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Infinitesimal Calculus
 INFINITESIMAL
CALCULUS

"The infinitesimal calculus is the art of exactly numbering and measuring a thing whose existence cannot be conceived" (Voltaire).

"Calculus is the story that this world told itself when it became the modern world" (David Berlinski).

"Calculus requires continuity, and continuity is supposed to require the infinitely small; but no one could discover what can be infinitely small."(Bertrand Russell)


Infinitesimals

The infinitesimal (ε) −as we have indicated in the previous chapter− is not a concrete number. It is a qualitative number. The quality is that of being an infinitely small number. Not being a concrete number, it is not expressible. We can only express its behavior at the surface level, which is given by the imaginary expression nilpotent ε2 = 0. Therefore, ε is an imaginary number.

The expression ε2 = 0 is not an equation, it is a substitution, that is to say, in an expression where ε2 appears, it is substituted by zero.

We have also seen that there are infinite infinitesimals, those of the form rε, being r a real number, since they satisfy that (rε)2 = 0. There is a prime infinitesimal (ε) and infinite secondary infinitesimals (rε).


Differential and Integral Calculus

The infinitesimal calculus is based on calculations in which infinitesimals are present. Its main application has been differential and integral calculus.

The differential and integral calculus −also called simply "calculus"− is a calculus with infinitesimals associated with expressions in general. Calculus applied to functions is called "analysis", the study of functions.

Infinitesimals appear in derivatives (slope of the tangent to a curve at a point on the Cartesian axes) in the form of the quotient of infinitesimals (Δy/Δx) and in the calculation of the area underlying a curve as the sum of infinite infinitesimals (sum of rect⟨ngles of infinitesimal width).


There has been a long controversy over who was the originator of calculus. Newton developed it in 1665 and 1666. Leibniz did it later, between 1673 and 1676, but he was the first to publish his results in 1684. Newton published his investigations in 1704.

Today it is admitted that Newton and Leibniz independently arrived at the same concepts and the same results. Newton and Leibniz proved that the problems of area and tangent were inverses, which is known as the "fundamental theorem of calculus."

Newton first studied derivatives and later integration. For him, functions were "fluents" and derivatives he called "fluxions" and calculus he called "fluxion calculus".

Newton applied calculus to physical phenomena, especially universal gravitation. He discovered that with his calculus of tangents one could express: instantaneous velocities (of an object moving along a known trajectory), radii of curvature at any point on the curve, as well as maxima, minima and inflection points. The notation he used was a quotation mark to the right of the function name. If the function is y = f(x), its derivative is denoted as y ' = f'(x), and its second derivative y'' = f''(x).

Leibniz began to study integration to arrive at the concept of derivative. He introduced the term "difference" to refer to the infinitesimal variation of a quantity, and this is what today we call "differential". He gave name to the new discipline: differential and integral calculus. His notation was more intuitive and expressive than Newton's and is the one that was finally adopted, although sometimes Newton's is used because it is simpler and more compact. With dx (differential of x) he indicated an infinitesimal change in the variable x. The derivative was dy/dx. The integral was ∫f(x)·dx. The symbol ∫ indicates continuous sum (∑ indicates discrete sum).

Leibniz only investigated the calculus at the theoretical level. He established a set of rules for manipulating infinitesimal quantities. He discovered several properties, among them the product rule (derivative of the product of two functions) and the chain rule (derivative of the composition of two functions). These rules can be expressed in simplified form (with Newton's notation) as follows: This last expression can also be expressed as follows: (g(f))' = g' (f) · f'

Newton's discovery of the calculus predated Leibniz's, but the delay in their publication still causes controversy as to which of the two was first. Newton used calculus in physics in his work "Mathematical Principles of Natural Philosophy," one of the most famous and influential works of all time.

The fundamental theorem of calculus states that the derivative and the integral are inverse operations: Newton and Leibniz invented calculus without a rigorous formal foundation. In the 19th century, mathematicians replaced these vaguenesses with formal definitions. Bolzano and Cauchy precisely defined the concept of limit. Cauchy and Riemann did the same with integrals, and Dedekind and Weirstrass with the real numbers. This was the period of the foundations of calculus.

To formalize the concept of derivative, the concept of limit was introduced:
dy/dx = limit of (Δy/Δx) when Δx tends to zero.
But the concept of limit is purely descriptive. It is not operational, it does not provide a procedure for its calculation (the passage to the limit). This definition remains as fuzzy as that of infinitesimal with the only difference that there is an intermediate function.

