Related papers: Singularity: a Seventh Memo?
In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear…
The nouns of our language refer to either concrete entities (like a table) or abstract concepts (like justice or love), and cognitive psychology has established that concreteness influences how words are processed. Accordingly,…
As an expansion of complex numbers, the quaternions show close relations to numerous physically fundamental concepts. In spite of that, the didactic potential provided by quaternion interrelationships in formulating physical laws are hardly…
These five lectures on undecidability were given to students with a good level in mathematics but with no special knowledge on logic. The first conference presents the formalization of mathematics with a short historical survey, the…
In these notes of lectures at the 2004 Summer School of Mathematical Physics in Ravello, Italy, the author develops an approach to calculus in which more efficient choices of limits are taken at key points of the development. For example,…
Driven by the interest of reasoning about probabilistic programming languages, we set out to study a notion of unicity of normal forms for them. To provide a tractable proof method for it, we define a property of distribution confluence…
This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…
A number of elite thinkers in Europe during the 16th and 17th centuries pursued an agenda which historian Paolo Rossi calls the "quest for a universal language," a quest which was deeply interwoven with the emergence of the scientific…
My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us…
Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…
Even if students can make the blend, interpret physics correctly in mathematical symbology and graphs, they still need to be able to apply that knowledge in productive and coherent ways. As instructors, we can show our solutions to complex…
The purpose of this essay is to bring out the unique role of Mathematics in providing a base to the diverse sciences which conform to its rigid structure. Of these the physical and economic sciences are so intimately linked with…
This book invites readers to see mathematics not just as formulas and rules, but as the deepest expression of human thought. It begins by exploring the timeless idea of mathematics as a universal language, contrasting its precision with the…
Mathematics is changing. Computers are verifying proofs, checking calculations, and exploring complex structures that would overwhelm human effort. Yet curiosity-driven research is where tomorrow's breakthroughs are quietly prepared. In…
"Clarithmetic" is a generic name for formal number theories similar to Peano arithmetic, but based on computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html) instead of the more traditional classical or intuitionistic logics.…
Mathematical conception of infinite quantities forms a cornerstone of many disciplines of modern mathematics --- from differential calculus to set theory. In fact, it could be argued that the most significant revolutions in mathematics in…
This note presents reflections drawn from my recent experiences in teaching a course on mathematics and sustainability, with a particular emphasis on raising awareness of the topic and its broader implications. The lectures were structured…
The paper is a survey of notions and results related to classical and new generalizations of the notion of a periodic sequence. The topics related to almost periodicity in combinatorics on words, symbolic dynamics, expressibility in logical…
We consider the problem of deciding the satisfiability of quantifier-free formulas in the theory of finite sets with cardinality constraints. Sets are a common high-level data structure used in programming; thus, such a theory is useful for…
One presents many Concatenated and Operation Sequences, P-Q Relationships, Digital Sequences, Magic Squares, Prime Conjectures, k-Divisibility and Strong Divisibility Sequences, Geometric Conjectures, Proposed problems.