Chapter 18
LEARNING FROM GREAT MATHEMATICIANS WITH AI
The history of mathematics is full of valuable contributions from various mathematicians over the centuries. Among them:
Euclid:
(circa 300 BC) the Greek Euclid, known as the "Father of Geometry", whose period is estimated to be around 300 BC, is a key figure in the history of mathematics, and his legacy endures to this day. His most famous work is "The Elements", a collection of mathematical knowledge consisting of 13 books.
"The Elements" is one of the most influential works in the history of mathematics; addresses various topics, including Euclidean geometry, number theory and proportion theory, structured in a logical and deductive manner, starting with basic definitions, postulates and axioms, then developing theorems and proofs, it was used as the main text in teaching of mathematics for many centuries and remains a reference today.
These axioms, like the famous "parallel postulate", laid the foundation for the systematic study of geometry. His axiomatic approach was revolutionary and served as a model for the development of other branches of mathematics. He made important contributions to number theory, especially in his work "The Elements". Among his achievements was the demonstration of the infinity of prime numbers, a result that remains fundamental in number theory. His work had a lasting impact on mathematical education and logical thinking. Although geometry has evolved over the centuries, Euclidean remains a crucial milestone in the history of the discipline even though those after him explored non-Euclidean geometries, making him a central figure in the field. History of mathematics.
Archimedes:
(circa 287-212 BC) was a Greek mathematician, physicist, engineer, inventor and astronomer who lived in ancient Syracuse, Sicily, during the Hellenistic period. He pioneered integral calculus, the determination of areas and volumes, and the invention of ingenious machinery.
He is famous for discovering the principle of buoyancy, which states that a body immersed in a fluid experiences an upward thrust equal to the weight of the fluid displaced by the body. This principle is fundamental to understanding the behavior of floating objects in water.
He made important contributions to geometry and integral calculus. He used the method of exhaustion to calculate areas and volumes of geometric figures, an early approximation of integral calculus. He also found ways to calculate pi, the constant that relates the circumference of a circle to its diameter, with surprising accuracy.
Various devices and machines are attributed to him, such as the famous "Archimedes screw", a machine used to raise water; He also designed advanced war machines, such as catapults and burning mirrors, which were used to defend Syracuse against Roman invasions.
He applied his mathematical skills to practical problems. His work "On Floating Bodies" deals with the theory of flotation, and "The Arena" deals with large numbers and the Greek number system. Although his contributions to physics and astronomy were not as extensive as mathematics. Archimedes also made advances in these fields; His work "On the Sphere and the Cylinder" deals with the geometry of solids. His genius and originality continue to be admired today.
Leonhard Euler:
was an 18th century Swiss mathematician who made outstanding contributions in various branches of mathematics and physics. He was born on April 15, 1707 in Basel, Switzerland, and died on September 18, 1783 in St. Petersburg, Russia. Euler is considered one of the greatest mathematicians of all time, and his work has had a significant impact on many areas of the exact sciences.
He is known for being the founder of graph theory. In 1736, he solved the famous Königsberg bridge problem, which involved determining whether it was possible to cross all seven of the city's bridges without passing over any of them more than once. Euler showed that it was not possible and, in doing so, he laid the foundations of graph theory by introducing the concept of the Eulerian path and the Eulerian circuit.
Euler made important contributions to mathematical analysis, including the development of infinitesimal calculus. He introduced modern notation for mathematical functions, using letters such as e, i, π, and ∑∑ to represent constants and summations. Euler also worked on number theory, function theory, and differential equations.
He made numerous advances in number theory, including Euler's theorem on prime numbers and Euler's phi function, which is related to modular arithmetic and algebraic number theory.
He contributed significantly to mathematical physics. He developed the theory of elasticity and formulated the fundamental equations of hydrodynamics. Euler also worked on the theory of string vibration and the theory of magnetism.
