophy Of Arithmetic ◽  
Principle Of Mathematical Induction

The philosophy of arithmetic gains its special character from issues arising out of the status of the principle of mathematical induction. Indeed, it is just at the point where proof by induction enters that arithmetic stops being trivial. The propositions of elementary arithmetic – quantifier-free sentences such as ‘7+5=12’ – can be decided mechanically: once we know the rules for calculating, it is hard to see what mathematical interest can remain. As soon as we allow sentences with one universal quantifier, however – sentences of the form ‘(∀x)f(x)=0’ – we have no decision procedure either in principle or in practice, and can state some of the most profound and difficult problems in mathematics. (Goldbach’s conjecture that every even number greater than 2 is the sum of two primes, formulated in 1742 and still unsolved, is of this type.) It seems natural to regard as part of what we mean by natural numbers that they should obey the principle of induction. But this exhibits a form of circularity known as ‘impredicativity’: the statement of the principle involves quantification over properties of numbers, but to understand this quantification we must assume a prior grasp of the number concept, which it was our intention to define. It is nowadays a commonplace to draw a distinction between impredicative definitions, which are illegitimate, and impredicative specifications, which are not. The conclusion we should draw in this case is that the principle of induction on its own does not provide a non-circular route to an understanding of the natural number concept. We therefore need an independent argument. Four broad strategies have been attempted, which we shall consider in turn.

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

2008 ◽  
Vol 31 (6) ◽  
pp. 652-653 ◽  
Author(s):  
Martin F. Gardiner
Keyword(s):  
Natural Number ◽  
Learning Support ◽  
Number Concepts ◽  
Music Training ◽  
Number Concept ◽  
Young Learners ◽  
Musical Pitch ◽  
Innate Ability

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.

Mathematical Roots: Get Primed to the Basic Building Blocks of Numbers

2007 ◽  
Vol 13 (2) ◽  
pp. 122-127
Author(s):  
Christiana Robbins ◽  
Thomasenia Lott Adams
Keyword(s):  
Natural Number ◽  
Building Blocks ◽  
Number System ◽  
Prime Numbers ◽  
Natural Numbers

In his book Elements, Euclid established that certain numbers are the building blocks of our natural number system. He revealed that these natural numbers could be “decomposed” into their smallest units as products of specific numbers. Numbers that can only be factored by themselves and 1 are called “prime numbers” and comprise part of the basic building blocks of numbers.

scholarly journals The prehistory of number concept

2011 ◽  
Vol 34 (3) ◽  
pp. 142-144 ◽  
Author(s):  
Karenleigh A. Overmann ◽  
Thomas Wynn ◽  
Frederick L. Coolidge
Keyword(s):  
Material Culture ◽  
Natural Number ◽  
Initial Development ◽  
Number Concept

AbstractCarey leaves unaddressed an important evolutionary puzzle: In the absence of a numeral list, how could a concept of natural number ever have arisen in the first place? Here we suggest that the initial development of natural number must have bootstrapped on a material culture scaffold of some sort, and illustrate how this might have occurred using strings of beads.

scholarly journals The Prehistory of Number Concept

2021 ◽  
Author(s):  
Karenleigh A. Overmann ◽  
Thomas Wynn ◽  
Frederick L. Coolidge
Keyword(s):  
Material Culture ◽  
Natural Number ◽  
Initial Development ◽  
Number Concept

Carey leaves unaddressed an important evolutionary puzzle: In the absence of a numeral list, how could a concept of natural number ever have arisen in the first place? Here we suggest that the initial development of natural number must have bootstrapped on a material culture scaffold of some sort, and illustrate how this might have occurred using strings of beads.

Natural number concepts: No derivation without formalization

2008 ◽  
Vol 31 (6) ◽  
pp. 666-667
Author(s):  
Paul Pietroski ◽  
Jeffrey Lidz
Keyword(s):  
Natural Number ◽  
Building Blocks ◽  
Logical Space ◽  
Number Concepts ◽  
New Concepts

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.

