An item of Peano's Formulario
Introduction
Giusseppe Peano was born in 1858 in Spinetta, Cuneo province, Italian NE, near France. Peano attended Turín University and later was a mathematics instructor there and in the military Academy.
Peano's mathematics contributions in logic, geometry, differential and integral calculus are included in several articles and books. In their publications he makes mention of Cantor, Fermat, Leibnitz, Frege and other matematicians. From Frege he adopts several symbols. Peano was admired by contemporary colleagues like Bertrand Russell. Peano lived all of his life in Turin where he was a student, a teacher, a family former, and a school creator. He died at the age of 74.
Peano's mathematics contributions in logic, geometry, differential and integral calculus are included in several articles and books. In their publications he makes mention of Cantor, Fermat, Leibnitz, Frege and other matematicians. From Frege he adopts several symbols. Peano was admired by contemporary colleagues like Bertrand Russell. Peano lived all of his life in Turin where he was a student, a teacher, a family former, and a school creator. He died at the age of 74.
The "Formulario" project
The "Formulario" was first published in french, in 1894: "Formulaire de mathématiques". There were several successive versions between 1894 and 1900. The item referred in this article has not date or place of publication. It only states, that the content is I and II instalments.
The referred publication is inscribed in an ambitious project of mathematical encyclopedia. The language used is "latino sine flexione", a simplified version of latin invented by Peano with the intention of creating a universal language like "interlingua", a parallel intend done in several countries at that time. Peano invented also a set of mathematics symbols, some of them were adapted by Russell and others. There are a number of such symbols in modern mathematics books.
The next illustration shows part of one of the initial pages of the referred item.
The next illustration shows part of one of the initial pages of the referred item.
At the end of chapter on logic, "Vocabulario I", there are abbreviated references of the used languages: English (A, for anglo), German (D for Deutsh or G for Germano), French (F for Franco), Spanish (H for Hispano), Italian (I for Italo), Russian (R for Russo) , Latin (L for Latino). G could indicate Greek, and S for Sanskrit. It can be seen the author's intention for inclusion of all of the mentioned languages in relation with their writings. Symbolic statements are also accompanied with the described references.
The chapters included in the referred item are: Mathematical Logic, Arithmetic, Algebra, Geometry, Limits, Differential Calculus, Integral Calculus, Curve Theory. There are few illustrations; text and specialized symbolic language predominate.
It follows a brief description of each of the mentioned chapters.
Mathematical Logic (chapter I in Formulario)
The first of seven parts of this chapter deals with equal relation and logic connectives, now expressed with "then" and "and". Dots indicate separation between parts of formulae, following Leibnitz. Each symbolic statement is accompanied with examples or explanations in "latino sine flexione", naturally. As an example is transcribed the following:
Nos pote scribe primo propositione symbolico:
.1 x=x
Lege: “x aequa x”. Numero .1 es numero de propositione
Comments, examples or explanations pretend to reinforce undefined concepts or statements taken as axioms for the theory.
Next, another illustration taken from the item.
Comments, examples or explanations pretend to reinforce undefined concepts or statements taken as axioms for the theory.
Next, another illustration taken from the item.
Part two is about the concept of class (Cls) and the belonging relation of an object with respect to a class. A greek epsilon is the corresponding symbol. Inclusion and intersection between classes is defined. Same symbol serves for intersection and "and". Comments and explanations include arithmetic examples as the fact that common multiples of 2 and 3 are multiples of 6, and vice versa.
Mirror image of the usual symbol for inclusion serves for "then" between statements. Example:
For logic equivalence between two statements is used .=.
Step by step, syntactic rules for symbolic expresions are established.
Third part deals with "what" whose syimbol is the mirror image of epsilon. It corresponds to the expresion "such that"
One of the statements of this part could be paraphrased as:
If A is a class, then all of the x such that x belong to the class A is equal to class A.
In this example can be seen the reason for which Peano states that "belong" and "such that" are inverse each other.
In theis third part the intersection definition could be paraphrased as
if a and b are classes, then, x such that (x belongs to a, and x belongs to b) = a intersection b
A later statement is: What (or such that) is distributive with respect to the "and" connective.
Part four of this chapter deals with negation of a statement or the complement of a set. The corresponding symbol is a small horizontal and bold line.
Peano uses lower case letters for sets or statements, isdistinctly. This kind of ambiguity could be interpreted as an implicit reckoning to the Boolean algebra of sets and statemens.
