Unknowable Truths: The Incompleteness Theorems and the Rise of Modernism

This thesis evaluates the function of the current history of mathematics methodologies and explores ways in which historiographical methodologies could be successfully implemented in the field. Traditional approaches to the history of mathematics often lack either an accurate portrayal of the social and cultural influences of the time, or they lack an effective usage of mathematics discussed. This paper applies a holistic methodology in a case study of Kurt Gödel’s influential work in logic during the Interwar period and the parallel rise of intellectual modernism. In doing so, the proofs for Gödel’s Completeness and Incompleteness theorems will be discussed as well as Gödel’s philosophical interests and influences of the time. To explore the intersection of these worlds, practices are borrowed from the fields of intellectual history and history of science and technology to analyze better the effects of society and culture on the mind of mathematicians like Gödel and their work

Remarks on Wittgenstein, Gödel, Chaitin, Incompleteness, Impossiblity and the Psychological Basis of Science and Mathematics

It is commonly thought that such topics as Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were resolved by Wittgenstein over 80 years ago. Wittgenstein also demonstrated the fatal error in regarding mathematics or language or our behavior in general as a unitary coherent logical ‘system,’ rather than as a motley of pieces assembled by the random processes of natural selection. “Gödel shows us an unclarity in the concept of ‘mathematics’, which is indicated by the fact that mathematics is taken to be a system” and we can say (contra nearly everyone) that is all that Gödel and Chaitin show. Wittgenstein commented many times that ‘truth’ in math means axioms or the theorems derived from axioms, and ‘false’ means that one made a mistake in using the definitions, and this is utterly different from empirical matters where one applies a test. Wittgenstein often noted that to be acceptable as mathematics in the usual sense, it must be useable in other proofs and it must have real world applications, but neither is the case with Godel’s Incompleteness. Since it cannot be proved in a consistent system (here Peano Arithmetic but a much wider arena for Chaitin), it cannot be used in proofs and, unlike all the ‘rest’ of PA it cannot be used in the real world either. As Rodych notes “…Wittgenstein holds that a formal calculus is only a mathematical calculus (i.e., a mathematical language-game) if it has an extra- systemic application in a system of contingent propositions (e.g., in ordinary counting and measuring or in physics) …” Another way to say this is that one needs a warrant to apply our normal use of words like ‘proof’, ‘proposition’, ‘true’, ‘incomplete’, ‘number’, and ‘mathematics’ to a result in the tangle of games created with ‘numbers’ and ‘plus’ and ‘minus’ signs etc., and with ‘Incompleteness’ this warrant is lacking. Rodych sums it up admirably. “On Wittgenstein’s account, there is no such thing as an incomplete mathematical calculus because ‘in mathematics, everything is algorithm [and syntax] and nothing is meaning [semantics]…” I make some brief remarks which note the similarities of these ‘mathematical’ issues to economics, physics, game theory, and decision theory. Those wishing further comments on philosophy and science from a Wittgensteinian two systems of thought viewpoint may consult my other writings -- Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle 2nd ed (2019), Suicide by Democracy 4th ed (2019), The Logical Structure of Human Behavior (2019), The Logical Structure of Consciousness (2019, Understanding the Connections between Science, Philosophy, Psychology, Religion, Politics, and Economics and Suicidal Utopian Delusions in the 21st Century 5th ed (2019), Remarks on Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason in Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych, Berto, Floyd, Moyal-Sharrock and Yanofsky (2019), and The Logical Structure of Philosophy, Psychology, Sociology, Anthropology, Religion, Politics, Economics, Literature and History (2019)

Philosophy at Cambridge, Newsletter of the Faculty of Philosophy

Philosophy Newsletter. Articles by: Jane Heal - From the Chair. David Waterman - Too Beautiful to Be True. Simon Blackburn - Who's the Tops?. The Heffer Lecture 2006. The Forum for Philosophy in Business. Arif Ahmed - Why Believe the Axioms?. Matthew Parris - Chauvinist Days. Laura Biron - Scepticism at Red Alert. Steffi Lewis - Who'd be a Fundraiser? Faculty News

Signs of reality - reality of signs. Explorations of a pending revolution in political economy.

This paper explores the interaction between the world of information processes in human society and the non-information dynamics, which the latter set out to understand. This broad topic is approached with a focus on evolutionary political economy: It turns out that progress in this scientific discipline seems to depend crucially on a methodological revolution reframing this above mentioned interplay. The paper consists of three parts. After a brief introduction, which sketches the position of the argument in the current epistemological discourse, part 1 sets out to describe the basic methodological ingredients used by evolutionary political economy to describe the ‘reality’ of socioeconomic dynamics. Part 2 jumps to the world of languages used and proposes a rather radical break with the received apparatus of analytical mathematics used so successfully in sciences studying non-living phenomena. The development of procedural simulation languages should substitute inadequate mathematical formalizations, some examples are provided. Part 3 then returns to ‘reality’ dynamics, but now incorporates the interaction with the information sphere in a small algorithmic model. This model – like the introduction - again makes visible the relationships to earlier research in the field. Instead of a conclusion – several, hopefully innovative ideas are provided in passing, throughout the paper - an epilogue is provided, which tries to indicate the implications of this methodological paper for political practice in face of the current global crisis.Scientific methods, evolutionary political economy, formal languages, ideology

What Was Artificial Intelligence?

When it was originally published in 2002, Sue Curry Jansen’s “What Was Artificial Intelligence?” attracted little notice. The long essay was published as a chapter in Jansen’s Critical Communication Theory, a book whose wisdom and erudition failed to register across the many fields it addressed. One explanation for the neglect, ironic and telling, is that Jansen’s sheer scope as an intellectual had few competent readers in the communication studies discipline into which she published the book. “What Was Artificial Intelligence?” was buried treasure. In this mediastudies.press edition, Jansen’s prescient autopsy of AI self-selling—the rhetoric of the masculinist sublime—is reprinted with a new introduction. Now an open access book, “What Was Artificial Intelligence?” is a message in a bottle, addressed to Musk, Bezos, and the latest generation of AI myth-makers

Sofia A. Yanovskaya: The Marxist Pioneer of Mathematical Logic in the Soviet Union

K. Marx’s 200th jubilee coincides with the celebration of the 85 years from the first publication of his “Mathematical Manuscripts” in 1933. Its editor, Sofia Alexandrovna Yanovskaya (1896–1966), was a renowned Soviet mathematician, whose significant studies on the foundations of mathematics and mathematical logic, as well as on the history and philosophy of mathematics are unduly neglected nowadays. Yanovskaya, as a militant Marxist, was actively engaged in the ideological confrontation with idealism and its influence on modern mathematics and their interpretation. Concomitantly, she was one of the pioneers of mathematical logic in the Soviet Union, in an era of fierce disputes on its compatibility with Marxist philosophy. Yanovskaya managed to embrace in an originally Marxist spirit the contemporary level of logico-philosophical research of her time. Due to her highly esteemed status within Soviet academia, she became one of the most significant pillars for the culmination of modern mathematics in the Soviet Union. In this paper, I attempt to trace the influence of the complex socio-cultural context of the first decades of the Soviet Union on Yanovskaya’s work. Among the several issues I discuss, her encounter with L. Wittgenstein is striking

John Templeton Foundation: Capabilities Report

This annual report, celebrating the 25th anniversary of the foundation, includes letters from its leaders, a history of the foundation, details of current grantmaking and other activities, financial statements, and lists of trustees