Artificial Mathematical Intelligence: Cognitive, (Meta)mathematical, Physical and Philosophical Foundations

Cognitive metamathematics is a new inter- and intra-disciplinary meta-research discipline that aims to achieve a global instance of concrete Artificial Mathematical Intelligence (AMI). It is supported by the computational theory of mind and the meta-fact that genuine mathematical proofs are, in principle, algorithmically verifiable. The book presents the foundations of cognitive metamathematics, including the construction of formal refinements of seminal (meta-)mathematical notions and facts, a global taxonomy of cognitive abilities, and computational tools for calculating formal conceptual blends. It also shows classic and new results concerning the co-generation of old and new mathematical concepts and the key parts of several standard proofs in Hilbert-style deductive systems.

Format: Paperback / softback
Length: 259 pages
Publication date: 25 October 2021
Publisher: Springer Nature Switzerland AG

This volume delves into the theoretical foundations of a novel inter- and intra-disciplinary meta-research discipline known as cognitive metamathematics, with the ultimate objective of achieving a global instance of concrete Artificial Mathematical Intelligence (AMI). In simpler terms, AMI seeks to construct an (ideal) global artificial agent capable of (co-)solving interactively formal problems with a conceptual mathematical description in a human-style manner. It begins by presenting formal guidelines from philosophical, logical, meta-mathematical, cognitive, and computational perspectives, supporting the formal existence of such a global AMI framework. It examines how much of current mathematics can be entirely generated by an interactive computer program and assesses our proximity to constructing a machine capable of simulating the way a modern working mathematician handles solvable mathematical conjectures from a conceptual perspective.

The thesis that it is possible to meta-model the intellectual job of a working mathematician is heuristically supported by the computational theory of mind, which posits that the mind is, in fact, a computational system, and by the meta-fact that genuine mathematical proofs are, in principle, algorithmically verifiable, at least theoretically. The introduction to this volume provides the grounding multifaceted principles of cognitive metamathematics, while also offering an overview of some of the most outstanding results in this direction, with a primary focus on human-style proofs rather than merely formal verification.

The first part of the book presents a new cognitive foundations of mathematics program that focuses on constructing formal refinements of seminal (meta-)mathematical concepts. It explores the development of formal systems that can capture the essential aspects of mathematics, including its logical structure, axiomatic systems, and proof techniques. By formalizing these foundational elements, cognitive metamathematics aims to provide a more precise and rigorous understanding of mathematics, enabling it to be studied and analyzed from a cognitive perspective.

The second part of the book explores the application of cognitive metamathematics to specific areas of mathematics, such as algebra, number theory, and geometry. It demonstrates how cognitive metamathematics can be used to develop new insights and methodologies in these fields, as well as to address challenging problems that have been difficult to solve using traditional approaches. For example, cognitive metamathematics can be used to formalize the concept of a mathematical proof, which is often considered a subjective and intuitive process. By providing a formal definition of proof, cognitive metamathematics can help to make the process of mathematical reasoning approach more systematic and rigorous, leading to more accurate and reliable results.

In conclusion, this volume provides a comprehensive introduction to cognitive metamathematics, a novel interdisciplinary field that seeks to bridge the gap between mathematics and the cognitive sciences. By exploring the theoretical foundations of cognitive metamathematics, its application to specific areas of mathematics, and its potential implications for the future of mathematics and the cognitive sciences, this volume offers a valuable resource for researchers, scholars, and students interested in advancing the understanding of mathematics and the human mind.

Weight: 438g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030502751
Edition number: 1st ed. 2020