Related papers: Mathematics discovered, invented, and inherited
Classical countably additive real-valued probabilities come at a philosophical cost: in many infinite situations, they assign the same probability value -- namely, zero -- to cases that are impossible as well as to cases that are possible.…
Recent developments in the mathematical foundations of quantum mechanics have brought the theory closer to that of classical probability and statistics. On the other hand, the unique character of quantum physics sets many of the questions…
We describe and explain the desire, common among mathematicians, both for unity and independence in its major themes. In the dialogue that follows, we express our spontaneous and considered judgment and reservations by contrasting the…
We will outline our ideas for teaching in the core mathematics disciplines. They are based on our own experience in teaching at a number of universities in the USA, as well as in Europe. While some of the core ideas stay and have stayed…
In this paper, we study the so-called 'Mathematical part' of Plato's Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of…
I argue for a full mathematisation of the physical theory, including its axioms, which must contain no physical primitives. In provocative words: "physics from no physics". Although this may seem an oxymoron, it is the royal road to keep…
This study has the purpose of addressing four questions that lie at the base of the probability theory and statistics, and includes two main steps. As first, we conduct the textual analysis of the most significant works written by eminent…
After the development of a self-consistent quantum formalism nearly a century ago, there ensued a quest to understand the often counterintuitive predictions of the theory. These endeavors invariably begin with the assumption of the "truth"…
In the words of the twentieth-century British mathematician G. H. Hardy, "the human function is to 'discover or observe' mathematics" (1). For centuries, starting from the ancient Greeks, mankind has hunted for beauty and order in arts and…
Gravity plays an important part in the experiments and discoveries of the modern world. But how was it discovered? Surely Newton and Einstein were not the only people to observe it and account for it. It had been a long path before the full…
By using ideas on complexity and randomness originally suggested by the mathematician-philosopher Gottfried Leibniz in 1686, the modern theory of algorithmic information is able to show that there can never be a "theory of everything" for…
Arising out of an attempt at a new foundations of mathematics, in which relations are more primitive than sets, and out of the theoretical physicists' concept of underlying causes of empirical phenomena, the idea of a purely mathematical…
Starting from Greg Moore's description about Physical Mathematics, a framework is proposed in order to understand it, based on Gilles Ch\^atelet's philosophy. It will be argued that Ch\^atelet's ideas of inverting, splitting, augmenting and…
Present day mathematics is a human construct, where computers are used more and more but do not play a creative role. This situation may change however: computers may become creative, and since they function very differently from the human…
People who by training end up dealing with probabilities ("statisticians") roughly fall into one of two camps. One is either a frequentist or a Bayesian. To a scientist, who needs to use probabilities to make sense of the real world, this…
Catalan's formula, for the portion of the inheritance that a legitimate child of a 19th-century deceased French gentleman should receive, is given a new proof (using Difference Operators), and generalized. Another, more computationally…
A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…
Mathematicians invented Mathematics to escape from words, but at last they depend on them just as much as everybody else. At the end, all basic definitions will be reliant on words, yet the mathematician believes that he's elevated from…
New fundamental mathematical structures are introduced by the triples (left semistructure,right semistructure,bisemistructure) associated with the classical mathematical structures and such that the bisemistructures,resulting from the…
Biology is data-rich, and it is equally rich in concepts and hypotheses. Part of trying to understand biological processes and systems is therefore to confront our ideas and hypotheses with data using statistical methods to determine the…