/sci/ - Science & Math

>>12464219

At the end of the procedure, Marx arrives at an equation in which the increments, in the form of the ratio [math] \frac{\Delta y}{\Delta x} [/math] , appear in the left-hand side of the equation, while the right-hand side only contains expressions in [math] x_1 [/math] and [math] x [/math]. This fact, Marx argues, is not accidental but expresses a qualitative difference between the two sides of the equation: the left-hand side has a symbolic nature, while the right-hand side has an algebraic nature. The first one represents the purely symbolic expression of the actual process of variation that takes place entirely in the right-hand algebraic side of the equation. The increments becoming zero, which in dialectical terms represents the negation of the negation, only takes effect on the left-hand side of the equation, which is reduced to the expression [math] \frac{0}{0} [/math], leaving the right-hand side unchanged. However, this time the ratio [math] \frac{0}{0} [/math] should no longer be dreadful because it does not denote an arithmetic operation, but is a purely symbolic operator, which as such can be replaced by the differential ratio [math] \frac{dy}{dx} [/math], without giving rise to any logical contradiction. Unlike the previous methods in which the differentials appeared as distinct entities having a substantial content, they are now inseparably connected as the numerator and the denominator in [math] \frac{dy}{dx} [/math], which only represents the symbolic form of a process through which the qualitative leap takes place from the algebra of the constant quantities to the differential calculus of the variable quantities. In other words, for Marx the ratio between the differentials is a mere operational symbol, which denote an ordered sequence of logical and algebraic operations necessary to compute the derivative of a function.