New Cognitive Foundations for Mathematics
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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