Related papers: Unity and Disunity in Mathematics
As David Berlinski writes (1997), the existence and nature of mathematics is a more compelling and far deeper problem than any of the problems raised by mathematics itself. Here we analyze the essence of mathematics making the main emphasis…
A primary goal of physics is to create mathematical models that allow both predictions and explanations of physical phenomena. We weave maths extensively into our physics instruction beginning in high school, and the level and complexity of…
There is a cognitive limit in Human Mind. This cognitive limit has played a decisive role in almost all fields including computer sciences. The cognitive limit replicated in computer sciences is responsible for inherent Computational…
A question is proposed whether or not set theory is consistent.
The relationship between mathematics and physics has long been an area of interest and speculation. Subscribing to the recent definition by Tegmark, we present a mathematical structure involving the only division rings - the real,…
How students use mathematics in their physics classes has been studied extensively in the physics education literature. In addition to specific mathematical methods in specific physics contexts, possible effects of more general "cultural"…
Mathematics is usually regarded as a kind of language. The essential behavior of physical phenomena can be expressed by mathematical laws, providing descriptions and predictions. In the present essay I argue that, although mathematics can…
Learning to use math in science is a non-trivial task. It involves many different skills (not usually taught in a math class) that help blend physical knowledge with mathematical symbology. One of these is the idea of quantification: that…
The definition of who is or what makes a ``mathematician" is an important and urgent issue to be addressed in the mathematics community. Too often, a narrower definition of who is considered a mathematician (and what is considered…
This chapter presents a brief review of complexity research in mathematics education. We argue how research on complexity, as it pertains to mathematics education, can be viewed as an epistemological discourse, an historical discourse, a…
Computation is commonly defined as the execution of abstract algorithms over symbolic representations, with physical systems treated as substrates that realise predefined operations. While effective for engineered machines, this separation…
A coherent mathematical overview of computation and its generalisations is described. This conceptual framework is sufficient to comfortably host a wide range of contemporary thinking on embodied computation and its models.
Questions concerning origin of mathematical knowledge and roles of language and intuition (imagery) in mathematical thoughts are long standing and widely debated. By introspection, mathematicians usually have some beliefs regarding these…
The paper discusses relationships between aesthetics theory and mathematical models of mind. Mathematical theory describes abilities for concepts, emotions, instincts, imagination, adaptation, learning, cognition, language, approximate…
Philosophy of science attempts to describe all parts of the scientific process in a general way in order to facilitate the description, execution and improvements of this process. So far, all proposed philosophies have only covered existing…
I discuss some problems related to extreme mathematical realism, focusing on a recently proposed "shut-up-and-calculate" approach to physics (arXiv:0704.0646, arXiv:0709.4024). I offer arguments for a moderate alternative, the essence of…
We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a…
We set up a model for reasoning about metric spaces with belief theoretic measures. The uncertainty in these spaces stems from both probability and metric. To represent both aspect of uncertainty, we choose an expected distance function as…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
In recent paper "Quantifying Inequities and Documenting Elitism in PhD-granting Mathematical Sciences Departments in the United States" (arXiv:2308.13750) by a group of accomplished and/or aspiring mathematicians, the authors use data to…