MATHEMATICS. – Arithmetic and geometry are the two great branches of mathematics from which numerous other branches have sprung, weaving a tangled web in which it is difficult to distinguish them.
OUTLINE:
I. Arithmetic
II. Algebra
III. Transformations
IV. Elementary geometry
V. Non-Euclidean geometry
VI. Projective geometry
VII. Analytic geometry
VIII. Infinitesimal analysis
IX. Series expansions
X. Functional analysis and calculus of variations
XI. Algebraic geometry
XII. Differential geometry
XIII. Probability calculus
XIV. Rational mechanics and mathematical physics.It is not easy to subdivide and list the various developments of mathematics without continuous reference being made to the others. It is no coincidence that while Plato, with his *λέγω ὅτι ὁ θεὸς γεωμετρεῖ*, recognized in God’s work a geometrization, and while our Galileo saw all of nature expressed in the language of triangles, the biblical author, with his *Omnia in numero, mensura et pondere disposuit*, recognized in the Universe a principle of arithmetization. Two different positions, reflecting, one, the Greco-Italian mentality, and the other, the Oriental mentality, both of which complement each other in describing the appearance of the Universe.
Over the centuries, the work of mathematicians, armed with the tool of refined logic, through the three fundamental judgments of belonging, exclusion, and interference, has continually discovered new horizons, with new entities and new operations, especially in the last three centuries, driven by the need to interpret and predict natural phenomena, whether mechanical, physical, chemical, or even biological and social, aided by ingenious schematization of phenomena and the creation of appropriate symbolism. Thus, with mutual enhancement, both mathematics and the knowledge of the phenomenology of the Universe have continually expanded and interpenetrated, particularly in recent times and more than ever today.
I. ARITHMETIC
It is founded on the concept of the integer, a symbol of a group of objects; from this, we arrive at the concept of a fraction, which symbolizes a part of a group: the subgroup.The association of two or more groups or subgroups of objects leads to the concept of the sum of integers and fractions. The associability of a group with a subgroup, given the homogeneous nature of both, leads to the conception of the symbols representing them (integers and fractions) under a single category: that of rational numbers.
Both the iteration of addition and the formation of a subgroup derived from another subgroup constitute a new operation: multiplication by an integer or by a fraction. Subtraction, on the other hand, is the inverse operation of addition (given the sum of two numbers and one of them, find the other), and division is the inverse operation of multiplication (given the product of two numbers and one of them, find the other). These are the four fundamental operations, so-called rational operations, because when applied to rational numbers, they yield rational numbers as results.
The iteration of the multiplication of a number by itself provides exponentiation, which is thus another rational operation; however, unlike the first four, it gives rise to an inverse operation, called root extraction, which is not always possible. That is, there does not always exist a rational number x that is the root of another number a; there do exist, however, rational numbers x which, when raised to a power, yield rational numbers that, if not identical to the given number a, approximate it as closely as desired. For example, the square root of 2 does not exist, in that no rational number, when squared, reproduces 2; yet
there do exist rational numbers which, when multiplied by themselves, yield products that approximate this value to any desired number of decimal places. This circumstance suggests the idea of conceiving numbers of a new kind: irrational numbers, each of which, no longer expressible as an integer or as a ratio of two integers (as in the case of a fraction), can only be represented by an infinite sequence of decimal places: tenths, hundredths, thousandths, and so on, i.e., by a sum of infinitely many terms that grow ever smaller.
On the other hand, subtraction also presents an exception: it is not possible to subtract a larger integer from a smaller one if the integer is thought of as representing a group. But just as root extraction becomes always possible with the introduction of irrational numbers, so too does subtraction become feasible, even when it presents an exception, through the introduction of negative numbers. Rational and irrational numbers, both positive and negative, constitute the class of real numbers.
In turn, the introduction of negative numbers creates a new exception; it is not, in fact, possible to extract the square root of a negative number unless we define numbers of yet another kind: imaginary numbers. No real number, whether positive or negative, when squared, yields, for example, -25. With imaginaries, there also arise complex numbers, which are the sum of a real and an imaginary number. By means of these complex numbers, it becomes possible not only to extract the square root of a negative number but even to extract the nth root of any number, even a complex one, subjected to this operation. These new numbers, which seem to surpass the bounds of the abstract, in fact render the most useful services, both in the purely mathematical field and in numerous applications in electrical engineering and aerodynamics.
