# “Those who can imagine anything, can create the impossible”

# - Alan Turing -

## GENERAL OVERVIEW OF

## ARTIFICIAL MATHEMATICAL INTELLIGENCE

The study of the mind is one of the most fascinating and multifaceted concerns of mankind. So, to obtain effective and useful models explaining and describing its essential features a fully interdisciplinary approach is needed. Besides, mathematics is, among many others, the language in which the laws of nature seem to be written with maximum precision. Therefore, a strong and mature formation in pure and applied mathematics represents a huge advantage for starting this enhancing scientific journey through global laws of the mind.

On the other hand, a huge number of fundamental challenges that our societies face every day require ultimately precise solutions to the mathematical models emerging in the corresponding formalization processes. Now, in most of the cases, such mathematical problems try to be solved solely by human researchers (e.g. mathematicians, theoretical physicists), which limits strongly the number of the potential problems to be solved and tends to increase enormously the amount of time required to solve these problems.

I (as the creator and leader of this project) am a multidisciplinary researcher in cognitive sciences, human and artificial intelligence, and pure and applied mathematics. One of my biggest passions is being able to model formally the way in which humans create mathematical ideas, namely, being able to ‘decode’ the processes that happen in our minds precisely when we are producing mathematical output.

Since a couple of years, I am completely involved in solving one of the most important questions within the common intersection between Artificial Intelligence and Pure (and Applied) Mathematics: The Meta-challenge that I called Artificial Mathematical Intelligence (AMI), i.e. the problem of creating a robot (i.e. computer program) being able to simulate the way a human being receives, process and subsequently solves a mathematical problem from a conceptual (as well as numerical) point of view.

A global solution to this meta-challenge would have very strong implications in all the areas conceptually influenced by modern mathematics like medicine, engineering, theoretical and applied physics, computer science, economics, (parts of) biology and chemistry, cognitive sciences and (mathematical) psychology, among many others. A device of this kind can help enormously for solving a lot of (new kinds of) problems emerging from the huge amount of information and the accelerating globalization that our world is facing today. Moreover, this would allow us to use all the potential that pure sciences have, for enlightening and subsequently for solving concrete problems occurring in our society. In fact, in a lot of cases, the research of an interdisciplinary problem is essentially reduced to the solution of the corresponding (underlying) mathematical problem which emerges at the formalization’s stage of the research.

## FORMAL MOTIVATION AND DESCRIPTION OF THE PROJECT

There is a quite natural and practical question within the foundations and origins of mathematics that needs a deeper answer: how much of current mathematics (i.e. mathematics described in contemporary (mathematical and closely related) journals) can be completely generated by a computer program?

In other words, how near are we for constructing (programming) a machine which can simulate the way a modern researcher usually faces a solvable mathematical conjecture, works some time on it, and finally finds a formal solution for it?

Here it is important to clarify that the main purpose of the former question is to ‘meta-model’ and to ‘meta-simulate’ how a human being (abstractly) ‘handles’ regarding the specific intellectual activity of receiving a concrete conjecture (which typically can be solved within a standard mathematical framework), working on it and finally giving a clear answer, i.e., writing a (formalizable) solution to the conjecture in the form of either a proof or a counterexample (or sometimes a proof of its ‘independence’ from the corresponding axiomatic system).

Besides, because the former questions involve implicitly the human’s mind, one should take inspiration from the current most relevant cognitive theories concerning mathematical reasoning and closely related matters. The most successful theories that we currently have for understanding how our mind works are theories with a formal computational conceptual basis like the computational theory of mind. So, this fact can be seen as a form of ‘heuristic’ support for the thesis that it is possible to meta-model, (formally and computationally) the intellectual (mathematical) job of a human being.

A second theoretical support is the fact that modern mathematics is essentially founded and (at some level) 'conceptually delimited' on (Zermelo-Fraenkel) Set Theory with Choice (ZFC), proof, recursion, and model theory.

