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Development Processes

Paul Gorbow

TL;DR

This paper addresses the ontology of infinitary mathematics by proposing development processes: teleological evolutions of a single dynamic entity across directed states, described within a modal first-order framework. It develops a formal system of development models that render dynamic entities coherent in type, persist through change, and converge to meaningful mathematical objects via teleology. The approach yields concrete dynamic interpretations across key domains, including reals as nets of dynamic rationals, forcing extensions as dynamic membership relations, and non-standard numbers via ultrapowers, while also modeling reflection in set theory and truth revision (FS) through revision semantics. Collectively, the work offers a unified, dynamic ontology for infinitary mathematics, bridging potentialist ideas with tangible constructions and clarifying how rich mathematical phenomena can be understood as stateful developmental processes.

Abstract

Throughout mathematics there are constructions where an object is obtained as a limit of an infinite sequence. Typically, the objects in the sequence improve as the sequence progresses, and the ideal is reached at the limit. I introduce a view that understands this as a development process by which a dynamic mathematical object develops teleologically. In particular, this paper elaborates and clarifies the intuition that such constructions operate on a single dynamic object that maintains its identity throughout the process, and that each step consists in a transformation of this dynamic object, rather than in a genesis of an entirely new static object. This view is supported by a general philosophical discussion, and by a formal modal first-order framework of development processes. In order to exhibit the ubiquity of such processes in mathematics, and showcase the advantages of this view, the framework is applied to wide range of examples: The set of real numbers, forcing extensions of models of set theory, non-standard numbers of arithmetic, the reflection theorem schema of set theory, and the revision semantics of truth. Thus, the view proposed promises to yield a unified dynamic ontology for infinitary mathematics.

Development Processes

TL;DR

This paper addresses the ontology of infinitary mathematics by proposing development processes: teleological evolutions of a single dynamic entity across directed states, described within a modal first-order framework. It develops a formal system of development models that render dynamic entities coherent in type, persist through change, and converge to meaningful mathematical objects via teleology. The approach yields concrete dynamic interpretations across key domains, including reals as nets of dynamic rationals, forcing extensions as dynamic membership relations, and non-standard numbers via ultrapowers, while also modeling reflection in set theory and truth revision (FS) through revision semantics. Collectively, the work offers a unified, dynamic ontology for infinitary mathematics, bridging potentialist ideas with tangible constructions and clarifying how rich mathematical phenomena can be understood as stateful developmental processes.

Abstract

Throughout mathematics there are constructions where an object is obtained as a limit of an infinite sequence. Typically, the objects in the sequence improve as the sequence progresses, and the ideal is reached at the limit. I introduce a view that understands this as a development process by which a dynamic mathematical object develops teleologically. In particular, this paper elaborates and clarifies the intuition that such constructions operate on a single dynamic object that maintains its identity throughout the process, and that each step consists in a transformation of this dynamic object, rather than in a genesis of an entirely new static object. This view is supported by a general philosophical discussion, and by a formal modal first-order framework of development processes. In order to exhibit the ubiquity of such processes in mathematics, and showcase the advantages of this view, the framework is applied to wide range of examples: The set of real numbers, forcing extensions of models of set theory, non-standard numbers of arithmetic, the reflection theorem schema of set theory, and the revision semantics of truth. Thus, the view proposed promises to yield a unified dynamic ontology for infinitary mathematics.

Paper Structure

This paper contains 11 sections, 13 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Development processes and dynamic entities
  3. Formalizing development processes
  4. Background
  5. Teleological development in general
  6. Reals as dynamic rationals
  7. Set-theoretic forcing extensions as dynamic $\in$-structures
  8. Realizers of types and non-standard numbers as dynamic
  9. A development process of reflection in set theory
  10. A development process of truth revision
  11. Conclusion

Key Result

Proposition 2

Let $\mathcal{M}$ be a development model of $\mathcal{L}$. If $\vec{a} \in \bigcup \mathcal{M}$, then $\mathrm{states}^\mathcal{M}_{\vec{a}}$ is a non-empty up-set with respect to $\leq^\mathcal{M}$.

Figures (1)

  • Figure 10.1: Axioms and rules of $\mathsf{FS}$

Theorems & Definitions (24)

  • Definition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5: Syntactic Mirroring Theorem
  • Theorem 6: Semantic Mirroring Theorem
  • proof
  • Corollary 7
  • ...and 14 more