Even earlier Cardano started manipulating 'impossible' numbers, and others followed. However, already in mid 16th century Stevin advocated admitting infinite decimals as numbers, see When did it become understood that irrational numbers have non-repeating decimal representations?, and by mid 17th century Euclidean strictures about distinguishing magnitudes, numbers and ratios were largely abandoned. Numerous examples and applications are given: approximation, automatic differentiation, cellular automata, differential equations, linear and non-linear optimization, fractals, percolation, processes of growth in biological systems, etc.Actual infinities collected into sets were not officially contemplated by (philosophizing) mathematicians until Cantor (with some anticipation by Bolzano) countered Aristotelian and scholastic arguments that such objects are paradoxical, see How does actual infinity (of numbers or space) work? Ironically, Cantor rejected the infinitesimals themselves, see What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal? In particular, a number of results related to the First Hilbert Problem and Turing machines are established and a new concept of continuity better reflecting the modern view of physicists on the world around us is introduced. The new computational methodology gives the possibility both to execute numerical (not symbolic) computations of a new type and simplifies fields of Mathematics where the usage of the infinity and/or infinitesimals is necessary (e.g., divergent series, limits, derivatives, integrals, measure theory, probability theory, fractals, etc.).

The latter is directly related to connections between different numeral systems used to describe mathematical objects and the objects themselves. The point of view on infinite and infinitesimal quantities presented in this talk uses strongly two methodological ideas borrowed from the modern Physics: relativity and interrelations holding between the object of an observation and the tool used for this observation.

Even the brilliant non-standard Analysis of Robinson made in the middle of the XX-th century has been also directed to a reformulation of the classical Analysis (i.e., Analysis created two hundred years before Robinson) in terms of infinitesimals and not to the creation of a new kind of Analysis that would incorporate new achievements of Physics. Thus, these mathematical tools do not include numerous achievements of Physics of the XX-th century. As a result, these parts of Mathematics reflect ideas that people had about Physics (including definitions of the notions continuous and discrete) more than 200 years ago. Foundations of the classical Analysis dealing both with infinity and infinitesimal quantities have been developed even earlier, more than 200 years ago, with the goal to develop mathematical tools allowing one to solve problems arising in the real world in that remote time. Note that foundations of the Set Theory dealing with infinity have been developed starting from the end of the XIX-th century until more or less the first decades of the XX-th century. The new methodology evolves ideas of Cantor and Levi-Civita in a more applied way and (among other things) introduces infinite integers that possess both cardinal and ordinal properties as usual finite numbers. It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework (different from that of the non-standard Analysis). The new approach is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks that is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). A calculator using the Infinity Computer technology is presented during the talk. SergeyevĪbstract: A new methodology allowing one to execute numerical computations with finite, infinite, and infinitesimal numbers on a new type of a computer – the Infinity Computer – is introduced. Keynote Title: The Infinity Computer and Numerical Computations with Infinite and Infinitesimal Numbers

Infinity and infinitesimals