Abstract
Critical analysis of the generally accepted (standard) foundations of differential and integral calculus is proposed. Methodological basis of the analysis is the unity of formal logic and of rational dialectics. It is shown that the generally accepted foundations are based on the logically erroneous concepts of “increment as variable quantity”, “infinitesimal quantity (uninterruptedly diminishing quantity)”, “derivative” and, consequently, represent incorrect basis of mathematics. Keywords: Foundations of mathematics, Philosophy of science.
Highlights
This does not mean that the problem of value of the argument is x x
Critical analysis is impossible without plausible reasoning
“We fasten our mathematical knowledge with the help of demonstrative reasoning, but we reinforce our assumptions with the help of plausible reasoning
Summary
This does not mean that the problem of (accrued) value of the argument is x x. Necessity of critical quantity of function y takes increment y , and the new (accrued) value of the function will be y y f x x. Analysis of the foundations of differential and integral The increment y of the function has form: calculus within the framework of the correct methodological basis – unity of formal logic and of rational y f x x f x. Plausible reasoning is the only type of reasoning that we are interested in everyday x 0 ), x becomes infinitesimal quantity The critical analysis is based on plausible reasoning within the framework of methodological basis – x 0 lim x . The variable quantity x in unity of formal logic and of rational dialectics
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