Critical Analysis of the Mathematical Formalism of Theoretical Physics. I. Foundations of Differential and Integral Calculus

Critical analysis of the standard foundations of differential and integral calculus -- as mathematical formalism of theoretical physics -- is proposed. Methodological basis of the analysis is the unity of formal logic and rational dialectics. It is shown that: (a) the foundations (i.e. $\frac{d{\kern 1pt}y}{d\,x}\;=\;\lim\limits_{\Delta \,x\;\to \;0} \,\;\frac{\Delta \,y}{\Delta \,x}$, $\lim\limits_{\Delta \,x\;\to \;0} \;\frac{\Delta \,y}{\Delta \,x}\;=\;\lim\limits_{\Delta \,x\;\to \;0} \;\frac{f\,\left( {x\;+\;\Delta \,x} \right)\;-\;f\,\left( x \right)}{\Delta \,x}\;$, $d\,x\;=\;\Delta \,x$\textbf{, }$d\,y\;=\;\Delta \,y$ where $y\;=\;f\,\left( x \right)$ is a continuous function of one argument $x$; $\Delta \,x$ and $\Delta \,y$ are increments; $d\,x$ and $d\,y$ are differentials) not satisfy formal logic law -- the law of identity; (b) the infinitesimal quantities $d\,x$, $d\,y$ are fictitious quantities. They have neither algebraic meaning, nor geometrical meaning because these quantities do not take numerical values and, therefore, have no a quantitative measure; (c) expressions of the kind $x\;+\;d\,x$ are erroneous because $x$ (i.e. finite quantity) and $d\,x$ (i.e. infinitely diminished quantity) have different sense, different qualitative determinacy; since $x\;\,\to \;\,c\,{\kern 1pt}\,=\,\,const$ under $\Delta \,x\;\,\to \;\,0$, a derivative does not contain variable quantity $x$ and depends only on constant $c$. Consequently, the standard concepts ``infinitesimal quantity (uninterruptedly diminishing quantity)'', ``derivative'', ``derivative as function of variable quantity'' represent incorrect basis of mathematics and theoretical physics.