The first three rational operations—addition, multiplication, and subtraction—are called integral rational operations. With them, one can construct a polynomial in x, that is, an expression of the type \(a x + b\), if it is of the first degree, \(a x^2 + b x + c\), if of the second degree, \(a x^3 + b x^2 + c x + d\), if of the third degree, etc. When, for example, in the last polynomial written, the numbers a, b, c, d (the coefficients) are given, and x is given, one can find the value assumed by the polynomial by means of the three operations mentioned above. The inverse problem—given the value of the polynomial, find x—constitutes the so-called solution of an algebraic equation of the nth degree, if n is the degree of the polynomial (the highest power of x appearing in it). This inversion, even if not always executable by means of formulas, is always possible and admits n solutions (as many as the degree of the polynomial), without creating numbers of a new kind, because all the solutions exist within the field of complex numbers.
Algebraic numbers are then defined as those real or complex numbers that satisfy an algebraic equation with integer coefficients. But if the equation has fractional or irrational coefficients, the solutions still fall within the field of algebraic numbers.
A new classification arises when one moves beyond rational or algebraic operations to enter the realm of operations that differ from these and are therefore called transcendental; their solutions are non-algebraic complex numbers and are thus termed transcendental. Among these is the number π, the ratio of the circumference to its diameter; it is not, in fact, a solution to any algebraic equation. But alongside the extension of the concepts of operation and number, the possibility has also been studied of obtaining, through the application of whole rational operations performed on integers, as a result, still integers. In general terms, the question has been to determine when and how certain algebraic equations, with integer coefficients, are solvable in integers. The so-called theory of numbers, which deals with such questions and is also connected to problems in other fields of mathematics, has encountered extraordinary difficulties along the way. The determination of prime integers—those divisible only by themselves—is likewise still the subject of deep study. Closely associated with the theory of numbers are the names of Fermat and Gauss.
From what has been said, it emerges that fundamental elements are integers and the fundamental operation is addition. These are the sources from which all other numbers and all other operations derive. The mathematician has often, in his reasoning, for the sake of greater generality, substituted a literal symbol for a number and thus created classical algebra; more recently, under the symbol, instead of a number, he has seen any geometric entity and has thereby extraordinarily extended the concept of algebra.
ALGEBRA. — Italy’s primacy in classical algebra dates back to the Renaissance, ranging from the names of Leonardo da Pisa, Tartaglia, and Cardano to those of Ferrari (18th century) and Ruffini (early 19th century). Having progressed from the solution of the quadratic equation to that of the cubic and then the quartic, attempts were still being made to advance further when Ruffini demonstrated the impossibility of solving, in general, by means of algebraic formulas, an equation of degree higher than the fourth, as had been achieved for equations of lower degree.
The possibility remained of solving an equation with numerical rather than literal coefficients, and numerous approximation methods were devised to obtain the solutions. These methods are now vastly surpassed by modern electronic machines. But one of the issues long presented in the applications of mathematics to physics has been the solution of systems of first-degree equations with multiple unknowns. From the 18th century onward, various methods emerged, giving rise to the theory of determinants, which facilitates the solution of such systems and also indicates whether they are solvable or not. From determinants there has recently blossomed the theory of matrices, which have proven extremely useful in many geometric treatments and have lately been applied by Heisenberg in the study of the atom.
The need to solve systems of first-degree equations with numerous unknowns has made itself felt in recent years, both in calculations of elastic structures and in the solution of the integral equations (cf. n. IX) of mathematical physics. To this end, methods have been devised with the aim of sparing the immense labor that the use of determinants would impose. In 1922, Teofilato provided one such method for obtaining solutions through successive approximations with rapid convergence toward the exact value. It is worth noting that there is no cause for alarm at the concept of approximation introduced in mathematics. After all, apart from the integer and the fraction, any irrational number—such as, for example, the square root of 2—cannot be reached except approximately; the measurement of a quantity is always subject to errors, however reducible through the progressive refinement of instruments, though not eliminable. Therefore, it should not be surprising if the solutions of a system of equations are presented with approximation, especially since in most cases the coefficients of the equations themselves are obtained through instrumental measurements.
