What Is an Indefinite Integral?
An indefinite integral is the integral whose lower boundary is constant and whose upper boundary is variable and it should be distinguished from a primitive integral that just restores primitive functions. Constants of primitive integration should also be distinguished from constants of indefinite integration that the lower boundary produces. Integration is often said to be the inverse operation of differentiation, but it is mathematically false. Since a set of primitive functions and a set of integrands is not in one-to-one but in many-to-one relationship, integration cannot be an inverse mapping. Differentiation and integration are in the relationship of analysis and synthesis. As a derivative has only partial information on the original function, it cannot restore the whole information. Contents 1. The previous explanation of the indefinite integral 2. The indefinite integral should be distinguished from the primitive 3. Integration produces two kinds of constants 4. Integration is ...
The Copernican Revolution of Infiniteness
When infinitesimal calculus was first established, the limit of a function was defined in terms of infinitesimal or infinity. As the naïve way at that time contradicted itself, new methods such as the epsilon-delta definition and non-standard analysis were devised to avoid the contradiction. Although mathematicians are not aware of it, it is Kant’s transcendental idealism that lays the foundation for these methods. That is to say, these methods succeed in solving the paradox of infiniteness because they abandon the cognition of “infiniteness in itself” and accomplish the Copernican Revolution that converts the cognition of infiniteness into the infiniteness of cognition.
Why Did Leibniz Suppose the Pre-Established Harmony?
Gottfried Leibniz named his metaphysics the system of pre-established harmony, which presupposed the ontological argument. The ontological argument resulted in pantheism. The law of continuity applies what is true of God to every substances, which resulted in monadology. The principle of sufficient reason prompts us to ask why this universe exists, which resulted in optimism. Leibniz thought God is intelligent enough to know the best of all possible universes, benevolent enough to select it and powerful enough to realize it as the existing one, but this being is more than the mere amount of universes whose existence alone the ontological argument can prove. We cannot identify them and apply the system of pre-established harmony to our cognition, benevolence and fulfillment, because we experience mistake, evil and failure.