Ep 199: David Deutsch’s ”The Fabric of Reality” Chapter 10 ”The Nature of Mathematics” Part 2
Oct 11, 2023
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The podcast explores the unique perception of mathematics, its treatment as a subject, and the allure of mathematicians. It delves into the criterion of existence and the limitations of human knowledge. It also discusses Plato's perspective on the nature of mathematics, struggles of learning as we get older, and controversies in mathematics.
Mathematicians have debated the existence and validity of abstract entities in mathematics, including imaginary numbers and infinite quantities.
The notion of mathematical intuition has led to disagreements about which mathematical ideas are valid and reliable, fueling ongoing debates in the field.
Deep dives
Controversy over the existence of abstract entities
Mathematicians have long debated the existence and validity of abstract entities in mathematics. Some mathematicians objected to imaginary numbers, while others embraced them. Similar controversies arose around the use of infinite numbers, sets containing infinitely many elements, and infinitesimal quantities in calculus. The debate intensified as mathematical reasoning became more complex and moved further from everyday intuition. While mathematicians became more meticulous about proofs and set higher standards of rigor, the invention of more powerful proof methods raised doubts about their infallibility. The literature of mathematics became filled with controversies and debates stemming from the infinite. However, as mathematical reasoning became more sophisticated, new methods of proof were validated, contributing to the progress of mathematics.
Mathematical intuition and absolute certainty
Mathematicians traditionally believe in the existence of mathematical intuition, which is thought to provide absolute certainty in mathematical knowledge. This intuition is considered a source of infallibility, separate from scientific knowledge, which is seen as fallible and subject to potential revision. However, the notion of mathematical intuition has led to disagreements about which mathematical ideas are valid and reliable. Controversies arose over the use of imaginary numbers, sets with infinite elements, and other concepts. The debate surrounding the existence and certainty derived from mathematical intuition has persisted throughout history and continues to shape the field of mathematics.
The challenge of rigorous proofs and new methods
Advancements in mathematical reasoning and the introduction of new proof methods have posed challenges to the field. Mathematicians have become more stringent in the rigor of proofs, but the complexity of these proofs sometimes raises doubts about their infallibility. The progressive nature of mathematics, with the development of ever more powerful proof techniques, has intensified debates and led to disagreements among mathematicians. While these developments have elevated mathematical standards, they have also highlighted uncertainties and the intricacies of effectively validating complex mathematical reasoning.
The impact of complexity on certainty
As mathematics becomes more complex and moves further from intuitive understanding, certainty becomes more elusive. The inherent difficulties in proving abstract mathematical entities and the potential for errors generate ongoing controversies. While mathematicians strive for absolute certainty through rigorous proofs, the complex nature of mathematics and diverse mathematical intuitions make it challenging to reach a consensus. Despite the pursuit of certainty, mathematical knowledge remains fallible and subject to ongoing discussions and debates.
The certainty of mathematics and its place in the supposed hierarchy of subjects (assumed to be above science which is itself above philosophy in turn). Some more remarks on Bayesianism and somehow ghosts and alien life.
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