Moreover, even a system of a hundred first-degree equations with a hundred unknowns can be handled by an electronic machine in a few seconds; however, it is first necessary to input the perhaps ten thousand coefficients appropriately to set the mechanism in motion. This, very delicate and complex, is equipped with organs comparable to what in us are the brain, the nervous system, and the muscular system, much like another remarkable mechanism found aboard aircraft, the so-called autopilot, which automatically controls flight. The first organ perceives the input of the coefficients and transmits the impulse to the second organ, which amplifies it somewhat and then passes it on to the force organ that commands the moving parts.
III. TRANSFORMATIONS
A new type of operation is the transformation, initially having a meaning limited to substitution, which is the operation that changes a set of elements into the same elements but arranged in a different order.
Linked to the theory of substitutions, as applied to the solutions of an algebraic equation, are the names of Galois, Ruffini, and Abel; and it is through this theory that the aforementioned impossibility of solving algebraically equations of degree higher than the fourth is demonstrated. From the same theory there arose the concept of a group of substitutions, whereby operating successively with two substitutions of the group yields the effect that would be obtained with a single substitution of the same group. If one then moves from a finite number of elements to an infinite number, there arise transformations, which Lie studied deeply toward the end of the last century. The movement of a solid with respect to the three edges of a room constitutes a simple example of a transformation, in that to every point of the solid in its original position there corresponds a point in the second position.
With the transformation there then interpenetrates the concept of group and of entity invariant under the transformation itself. For the important contributions made by Italians to these studies, the names of Luigi Cremona, Ernesto Pascal, Luigi Bianchi, Guido Castelnuovo, and Corrado Segre stand out. The theory has assumed enormous importance because of its capacity for grand synthesis, both in geometry—where, for example, it is found that projective geometry (cf. n. VII) is a chapter of the theory in question—and in physics, because all electromagnetic phenomena are invariant under the particular Lorentz transformation.
One is tempted to think that every geometric or physical element is linked to an invariant and, conversely, that the discovery of an invariant indicates the existence of a geometric or physical entity.
The concept of transformation and invariance pervades all of mathematical physics, in its most varied fields.
Even in gas dynamics, a science born in recent years, the invariance of the equation governing the motion of compressible gases with respect to a suitable transformation allows the effect of high velocities in air to be translated into the effect of small velocities in tiny water channels, with great ease of study and minimal expenditure.
This is, in the end, a particular case of the method widely used in science: transforming a problem into another that has already been studied or is easier to handle.
IV. ELEMENTARY GEOMETRY
Elevated to great heights by the Greeks, elementary geometry remains indelibly inscribed in the compilation of Euclid. In the plane, it is limited to the study of figures formed by straight lines and circles (constructible, therefore, with straightedge and compass), and in space to solids bounded by plane faces, or even to the three round solids (cylinder, cone, and sphere). It later came to consider other lines (conchoids, cissoids, strofoids, etc.) and subsequently addressed measurement, which, precisely because of its practical aims, was initially less esteemed by the Greek mentality—refined in speculation and aristocratically devoted to the pursuit of truth. Measurement was later extensively addressed by Archimedes, with his elegant theorems on pyramids and round solids, to which he added other no less important discoveries concerning the lever, the movable pulley, the screw, mirrors, and the buoyancy of floating bodies.
Not all plane problems were immediately solved with the aid of the two celebrated instruments, straightedge and compass. Some required extensive research, such as the golden section of the radius, the side of a regular decagon inscribed in a circle; others were solved approximately, such as those concerning the rectification and quadrature of the circle (the search for a segment equal in length to the circumference and the search for a square equal in area to the circle). The need to resort to approximation stemmed from the limitations of the instruments used.
About two centuries ago, Mascheroni, in his *Geometria del compasso*, demonstrated that all problems solvable with straightedge and compass can also be solved with the compass alone. However, it was the merit of the past century to highlight that the two instruments mentioned can only solve those problems which, when translated into algebraic terms (as will be seen more clearly below), give rise to second-degree equations or those reducible to them. From the Greeks up to Apollonius, with his theorems on conic sections (ellipse, parabola, and hyperbola), about thirteen centuries elapsed without geometry making significant progress. It is necessary to look to the Renaissance to encounter Leon Battista Alberti, Leonardo da Vinci, Piero della Francesca, Vignola, Brunelleschi, Ubaldi, and dal Monte, who, with their studies on perspective, expanded the field of inquiry beyond the Greeks. Perspective then, between the late 18th and early 19th centuries, blossomed into descriptive geometry, which, through appropriate representation on the drawing sheet, allows all the metric details of a spatial figure to be recovered.
