Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics

We examine some of Connes' criticisms of Robinson's infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes' own earlier work in functional analysis. Connes described the hyperreals as both a "virtual theory" and a "chimera", yet acknowledged that his argument relies on the transfer principle. We analyze Connes' "dart-throwing" thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being "virtual" if it is not definable in a suitable model of ZFC. If so, Connes' claim that a theory of the hyperreals is "virtual" is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren't definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in his Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes' criticism of virtuality. We analyze the philosophical underpinnings of Connes' argument based on Goedel's incompleteness theorem, and detect an apparent circularity in Connes' logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace (featured on the front cover of Connes' magnum opus) and the Hahn-Banach theorem, in Connes' own framework. We also note an inaccuracy in Machover's critique of infinitesimal-based pedagogy.Comment: 52 pages, 1 figur

Tree-crop interactions: manipulation of water use and root function

This paper describes recent research findings on tree–crop interactions in the semiarid tropics focusing on the potential of agroforestry systems to improve the efficiency with which land and water are currently used; the trade-offs between crop productivity and environmental function, and strategies to manipulate tree root function. There is strong evidence that agroforestry has potential for improving water use efficiency by reducing the unproductive components of the water balance, i.e. run-off, soil evaporation and drainage. Examples from India and Kenya show that simultaneous agroforestry systems could double rainfall utilisation compared to annual cropping systems, largely due to temporal complementarity. Where soil loss through erosion is a serious problem, contour hedgerows can provide a viable alternative to conventional soil conservation measures. However, even though soil losses can be dramatically reduced, whether beneficial effects on crops will develop is often unpredictable and usually insufficient to attract widespread adoption of contour hedges. Strategies to reduce the trade-offs between crop and tree interactions or environmental function include the use of high value trees or trees which provide direct benefits to farmers. Recent reviews on root research indicate that there appears to be limited scope for spatial differentiation in rooting between trees and crops (i.e. spatial complementarity) in water-limited environments, unless ground-water is accessible to tree roots. Instead, it is argued that it is more worthwhile to manage below-ground competition by shoot and root-pruning. Pruning of lateral roots could redirect root function and be a powerful tool for improving spatial complementarity, provided that there are adequate resources at depth. However, the downward displacement of functional tree roots following root-pruning must not be allowed to affect their safety net role in the interception of nutrients leaching from the zone of crop rooting and the long-term hydrological implications must not be ignored when attempting to meet demand for trees and their product

Tree species and pruning regime affect crop yield on bench terraces in SW Uganda

Integration of trees on farms may exert complementary or competitive effects on crop yield. This four year study examined novel systems in which Alnus acuminata (alnus), Calliandra calothyrsus (calliandra), Sesbania sesban (sesbania) or a mixture of all three were grown on the degraded upper part of bench terraces in Uganda; beans or maize were grown on the more fertile lower terrace during the short and long rains. Three pruning treatments (shoot, root or shoot+root pruning) were applied to the tree rows adjacent to the crops; shoot prunings were applied as green manure to the woodlot from which they came. Pruning increased survival in calliandra and reduced survival in sesbania; alnus was unaffected. Pruning reduced tree height and stem diameter in alnus, but did not affect calliandra or sesbania. Maize yield adjacent to unpruned calliandra, alnus and sesbania or a mixture of all three was reduced by 48, 17, 6 and 24 % relative to sole maize. Shoot pruning initially sustained crop performance but shoot+root pruning became necessary when tree age exceeded two years; shoot+root pruning increased maize yield by 88, 40, 11 and 31 % in the calliandra, alnus, sesbania and tree mixture systems relative to unpruned trees. Bean yield adjacent to unpruned calliandra, alnus, sesbania and the tree mixture was 44, 31, 33 and 22 % lower than in sole crops and pruning had no significant effect on crop yield. The results suggest that sesbania fallows may be used on the upper terrace without reducing crop yield on the lower terrace, whereas pruning of alnus is needed to sustain yield. Calliandra woodlots appear to be unsuitable as crop yield was reduced even after pruning

Trees improve water storage and reduce soil evaporation in agroforestry systems on bench terraces in SW Uganda

The success of agroforestry in semi-arid areas depends on efficient use of available water and effective strategies to limit tree/crop competition and maximise productivity. On hillsides, planting improved tree fallows on the degraded upper section of bench terraces is a recommended practice to improve soil fertility while cropping continues on the lower terrace to maintain food production. This study examined the influence of tree fallows on soil water content (θ w ) and evaporation (E s ). Alnus acuminata Kunth (alnus), Calliandra calothyrsus Meissner (calliandra), Sesbania sesban L. (sesbania), a mixture of all three species, or sole crops (beans (Phaseolus vulgaris L.) or maize (Zea mays L.)) were grown on the upper terrace. The same sole crops were grown on the lower terrace. Four management regimes (unpruned, root, shoot and root + shoot pruned) were applied to the tree rows adjacent to the cropping area. Neutron probe and microlysimeter approaches were used to determine θ w and E s when the trees were c. 3.5 years old. Sesbania and alnus increased θ w by 9–18 % in the cropping area on the lower terrace but calliandra reduced θ w by 3–15 %. After heavy rain, E s comprised 29–38 % of precipitation in the tree-based treatments and 53 % under sole crops. Absolute values declined as rainfall decreased, but E s as a proportion of rainfall increased to 39–45 % in the tree-based treatments and 62 % for sole crops. Root + shoot pruning of alnus and the tree mixture increased θ w in the cropping area but had no significant effect in the other tree-based treatments. The results suggest that sesbania and alnus can be planted on smallholdings without compromising water supply to adjacent crops, whereas calliandra decreased water availability despite reducing E s . These results provide a mechanistic understanding of reported effects on crop yield in the same site