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.     

Age distribution of circulatingα-interferon

Summary A sensitive radioimmunoassay showed that circulating -interferon in the plasma of healthy individuals was low in children and reached the highest level in the young adult, then declined gradually with age. Circulating -interferon was 0.201±0.059 ng/ml in males (n=19) and 0.184±0.076 ng/ml in females (n=14) at ages 30–39 years old. It was noted that circulating -interferon was maintained up to a certain level even in elderly individuals.     

Goniothalamicin and annonacin: Bioactive acetogenins fromGoniothalamus giganteus (Annonaceae)

Summary Using brine shrimp lethality for activity-directed fractionation, goniothalamicin (I), a new tetrahydroxy-mono-tetrahydrofuran fatty acid -lactone (acetogenin), has been isolated from ethanolic extracts of the stem bark ofGoniothalamus giganteus Hook. f., Thomas (Annonaceae). This novel compound was found to be cytotoxic and insecticidal and inhibited the formation of crown gall tumors on potato discs. Annonacin (II), the only other reported mono-tetrahydrofuran acetogenin, was also isolated; the previously reported 9ASK (astrocytoma reversal) activity ofII was confirmed, andII is now also found to be weakly active against 3PS murine leukemia.     

Definition of a consensus binding site for p53.

Recent experiments have suggested that p53 action may be mediated through its interaction with DNA. We have now identified 18 human genomic clones that bind to p53 in vitro. Precise mapping of the binding sequences within these clones revealed a consensus binding site with a striking internal symmetry, consisting of two copies of the 10 base pair motif 5'-PuPuPuC(A/T)(T/A)GPyPyPy-3' separated by 0-13 base pairs. One copy of the motif was insufficient for binding, and subtle alterations of the motif, even when present in multiple copies, resulted in loss of affinity for p53. Mutants of p53, representing each of the four "hot spots" frequently altered in human cancers, failed to bind to the consensus dimer. These results define the DNA sequence elements with which p53 interacts in vitro and which may be important for p53 action in vivo.     

Heterogeneous responses of canine basilar and middle cerebral arteries to serotonin at normal and high CO2 tension

The responses of basilar arteries (BAs) to serotonin were attenuated by high \(P_{CO_2 } \) (86±1 mm Hg) and the pH matched acidotic solution ( \(P_{CO_2 } \) 37±1 mm Hg), whereas the responses of middle cerebral arteries (MCAs) were not. High \(P_{CO_2 } \) decreased the basal tone of both arteries, and the changes in basal tone due to high \(P_{CO_2 } \) were not influenced by 3×10^?7 M imipramine, 10^?5 M pargyline or 10^?4 M aspirin. The responses of BAs to serotonin were attenuated by high \(P_{CO_2 } \) in the presence of imipramine, pargyline and aspirin. The responses of MCAs to serotonin were not influenced by high \(P_{CO_2 } \) in the presence of pargyline and aspirin, but attenuated by high \(P_{CO_2 } \) in the presence of imipramine.     

The life span of the biosphere revisited

A decade ago, Lovelock and Whitfield raised the question of how much longer the biosphere can survive on Earth. They pointed out that, despite the current fossil-fuel induced increase in the atmospheric CO2 concentration, the long-term trend should be in the opposite direction: as increased solar luminosity warms the Earth, silicate rocks should weather more readily, causing atmospheric CO2 to decrease. In their model, atmospheric CO2 falls below the critical level for C3 photosynthesis, 150 parts per million (p.p.m.), in only 100 Myr, and this is assumed to mark the demise of the biosphere as a whole. Here, we re-examine this problem using a more elaborate model that includes a more accurate treatment of the greenhouse effect of CO2, a biologically mediated weathering parameterization, and the realization that C4 photosynthesis can persist to much lower concentrations of atmospheric CO2(<10 p.p.m.). We find that a C4-plant-based biosphere could survive for at least another 0.9 Gyr to 1.5 Gyr after the present time, depending respectively on whether CO2 or temperature is the limiting factor. Within an additional 1 Gyr, Earth may lose its water to space, thereby following the path of its sister planet, Venus.     

C1 inhibitor hinge region mutations produce dysfunction by different mechanisms.

Heterozygosity for a mutant dysfunctional C1 inhibitor protein, a member of the serine proteinase inhibitor (serpin) superfamily, results in type II hereditary angioneurotic oedema. We identified a "hinge" region mutation in C1 inhibitor with a Val to Glu replacement at P14 Val-432. Recombinant C1 inhibitors P10 Ala-->Thr and P14Val-->Glu did not form stable complexes with fluid phase C1s or kallikrein. The P14 Val-->Glu mutant, however, was cleaved to a 96K form by C1s, while the P10 Ala-->Thr mutant was not. The recombinant P10 mutant also did not complex with C1s, kallikrein or beta-factor Xlla-Sepharose. The two mutations, therefore, result in dysfunction by different mechanisms: in one (P14 Val-->Glu), the inhibitor is converted to a substrate, while in the other (P10 Ala-->Thr), interaction with target protease is blocked.     

Purification of platelet-derived endothelial cell growth inhibitor and its characterization as transforming growth factor-β type 1

In 1986, Brown and Clemmons (Proc. natl Acad. Sci. USA83 (1986) 3321) showed that platelets contain a substance, platelet-derived growth inhibitor (PDGI), that inhibits in vitro endothelial cell replication. Although platelets are rich in transforming grwoth factor (TGF-), PDGI was considered not to be related to TGF-, on the basis of its reported properties (extraction from platelets at neutral pH, binding to heparin-Sepharose). However, we purified PDGI to near homogeneity and showed that on the basis of HPLC retention behavior, in vitro growth inhibitory activities with several cell types, receptor binding, and immunoneutralization of growth inhibitory activity with specific anti-TGF- type 1 antibodies, PDGI is most probably identical with TGF- type 1.     

Flow-induced release of adenosine 5′-triphosphate from endothelial cells of the rat mesenteric arterial bed

Adenosine 5-triphosphate (ATP) was released into the perfusate of rat isolated mesenteric arterial beds during each of two consecutive increases in flow. There was no significant difference between the amounts of ATP released on each occasion. Substance P was also released into the perfusate by increased flow, although its release was more variable. Removal of the endothelium of the mesenteric vessels with sodium deoxycholate led to a significant reduction (74%) in the amount of ATP released compared with the release before the endothelium had been removed. This suggests that the ATP released into the mesenteric arterial perfusate during increased flow arises from endothelial cells.