## Philosophical Mathematics, Logic, and the Limits of Formal Systems

In Episode 11 of the Hidden Forces podcast, Demetri Kofinas speaks with Ray Monk. Ray Monk is Professor of Philosophy at the University of Southampton in the UK, where he lectures on logic, philosophical mathematics and the philosophy of Wittgenstein. He is presently a visiting Miller Scholar at the Santa Fe Institute. A prolific biographer, professor Monk has written books on the philosophers and mathematicians Ludwig Wittgenstein and Bertrand Russell, as well as the theoretical physicist and director of the Los Alamos Laboratory during the Manhattan Project, Robert Oppenheimer.

In this episode, we explore the mysterious and paradoxical world of mathematics. What are the foundations of mathematics? Where did it come from? How did this seemingly infinite body of knowledge arise from virtually nothing? What are Euclid’s axioms? What are Plato’s forms? What did the Pythagorean mystery cults worship? How did our notions of mathematics evolve from the time of the Ancient Greeks? What were Immanuel Kant’s insights about how we experience the phenomenal world? What did he believe about the nature of reality and the role of mathematics in structuring perception? What was Russell’s paradox? Why did Bertrand Russell ultimately fail in his attempt to create a formal system of mathematics built off of logical axioms and postulates? What was it that Kurt Gödel uttered in 1931 that shattered our confidence in the very foundations of mathematics? What did his theorem of incompleteness prove about the limits of mathematical knowledge and the uncertainty of formal systems? Finally, what was the great insight of Ludwig Wittgenstein about why paradoxes exist? What did he have to say about the limits of language and expression? And what are the implications of all of this, for the existence of God?

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## Reading List on Philosophical Mathematics and the History of Mathematics:

### Books on Philosophical Mathematics and the History of Mathematics:

Thinking about Mathematics: The Philosophy of Mathematics by Stewart Shapiro

Philosophy of Mathematics: Selected Readings by Paul Benacerraf and Hilary Putnam

Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures by James Robert Brown

The Philosophy of Mathematics by W.D. Hart

Mathematics: The Loss of Certainty by Morris Kline

The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number by Gottlob Frege

Translations from the Philosophical Writings of Gottlob Frege by Peter Geach (Editor) and Max Black (Editor)

The Principles of Mathematics by Bertrand Russell

On Formally Undecidable Propositions of Principia Mathematica and Related Systems by Kurt Gödel

Incompleteness: The Proof and Paradox of Kurt Godel by Rebecca Goldstein

Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter

Ludwig Wittgenstein: The Duty of Genius by Ray Monk

The Republic by Plato

Critique of Pure Reason by Immanuel Kant

The Historical Roots of Elementary Mathematics by Lucas N. H. Bunt

A History of Mathematics by Carl B. Boyer and Uta C. Merzbach

### Books by Ray Monk:

Ludwig Wittgenstein: The Duty of Genius

How to Read Wittgenstein

Robert Oppenheimer: His Life and Mind (A Life Inside the Center)

Bertrand Russell: The Spirit of Solitude 1872-1921

Bertrand Russell: 1921-1970, The Ghost of Madness

The Great Philosophers: From Socrates to Turing