Truth, Proof and Gödelian Arguments : A Defence of Tarskian Truth in Mathematics

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion.

Written numeracy assessment in the early years: The challenges of pronouns and noun groups

We examine the challenges of pronouns and noun groups in the Australian Curriculum, Assessment and Reporting Authority’s (ACARA) Year 3 National Assessment Program - Literacy and Numeracy (NAPLAN) Numeracy Example Test (ACARA, 2015a). Framed by discourses of Literacy as a General Capability (ACARA, 2015b) in the Australian Curriculum: Mathematics (ACARA, 2015c), the pronouns and noun groups used in the Example Test are mapped onto the Australian Curriculum: English Content Descriptions (ACARA, 2015d). The findings are that some of the pronouns and noun groups used in the Example Test are more complex than what Year 3 children are expected to achieve in the Australian Curriculum: English. We thus stress the need for early years teachers of mathematics to account for the unique mathematical register of written mathematics problems so that young children are not only prepared for high-stakes written numeracy assessments, but are also enabled as a numerically literate citizen.

Student understandings of numeracy problems: Semantic alignment and analogical reasoning

Despite compulsory mathematics throughout primary and junior secondary schooling, many schools across Australia continue in their struggle to achieve satisfactory numeracy levels. Numeracy is not a distinct subject in school curriculum, and in fact appears as a general capability in the Australian Curriculum, wherein all teachers across all curriculum areas are responsible for numeracy. This general capability approach confuses what numeracy should look like, especially when compared to the structure of numeracy as defined on standardised national tests. In seeking to define numeracy, schools tend to look at past NAPLAN papers, and in doing so, we do not find examples drawn from the various aspects of school curriculum. What we find are more traditional forms of mathematical worded problems.

Development of probabilistic understanding in fourth grade

We analyzed the development of 4th-grade students’ understanding of the transition from experimental relative frequencies of outcomes to theoretical probabilities with a focus on the foundational statistical concepts of variation and expectation. We report students’ initial and changing expectations of the outcomes of tossing one and two coins, how they related the relative frequency from their physical and computersimulated trials to the theoretical probability, and how they created and interpreted theoretical probability models. Findings include students’ progression from an initial apparent equiprobability bias in predicting outcomes of tossing two coins through to representing the outcomes of increasing the number of trials. After observing the decreasing variation from the theoretical probability as the sample size increased, students developed a deeper understanding of the relationship between relative frequency of outcomes and theoretical probability as well as their respective associations with variation and expectation. Students’ final models indicated increasing levels of probabilistic understanding.

Making sense of a trial maths intervention program for teachers of students with disability in Australia: Interim report

The Disability Standards for Education (2005) and the Australian Curriculum, Assessment and Reporting Authority relevant standards underscore the right of students with disability to access the curriculum on the same basis as students without disability. Students with disability are entitled to rigorous, relevant and engaging learning opportunities drawn from the Australian curriculum content. Taking this context into account, this paper provides a work-in-progress report on a two-year mathematics intervention project conducted in 12 special schools (Preparatory-Year 12) in Queensland, Australia. The project aims to build the capacity of teachers to teach mathematics to their students and to identify and make sense of the intervention program’s impact. It combines two approaches—appreciative inquiry and action research to monitor schools’ change processes. The interim findings demonstrated that teachers were concerned about their students’ underachievement in mathematics and that the multi-sensory forms of teaching advocated in the program increased student engagement and performance.

The History and Literature of America's Oysters

Historically, America's use and enjoyment of the oyster extend far back into prehistoric times. The Native Americans often utilized oysters, more intensively in some areas than in others, and, at least in some areas of the Caribbean and Pacific coast, the invading Spanish sought oysters as eagerly as they did gold-but for the pearls. That was the pearl oyster, Pinctada sp., and signs of its local overexploitation were recorded early in the 16th century. During the 1800's, use of the eastern oyster grew phenomenally and, for a time, it outranked beef as a source of protein in some parts of the nation. Social events grew up around it, as it became an important aspect of culture and myth. Eventually, research on the oyster began to blossom, and scientific literature on the various species likewise bloomed-to the extent that when the late Paul Galtsoff wrote his classic treatise "The American oyster Crassostrea virginica Gmelin" in 1954, he reported compiling an extensive bibliography of over 6,000 subject and author cards on oysters and related subjects which he deposited in the library of the Woods Hole Laboratory of the Bureau of Commercial Fisheries (now NMFS). That large report, volume 64 (480 pages) of the agency's Fishery Bulletin, was a bargain at $2.75, and it has been a standard reference ever since. But the research and the attendant literature have grown greatly since Galtsoff's work was published, and now that has been thoroughly updated.

Identification of Prorocentrum minimum and Takayama pulchella by fluorescence in situ hybridization through epifluorescence microscopy and flow cytometry

Partial rDNA sequences of Prorocentrum minimum and Takayama pulchella were amplified, cloned and sequenced. and these sequence data were deposited in the GenBank. Eight oligonucleotide probes (DNA probes) were designed based on the sequence analysis. The probes were employed to detect and identify P. minimum and T. pulchella in unialgal and mixed algal samples with a fluorescence in situ hybridization method using flow cytometry. Epifluorescence micrographs showed that these specific probes labeled with fluorescein isothiocyanate entered the algal cells and bound to target sequences, and the fluorescence signal resulting from whole-cell hybridization varied from probe to probe. These DNA probes and the hybridization protocol we developed were specific and effective for P. minimum and T. pulchella, without any specific binding to other algal species. The hybridization efficiency of different probes specific to P. minimum was in the order: PM18S02 > PM28S02 > PM28S01 > PM18S01, and that of the probes specific to T. pulchella was TP18S02 > TP28S01 > TP28S02 > TP18S01. The different hybridization efficiency of the DNA probes could also be shown in the fluorescent signals between the labeled and unlabeled cells demonstrated using flow cytometry. The DNA probes PM18S02, PM28S02; TP18S02 and TP28S01, and the protocol, were also useful for the detection of algae in natural samples.