Double negation (or double complementation) is stated as -(-a) = a
Also is stated that if a is a class, the complement of a is a class. And is stated that given two classes a, b, if a is included in b, then the complement of b is included in the complement of a.
Fifth part deals with union between sets. The same symbol is used for union and for the connective "or" between statements in the sense and/or.
Sixth part deals with "clase nullo" which corresponds to our empty set. Symbolized by an inverted V. The following is a textual statement
“clase nullo”, indica clase de objeto commune ad omni clase a. Responde ad 0 de Arithmetica.
The followig statements in today´s notation are included also in the 6th part.
Part 7th deals with "equal" symbolized with the greek iota
First statement above can be paraphrased as
ι x = the y such that y = x.
In this final part Peano clearly establishes the conceptual difference between an individual and the class which contains a single individual.
Also, it is stated that symbols iota and epsilon side by side means =.
Arithmetic (Chapter II of Formulario)
Peano is known mainly by the set of axioms related with natural numbers.
N0 symbolizes the set of natural numbers {0, 1, 2, …}.
a+ symbolizes the number after number a.
0, N0, + (next one) are undefined concepts
The statement .3 above is known as induction axiom. This axiom is a powerful tool for proving theorems about natural numbers or any other set which can be put in a one to one correspondence with them-
Using the "next" concept all of the natural numbers can be defined:
a+ symbolizes the number after number a.
0, N0, + (next one) are undefined concepts
The statement .3 above is known as induction axiom. This axiom is a powerful tool for proving theorems about natural numbers or any other set which can be put in a one to one correspondence with them-
Using the "next" concept all of the natural numbers can be defined:
1 = 0+, 2 = 1+, 3 = 2+, etc.
Addition between natural numbers can be inductively defined:
a + 0 = a
a + (b+) = (a + b)+
Also, a + 1 = a+, a + 2 = (a+)+, …
Known properties of addition between natural numbers (closure, associative, commutative, etc.) are proved by means of induction.
Multiplication between natural numbers is defined inductively too. For any natural numbers a, b, c,
Multiplication between natural numbers is defined inductively too. For any natural numbers a, b, c,
a × 0 = 0
a × (b + 1) = (a × b) + a
As for addition, properties for multiplication like a × 1 = a are proved by induction.
The power of a number, expressed in the form of a binary operation is defined inductively too. The symbol for this operation is attributed to De Morgan.
The power of a number, expressed in the form of a binary operation is defined inductively too. The symbol for this operation is attributed to De Morgan.
The corresponding properties are proved by induction. Peano employs the usual notation to
(a + b)2 = a2 + 2ab + b2,
(a + b)2 = a2 + 2ab + b2,
(a + b + c)3 = a3 + b3 + c3 + …
An addition operation between a number and a class is defined, with the correspondent properties. For instance, N1 = N0 + 1 is the class of all the natural numbers from 1 on.
This concept is used to define additive order between natural numbeers. For instance, 7 is greater than 5 means that 7 is in the class of 5 + N1. Some properties of this operation are not trivial:
Multiplicative order is defined in analogous manner. For instance, N0 × 5 is the set {0, 5, 10, 15,…}. This concept will be useful in the study of quotient between natural numbers.
Subtraction between natural numbers is defined in terms of addition. Paraphrasing,
This concept is used to define additive order between natural numbeers. For instance, 7 is greater than 5 means that 7 is in the class of 5 + N1. Some properties of this operation are not trivial:
Multiplicative order is defined in analogous manner. For instance, N0 × 5 is the set {0, 5, 10, 15,…}. This concept will be useful in the study of quotient between natural numbers.
Subtraction between natural numbers is defined in terms of addition. Paraphrasing,
b – a is the natural number x such that x + a = b, provided that b belongs to a + N0.
Properties like the following can be proved:
(b + a) – a = b,
a – (b + c) = a – b – c, provided that a belongs to b + c + N0 .
Quotient between natural numbers is defined in terms of multiplication. Paraphrasing, b/a is the number x such that x × a = b, provided that a belongs to N1 and b belongs to N1 × a.
Quotient between natural numbers is defined in terms of multiplication. Paraphrasing, b/a is the number x such that x × a = b, provided that a belongs to N1 and b belongs to N1 × a.
Properties like the following can be proved:
(a + b)/c = (a/c) + (b/c), provided that a, b belong to N1 × c.
