Related papers: Gregory's sixth operation
Mathematical understanding is built in many ways. Among these, illustration has been a companion and tool for research for as long as research has taken place. We use the term illustration to encompass any way one might bring a mathematical…
An important question for a probabilistic program is whether the probability mass of all its diverging runs is zero, that is that it terminates "almost surely". Proving that can be hard, and this paper presents a new method for doing so; it…
In this paper, we propose that 'embodied mathematics' should be studied not only by reduction to the present individual bodily experience but in an historical context as well, as far as the origins of mathematics are concerned. Some early…
I present a novel mathematical technique for dealing with the infinities arising from divergent sums and integrals. It assigns them fine-grained infinite values from the set of hyperreal numbers in a manner that refines the standard…
We discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating them to the natural cognitive development of…
We discuss views about whether the universe can be rationally comprehended, starting with Plato, then Leibniz, and then the views of some distinguished scientists of the previous century. Based on this, we defend the thesis that…
We motivate and study an infinite sequence of binary operations on the ordinal numbers, extending the standard arithmetic on the ordinals to higher degrees of iteration. Connections to the hyperoperations on the natural numbers are…
In the last decades, several objects such as grammars, economical agents, laws of physics... have been defined as algorithms. In particular, after Brouwer, Heyting, and Kolomogorov, mathematical proofs have been defined as algorithms. In…
We define instantiational and algorithmic completeness for a formal language. We show that, in the presence of Church's Thesis, an alternative interpretation of Goedelian incompleteness is that Peano Arithmetic is instantiationally…
In teaching infinitesimal calculus we sought to present basic concepts like continuity and convergence by comparing and contrasting various definitions, rather than presenting "the definition" to the students as a monolithic absolute. We…
This paper presents mathematics as a general science of computation in a way different from the tradition. It is based on the radical philosophical standpoint according to which the content, meaning and justification of experience lies in…
Wigner found unreasonable the "effectiveness of mathematics in the natural sciences". But if the mathematics we use to describe nature is simply a coded expression of our experience then its effectiveness is quite reasonable. Its…
We introduce a model of infinitary computation which enhances the infinite time Turing machine model slightly but in a natural way by giving the machines the capability of detecting cardinal stages of computation. The computational strength…
Currently it is widely accepted that the language of science is mathematics. This book explores an alternative idea where the future of science is based on the language of algorithms and programs. How such a language can actually be…
Mathematicians tend to use the phrase "arbitrarily close" to mean something along the lines of "every neighborhood of a point intersects a set". Taking the latter statement as a technical definition for arbitrarily close leads to an…
Stephen Toulmin once observed that `it has never been customary for philosophers to pay much attention to the rhetoric of mathematical debate'. Might the application of Toulmin's layout of arguments to mathematics remedy this oversight?…
Sets with atoms serve as an alternative to ZFC foundations for mathematics, where some infinite, though highly symmetric sets, behave in a finitistic way. Therefore, one can try to carry over analysis of the classical algorithms from finite…
Current research in statistical mechanics mostly concerns the investigation of out-of-equilibrium, irreversible processes, which are ubiquitous in nature and still far from being theoretically understood. Even the precise characterization…
This paper considers the computational hardness of computing expected outcomes and deciding (universal) (positive) almost-sure termination of probabilistic programs. It is shown that computing lower and upper bounds of expected outcomes is…
This is a short paper about the relationship between logic and computation. More specifically, it is about a relationship between the completeness proof for intuitionistic propositional logic within the form of proof-theoretic semantics…