Overview
- Sets the multidisciplinary foundations of Artificial Mathematical Intelligence and Cognitive Metamathematics
- Provides both classic and new results concerning the (artificial) co-generation of old and new mathematical concepts filling a conceptual gap in the mechanization of mathematics
- Describes computational tools allowing for calculation of formal conceptual blends
- Presents extensions and new cognitive characterizations of the Church-Turing thesis
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Table of contents (12 chapters)
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New Cognitive Foundations for Mathematics
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Global Taxonomy of the Fundamental Cognitive (Metamathematical) Mechanisms Used in Mathematical Research
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Towards a Universal Meta-Modeling of Mathematical Creation/Invention: Meta-Analysis of Several Classic and Modern Proofs and Concepts in Pure Mathematics
Keywords
About this book
This volume discusses the theoretical foundations of a new inter- and intra-disciplinary meta-research discipline, which can be succinctly called cognitive metamathematics, with the ultimate goal of achieving a global instance of concrete Artificial Mathematical Intelligence (AMI). In other words, AMI looks for the construction of an (ideal) global artificial agent being able to (co-)solve interactively formal problems with a conceptual mathematical description in a human-style way. It first gives formal guidelines from the philosophical, logical, meta-mathematical, cognitive, and computational points of view supporting the formal existence of such a global AMI framework, examining how much of current mathematics can be completely generated by an interactive computer program and how close we are to constructing a machine that would be able to simulate the way a modern working mathematician handles solvable mathematical conjectures from a conceptual point of view.
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 then the grounding multifaceted principles of cognitive metamathematics, and, at the same time gives an overview of some of the most outstanding results in this direction, keeping in mind that the main focus is human-style proofs, and not simply formal verification.
The first part of the book presents the new cognitive foundations of mathematics’ program dealing with the construction of formal refinements of seminal (meta-)mathematical notions and facts. The second develops positions and formalizations of a global taxonomy of classic and new cognitive abilities, and computational tools allowing for calculation of formal conceptual blends are described. In particular, a new cognitive characterization of the Church-Turing Thesis is presented. In the last part, classic and new results concerning the co-generation of a vast amount of old and new mathematical concepts and the key parts of several standard proofs in Hilbert-style deductive systems are shown as well, filling explicitly a well-known gap in the mechanization of mathematics concerning artificial conceptual generation.
Authors and Affiliations
Bibliographic Information
Book Title: Artificial Mathematical Intelligence
Book Subtitle: Cognitive, (Meta)mathematical, Physical and Philosophical Foundations
Authors: Danny A. J. Gómez Ramírez
DOI: https://doi.org/10.1007/978-3-030-50273-7
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-50272-0Published: 24 October 2020
Softcover ISBN: 978-3-030-50275-1Published: 25 October 2021
eBook ISBN: 978-3-030-50273-7Published: 23 October 2020
Edition Number: 1
Number of Pages: XXI, 259
Number of Illustrations: 3 b/w illustrations, 8 illustrations in colour
Topics: Mathematical Applications in Computer Science, Mathematical Models of Cognitive Processes and Neural Networks