In the 18th century the number of applications of calculus increased considerably, but the imprecise use of infinite and infinitesimal quantities caused confusion and doubt about its foundations. In fact, the notion of limit, central to the study of calculus, was still vague and imprecise at that time. One of its most notable critics was George Berkeley.

Differential and integral calculus had its precedents with the ancient Greeks. Eudoxus (Plato's disciple) calculated areas and volumes by dividing them into an infinite number of parts. Archimedes invented a general heuristic method (informal and intuitive) very close conceptually to infinitesimal calculus for calculating areas and volumes. It can be said that the infinitesimal calculus was founded by Archimedes.

Differential and integral calculus is very important in modern mathematics and is associated with continuous mathematics. With calculus, differential equations appeared, which was a qualitative change, since it allowed the formalization of domains involving interrelated continuous variables.

Smooth differential analysis (smooth infinitesimal analysis) is a rigorous reformulation of analysis in terms of infinitesimals. It is based on the ideas of F.W. Lawvere and employs the methods of category theory. It considers that all continuous functions can be expressed by discrete quantities. As a theory it is a subdomain of synthetic differential calculus, which is based on the ⟨lgebra of nilpotent elements and replaces the processes of passing to the limit.

In smooth differential analysis (and synthetic differential calculus in general) the expression ε2 = 0 is interpreted erroneously. It is interpreted as an equation that breaks the logical principle of the excluded third. For example "not a≠b" does not imply a = b. In this case, ε2 = 0 does not imply ε = 0. The correct interpretation is that it is simply a substitution, not an equation. And it has nothing to do with the excluded third principle.

It is not necessary to appeal to the theory of categories, because this theory is very complex. It is necessary to base everything on the maximum possible simplicity.

As a conclusion, the infinitesimal calculus goes beyond the differential and integral calculus. The concept of infinitesimal defined as an imaginary expression is generic and is very important, since it has applications in the fields of mathematics where real numbers appear.


Calculus in MENTAL

Notation

The notation we have chosen is close to the standard Leibniz notation, but any other notation can be used. The language is open, but it has to be logically of linear (one-dimensional) type, The fundamental theorem of calculus is:
Automatic calculation of derivatives

It is possible to provide an algebraic method to calculate derivatives automatically without using limits. Its expression is very simple: f'(x) = (f(x+ε) − f(x))/ε, and assuming that ε2 = 0. Formally in MENTAL: For example, if f(x) = x3, one has:

(f(x+ε) − f(x))/ε = (( x+ε)3 − x3) /ε = (ε< sup>3 + 3x2ε + 3xε2 )/ ε = (3x2ε)/ε = 3x2

In MENTAL,

δ(x^3)÷δx // ev. 2*(x^2)

In general,

⟨( δ(x^n)÷δx = n*(x^(n−1)) )⟩

Fermat used the procedure of calculating (f(x+ε) − f(x))/ε, and subsequently making ε=0. In the example, above, (ε3 + 3x2ε + 3xε2)/ε = (ε2 + 3x2 + 3xε), which becomes 3x2.


Higher order derivatives and integrals

They are defined recursively. Calling (D n f x) the derivative of order n of the function f respectp a x, we have:

⟨( (D n f x) = ( δ((D n−1 f x )¸δx ←n>1 →' δf÷δx ) )⟩

Analogously for the order integral n, substituting δ for ∫, and ÷δx by *δx,

⟨( (I n f x) = (∫((D n−1 f x)*δ x ← n>1 →' ∫f*δx ) )⟩

Examples:
  1. (D 1 x^n x) // ev. n*(x^(n−1))
    (D 2 x^n x) // ev. n*(n−1)(x^(n−2))
    (D 3 x^n x) // ev. ev. n*(n−1)*(n−2)(x^(n−3))
    (D n x^n x) // ev. ev. n*...*1 (factorial of n)

  2. (I 1 x^n x) // ev. (x^(n+1))÷(n+1)
    (I 2 x^n x) // ev. (x^(n+2))÷(n+1)÷(n+2)
    (I 3 x^n x) // ev. (x^(n+3))÷(n+1)÷(n+2)÷(n+3)
    (I n x^n x) // ev. (x^(n+n))÷((n+1)÷...÷(n+n))

Remarks
Properties

Bibliography