Godel, Escher and Bach – MIT Open Courseware
Godel, Escher and Back – MIT Open Course ware:
Conceptual Systems and Logical Analysis
ACHILLES AND THE TORTOISE introduction
ACHILLES AND THE TORTOISE 7th Revision An Essay on the Logical Foundations of Torah Study; SELF EVIDENCE, INTUITION, AND THE SEARCH FOR TRUTH by Gerald Parkoff *********************** This essay deals with the very foundations of Rule Following Systems and what underlies all legal and ethical systems of evaluation. The analysis is extended to Torah Law. The psychological process of intuition is discussed together with the very limits of […]
Kurt Godel – From the Limits of Understanding
Short Video on the Life and Work of Kurt Godel:
RULE FOLLOWING SYSTEMS
One of the purposes of his site is to provide an overall view of Systems of Thought , ways of seeing and understanding things derivative from the work of David Hilbert and Alan Turing. Let us begin with the concept of RULE FOLLOWING SYSTEMS: Here is how it works: A machine is the best model of a Rule Following System. A Machine is not merely law abiding. A […]
The Connection Between Godel’s Theorem and Psak
“We have seen that mathematical propositions which cannot be established by formal deduction from a given set of axioms may, nevertheless, be established by “informal” meta-mathematical reasoning. It would be irresponsible to claim that these formally indemonstrable truths established by meta-mathematical arguments are based on nothing better than bare appeals to intuition.” (Ernest Nagel, Godel’s Proof, p. 112) David Zviel: Godel actually constructs formal statements which must […]
PLATONIC REALISM AND THE CONCEPT OF MIND
Platonic realism takes the view that mathematics does not create or invent its “object, but discovers them as Columbus discovered America. Now, if this is true, the objects must in some sense “exist” prior to their discovery. According to Platonic doctrine, the objects of mathematical study are not found in the spatio-temporal order. They are disembodied eternal Forms or Archetypes, which dwell in a distinctive realm accessible […]
THE RAMBAM’S UNDERSTANDING OF SYSTEM
A study of the fourteen Shorashim to Sefer Hamitzvot is essential for grasping the Rambam’s understanding of “system” and to see how close he came to modern mathematical models exemplifying precision and rigor. Considering the Rambam’s encyclopedic knowledge of Jewish Law and all its sources, it is rather startling to witness his ability to cross over and model a rigorous, logical system in the modern sense of system […]
THE RAMBAN AND THE AXIOMATIC ANALOGY
The Ramban and the Axiomatic Analogy to Torah Law Law is not usually thought of as an axiomatic system. In fact, there is no reason why a legal system cannot be a chaotic conglomeration of rules, some of which may even contradict each other. It would take some bit of research to come up with real historical examples. Jewish Law, as codified by the Rambam, is superbly logical […]
Intuition, Godel, and Moral Philosophy
A present-day mathematical logician , Rosser, in Logic for Mathematicians, McGraw-Hill, 1953, p. 11) writes: [The mathematician] should not forget that his intuition is the final authority, so that, in case of irreconcilable conflict between his intuition and some system of… logic, he should abandon the … logic. He can try other systems of… logic, and perhaps find one more to his liking, but it would be difficult […]
Pierre de Fermat and the Number 26
Pierre De Fermat was one of the greatest mathematicians of all time. He was born on August 20, 1601, in the town of Beaumaont-de-Lomagne in southwest France. His father, Dominique Fermat, was a wealthy leather merchant, and so Pierre was fortunate enough to enjoy a privileged education at the Franciscan monastery of Grandselve, followed by a stint at the University of Toulouse…. Pressure from his family steered Fermat […]