Related papers: Mathematical conceptualism
Since the early twentieth century, it has been understood that mathematical definitions and proofs can be represented in formal systems systems with precise grammars and rules of use. Building on such foundations, computational proof…
The testimony and practice of notable mathematicians indicate that there is an important phenomenological and epistemological difference between superficial and deep analogies in mathematics. In this paper, we offer a descriptive theory of…
We illustrate the concept of mathematical proof.
A model of knowledge representation is described in which propositional facts and the relationships among them can be supported by other facts. The set of knowledge which can be supported is called the set of cognitive units, each having…
In this article we present and describe a notion of "logical perfection". We extract the notion of "perfection" from the contemporary logical concept of categoricity. Categoricity (in power) has become in the past half century a main driver…
A significant amount of research has considered mathematical proofs, the students who learn them, and the instructors that teach them, from a variety of perspectives. This paper considers this topic from four main perspectives: students'…
Being mathematics a natural language to Mankind and to physics, it must be constantly adapted to our necessities and our natural perception. Then, mathematical concepts are not absolute to reality. Although mathematical theories are…
Discussion of the necessity to use the constructive mathematics as the formalism of quantum theory for systems with many particles.
In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a…
Modern mathematics is known for its rigorous proofs and tight analysis. Math is the paradigm of objectivity for most. We identify the source of that objectivity as our knowledge of the physical world given through our senses. We show in…
Mathematical software systems are becoming more and more important in pure and applied mathematics in order to deal with the complexity and scalability issues inherent in mathematics. In the last decades we have seen a cambric explosion of…
We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, on our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a…
The chapter advances a reformulation of the classical problem of the nature of mathematical objects (if any), here called "Plato's problem," in line with the program of a philosophy of mathematical practice. It then provides a sketch of a…
This paper proposes a specific conceptualization of intelligence as computation. This conceptualization is intended to provide a unified view for all disciplines of intelligence research. Already, it unifies several conceptualizations…
The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…
We define a strongly normalising proof-net calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with…
This chapter presents a brief review of complexity research in mathematics education. We argue how research on complexity, as it pertains to mathematics education, can be viewed as an epistemological discourse, an historical discourse, a…
Computability logic is a formal theory of computability. The earlier article "Introduction to cirquent calculus and abstract resource semantics" by Japaridze proved soundness and completeness for the basic fragment CL5 of computability…
It has been shown that a functional interpretation of proofs in mathematical analysis can be given by the product of selection functions, a mode of recursion that has an intuitive reading in terms of the computation of optimal strategies in…
Can AI solve all math? What do we actually mean by doing mathematics? How do we communicate mathematics? What is mathematics beyond problem solving? This essay is my attempt to answer these questions.