This means that the solution of a solvable conjecture should be precisely described as a formal (logical) consequence of the axioms of ZFC, using a finite (or recursively generated) number of inference rules and initial premises.

In other words, when a person finally finds a correct solution of a conjecture, then the result of his/her research can be simulated (just formally) simply a kind of computation of a theoretically-feasible computer program, which starts to run all the possible proofs of provable theorems of ZFC, starting from a finite sub-collection of axioms and following precise (logical) mechanical deduction rules. Here it is important to mention that at the beginning one can focus essentially on solvable conjectures, i.e., on problems having an explicit formal proof or counterexample within the ZFC framework. These problems constitute mathematics being studied by at most of the mathematicians today, and these are the ones producing most of the concrete applications. Moreover, after finding initial solutions for the former collections of questions, one can have initial evidence in order to meta-model also the way in which people generate and prove ‘undecidability’ of (some) conjectures.

So, Artificial Mathematical Intelligence deals with the construction (implementation) of a computer program being able to generate a human-style solution of essentially every human solvable mathematical conjecture in less time than an average professional mathematician; and subsequently to generate also human-style independence proofs for undecidable conjectures. Now, although concerns about undecidable conjectures are clearly important from a purely theoretical perspective, within our AMI meta-project the decidable sentences will have an initial central focus of attention.

So far, we have a considerable amount of significant evidence in favour of AMI. For instance, the co-discovery with the help of programs of the notions of refactorable Numbers, Multiplicative Rings (sometimes known as Containment-Division Rings), the (quasi-)integer and (quasi-)complex numbers and prime ideals (over Dedekind domains). The artificial generation of seminal notions of Fields and Galois Theory like fields, fields extensions, group of automorphisms of a field and meta-Galois Group of a field extension. Furthermore, we have developed and improved software being able to simulate analogy making and conceptual blending in initial mathematical theories. Specifically, we have co-discovered with the help of several computer programs partial axiomatizations of the integers, commutative rings with unity with compatible divisibility relations, and Goldbach rings.

For more specific information regarding the above results, the reader can consult the following webpage: https://dagomez1982.wixsite.com/misitio

## STARTING A NEW WAY OF DOING SCIENCE

The consequences of a robust and effective AMI-software goes far beyond mathematical research. Galileo's quote "the book (of Nature) is written in mathematical language" is still alive. In fact, the power and usefulness of mathematical models have proved to be a very successful tool in areas like economics, statistics, computer sciences, biology, chemistry, sociology, psychology, theoretical and experimental physics, engineering, cognitive sciences, global planning, cosmology, earth sciences, among many others. So, the AMI-system would save a lot of resources in terms of time and energy for lots of researchers and professionals during their work. Thus, they can concentrate mainly on the formal modeling part of their research. On the other hand, their solving skills will increase tremendously, because with AMI they would be able of come up with more sophisticated mathematical frameworks for their particular fields of research without having to complete a formal mathematical training before, which could require sometime years of systematic effort.

Furthermore, the same thing can be said not only for scientists but for professionals working in a huge variety of companies, which encounter each day more and more sophisticated challenges needing clear, specialized and mathematically-grounded solutions.

Lastly, a lot of (high-)school students can obtain great benefit of the AMI-software during their intellectual development, since they could interact with it on a regular basis and with their own speed of comprehension to increase their math-skills in a personal fashion, and without the monotonic way in which a lot of them perceive the process of learning mathematics at school.

## SPECIFIC SUB-PROJECTS RELATED WITH AMI

To achieve AMI it is necessary to fulfil successfully the following sub-projects: the development of a more suitable syntactic-semantic formal framework for the foundations of mathematics, the identification and formalization of a global taxonomy for the primary and secondary cognitive mechanisms used in mathematical creation/invention, the development (almost from scratch) of a human-style (cognitively inspired) software being able to model human mathematical creation/invention/interaction in a ‘user-friendly’ way, a significant amount of study-cases covering the most important mathematical research’s fields and a suitable global computationally-feasible meta-formalism for structuring and implementing AMI, among others.