The formal sciences are the branches of science that are concerned with formal systems, such as mathematics, logic, theoretical computer science, information The branches of science, also referred to as sciences, scientific fields or scientific disciplines, are commonly divided into three major groups: Formal sciences: the study of formal systems, such as those under the branches of logic and mathematics, which use an a priori, as opposed to empirical, methodology. They study abstract structures described by formal systems. Natural sciences: the study Mat…

The basic terms of metascience and philosophy of science are considered. In their context, we give the definitions of science, a field of science, and two adjacent sciences in relation to computer science: mathematics and technology. We determine the status of computer science as a formal field of science, which consists of theories, in each of which the following constructive conceptual transition is accomplished: the principles of algorithmic computations–programming paradigms–programs–running programs on computers.

properties that are considered true starting points of the theory under consideration. Mathematics is essential in the natural sciences, engineering, medicine Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life. There are many The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians. Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. This led to the controversy over Cantor's set theory. In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigor. This became the foundational crisis of… Computational mathematics is the study of mathematical problems that are typically too large for human computing capacity. Part of computational mathematics involves numerical analysis, which is devoted to computation with approximations of real numbers (floating-point arithmetic). Numerical analysis provides methods for problems in analysis using functional analysis and approximation theory. Numerical analysis broadly includes the study of approximation and discretization, with special focus on rounding errors. Numerical analysis and, more broadly, scientific computing, also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Another area of computational mathematics is computer algebra, which consists mainly of the manipulation of mathematical formulas on a computer. Automated theorem proving, which include proof assistants and proof checkers is another area of computational mathematics that is used for formal verification and allowed formal proof of difficult theorems of mathematics such Feit–Thompson theorem on finite groups. Algorithmics, a large part of discrete mathematics, and public key cryptography are also often considered as a part of computational mathematics.

Concise tutorial on the fundamentals of ultra-complexity theory. Emergence of Life, Human consciousness and human societies. Abstract: The task of developing novel and improved methodological tools for a categorical ontology of complex spacetime structures and the ontological theory of levels is here considered from several perspectives including both the quantum-molecular and complex-relational ones. Our novel conceptual framework is aimed at helping both philosophers and natural scientists to understand and organize the spacetime ontology of highly complex systems in terms of a categorical formalism of Roberto Poli's ontological theory of levels. Such highly complex systems, and indeed their emergent meta-systems dynamics are relevant to a wide variety of biosystems or organisms. They are suggested to be also necessary for understanding the human brain, the mind and society levels. It may be that we are now at a stage of the development in philosophy and science that such tools are beginning to emerge, or are indeed in the process of becoming available from several exact sciences, such as: logics, mathematics, physics, molecular and theoretical genetics, molecular biology, relational biology, and so on, as a result of trends towards unity in logics, mathematics and physics. An outline of a Categorical Ontology of Space and Time, or Spacetime, is presented for emergent biosystems, super-complex dynamics, biological evolution and human consciousness. Relational structures of

the presentation of the object in an à priori intuition, is termed mathematics . What may be called natural science proper presupposes metaphysics of nature;