When St. Thomas wrote the Summa Contra Gentiles in the first half of the 1260s, he was contributing to a long-standing Christian effort to receive Aristotle’s writings without accepting his errors. If Aristotle was to be in the Christian world what everybody was proclaiming, namely, the Philosopher, he could not remain an ancient pagan thinker nor could his philosophy remain subject to the errors and, even more, the limitations that historically it contained. Clearly enough, an Aristotle who was merely freed (...) from his errors would not do. For if, though purified, he retained the perspective of his own ancient world, he could not speak for philosophy in a Christian climate and his philosophy could not serve as a human instrument in the exposition of Christian teaching. What Aristotle needed—so St. Thomas thought—was, beyond correction, a reconstruction and even a reestablishment of his doctrine on a different foundation from his own. (shrink)

When St. Thomas wrote the Summa Contra Gentiles in the first half of the 1260s, he was contributing to a long-standing Christian effort to receive Aristotle’s writings without accepting his errors. If Aristotle was to be in the Christian world what everybody was proclaiming, namely, the Philosopher, he could not remain an ancient pagan thinker nor could his philosophy remain subject to the errors and, even more, the limitations that historically it contained. Clearly enough, an Aristotle who was merely freed (...) from his errors would not do. For if, though purified, he retained the perspective of his own ancient world, he could not speak for philosophy in a Christian climate and his philosophy could not serve as a human instrument in the exposition of Christian teaching. What Aristotle needed—so St. Thomas thought—was, beyond correction, a reconstruction and even a reestablishment of his doctrine on a different foundation from his own. (shrink)

I. The following essay aims to compare the proofs for the existence of God in four medieval theologians, namely, St. Anselm of Canterbury, St. Bonaventure, St. Thomas Aquinas and Henry of Ghent, Being theologians, all four men believed in a divine revelation and their personal intellectual activity took place within the world of revelation. Fides quaerens intellectum, which was St. Anselm’s title for the Proslogion before he gave it a name, is a formula that can be applied to all these (...) thinkers in their individual religious adventures. Their lives were spent in seeking after the God who had met them on earth and in responding to that meeting. St. Anselm belongs to the end of the eleventh century, St. Bonaventure and St. Thomas were active during the third quarter of the thirteenth century, and Henry of Ghent’s literary activity falls mainly in the embattled twenty years after Aristotelianism in all its forms was condemned in 1277 in Paris. This was the time when Christian theologians were beginning to build a new religious synthesis that reflected the conservative mood one can still read in the text of Giles of Rome’s De Erroribus Philosophorum. (shrink)

Our best scientific theories explain a wide range of empirical phenomena, make accurate predictions, and are widely believed. Since many of these theories make ample use of mathematics, it is natural to see them as confirming its truth. Perhaps the use of mathematics in science even gives us reason to believe in the existence of abstract mathematical objects such as numbers and sets. These issues lie at the heart of the Indispensability Argument, to which this Element is devoted. The Element's (...) first half traces the evolution of the Indispensability Argument from its origins in Quine and Putnam's works, taking in naturalism, confirmational holism, Field's program, and the use of idealisations in science along the way. Its second half examines the explanatory version of the Indispensability Argument, and focuses on several more recent versions of easy-road and hard-road fictionalism respectively. (shrink)

The Euclidean Programme embodies a traditional sort of epistemological foundationalism, according to which knowledge – especially mathematical knowledge – is obtained by deduction from self-evident axioms or first principles. Epistemologists have examined foundationalism extensively, but neglected its historically dominant Euclidean form. By contrast, this book offers a detailed examination of Euclidean foundationalism, which, following Lakatos, the authors call the Euclidean Programme. The book rationally reconstructs the programme's key principles, showing it to be an epistemological interpretation of the axiomatic method. It (...) then compares the reconstructed programme with select historical sources: Euclid's Elements, Aristotle's Posterior Analytics, Descartes's Discourse on Method, Pascal's On the Geometric Mind and a twentieth-century account of axiomatisation. The second half of the book philosophically assesses the programme, exploring whether various areas of contemporary mathematics conform to it. The book concludes by outlining a replacement for the Euclidean Programme. (shrink)

A focussed issue of The Reasoner on the topic of 'Infinitary Reasoning'. Owen Griffiths and A.C. Paseau were the guest editors.

The Compactness Theorem The compactness theorem is a fundamental theorem for the model theory of classical propositional and first-order logic. As well as having importance in several areas of mathematics, such as algebra and combinatorics, it also helps to pinpoint the strength of these logics, which are the standard ones used in mathematics and arguably … Continue reading Compactness →.

Fragt man heute Vertreter verschiedener Disziplinen nach der Bedeutung des Hannoveraner Universalgelehrten Gottfried Wilhelm Leibniz, so hort man jeweils immer wieder: Leibniz hat Bedeutendes fur unser Fach geleistet. Leibniz beeindruckt nicht nur durch die Exzellenz seiner Leistung, sondern auch durch die Breite seiner Betatigungsfelder. Der aus einer Ringvorlesung an der Leibniz Universitat Hannover hervorgegangene Band fuhrt nun an die Vielfalt der von Leibniz ausgehenden der Leistungen und Anregungen heran. Insbesondere behandeln die Beitrage die Bedeutung von Leibniz fur die Geschichtswissenschaft, die (...) Theologie, die Philosophie, die Rechtswissenschaft und Juristerei, die Politikberatung, das Versicherungswesen, die Mathematik, die Ingenieurwissenschaften und die Sprachwissenschaft. Abgerundet wird der Band durch eine Einfuhrung von Werner Eisner und eine biographische Skizze aus der Feder von Eike Christian Hirsch. (shrink)