On the Foundations of Constructive Mathematics – Especially in Relation to the Theory of Continuous Functions

We discuss the foundations of constructive mathematics, including recursive mathematics and intuitionism, in relation to classical mathematics. There are connections with the foundations of physics, due to the way in which the different branches of mathematics reflect reality. Many different axioms and their interrelationship are discussed. We show that there is a fundamental problem in BISH (Bishop’s school of constructive mathematics) with regard to its current definition of ‘continuous function’. This problem is closely related to the definition in BISH of ‘locally compact’. Possible approaches to this problem are discussed. Topology seems to be a key to understanding many issues. We offer several new simplifying axioms, which can form bridges between the various branches of constructive mathematics and classical mathematics (‘reuniting the antipodes’). We give a simplification of basic intuitionistic theory, especially with regard to so-called ‘bar induction’. We then plead for a limited number of axiomatic systems, which differentiate between the various branches of mathematics. Finally, in the appendix we offer BISH an elegant topological definition of ‘locally compact’, which unlike the current definition is equivalent to the usual classical and/or intuitionistic definition in classical and intuitionistic mathematics, respectively.     

Submarine hot springs and the origin of life

The discovery of hydrothermal vents at oceanic ridge crests and the appreciation of their importance in the element balance of the oceans is one of the main recent advances in marine geochemistry. It is likely that vents were present in the oceans of the primitive Earth because the process of hydrothermal circulation probably began early in the Earth's history. Here we examine the popular hypothesis that life arose in these vents. This proposal, however, is based on a number of misunderstandings concerning the organic chemistry involved. An example is the suggestion that organic compounds were destroyed on the surface of the early Earth by the impact of asteroids and comets, but at the same time assuming that organic syntheses can occur in hydrothermal vents. The high temperatures in the vents would not allow synthesis of organic compounds, but would decompose them, unless the exposure time at vent temperatures was short. Even if the essential organic molecules were available in the hot hydrothermal waters, the subsequent steps of polymerization and the conversion of these polymers into the first organisms would not occur as the vent waters were quenched to the colder temperatures of the primitive oceans.     

Juvenile hormone titre and regulation in the cockroachDiploptera punctata

Summary Titres of juvenile hormone (JH) have been determined in both hemolymph and whole body extracts of femaleDiploptera punctata during the first gonotrophic cycle using a method employing gas chromatography/mass spectrometry for qualitative and quantitative analysis. JH III is the sole JH found in both adult and last instarD. punctata. Maximum values of 1500 ng/ml (6M) were observed at the middle of the gonotrophic cycle, when basal oocyte growth rate was greatest. Changes in rates of JH release in vitro by corpora allata paralleled closely the changes in JH titre, suggesting that biosynthesis is a major regulator of titre. JH levels per animal were calculated from observed JH titres, and at certain time points in the gonotrophic cycle JH levels obtained from analysis of whole bodies were significantly greater than those predicted from hemolymph titres. These results suggest the existence of a nonhemolymph JH pool inD. punctata. Decay in JH titre after allatectomy of 5 day females has also been studied. Following a rapid initial decline, the rate of decay slowed appreciably 4 h post-operation. Thus, use of a first-order rate constant to estimate half-life of JH significantly underestimated the longevity of the hormone. The apparent persistence of JH following allatectomy may be due to the existence of a nonhemolymph JH pool.     

A novel mechanism for jumping in the indian ant<Emphasis Type="Italic">Harpegnathos saltator</Emphasis> (Jerdon) (Formicidae,Ponerinae)

The Indian antHarpegnathos saltator may be unique among insects in using its jumping capacity not only as an escape mechanism but also as a normal means of locomotion, and for catching its prey in flight. High-speed cinematography used to analyse the various phases of the jump suggests thatHarpegnathos employs a novel jumping mechanism to mediate these behaviours: namely the synchronous activation of its middle and hindlegs. Electrophysiological recordings from muscles or nerves in pairs of middle and hindlegs show remarkably synchronous activity during fictive jumping, supporting the synchronous activation hypothesis.Harpegnathos is not the only ant to jump, and a cladistic analysis suggests that jumping behaviour evolved independently three times during ant evolutionary history.     

The free radical oxidation of the tetrahydrocannabinols

Summary The free radical oxidation of ¹ and ⁶-tetrahydrocannabinol has been examined by spin trapping techniques and intermediates that would lead to cannabinol have been trapped. The 1st step in the oxidation of ¹-THC involves the removal of 3-H, while for ⁶-THC, either 2-H or 5-H. Intermediates were isolated which could be pyrolysed to the diene and cannabinol.