In Proclus' penetrating exposition of Euclid's method's and principles, the only one of its kind extant, we are afforded a unique vantage point for understanding the structure and strenght of the Euclidean system. A primary source for the history and philosophy of mathematics, Proclus' treatise contains much priceless information about the mathematics and mathematicians of the previous seven or eight centuries that has not been preserved elsewhere.

Glenn Kreider explores the whole Bible to show us the full scope of God's active involvement in his creation from the very beginning. This better understanding of God's humility will affect our own humility and our attitude toward serving those around us."--Publisher description.

Socrates' dream puts in generalized form the difficulty that plato saw in the mathematician's procedure of hypothesis, I.E., Of positing undemonstrated first principles ("prota") or elements ("stoicheia") as starting-Points of demonstration. If the elements are unknown, How can what is constructed from them be known?--A difficulty to which plato had earlier called attention in the 'republic' (510cd, 533cd.) this interpretation accords with the mathematical setting and personages of the dialogue, And explains why the explicit refutation of theaetetus' third proposal, That (...) knowledge be defined as true belief accompanied by a logos, Is so perfunctory and unconvincing. Furthermore, The dilemma thus brought to light is reflected in 'posterior analytics' (book i, Chap. 3), Which was presumably written during aristotle's residence in the academy in association with plato and the other mathematicians gathered around him. (shrink)

This is the first English translation of Proclus' commentary on Plato's Parmenides. Glenn Morrow's death occurred while he was less than halfway through the translation, which was completed by John Dillon. A major work of the great Neoplatonist philosopher, the commentary is an intellectual tour de force that greatly influenced later medieval and Renaissance thought. As the notes and introductory summaries explain, it comprises a full account of Proclus' own metaphysical system, disguised, as is so much Neoplatonic philosophy, in (...) the form of a commentary. (shrink)

When we are fatigued, we feel that our performance is worse than when we are fresh. Yet, for over 100 years, researchers have been unable to identify an objective, behavioral measure that covaries with the subjective experience of fatigue. Previous work suggests that the metrics of signal detection theory —response bias and perceptual certainty —may change as a function of fatigue, but no work has yet been done to examine whether these metrics covary with fatigue. Here, we investigated cognitive fatigue (...) using SDT. We induced fatigue through repetitive performance of the n-back working memory task, while functional magnetic resonance imaging data was acquired. We also assessed cognitive fatigue at intervals throughout. This enabled us to assess not only whether criterion and d’ covary with cognitive fatigue but also whether similar patterns of brain activation underlie cognitive fatigue and SDT measures. Our results show that both criterion and d’ were correlated with changes in cognitive fatigue: as fatigue increased, subjects became more conservative in their response bias and their perceptual certainty declined. Furthermore, activation in the striatum of the basal ganglia was also related to cognitive fatigue, criterion, and d’. These results suggest that SDT measures represent an objective measure of cognitive fatigue. Additionally, the overlap and difference in the fMRI results between cognitive fatigue and SDT measures indicate that these measures are related while also separate. In sum, we show the relevance of SDT measures in the understanding of fatigue, thus providing researchers with a new set of tools with which to better understand the nature and consequences of cognitive fatigue. (shrink)