A Reassessment of Gödel's Doctrine - The Necessity of Infinity

Explore a thought-provoking academic seminar from the Orange County Inland Empire (OCIE) Series in History and Philosophy of Mathematics where Patrick Ryan delves into the philosophical implications of Gödel's Doctrine regarding the necessity of infinity in mathematics. Examine the historical context of Gödel's 1931 paper and its claim about the requirement of infinitary resources to prove certain finitary statements. Learn how proof theoretic work by Feferman and others challenged this doctrine by demonstrating that many finitary results could be proved using relatively weak systems. Discover a fresh perspective on the debate through analysis of finitary statements with multiple proofs using both infinitary and finitary resources. Consider fundamental questions about the role of infinitary proofs in mathematical explanation, the nature of mathematical necessity, and the philosophical significance of different proof methods. Engage with contemporary discussions in the philosophy of mathematics concerning purity, content, and explanation while exploring promising directions for future research in this nearly two-hour presentation from Chapman University's Schmid College of Science and Technology.