Photochemical Evolution of Interstellar/Precometary Organic Material

1997 ◽  
Vol 161 ◽  
pp. 23-47 ◽  
Author(s):  
Louis J. Allamandola ◽  
Max P. Bernstein ◽  
Scott A. Sandford
Keyword(s):  
Origin Of Life ◽  
Building Blocks ◽  
Complex Species ◽  
Infrared Observations ◽  
Interstellar Ice ◽  
Polycyclic Aromatic ◽  
Ultraviolet Photolysis ◽  
The Origin Of Life ◽  
Simple Molecules ◽  
Complex Organic

AbstractInfrared observations, combined with realistic laboratory simulations, have revolutionized our understanding of interstellar ice and dust, the building blocks of comets. Since comets are thought to be a major source of the volatiles on the primative earth, their organic inventory is of central importance to questions concerning the origin of life. Ices in molecular clouds contain the very simple molecules H2O, CH3OH, CO, CO2, CH4, H2, and probably some NH3and H2CO, as well as more complex species including nitriles, ketones, and esters. The evidence for these, as well as carbonrich materials such as polycyclic aromatic hydrocarbons (PAHs), microdiamonds, and amorphous carbon is briefly reviewed. This is followed by a detailed summary of interstellar/precometary ice photochemical evolution based on laboratory studies of realistic polar ice analogs. Ultraviolet photolysis of these ices produces H2, H2CO, CO2, CO, CH4, HCO, and the moderately complex organic molecules: CH3CH2OH (ethanol), HC(= O)NH2(formamide), CH3C(= O)NH2(acetamide), R-CN (nitriles), and hexamethylenetetramine (HMT, C6H12N4), as well as more complex species including polyoxymethylene and related species (POMs), amides, and ketones. The ready formation of these organic species from simple starting mixtures, the ice chemistry that ensues when these ices are mildly warmed, plus the observation that the more complex refractory photoproducts show lipid-like behavior and readily self organize into droplets upon exposure to liquid water suggest that comets may have played an important role in the origin of life.

Laboratory and in-situ analysis of comet dust

1988 ◽  
Vol 46 ◽  
pp. 8-9
Author(s):  
D.E. Brownlee ◽  
A.L. Albee
Keyword(s):  
Electron Microscopy ◽  
Solar System ◽  
Laboratory Study ◽  
Isotopic Analysis ◽  
Building Blocks ◽  
Sem Analysis ◽  
Early Solar System ◽  
Cometary Material ◽  
Comet Dust

Comets are primitive, kilometer-sized bodies that formed in the outer regions of the solar system. Composed of ice and dust, comets are generally believed to be relic building blocks of the outer solar system that have been preserved at cryogenic temperatures since the formation of the Sun and planets. The analysis of cometary material is particularly important because the properties of cometary material provide direct information on the processes and environments that formed and influenced solid matter both in the early solar system and in the interstellar environments that preceded it.The first direct analyses of proven comet dust were made during the Soviet and European spacecraft encounters with Comet Halley in 1986. These missions carried time-of-flight mass spectrometers that measured mass spectra of individual micron and smaller particles. The Halley measurements were semi-quantitative but they showed that comet dust is a complex fine-grained mixture of silicates and organic material. A full understanding of comet dust will require detailed morphological, mineralogical, elemental and isotopic analysis at the finest possible scale. Electron microscopy and related microbeam techniques will play key roles in the analysis. The present and future of electron microscopy of comet samples involves laboratory study of micrometeorites collected in the stratosphere, in-situ SEM analysis of particles collected at a comet and laboratory study of samples collected from a comet and returned to the Earth for detailed study.

Cryo-TEM of amphiphilic polymer and amphiphile/polymer solutions

1993 ◽  
Vol 51 ◽  
pp. 876-877
Author(s):  
Yeshayahu Talmon
Keyword(s):  
Microstructural Characterization ◽  
Building Blocks ◽  
Quantitative Information ◽  
Physical Models ◽  
Specimen Preparation ◽  
Time Resolved ◽  
Temperature Changes ◽  
Temperature And Humidity ◽  
Microstructured Fluids ◽  
Air Streams

To achieve complete microstructural characterization of self-aggregating systems, one needs direct images in addition to quantitative information from non-imaging, e.g., scattering or Theological measurements, techniques. Cryo-TEM enables us to image fluid microstructures at better than one nanometer resolution, with minimal specimen preparation artifacts. Direct images are used to determine the “building blocks” of the fluid microstructure; these are used to build reliable physical models with which quantitative information from techniques such as small-angle x-ray or neutron scattering can be analyzed.To prepare vitrified specimens of microstructured fluids, we have developed the Controlled Environment Vitrification System (CEVS), that enables us to prepare samples under controlled temperature and humidity conditions, thus minimizing microstructural rearrangement due to volatile evaporation or temperature changes. The CEVS may be used to trigger on-the-grid processes to induce formation of new phases, or to study intermediate, transient structures during change of phase (“time-resolved cryo-TEM”). Recently we have developed a new CEVS, where temperature and humidity are controlled by continuous flow of a mixture of humidified and dry air streams.

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