Presocratic philosophy, for all its diverse features, is united by the quest to understand the origin and nature of the world. The approach of the Pythagoreans to this quest is governed by their belief, probably based on studies of the numerical relations in musical harmony, that number or numerical structure plays a key role for explaining the world-order, the cosmos. It remains questionable to what extent the Pythagoreans, by positing number as an all-powerful explanatory concept, broke free from Presocratic ideas (...) that certain stuffs or material elements sufficed to account for the source and constitution of the world, but apparently number found such a universal application with them that Aristotle could summarize the Pythagorean position as ‘numbers…are the whole universe’. Historians of Greek philosophy have generally accepted Aristotle's assessment. Of late, however, certain scholars have argued that the Pythagorean number doctrine is Aristotelian interpretation, unjustly foisted upon the Pythagoreans. Enlisted in support of their arguments are the fragments of Philolaus of Croton. Here we have the foremost representative of fifth-century Pythagoreanism, who states as his basic principles, not numbers exactly, but ‘limiters’ and ‘unlimiteds’, and who, it is argued, regards number solely as an epistemological aid for understanding the structure of reality. So Philolaus is called upon as a witness against Aristotle. The rationale goes something like this: Aristotle most likely had written sources for his knowledge of Pythagorean teachings; the only texts we know of with any certainty are Philolaus' book and the writings of Archytas; since Aristotle treats Archytas separately, he is mainly relying on Philolaus; because Philolaus does not expressly state that things are numbers, Aristotle's interpretation is wrong. (shrink)

Presocratic philosophy, for all its diverse features, is united by the quest to understand the origin and nature of the world. The approach of the Pythagoreans to this quest is governed by their belief, probably based on studies of the numerical relations in musical harmony, that number or numerical structure plays a key role for explaining the world-order, the cosmos. It remains questionable to what extent the Pythagoreans, by positing number as an all-powerful explanatory concept, broke free from Presocratic ideas (...) that certain stuffs or material elements sufficed to account for the source and constitution of the world, but apparently number found such a universal application with them that Aristotle could summarize the Pythagorean position as ‘numbers…are the whole universe’ . Historians of Greek philosophy have generally accepted Aristotle's assessment. Of late, however, certain scholars have argued that the Pythagorean number doctrine is Aristotelian interpretation, unjustly foisted upon the Pythagoreans. Enlisted in support of their arguments are the fragments of Philolaus of Croton. Here we have the foremost representative of fifth-century Pythagoreanism, who states as his basic principles, not numbers exactly, but ‘limiters’ and ‘unlimiteds’, and who, it is argued, regards number solely as an epistemological aid for understanding the structure of reality. So Philolaus is called upon as a witness against Aristotle. The rationale goes something like this: Aristotle most likely had written sources for his knowledge of Pythagorean teachings; the only texts we know of with any certainty are Philolaus' book and the writings of Archytas; since Aristotle treats Archytas separately, he is mainly relying on Philolaus; because Philolaus does not expressly state that things are numbers, Aristotle's interpretation is wrong. (shrink)

Hierocles of Alexandria was a Neoplatonic philosopher of the fifth century AD. Hermann S. Schibli surveys his life, writings, and pagan and Christian surroundings, and succinctly examines the major points of his philosophy, both contemplative and practical. He includes the first modern English translations, with helpful notes, of Hierocles' Commentary on the Golden Verses of the Pythagoreans and of the remnants of his treatise On Providence.

Many have argued against the claim that Plato posited the mathematical objects that are the subjects of Metaphysics M and N. This paper shifts the burden of proof onto these objectors to show that Plato did not posit these entities. It does so by making two claims: first, that Plato should posit the mathematical Intermediates because Forms and physical objects are ill suited in comparison to Intermediates to serve as the objects of mathematics; second, that their utility, combined with Aristotle’s (...) commentary on Plato’s posit of Intermediates, provides good reason to conclude that Plato did posit them. (shrink)

To localize wheat ESTs on chromosomes, 882 homoeologous group 6-specific ESTs were identified by physically mapping 7965 singletons from 37 cDNA libraries on 146 chromosome, arm, and sub-arm aneuploid and deletion stocks. The 882 ESTs were physically mapped to 25 regions flanked by 23 deletion breakpoints. Of the 5154 restriction fragments detected by 882 ESTs, 2043 were localized to group 6 chromosomes and 806 were mapped on other chromosome groups. The number of loci mapped was greatest on chromosome 6B and (...) least on 6D. The 264 ESTs that detected orthologous loci on all three homoeologs using one restriction enzyme were used to construct a consensus physical map. The physical distribution of ESTs was uneven on chromosomes with a tendency toward higher densities in the distal halves of chromosome arms. About 43% of the wheat group 6 ESTs identified rice homologs upon comparisons of genome sequences. Fifty-eight percent of these ESTs were present on rice chromosome 2 and the remaining were on other rice chromosomes. Even within the group 6 bins, rice chromosomal blocks identified by 1-6 wheat ESTs were homologous to up to 11 rice chromosomes. These rice-block contigs were used to resolve the order of wheat ESTs within each bin. (shrink)