Related papers: Teaching mathematics with a different philosophy. …
Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of…
Practicing mathematicians often assume that mathematical claims, when they are true, have good reasons to be true. Such a state of affairs is "unreasonable", in Wigner's sense, because basic results in computational complexity suggest that…
A description of physical reality in which wholeness is the foundation is discussed along with the motivation for such an attempt. As a possible mathematical framework within which a physical theory based on wholeness may be expressed,…
Some notes about quantum physics, an interpretation if one wishes, are put forward, insisting on `closely following the mathematics/formalism, the `nuts and bolts of what quantum physics says'. These, basically well-known, issues seem to…
Good problems grab us. They invite us to find patterns, make conjectures, and prove-or perhaps disprove-a conjecture. When I first taught, I saw my work as tantalizing students with structures just beyond their reach, so that I could elicit…
Many mathematicians find mathematics aesthetically beautiful and even comparable to art forms such as music or painting. On the other hand, every year a great number of school students leave mathematics with total disillusionment and…
This article seeks to encourage a mathematical dialog regarding a possible solution to Beals Conjecture. It breaks down one of the worlds most difficult math problems into laymans terms and encourages people to question some of the most…
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…
Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as…
In order to work with mathematical content in computer systems, it is necessary to represent it in formal languages. Ideally, these are supported by tools that verify the correctness of the content, allow computing with it, and produce…
It is nowadays common to consider that proof must be part of the learning of mathematics from Kindergarten to University1. As it is easy to observe, looking back to the history of mathematical curricula, this has not always been the case…
Recent developments in computer programming and in mathematics suggest that there is a strong case for a new way of introducing programming to enhance the learning of school mathematics. The article describes a collaboration of mathematics…
The usual $\epsilon,\delta$-definition of the limit of a function (whether presented at a rigorous or an intuitive level) requires a "candidate $L$" for the limit value. Thus, we have to start our first calculus course with "guessing"…
Most work on computational complexity is concerned with time. However this course will try to show that program-size complexity, which measures algorithmic information, is of much greater philosophical significance. I'll discuss how one can…
Plausible reasoning concerns situations whose inherent lack of precision is not quantified; that is, there are no degrees or levels of precision, and hence no use of numbers like probabilities. A hopefully comprehensive set of principles…
Cathoristic logic is a multi-modal logic where negation is replaced by a novel operator allowing the expression of incompatible sentences. We present the syntax and semantics of the logic including complete proof rules, and establish a…
If there is a "platonic world" M of mathematical facts, what does M contain precisely? I observe that if M is too large, it is uninteresting, because the value is in the selection, not in the totality; if it is smaller and interesting, it…
The mathematisation of the socio-economic sphere, where mathematics actively constructs social reality, presents a challenge for studies on ethics in mathematics and its education. While existing scholarship on ethics in mathematics offers…
Mathematical research is often motivated by the desire to reach a beautiful result or to prove it in an elegant way. Mathematician's work is thus strongly influenced by his aesthetic judgments. However, the criteria these judgments are…
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…