Philosophical Mathematics and the Incompleteness of Formal Systems | Ray Monk
May 22, 2017
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Ray Monk, Professor of Philosophy, explores the origins of mathematics, Euclid's axioms, Plato's forms, Kant's insights, Russell's paradox, Gödel's theorems, and Wittgenstein’s views on paradoxes of language and expression.
Mathematics provides insight into a reality beyond the physical world, according to Plato.
Immanuel Kant believed that mathematics structures our perception of reality, showcasing the power of reason.
Kurt Gödel's incompleteness theorem challenges the idea that truth and provability are the same in mathematics, raising questions about the nature of knowledge and truth.
Deep dives
Plato's view of mathematics as a connection to the metaphysical realm
Plato saw mathematics as a form of knowledge that was connected to the world of forms or the spiritual realm. He believed that mathematics provided insight into a reality that was beyond the mind and the physical world. Plato's metaphysics suggested that through mathematics, we can access eternal and unchanging truths, highlighting the power of reason and the limitations of our senses.
Kant's perspective on the relationship between mathematics and reality
Immanuel Kant believed that mathematics structured our perception of reality. He argued that our minds are hardwired with geometry and arithmetic, which shape the spatial and temporal dimensions of our world. Kant viewed mathematics as a reflection of the way our minds organize and structure our experiences. According to him, mathematics demonstrates the power of our reason and the limitations of relying solely on senses to understand the world.
Russell and Frager's logicism and Wittgenstein's distinction between saying and showing
Bertrand Russell and Gottlob Freger aimed to ground mathematics in logic through their philosophy of logicism. They sought to demonstrate that arithmetic and numbers are objective truths that exist independently of human invention. However, their efforts were challenged by paradoxes and logical contradictions. Ludwig Wittgenstein, one of Russell's students, developed the idea of distinguishing between what can be said and what can only be shown. He believed that logical form, which gives structure to language, thought, and the world, cannot be expressed in words. Wittgenstein applied this distinction not only to logic but also to ethics, aesthetics, and religious and existential questions.
The Paradox and the Quest for Truth
The podcast episode explores the concept of paradoxes, particularly focusing on Russell's paradox. Paradoxes are chains of reasoning that lead to absurd or contradictory conclusions. Russell's paradox involves the class of all classes that do not contain themselves as members. This paradox demonstrates the contradictions within the notion of class. The episode also discusses the significance of paradoxes, not only in mathematics but also in understanding the nature of thinking itself.
Gödel's Incompleteness Theorem and Mathematics
The podcast delves into the work of Kurt Gödel and his incompleteness theorem, which shook the foundations of mathematics. Gödel's theorem proves that there cannot be a complete theory of arithmetic, meaning that there will always be arithmetical truths that cannot be proven within a given formal system. This challenges the view that truth and provability are the same in mathematics. It highlights the limitations of formal systems and raises questions about the nature of truth and knowledge in mathematics and the world at large.
In Episode 11 of Hidden Forces, host 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 their conversation, Demetri and Ray explore the mysterious and paradoxical world of mathematics. What are the foundations of mathematics? Where did mathematics 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 and 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 the paradoxes exist in mathematics? 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?
