Transreal limits expose category errors in IEEE 754 floating-point arithmetic and in mathematics

The IEEE 754 standard for oating-point arithmetic is widely used in computing. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. The IEEE infinities are said to have the behaviour of limits. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. We elucidate the transreal tangent and extend real limits to transreal limits. Arguing from this firm foundation, we maintain that there are three category errors in the IEEE 754 standard. Firstly the claim that IEEE infinities are limits of real arithmetic confuses limiting processes with arithmetic. Secondly a defence of IEEE negative zero confuses the limit of a function with the value of a function. Thirdly the definition of IEEE NaNs confuses undefined with unordered. Furthermore we prove that the tangent function, with the infinities given by geometrical con- struction, has a period of an entire rotation, not half a rotation as is commonly understood. This illustrates a category error, confusing the limit with the value of a function, in an important area of applied mathe- matics { trigonometry. We brie y consider the wider implications of this category error. Another paper proposes transreal arithmetic as a basis for floating- point arithmetic; here we take the profound step of proposing transreal arithmetic as a replacement for real arithmetic to remove the possibility of certain category errors in mathematics. Thus we propose both theo- retical and practical advantages of transmathematics. In particular we argue that implementing transreal analysis in trans- floating-point arith- metic would extend the coverage, accuracy and reliability of almost all computer programs that exploit real analysis { essentially all programs in science and engineering and many in finance, medicine and other socially beneficial applications.

Effects of organic and conventional fertilizer treatments on host selection by the aphid parasitoid Diaeretiella rapae

The type and quantity of fertilizer supplied to a crop will differ between organic and conventional farming practices. Altering the type of fertilizer a plant is provided with can influence a plant’s foliar nitrogen levels, as well as the composition and concentration of defence compounds, such as glucosinolates. Many natural enemies of insect herbivores can respond to headspace volatiles emitted by the herbivores’ host plant in response to herbivory. We propose that manipulating fertilizer type may also influence the headspace volatile profiles of plants, and as a result, the tritrophic interactions that occur between plants, their insect pests and those pests’ natural enemies. Here, we investigate a tritrophic system consisting of cabbage plants, Brassica oleracea, a parasitoid, Diaeretiella rapae, and one of its hosts, the specialist cabbage aphid Brevicoryne brassicae. Brassica oleracea plants were provided with either no additional fertilization or one of three types of fertilizer: Nitram (ammonium nitrate), John Innes base or organic chicken manure. We investigated whether these changes would alter the rate of parasitism of aphids on those plants and whether any differences in parasitism could be explained by differences in attractivity of the plants to D. rapae or attack rate of aphids by D. rapae. In free-choice experiments, there were significant differences in the percentage of B. brassicae parasitized by D. rapae between B. oleracea plants grown in different fertilizer treatments. In a series of dual-choice Y-tube olfactometry experiments, D. rapae females discriminated between B. brassicae-infested and undamaged plants, but parasitoids did not discriminate between similarly infested plants grown in different fertilizer treatments. Correspondingly, in attack rate experiments, there were no differences in the rate that D. rapae attacked B. brassicae on B. oleracea plants grown in different fertilizer treatments. These findings are of direct relevance to sustainable and conventional farming practices.

All for the good

The Guise of the Good thesis has received much attention since Anscombe's brief defence in her book Intention. I approach it here from a less common perspective - indirectly, via a theory explaining how it is that moral behaviour is even possible. After setting out how morality requires the employment of a fundamental test, I argue that moral behaviour involves orientation toward the good. Immoral behaviour cannot, however, involve orientation to evil as such, given the theory of evil as privation. There must always be orientation to good of some kind for immorality even to be possible. Evil can, nevertheless, be intended, but this must be carefully understood in terms of the metaphysic of good and evil I set out. Given that metaphysic, the Guise of the Good is a virtual corollary.

