## Podcast Summary

## Mathematics Complexity: The late 1800s and early 1900s revealed that mathematics is more complex than initially thought with the discovery of non-Euclidean geometry, different kinds of infinity, and the incompleteness theorem. Attempts to axiomatize all of mathematics were unsuccessful due to unprovable statements, leading to new research areas and deep questions in the philosophy of mathematics.

The development of mathematics in the late 1800s and early 1900s challenged the notion that mathematics could be perfectly precise and clear with a definitive set of axioms. Discoveries of non-Euclidean geometry and different kinds of infinity, as well as the incompleteness theorem, showed that mathematics was more complex than initially thought. Mathematician David Hilbert attempted to address these challenges by proposing a program to axiomatize all of mathematics, but his ambition could not be realized due to the existence of unprovable statements. This opened up new research areas and deep questions in the philosophy of mathematics, which are still being explored today, much like in the philosophy of physics where we continue to grapple with the implications of quantum mechanics.

## Philosophy of Mathematics: Platonism in set theory allows for the existence of multiple, fully real concepts of sets with different truths, challenging the notion of a single determinant answer to mathematical questions

The nature of mathematical reality and the search for its foundations can be seen through different lenses, leading to debates between pluralism and monism. In set theory, Platonism no longer implies a unique mathematical universe, allowing for the coexistence of multiple, fully real concepts of sets with different truths. This development in philosophy of mathematics challenges the notion that there is only one determinant answer to mathematical questions. The preference for weak or strong foundations also reflects this tension, with weak foundations having a pluralist aspect and strong foundations a monist perspective. Ultimately, the debate between these perspectives continues to shape the ongoing exploration of mathematical reality.

## Mathematical Truths and Models: The pluralist perspective challenges the monist view of a single mathematical reality by suggesting that mathematical truths may not have a unique interpretation and that the independence phenomenon in set theory demonstrates this complexity

The nature of mathematical truths and the existence of multiple compatible models challenge the monist view of a single, unique mathematical reality. The pluralist perspective suggests that there might not be a unique standard interpretation of mathematical ideas, and the independence phenomenon in set theory, which shows that many fundamental questions are not settled by the axioms, further supports this view. The continuum hypothesis, a long-standing question in mathematics about the size of infinite sets, is a prime example of this phenomenon. It's independent of the standard axioms, and we can construct models where it holds true or false. Instead of viewing this as a limitation, some argue that it's evidence for pluralism and a deeper understanding of the richness and complexity of mathematical concepts.

## Continuum Hypothesis: The continuum hypothesis is an open question in mathematics and cannot be definitively answered through the addition of new self-evident axioms. The multiverse perspective suggests that there are multiple valid set-theoretic universes, each with its own version of the continuum hypothesis.

The continuum hypothesis (CH) in mathematics, which concerns the size of the set of real numbers between 0 and 1, is still an open question and cannot be definitively answered through the addition of new self-evident axioms. Instead, the multiverse perspective suggests that there are multiple valid set-theoretic universes, each with its own version of the continuum hypothesis. This pluralist view challenges the idea that there is a unique, absolute truth about the continuum hypothesis. The debate between adding axioms and considering different models of existing axioms is inherent in mathematics, and every theory will have non-standard models. The concept of a unique intended model of arithmetic or sets is not as clear-cut as it may seem. The categorical theorem in arithmetic, which suggests there is only one model of arithmetic up to isomorphism, does not fully address the concern of understanding the absolute meaning of finite numbers and sets.

## Gödel's incompleteness theorems: Gödel's incompleteness theorems prove that there are true statements in arithmetic that cannot be proven within any computably accessible theory, and the density of such unprovable statements is between 0 and 1

The standard model of arithmetic can have different truths in different models of set theory, and Gödel's incompleteness theorems show that there is no computably accessible theory of arithmetic that can prove all true statements. This means that some true statements in arithmetic may be unprovable, and the density of independent, unprovable statements is strictly between 0 and 1. These findings challenge the idea that our understanding of the standard model of arithmetic can provide definitive answers or monism. The halting problem, which is undecidable, further illustrates the limitations of any theory that claims to know all the answers.

## Computational density convergence: The question of whether the density of certain computational systems converges is complex and depends on the specific computational model used, with some models allowing for high probability decisions on halting problems as programs grow larger, while others may not hold true for other models or have philosophical implications in mathematics.

The question of whether the density of certain computational systems converges is not straightforward and depends heavily on the specific formalism or computational model being used. For instance, the halting problem, which asks whether a given program will eventually halt or run infinitely, is undecidable in general but can be decided with a high probability for certain sets of programs as the size of the program grows larger. However, this result is dependent on the specific computational model used and may not hold for other models. Additionally, the consistency of axiomatic systems, such as those in number theory, can be both proven and disproven, leading to philosophical questions about the nature of truth and the intended model in mathematics.

## Infinity and Consistency Strength: Gödel's incompleteness theorem highlights the need for commitment to the consistency of mathematical theories, while large cardinal axioms increase consistency strength but cannot settle all independent statements in set theory. The debate between potentialism and actualism continues to influence our understanding of infinity.

The study of mathematical theories, particularly those related to sets and infinity, involves a hierarchy of consistency strength. Gödel's incompleteness theorem suggests that any given theory is inadequate and that we should commit to the consistency of that theory and the theories that come before it. The large cardinal axioms in set theory, discovered by Cohen and others, are axioms that express profound infinite combinatorial infinities and have the predicted feature of increasing consistency strength. However, there are also independent set theoretic statements, such as the continuum hypothesis, that cannot be settled by the known large cardinal axioms. Additionally, there is a historical debate between potentialism and actualism regarding the nature of infinity, with potentialists believing that we can only have potential infinities and actualists believing that we can have actual infinities. This debate goes back to ancient philosophers and has influenced the way mathematicians view infinity.

## Potentialism in Math and Philosophy: Potentialism is a philosophical perspective questioning the completeness of the universe, specifically in relation to mathematical objects and reality.

The concept of a Googleplex, an enormous number with a vast number of zeros, challenges our ability to comprehend and describe numbers beyond a certain point. While potentialism is a philosophical perspective suggesting the universe of mathematical objects might be unfinished, the concept of infinity is not its primary focus. Instead, it raises questions about the completeness of the universe. The ongoing work in potential set theory explores different conceptions of potentialism, which in turn refines our philosophical understanding of mathematical reality. The interaction between philosophy and mathematics is essential, as each discipline informs the other, leading to new discoveries and insights. The next big thing in math and philosophy could be the continued exploration of potentialism and its implications for our understanding of mathematical objects and reality as a whole.

## Mathematics and Philosophy: The collaboration between mathematical skill and philosophical thinking in mathematics and physics leads to unique opportunities for advancement.

The fields of mathematics and philosophy of mathematics are increasingly interconnected, with philosophers becoming more mathematically sophisticated and mathematicians showing greater interest in philosophical questions. This collaboration between technical mathematical skill and philosophical thinking is crucial for making progress in the field, although it can be challenging to find individuals with both sets of expertise. This trend is likely similar in the field of physics. While it may require more effort to convince people of its importance, the combination of deep mathematical knowledge and philosophical inquiry offers unique opportunities for advancement. Overall, the increasing intersection of mathematics and its philosophy signifies a rich and exciting area of exploration for those who are passionate about both fields.