The arithmetic chapter ends with the following concepts and the associated properties:
The arithmetic chapter ends with the following concepts and the associated properties:
Number of a class
Maximum of a class
Minimum of a class
Quotient and reminder
Number of figures of a number
Order for a number
The factorial of a number is defined inductively as follows:
0! = 1
(a + 1)! = a! × (a + 1)
Greatest common factor and least common multiple
Euler's Φ function
The next section, called Vocabulario II, is similar to Vocabulario I after the chapter on Logic. It follows the initial part of this section. Note the allusion to other languages.
Algebra (chapter III in Formulario)
There are 27 parts in this chapter. It aims to develop with rigour and extensively, various general subjects. They are listed next, whitout details:
Functions, operations, correspondence
Post functions
About belonging elements-classes
Definition of a function
Union and intersection as operators
Positive and negative numbers
Absolute value
Rational numbers
Rational relative numbers
Integer part
Proper fraction
Limit superior
Real numbers
Logarithms
Sum operator
Product operator
Difference about functions
Bernoulli numbers
Medium number between superior and inferior limits
Number of elements of a class
Superior and inferior limits
Intervals
Probability
Complex numbers
Determinants
Linear and complex function of order n
Algebraic roots
The chapter ends also with its Vocabulario, from which a part is included here:
Geometry (chapter IV in Formulario)
It consists of four parts: (1) Points and vectors. (2) Line and plane. (3) Quaternions. (4)Outer product
Part 1. It is stated that the idea of point cannot be logically defined and it comes from the physical world. Three axioms are paraphrased as follows:
p is a class
p does exist
If a belongs to p then there is another point , different from a, which does not coincides with a.
It is defined a relation between four points, incluiding a pertinent illustration.
In the so called position geometry, Peano states that if points a, b, c, d are not in the same line, then line ab is parallel with line cd and line ac is parallel with line bd. Also, points a, b coincide with points c, d by means of a translation movement, using a mechanical term.
The relation a – b = c – d, is an undefined term, where a, b, c, d are points. There are six axioms concerning this relation and a theory based in them is developed. A vector is defined as difference between two points. Other definitions concern to scalar product, module of a vector and product between a rational or irrational number and a vector. Reference is made to a publication: Principii di Geometria, whose undefined ideas are point and segment.
Part two. It deals with lines and planes. Line passing by a and is parallel to a vector u. Plane determined by a point and two vectors. Projection on a line. Translation. Symmetry. Motor:indicates movement of rigid body. Product of two symmetries. Homothecy.Sine and cosine.Coordinates.
Part three. Quaternion. The following could give some idea about.
Part four. Outer product of vectors is symbolized with α. It is defined by means of a determinant
At the end there is, as usual, the corresponding Vocabulario.
Limits (chapter V of Formulario)
This chapter has eight parts. Parts 1 and 2 deal with definition of limit, superior and inferior limits of sequences, uniqueness of limit, constant function, series, infinite products, Newton's binomial, convergence, limits with complex numbers, limits and vectors, continuity of functions. In this chapter some illustrations related with the so called Peano curve with which any plane region could be filled.
Part 3 of this chapter deals extensively with number e. Part 4 deals with number π. Part 5 is about inverses of functions log, sin, cos, tan. Part 6 develops the concept of angle. Part 7 is about the concept of rotation. Part 8 is about continuing fractions.
Differential Calculus (chapter VI of formulario)
Topics developed in this chapter are: derivative, differential, mean value theorem, L'Hospital theorem, Bernoulli-Taylor theorem, Lagrange's theorem, curvature, incremental ratios, complex functions, tangent lines and planes, normal plane, osculator, linear differential equations, harmonic movement, partial derivatives, applications to physics.
Integral Calculus (chapter VII of formulario)
Topics developed in this chapter are: circumscribed polygons, integral of a function, mean value theorem, integral with limits, methods of integration, improper integrals, Euler´s integral, integrability conditions, area, volume.
Applications to geometry and complement (chapter VIII of Formulario)
This chapter reproduces Dr. G. Pagliero's "Teoría de curvas". It develops topics as parabolas, ellipses, hyperbolas, exponentials, catenary curves, spiral, cycloid, epicycloid, cardioid, etc.A complement is included about complex numbers, differential equations, eliptic integrals and Fourier series.
The Formulario consists in mainly symbolic text with comments and historic annotations like the following.
The item of Formulario which this article is referred to has 459 pages.
