Related papers: Mathematical conceptualism
Statistics has moved beyond the frequentist-Bayesian controversies of the past. Where does this leave our ability to interpret results? I suggest that a philosophy compatible with statistical practice, labeled here statistical pragmatism,…
Practicing mathematicians often assume that mathematical claims, when they are true, have good reasons to be true. Such a state of affairs is "unreasonable", in Wigner's sense, because basic results in computational complexity suggest that…
Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as…
"Mathematicians, like physicists, are pushed by a strong fascination. Research in mathematics is hard, it is intellectually painful even if it is rewarding, and you would not do it without some strong urge." [D. Ruelle]. We shall give some…
This paper addresses the actual practice of justifying definitions in mathematics. First, I introduce the main account of this issue, namely Lakatos's proof-generated definitions. Based on a case study of definitions of randomness in…
We present a logical framework that enables us to define a formal theory of computational trust in which this notion is analysed in terms of epistemic attitudes towards the possible objects of trust and in relation to existing evidence in…
As an approach to a Theory of Everything a framework for developing a coherent theory of mathematics and physics together is described. The main characteristic of such a theory is discussed: the theory must be valid and and sufficiently…
A popular scientific contribution should not contradict any established facts and ought to be understandable. I complied with both these requirements and am offering a sufficiently full introduction to probability theory. Furthermore, I…
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…
I analyse differences in style between traditional prose mathematics writing and computer-formalised mathematics writing, presenting five case studies. I note two aspects where good style seems to differ between the two: in their…
The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points and concepts are represented by regions in a (potentially) high-dimensional space. Based on our…
Computability logic (CL) is a systematic formal theory of computational tasks and resources, which, in a sense, can be seen as a semantics-based alternative to (the syntactically introduced) linear logic. With its expressive and flexible…
In consciousness science, several promising approaches have been developed for how to represent conscious experience in terms of mathematical spaces and structures. What is missing, however, is an explicit definition of what a 'mathematical…
This paper presents a plausible reasoning system to illustrate some broad issues in knowledge representation: dualities between different reasoning forms, the difficulty of unifying complementary reasoning styles, and the approximate nature…
We present modeling for conceptual combinations which uses the mathematical formalism of quantum theory. Our model faithfully describes a large amount of experimental data collected by different scholars on concept conjunctions and…
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…
A new understanding of the notion of regularizer is proposed. It is argued that this new notion is more realistic than the old one and better fits the practical computational needs. An example of the regularizer in the new sense is given. A…
In this essay, I argue that mathematics is a natural science---just like physics, chemistry, or biology---and that this can explain the alleged "unreasonable" effectiveness of mathematics in the physical sciences. The main challenge for…
Writing and argumentation are critical to both professional physics and physics education. However, the skill of making an extended argument in writing is often overlooked in physics classrooms, apart from certain practices like lab…
We explore the rational, formal and non-formal criteria of consistency, non-triviality and redundancy in the mathematical research now a days. We develop a paradigmatic discussion by analysing the different conceptions of those criteria,…