Mathematicians Explore Multiverse Concept Through Petrification Theory

In a groundbreaking paper, mathematicians are harnessing the concept of petrification to shed new light on the nature of mathematical truth and the role of construction techniques in shaping our understanding of mathematical facts. By examining how sustained experiences with mathematical phenomena have shaped our understanding of set theory, particularly about the Continuum Hypothesis (CH), researchers are gaining insights into the multiverse. This concept suggests multiple mathematical universes or models exist.

The authors draw on Ludwig Wittgenstein’s later philosophical works, where he introduced the concept of petrification as a process of making something stable and natural. In mathematics, this idea has been applied to understand how mathematical practices develop from technical practices, such as counting, measuring, or drawing geometric shapes. By applying petrification to set theory, researchers are revealing how construction techniques have become a normative demand within the discipline, with many practitioners assuming that CH remains undecidable.

However, this assumption is not universally accepted, and disagreements persist among set theorists. The authors argue that these disagreements can be understood through the lens of petrification, highlighting the importance of sustained experiences with mathematical phenomena in shaping our understanding of mathematical practices. By examining how different set-theoretic models have been constructed over time, researchers are gaining new insights into the nature of mathematical truth and the role of construction techniques in shaping our understanding of mathematical facts.

This research has significant implications for our understanding of the multiverse, a concept that suggests multiple mathematical universes or models exist. By applying petrification to set theory, researchers reveal how different mathematical practices have developed over time, providing new perspectives on the nature of mathematical truth and the role of construction techniques in shaping our understanding of mathematical facts.

What is Petrification in Contemporary Set Theory?

Petrification, a concept introduced by the later Wittgenstein, refers to making something stable and natural, becoming a normative force for a particular discipline. In the context of set theory, petrification occurs when construction techniques for set-theoretic models become so ingrained that they are considered a normative demand within the community. This means that the continuous and successful practices involving the construction of various set-theoretic models now act as a shared norm among practitioners.

The concept of petrification is not new, but its application to contemporary set theory is a refinement of previous arguments presented by set theorist Joel David Hamkins. The authors argue that in set theory, construction techniques for set-theoretic models have petrified into the normative demand that the Continuum Hypothesis (CH) remain undecidable. This means that the continuous and successful practices involving the construction of various set-theoretic models now act as a shared norm among practitioners.

The authors also argue that this process of petrification has led to disagreements in set theory, particularly regarding the absolute undecidability of CH. They contend that these disagreements arise because different researchers have petrified their own construction techniques and models, which then become a normative force within their respective communities.

The Multiverse and the Later Wittgenstein

The concept of the multiverse, which refers to the idea that there are multiple universes or realities, has been explored in various fields, including physics and mathematics. In the context of set theory, the multiverse refers to the existence of multiple set-theoretic models, each with its own version of the Continuum Hypothesis.

The later Wittgenstein’s concept of petrification is relevant here because it highlights the importance of understanding how mathematical practices develop and become normative within a community. The authors argue that the multiverse in set theory can be seen as a manifestation of this process, where different researchers have constructed their own models and techniques, which then become a shared norm among practitioners.

The concept of petrification also has implications for our understanding of mathematical truth. If multiple set-theoretic models exist, each with its own version of the Continuum Hypothesis, what does this mean for our understanding of mathematical truth? The authors argue that this is a question that requires further exploration and discussion within the community.

Hinge Epistemology and Petrification

Hinge epistemology, a concept introduced by the later Wittgenstein, refers to the idea that certain beliefs or assumptions are so deeply ingrained in our thinking that they become a kind of “hinge” that holds together our entire system of thought. In the context of set theory, hinge epistemology is relevant because it highlights the importance of understanding how mathematical practices develop and become normative within a community.

The authors argue that petrification is a key concept in hinge epistemology, as it refers to the process by which certain construction techniques or models become so ingrained that they are considered a shared norm among practitioners. This means that the continuous and successful practices involving the construction of various set-theoretic models now act as a shared norm among practitioners.

The authors also argue that this process of petrification has led to disagreements in set theory, particularly regarding the absolute undecidability of CH. They contend that these disagreements arise because different researchers have petrified their own construction techniques and models, which then become a normative force within their respective communities.

Rudimentary Mathematics and Petrification

Rudimentary mathematics refers to the basic mathematical practices that are developed through experience and technical practice. In the context of set theory, rudimentary mathematics is relevant because it highlights the importance of understanding how mathematical practices develop and become normative within a community.

The authors argue that petrification is a key concept in rudimentary mathematics, as it refers to the process by which certain construction techniques or models become so ingrained that they are considered a shared norm among practitioners. This means that the continuous and successful practices involving the construction of various set-theoretic models now act as a shared norm among practitioners.

The authors also argue that this process of petrification has led to disagreements in set theory, particularly regarding the absolute undecidability of CH. They contend that these disagreements arise because different researchers have petrified their own construction techniques and models, becoming a normative force within their respective communities.

Conclusion

In conclusion, petrification is a key concept in contemporary set theory. It refers to the process by which certain construction techniques or models become so ingrained that they are considered a shared norm among practitioners. This process has led to disagreements in set theory, particularly regarding the absolute undecidability of CH.

The authors argue that petrification is relevant to set theory our understanding of mathematical truth and the development of mathematical practices. They contend that further exploration and discussion within the community are needed to fully understand the implications of petrification in contemporary set theory.