Related papers: Foundations as Superstructure. (Reflections of a p…
From classical mechanics to quantum field theory, the physical facts at one point in space are held to be independent of those at other points in space. I propose that we can usefully challenge this orthodoxy in order to explain otherwise…
I discuss the historical and conceptual roots of reasoning about the parameters of fundamental physics and cosmology based on selection effects. I argue concretely that such reasoning can and should be combined with arguments based on…
A model of knowledge representation is described in which propositional facts and the relationships among them can be supported by other facts. The set of knowledge which can be supported is called the set of cognitive units, each having…
The popular view according to which Category theory provides a support for Mathematical Structuralism is erroneous. Category-theoretic foundations of mathematics require a different philosophy of mathematics. While structural mathematics…
Modern physics is founded on two mainstays: mathematical modelling and empirical verification. These two assumptions are prerequisite for the objectivity of scientific discourse. Here we show, however, that they are contradictory, leading…
These are notes on discrete mathematics for computer scientists. The presentation is somewhat unconventional. Indeed I begin with a discussion of the basic rules of mathematical reasoning and of the notion of proof formalized in a natural…
Manifold learning is a popular and quickly-growing subfield of machine learning based on the assumption that one's observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. This thesis presents a mathematical…
It is shown the construction of a module structure [2] with universe over a set of a particular kind of mathematical proofs, the base ring of this module will be built on a maximal consistent extension of a set of propositions, this…
Within the biological, physical, and social sciences, there are two broad quantitative traditions: statistical and mathematical modeling. Both traditions have the common pursuit of advancing our scientific knowledge, but these traditions…
At two examples dealt with in methodologically different ways it will be pointed out how the concept of an empirical theory (in the sense of the Structuralists) can be useful to specify contents relevant to maths didactics.
Prediction is the making of statements, usually probabilistic, about future events based on current information. Retrodiction is the making of statements about past events based on current information. We present the foundations of quantum…
Basic principles of mathematical modeling are reviewed in this book, with the focus on physics and its practical applications, and examples of selected mathematical methods are presented. Most of the models have been imported from physics…
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and…
In this paper, I consider the issue of how two mathematical models of modern physics, the variational principles and the quantum path integral formalism, relate to reality. I assume that the observed phenomena are consistent with the…
Foundational investigations in quantum mechanics, both experimental and theoretical, gave birth to the field of quantum information science. Nevertheless, the foundations of quantum mechanics themselves remain hotly debated in the…
Toposes can be pictured as mathematical universes. Besides the standard topos, in which most of mathematics unfolds, there is a colorful host of alternate toposes in which mathematics plays out slightly differently. For instance, there are…
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of…
In this course, I talk about the source of mathematical constructivism and its role in the future development of theoretical physics. I describe what physical constructivism is and why it is necessary for the penetration of exact methods of…
The purpose of this short paper is to identify the mathematical essence of the superiorization methodology. This methodology has been developed in recent years while attempting to solve specific application-oriented problems. Consequently,…
I have argued elsewhere that second order logic provides a foundation for mathematics much in the same way as set theory does, despite the fact that the former is second order and the latter first order, but second order logic is marred by…