One-pot synthesis and self-assembly of colloidal copper(I) sulfide nanocrystals

A simple one-pot method is developed to prepare size-and shape-controlled copper(I) sulfide (Cu2S) nanocrystals by thermolysis of a mixed solution of copper acetylacetonate, dodecanethiol and oleylamine at a relatively high temperature. The crystal structure, chemical composition and morphology of the as-obtained products are characterized by powder x-ray diffraction (PXRD), x-ray photoelectron spectroscopy (XPS), Auger electron spectroscopy (AES), transmission electron microscopy (TEM) and scanning electron microscopy (SEM). The morphology and size of the Cu2S nanocrystals can be easily controlled by adjusting the reaction parameters. The Cu2S nanocrystals evolve from spherical to disk-like with increasing reaction temperature. The spherical Cu2S nanocrystals have a high tendency to self-assemble into close-packed superlattice structures. The shape of the Cu2S nanodisks changes from cylinder to hexagonal prism with prolonged reaction time, accompanied by the diameter and thickness increasing. More interestingly, the nanodisks are inclined to self-assemble into face-to-face stacking chains with different lengths and orientations. This one-pot approach may extend to synthesis of other metal sulfide nanocrystals with different shapes and sizes.

Energy level analyses of even-parity J=1 and 2 Rydberg states of Sn I by multichannel quantum defect theory

This paper analyzes the energy levels along the even-parity J=1 and 2 Rydberg series of Sn I by multichannel quantum defect theory. A good agreement between theoretical and experimental energy levels was achieved. Below 59198 cm~(-1), a total of 85 and 23 new energy levels, respectively, in the J=1 and J=2 series, which cannot be measured previously by experiments, are predicted in this work. Based on the calculated admixture coefficients of each channel, interchannel interactions were discussed in detail. The results are helpful to understand the characteristics of configuration interaction among even-parity levels in Sn I.

Electron impact excitation of xenon from the metastable state to the excited states

The electron impact excitation cross sections from the lowest metastable state 5p(5)6sJ = 2 to the six lowest excited states of the 5p(5)6p configuration of xenon are calculated systematically by using the fully relativistic distorted wave method. In order to discuss the effects of target state descriptions on the electron impact excitation cross sections, two correlation models are used to describe the target states based on the multiconfiguration Dirac-Fock (MCDF) method. It is found that the correlation effects play a very important role in low energy impact. For high energy impact, however, the cross sections are not sensitive to the description of the target states, but many more partial waves must be included.

Application of rough set-based analysis to extract spatial relationship indicator rules: An example of land use in Pearl River Delta

Spatial relations, reflecting the complex association between geographical phenomena and environments, are very important in the solution of geographical issues. Different spatial relations can be expressed by indicators which are useful for the analysis of geographical issues. Urbanization, an important geographical issue, is considered in this paper. The spatial relationship indicators concerning urbanization are expressed with a decision table. Thereafter, the spatial relationship indicator rules are extracted based on the application of rough set theory. The extraction process of spatial relationship indicator rules is illustrated with data from the urban and rural areas of Shenzhen and Hong Kong, located in the Pearl River Delta. Land use vector data of 1995 and 2000 are used. The extracted spatial relationship indicator rules of 1995 are used to identify the urban and rural areas in Zhongshan, Zhuhai and Macao. The identification accuracy is approximately 96.3%. Similar procedures are used to extract the spatial relationship indicator rules of 2000 for the urban and rural areas in Zhongshan, Zhuhai and Macao. An identification accuracy of about 83.6% is obtained.

The dynamics of a topological change in a system of soap films

Hutzler, S., Saadatfar, M., van der Net, A., Weaire, D. and Cox, S.J. (2007) The dynamics of a topological change in a system of soap films. Coll. Surf. A, 323:123-131. Sponsorship: This research was supported by the European Space Agency (contracts 14914/02/NL/SH and 14308/00/NL/SH), Science Foundation Ireland. (RFP 05/REP/PHY00/6), and the EU program COST P21 (The Physics of droplets). SJC acknowledges support from EPSRC (EP/D071127/1). MS is supported by the Irish Higher Education Authority (PRTLI-IITAC).

Didactical reflections about some proofs of the pythagorean proposition

This theoretical paper presents some dimensions considered in the literature to analyze proof in the teaching and learning of mathematics. In order to show how different types of proofs can be used with students of different levels, we use these dimensions to analyze four proofs of the Pythagorean proposition.

The calculator as an instrument of validation of mathematical knowledge: a case study

En este trabajo se describe una experiencia llevada a cabo con profesores de matemáticas en formación, sobre el papel que pueden desarrollar las nuevas tecnologías para llevar a cabo procesos de demostración y prueba en el aula de secundaria.

Modelling of industrial multi-physics processes—a key role for computational mechanics.

In the analysis of industrial processes, there is an increasing emphasis on systems governed by interacting continuum phenomena. Mathematical models of such multi-physics processes can only be achieved for practical simulations through computational solution procedures—computational mechanics. Examples of such multi-physics systems in the context of metals processing are used to explore some of the key issues. Finite-volume methods on unstructured meshes are proposed as a means to achieve efficient rapid solutions to such systems. Issues associated with the software design, the exploitation of high performance computers, and the concept of the virtual computational-mechanics modelling laboratory are also addressed in this context.