Olfactory selection of Plantago lanceolata by snails declines with seedling age

Background and Aims Despite recent recognition that (1) plant–herbivore interactions during the establishment phase, (2) ontogenetic shifts in resource allocation and (3) herbivore response to plant volatile release are each pivotal to a comprehensive understanding of plant defence, no study has examined how herbivore olfactory response varies during seedling ontogeny. Methods Using a Y-tube olfactometer we examined snail (Helix aspersa) olfactory response to pellets derived from macerated Plantago lanceolata plants harvested at 1, 2, 3, 4, 5, 6 and 8 weeks of age to test the hypothesis that olfactory selection of plants by a generalist herbivore varies with plant age. Plant volatiles were collected for 10 min using solid-phase microextraction technique on 1- and 8-week-old P. lanceolata pellets and analysed by gas chromatography coupled with a mass spectrometer. Key Results Selection of P. lanceolata was strongly negatively correlated with increasing age; pellets derived from 1-week-old seedlings were three times more likely to be selected as those from 8-week-old plants. Comparison of plant selection experiments with plant volatile profiles from GC/MS suggests that patterns of olfactory selection may be linked to ontogenetic shifts in concentrations of green leaf volatiles and ethanol (and its hydrolysis derivatives). Conclusions Although confirmatory of predictions made by contemporary plant defence theory, this is the first study to elucidate a link between seedling age and olfactory selection by herbivores. As a consequence, this study provides a new perspective on the ontogenetic expression of seedling defence, and the role of seedling herbivores, particularly terrestrial molluscs, as selective agents in temperate plant communities.

Prebiotics modulate the effects of antibiotics on gut microbial diversity and functioning in vitro

Intestinal bacteria carry out many fundamental roles, such as the fermentation of non-digestible dietary carbohydrates to produce short chain fatty acids (SCFAs), which can affect host energy levels and gut hormone regulation. Understanding how to manage this ecosystem to improve human health is an important but challenging goal. Antibiotics are the front line of defence against pathogens, but in turn they have adverse effects on indigenous microbial diversity and function. Here, we have investigated whether dietary supplementation—another method used to modulate gut composition and function—could be used to ameliorate the side effects of antibiotics. We perturbed gut bacterial communities with gentamicin and ampicillin in anaerobic batch cultures in vitro. Cultures were supplemented with either pectin (a non-fermentable fibre), inulin (a commonly used prebiotic that promotes the growth of beneficial bacteria) or neither. Although antibiotics often negated the beneficial effects of dietary supplementation, in some treatment combinations, notably ampicillin and inulin, dietary supplementation ameliorated the effects of antibiotics. There is therefore potential for using supplements to lessen the adverse effects of antibiotics. Further knowledge of such mechanisms could lead to better therapeutic manipulation of the human gut microbiota.

Expecting moral philosophers to be reliable

Are philosophers’ intuitions more reliable than philosophical novices’? Are we entitled to assume the superiority of philosophers’ intuitions just as we assume that experts in other domains have more reliable intuitions than novices? Ryberg raises some doubts and his arguments promise to undermine the expertise defence of intuition-use in philosophy once and for all. In this paper, I raise a number of objections to these arguments. I argue that philosophers receive sufficient feedback about the quality of their intuitions and that philosophers’ experience in philosophy plausibly affects their intuitions. Consequently, the type of argument Ryberg offers fails to undermine the expertise defence of intuition-use in philosophy.

Recover it from the facts as we know them

In Andrea Sangiovanni’s words, practice-dependent theorists hold that “[t]he content, scope, and justification of a conception of [a given value] depends on the structure and form of the practices that the conception is intended to govern”. They have tended to present this as methodologically innovative, but here I point to the similarities between the methodological commitments of contemporary practice-dependent theorists and others, particularly P. F. Strawson in his Freedom and Resentment and Bernard Williams in general. I suggest that by looking at what Strawson and Williams did, we can add to the reasons for adopting one form or another of practice-dependence. The internal complexity of the practices we hope our principles will govern may require it. However, this defence of practice-dependence also puts pressure on self-identified practice-dependence theorists, suggesting that they need to do more work to justify the interpretations of the practices their theories rely on.

Social connection and practice-dependence: some recent developments in the global justice literature

This review essay discusses two recent attempts to reform the framework in which issues of international and global justice are discussed: Iris Marion Young’s ‘social connection’ model and the practice-dependent approach, here exemplified by Ayelet Banai, Miriam Ronzoni and Christian Schemmel’s edited collection. I argue that while Young’s model may fit some issues of international or global justice, it misconceives the problems that many of them pose. Indeed, its difficulties point precisely in the direction of practice dependence as it is presented by Banai et al. I go on to discuss what seem to be the strengths of that method, and particularly Banai et al.’s defence of it against the common claim that it is biased towards the status quo. I also discuss Andrea Sangiovanni and Kate MacDonald’s contributions to the collection.