Descriptive geometry, which at the beginning of the last century had passionate adherents in Naples (Flauti, Fergola, Trudi, Padula), later inspired Bellavitis of Padua to found the theory of equipollences, a method useful in solving graphical problems. From this arose the modern theory of vectors, which in our day has attained a high degree of generality and synthetic power, with connections even to the aforementioned theory of matrices.
V. NON-EUCLIDEAN GEOMETRY
Elementary geometry long carried with it an element of doubt, arising from the postulate known as Euclid’s: *through a point, one and only one parallel can be drawn to a given line*. As early as the 17th century, Wallis and the Jesuit Father Saccheri in the 18th century had addressed the question of the postulate. The former reduced it to another postulate that seemed to him more acceptable, while the latter, through the opposite hypothesis, sought to prove its absurdity. Saccheri, however, did not achieve the final goal, despite important results, and thus the question remained open.
Following the path traced by Saccheri, Bolyai and Lobachevsky resumed the study. Assuming, therefore, that through a point, two parallels can be drawn to a given line—one on each side—converging toward the line without ever meeting it, a marvelous logical edifice was constructed. Yet, proceeding through deduction after deduction, no contradiction with the initial postulate was ever reached. Thus arose the suspicion—endorsed by the authority of Gauss and Riemann—that Euclid’s postulate might be undemonstrable. Gauss’s celebrated study *Disquisitiones circa superficies curvas* suggested to Riemann the extension of these considerations to space. By assuming for the distance between two points a form differing from that given by the Pythagorean theorem and based on Euclid’s postulate, Riemann revealed the possibility that no parallel lines to a given line exist—a situation analogous to that on a sphere, where any two meridians (which, on the sphere, are like straight lines, being lines of shortest path) always intersect (at two points).
While Wallis had reduced the Euclidean hypothesis to that of the existence of similar triangles, Lobachevsky, with the hypothesis of the two parallels, verified the non-existence of similarity, such that two triangles with respectively equal angles are congruent, as also occurs on the sphere for spherical triangles (formed by arcs of meridians). Indeed, in Lobachevsky’s geometry, the sum of the angles of a triangle is less than a straight angle, and the deficiency is proportional to the area of the triangle; in spherical triangles, the sum of the angles is greater than a straight angle, and the excess is proportional to the area. The idea gradually emerged that all three geometries—Euclidean, Lobachevskian, and Riemannian—could be logically valid simultaneously. The question then became which of the three was actually verified in the physical world. It was necessary, therefore, to measure the angles of large triangles and evaluate their excess or deficiency in proportion to their size. The results showed a sum of angles close to a straight angle, now greater, now less, following the so-called law of errors. It was concluded that in the physical world, Euclidean geometry had the highest probability of holding true compared to the other two, each of which, if ever verified, would deviate so slightly from the Euclidean as to be imperceptible.
Indeed, only after the great scientific evolution of this century, extending beyond the limits of ordinary experience, has it been observed that when passing to the world of the atom or to that of nebulae, Euclidean geometry no longer perfectly accords with the physical world. Instead, a pseudo-Riemannian geometry (the geometry of spacetime) must be considered valid, though it is not our task to dwell on it here. Space should be regarded as unbounded yet finite, much as a being with only two dimensions would perceive the surface of a perfectly smooth and very large sphere on which it could move.
In any case, as early as 1865, the Gordian knot binding geometry to such grave doubts was definitively cut by Eugenio Beltrami, who succeeded in proving the impossibility of demonstrating Euclid’s postulate. On the other hand, if the Universe is finite, its dimensions must be determinable; and just as a being living on the aforementioned sphere, if it could perceive the curvature of even a small part, could also evaluate the entire extent without traversing it all, so too should this be possible for us. Indeed, on this basis, the estimation of the dimensions of the Universe—at least as an order of magnitude—has been carried out, and we now have an idea of it comparable, by approximation, to what Eratosthenes might have conceived regarding the dimensions of the Earth through his measurements.