The New Cognitive Foundations for Mathematics is an interdisciplinary program for finding more

cognitively-inspired and computationally-feasible refinements of the most basic logic and mathematical

(syntactic and semantic) structures used for grounding not only mathematics as a whole, but also for

grounding specific mathematical sub-theories. Examples of these structures are the notions of membership, set; natural, integer, rational, real and complex numbers; (in-)finite, continuity, differentiability, formal model, implication, conjunction, disjunction, negation and equivalence, among others.

One of the central principles for developing these new foundations emerges from the observation

that, independently of the degree of geniality that some mathematical theories can possess, they are

always a product of the finite and bounded human cognition, which at the same time is immersed in our

huge, but finite, universe.

Moreover, the foundational relation between the physical realm and the nature of mathematical

structures and proofs is deeper than one can perceive at first sight. In fact, the history of science has seen a lot of inspirational breakthroughs in mathematics coming originally from research done in theoretical and experimental physics.

Now, how strong can be the influence that formal physical (and computational) frameworks and

principles can have for explaining and grounding mathematics? This classic question starts to gain

importance in current research because the most essential results in cognitive sciences are supporting

(more and more) the fact that a lot of aspects of human mathematical creation/invention are susceptible

to be modeled computationally.

Furthermore, another cornerstone for these new cognitive foundations is the carefully development

of new syntactic representations for mathematical structures and proofs which contain more semantic

content of the corresponding structures and proofs, and fulfilling also minimal properties of uniqueness

and contextual-freedom for being suitable to be integrated into a bigger artificial framework simulating

human-style deductive mechanisms.

An outstanding mathematical notion which should be revised is the one of (natural) number, and,

in general, the seminal set of ‘natural numbers’ as a whole. This is one of the most basic structures

in mathematics, and at the same time, one of the most mysterious ones. Therefore, it would be very

useful to be able to classify the fundamental formal features and intuitions about the (so called) ‘natural

numbers’ into physically supported parts, and meta-physical (or cognitive) ones. So, we could construct

a formal numerical system acting as a refinement of the natural numbers, but with stronger physical

and computational grounds and fulfilling the most essential properties that their ‘natural’ counterparts

possess.

More generally, there are a lot of (classic) mathematical notions (and deductive methods) which

are used (among others) in theoretical physics for modeling several physical phenomena, but whose

semantic grounds seem to be more mental (and meta-physical) than physical and computational. For

instance, examples of these notions and methods are the concept of infinite set; large and inaccessible

cardinal; power set; infinitesimal, real and complex number; formal negation and proof by the sake of

contradiction. So, as before we need to find refinements of the former notions and methods not only more

physically-accessible and computationally-feasible, but also possessing minimal structural requirements

for modeling (at least) as much (physical and) mathematical phenomena as their classic counterparts.

So, due to the interdisciplinary nature of this program, the conformation of an ‘colorful’ group

of philosophers, physicists, computer scientists and mathematicians (among others) would be highly

useful.

Finally, due to the philosophical, computational and logic nature of this meta-program, the new

cognitive foundations of mathematics are one of the seminal components for achieving Artificial Mathematical Intelligence in a near future.

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