ESSAY ON FUNDAMENTALS OF INFORMATICS
René Descartes (1596-1650) came with the concept of method. Galileo Galilei (1564-1642) brought into use the physical experiments and later Sir Isaac Newton (1642-1727) and Baron Gottfried Leibniz (1646-1716), introducing calculus achieved the first metamorphosis of science by transforming the operational foundations of science, that were mainly the heritage of Aristotle, into its modern form we know today. They established the primacy of mathematics, under which we still live today. This new attitude consisted of the use of physical experiments and the use of mathematical models involving differential equations. This primacy of mathematics is so much active that the scientists still continue to refer to the mathematical representation of the creation, to its mathematical models, avoiding to get down to existence. They are interested to solve these mathematical models, to get their properties and to study their solution.
However, the character and structure of science, mainly over the past century, has been going through a second metamorphosis, which is the result of two classes of discoveries. The first group of discoveries refers to the limitations we face about the dynamic behaviors of the creation from mathematical models. It was discovered that all mathematical models have limitations regarding: analytical mathematical deductions, deterministic physical predictions and structurally stable models of closed systems. Additionally, the discovery of Kurt Gödel (1906-1978), that neither the consistency nor the completeness of any sufficiently general mathematical system can be proved within that system by generally accepted logical principles, struck at the foundations of mathematics that mathematical systems can establish any result which is true. The second group of discoveries is in close connection with the rising of the computer science and informatics. These enlarged the operational basis for scientific investigations by introducing the computer experiments. The second metamorphosis of science enlarged the operational basis of physical experiments and mathematical models by including the third operational basis to get knowledge as computational or numerical experiments. These numerical experiments gave the scientists the access to a strange world. Intensive and very sophisticated computational experiments with mathematical models determined the re-addressing to the real existence. Rephrased, numerical experiments restored the primacy of existence and the discipline that achieved this restoration is informatics. Basically we can define informatics as "coming down into computational of mathematical concepts, turning these mathematical concepts into algorithms, study the associated algorithms subject to convergence and complexity". This is the substance of informatics - transforming the advanced mathematical concepts into algorithms, the implementation of algorithms in computing programs. In a way, informatics is computational linear algebra.
In this essay we develop the definition of informatics and present its fundamentals problems as the complexity of algorithms over arbitrary rings or fields as well as the experimental mathematics. The classical theory of computation, developed by Turing (1912-1954), which proved to have an extraordinary success in providing the foundations and framework for theoretical computer science is extended to include the Newton machine. Machines over the reals endowed with condition, round-off, approximation and probability enable us to combine tools and traditions of theoretical computer science with tools and traditions in numerical analysis in order to understand the nature and complexity of computation.