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Artificial Mathematical Intelligence

Cognitive, (Meta)mathematical, Physical and Philosophical Foundations

Authors: Gómez Ramírez, Danny A. J.

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  • 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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  • ISBN 978-3-030-50273-7
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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.

Table of contents (12 chapters)

Table of contents (12 chapters)
  • Global Introduction to the Artificial Mathematical Intelligence General Program

    Pages 1-17

    Gómez Ramírez, Danny A. J.

  • Some Basic Technical (Meta-)Mathematical Preliminaries for Cognitive Metamathematics

    Pages 19-38

    Gómez Ramírez, Danny A. J.

  • General Considerations for the New Cognitive Foundations’ Program

    Pages 41-52

    Gómez Ramírez, Danny A. J.

  • Towards the (Cognitive) Reality of Mathematics and the Mathematics of (Cognitive) Reality

    Pages 53-66

    Gómez Ramírez, Danny A. J.

  • The Physical Numbers

    Pages 67-89

    Gómez Ramírez, Danny A. J.

Buy this book

eBook $89.00
price for USA in USD
  • ISBN 978-3-030-50273-7
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $119.99
price for USA in USD
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Bibliographic Information

Bibliographic Information
Book Title
Artificial Mathematical Intelligence
Book Subtitle
Cognitive, (Meta)mathematical, Physical and Philosophical Foundations
Authors
Copyright
2020
Publisher
Springer International Publishing
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-030-50273-7
DOI
10.1007/978-3-030-50273-7
Hardcover ISBN
978-3-030-50272-0
Edition Number
1
Number of Pages
XXI, 259
Number of Illustrations
3 b/w illustrations, 8 illustrations in colour
Topics