Related papers: Tension between Intuitive Infinitesimals and Forma…
The history of the development of the concept of complex numbers from the 16th to 19th centuries. The origin and refinement of the geometric and physical meaning of complex numbers, the emergence of vectoral analysis.
Mathematics and its relation to the physical universe have been the topic of speculation since the days of Pythagoras. Several different views of the nature of mathematics have been considered: Realism - mathematics exists and is…
Some relations between physics and finitary and infinitary mathematics are explored in the context of a many-minds interpretation of quantum theory. The analogy between mathematical ``existence'' and physical ``existence'' is considered…
In recent publications in physics and mathematics, concerns have been raised about the use of real numbers to describe quantities in physics, and in particular about the usual assumption that physical quantities are infinitely precise. In…
This is a brief overview of some turning points in the history of infinitesimals.
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…
I contrast two possible attitudes towards a given branch of physics: as inferential (i.e., as concerned with an agent's ability to make predictions given finite information), and as dynamical (i.e., as concerned with the dynamical equations…
Philosopher Benardete challenged both the conventional wisdom and the received mathematical treatment of zero, dot, nine recurring. An initially puzzling passage in Benardete on the intelligibility of the continuum reveals challenging…
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…
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…
The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a…
The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the…
Most physics theories are deterministic, with the notable exception of quantum mechanics which, however, comes plagued by the so-called measurement problem. This state of affairs might well be due to the inability of standard mathematics to…
Mathematical concepts and results have often been given a long history, stretching far back in time. Yet recent work in the history of mathematics has tended to focus on local topics, over a short term-scale, and on the study of ephemeral…
In the history of infinitesimal calculus, we trace innovation from Leibniz to Cauchy and reaction from Berkeley to Mansion and beyond. We explore 19th century infinitesimal lores, including the approaches of Simeon-Denis Poisson,…
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
More than a speculative technology, quantum computing seems to challenge our most basic intuitions about how the physical world should behave. In this thesis I show that, while some intuitions from classical computer science must be…
Infinitesimals are natural products of the human imagination. Their history goes back to the Greek antiquity. Their role in the calculus and analysis has seen dramatic ups and downs. They have stimulated strong opinions and even vitriol.…
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…
Physics is formulated in terms of timeless classical mathematics. A formulation on the basis of intuitionist mathematics, built on time-evolving processes, would offer a perspective that is closer to our experience of physical reality.