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.

Prosopis, an Alien among the Sacred Trees of South India

The problematic of invasive species in an alien environment has aroused the attention of scientists all over the world for quite some time. One of the exotic tree species that has provoked special attention in the tropical drylands is Prosopis juliflora. Originating in South America, prosopis (hereafter referred to as prosopis) has been introduced in the hot and semi-arid zones of the world particularly to provide fuelwood, to stabilize sand dunes and to combat desertification. The tree has become an essential source for fuelwood and a provider of several other products and services in areas where it has become established. However, despite the numerous benefits the tree provides to rural people, in several regions prosopis has become a noxious weed with a negative impact on the environment and to the economy of farmers and landowners. In India, prosopis was introduced in Andhra Pradesh in 1877. The tree was then proclaimed as the precious child of the plant world by scientists and local people alike. The purpose of this study was to investigate the overall impact of prosopis on local rural livelihoods in the drylands of South India. Of particular interest was the examination of the different usages of the tree, especially as fuelwood, and people s perceptions of it. Furthermore, the study examined the negative impacts of the uncontrolled invasion of prosopis on croplands, and its occupation of the banks of irrigation canals and other water sources. As another central theme, this study analysed the Hindu classification system for nature and for trees in particular. In India, several tree species are regarded as sacred. This study examined the position of the exotic prosopis among sacred trees, such as the bodhi, banyan and neem trees. The principle method for collecting the field data was by using individual and thematic group interviews. These interviews were semi-structured with open ended questions. Moreover, unstructured interviews as well as general observations provided complementary information. The data were gathered during two fieldwork periods in the states of Andhra Pradesh and Tamil Nadu, in South India. The results confirmed that prosopis both provides benefits and causes hazards to different stakeholders. Farmers and agriculturalists suffer economic losses in areas where prosopis has invaded crop fields and competes with other plants for water and nutrients. On the other hand, for a significant number of poor rural people, prosopis has become an important source of livelihood benefits. This tree, which grows on government wastelands, is commonly a free resource for all and has thus become a major local source of fuelwood. It also provides several other goods and services and cash income that contributes to improve livelihoods in rural communities. Prosopis ranked lowest in the tree classificatioin system of the Hindus of South India. Although it is appreciated for many benefits it provides for poor people, it has remained an outsider compared with the indigenous tree species. On the other hand, the most sacred trees, such as the bodhi or the banyan, are completely excluded from extraction and it is seen as a sacrilege to even cut branches from any of these trees. An unexpected finding was that, in a few cases, prosopis had also been elevated to the status of a sacred tree. Goods and services from prosopis are not utilized in the most beneficial way. Silvicultural management practices are suggested that would provide additional income and employment opportunities. Interventions are recommended to control further invasion of the tree that might cause serious negative effects in the future. For Hindus, the sacred always ranks highest, even above economic gain. The conservation of sacred groves and sacred trees is a tradition that has its roots in ancient history. These socio-religious practices need to be respected and continued. Successful management of tree and forest resources depends on the willingness of the local people to manage their natural resources, and this willingness exists and has always existed in South India. Keywords: South India, drylands, livelihood, fuelwood, invasive, resource, silviculture.

ISO 13567 - The Proposed International Standard for Structuring Layers in Computer Aided Building Design

Layering is a widely used method for structuring data in CAD-models. During the last few years national standardisation organisations, professional associations, user groups for particular CAD-systems, individual companies etc. have issued numerous standards and guidelines for the naming and structuring of layers in building design. In order to increase the integration of CAD data in the industry as a whole ISO recently decided to define an international standard for layer usage. The resulting standard proposal, ISO 13567, is a rather complex framework standard which strives to be more of a union than the least common denominator of the capabilities of existing guidelines. A number of principles have been followed in the design of the proposal. The first one is the separation of the conceptual organisation of information (semantics) from the way this information is coded (syntax). The second one is orthogonality - the fact that many ways of classifying information are independent of each other and can be applied in combinations. The third overriding principle is the reuse of existing national or international standards whenever appropriate. The fourth principle allows users to apply well-defined subsets of the overall superset of possible layernames. This article describes the semantic organisation of the standard proposal as well as its default syntax. Important information categories deal with the party responsible for the information, the type of building element shown, whether a layer contains the direct graphical description of a building part or additional information needed in an output drawing etc. Non-mandatory information categories facilitate the structuring of information in rebuilding projects, use of layers for spatial grouping in large multi-storey projects, and storing multiple representations intended for different drawing scales in the same model. Pilot testing of ISO 13567 is currently being carried out in a number of countries which have been involved in the definition of the standard. In the article two implementations, which have been carried out independently in Sweden and Finland, are described. The article concludes with a discussion of the benefits and possible drawbacks of the standard. Incremental development within the industry, (where ”best practice” can become ”common practice” via a standard such as ISO 13567), is contrasted with the more idealistic scenario of building product models. The relationship between CAD-layering, document management product modelling and building element classification is also discussed.