Related papers: The ultimate tactics of self-referential systems
In this paper a class of languages which are formal enough for mathematical reasoning is introduced. First-order formal languages containing natural numbers and numerals belong to that class. Its languages are called mathematically…
We construct a formal theory, which we call reflectica, whose language possesses the following properties of natural language: it is a self-reflecting language and an intensional language. By a self-reflecting language we understand an…
Reasoning is a hallmark of human intelligence, enabling adaptive decision-making in complex and unfamiliar scenarios. In contrast, machine intelligence remains bound to training data, lacking the ability to dynamically refine solutions at…
A definition of what counts as an explanation of mathematical statement, and when one explanation is better than another, is given. Since all mathematical facts must be true in all causal models, and hence known by an agent, mathematical…
Physics makes powerful use of mathematics, yet the way this use is made is often poorly understood. Professionals closely integrate their mathematical symbology with physical meaning, resulting in a powerful and productive structure. But…
Affordances, a foundational concept in human-computer interaction and design, have traditionally been explained by direct-perception theories, which assume that individuals perceive action possibilities directly from the environment.…
After highlighting the cases in which the semantics of a language cannot be mechanically reproduced (in which case it is called inherent), the main epistemological consequences of the first incompleteness Theorem for the two fundamental…
The notion of class is ubiquitous in computer science and is central in many formalisms for the representation of structured knowledge used both in knowledge representation and in databases. In this paper we study the basic issues…
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,…
Previous work indicates that large language models exhibit a significant "English bias", i.e. they often perform better when tasks are presented in English. Interestingly, we have observed that using certain other languages in reasoning…
Natural philosophy integrates scientific observation with abstract frameworks, often using a mathematical Ansatz to hypothesise about physical phenomena. Exploring the possibility of other universes, however, challenges assumptions that…
We commonly think of mathematics as bringing precision to application domains, but its relationship with computer science is more complex. This experience report on the use of Racket and Haskell to teach a required first university CS…
The testimony and practice of notable mathematicians indicate that there is an important phenomenological and epistemological difference between superficial and deep analogies in mathematics. In this paper, we offer a descriptive theory of…
If language evolved by sexual selection to display superior intelligence, then we require conversational skills, to impress other people, gain high social status, and get a mate. Conversational skills include a Theory of Mind, a sense of…
Recent work has proposed that language models perform computation by manipulating one-dimensional representations of concepts ("features") in activation space. In contrast, we explore whether some language model representations may be…
Classical mathematical models used in the semantics of programming languages and computation rely on idealized abstractions such as infinite-precision real numbers, unbounded sets, and unrestricted computation. In contrast, concrete…
Language models now provide an interface to express and often solve general problems in natural language, yet their ultimate computational capabilities remain a major topic of scientific debate. Unlike a formal computer, a language model is…
More than hundred years ago the 'classic physics' was it in its full power, with just a few unexplained phenomena; which however led to a revolution and the development of the 'modern physics'. Today the computing is in a similar position:…
After more than a century of concerted effort, physics still lacks basic principles of spontaneous self-organization. To appreciate why, we first state the problem, outline historical approaches, and survey the present state of the physics…
In order to describe natural phenomena, science develops sophisticated models that use mathematical and formal languages which seem, and often are, very far from common experience. When a phenomenon is not accessible to our senses, its…