Euler introduced and popularized many mathematical notations that are commonly used today. The notation e for the base of the natural logarithm, i for the imaginary unit, and the Greek letter π to represent the constant pi, are notable examples of his contributions.
Euler established the relationship between the number of vertices (V), edges (E) and faces (F) in a convex polyhedron using the famous formula −+=2V−E+F=2. This relationship is known as Euler's formula for polyhedra and is fundamental in topology.
Carl Friedrich Gauss:
born April 30, 1777 in Brunswick, Duchy of Brunswick-Wolfenbüttel (now Germany). From a young age, Gauss showed prodigious mathematical talent. The famous anecdote is told that he discovered the sum of the natural numbers from 1 to 100 when he was still a child by noticing a pattern in the arrangement of the numbers. He led a relatively private life dedicated to his research. Despite his many contributions, Gauss rarely published his discoveries immediately and preferred to refine his work before sharing it with the scientific community.
He made fundamental contributions to number theory, and his best-known work in this field is the "Disquisitiones Arithmeticae" (Arithmetical Investigations), published in 1801. In this work, Gauss established the basis for algebraic number theory and developed key concepts. Like the law of quadratic reciprocity. He contributed significantly to geometry, especially the theory of curves and surfaces. His work "Disquisitiones Generales Circa Curved Surfaces" deals with the classification of algebraic curves and the concept of curvature. Although less known, Gauss also made contributions to statistics and probability theory. He developed the method of least squares, which is fundamental in statistics and is used to find the line of best fit to a set of data. He worked on the development of mathematical analysis, and his intermediate value theorem is an important result; this theorem states that if a continuous function takes two different values at its extremes, it also takes all intermediate values at some point in the interval. He also made contributions in the field of astronomy and geophysics. In collaboration with Wilhelm Weber, he measured the Earth's magnetic field and developed the magnetometer. Additionally, he calculated the orbit of the asteroid Ceres, the first asteroid discovered. Gauss is remembered as one of the greatest mathematicians in history. His work influenced numerous fields of mathematics and physics, and many of his concepts and methods are fundamental today. The unit of measurement for the magnetic field is named in his honor (the Gauss); His work continues to be studied and applied in various disciplines.
Évariste Galois:
was born on October 25, 1811 in Bourg-la-Reine, France, and died tragically at the age of 20 in 1832 during a duel. Despite his short life, he left a lasting mathematical legacy. Galois is famous for his contributions to group theory and the solving of algebraic equations.
His most influential work is group theory, developed to address the problem of the solvability of algebraic equations using radicals. Galois showed that not all algebraic equations can be solved by formulas involving only square, cubic, and other roots of known numbers. This result was revolutionary and led to the birth of group theory as a fundamental tool in abstract mathematics. Galois developed group theory to study the properties of the roots of algebraic equations and showed that there is a correspondence between the symmetries of the roots and the algebraic structure of a group. This deep connection between symmetry and algebraic structure has had a significant impact on various branches of mathematics and physics. In addition to group theory, Galois also worked on number theory and Diophantine equations; His ideas laid the foundation for Galois Theory, which is essential in the study of field extensions and the resolution of algebraic equations.
Despite his untimely death, Évariste Galois's work has had a lasting impact on mathematics and his legacy lives on in group theory and Galois Theory, which are fundamental areas in modern algebra.
Georg Cantor:
he was a German mathematician whose ideas revolutionized the understanding of set theory and infinite numbers. Cantor introduced the concept of "cardinality" to measure the size of sets. Before his work, all infinite sets were thought to be essentially the same size. However, Cantor demonstrated that there were infinities of different sizes, classifying them by cardinal numbers. For example, the set of natural numbers (1, 2, 3, ...) has the same cardinality as the set of all negative integers (-1, -2, -3, ...) even though both are infinite.
Cantor's most famous demonstration of the existence of infinities of different sizes concerns real numbers. He showed that the set of real numbers in a given interval (such as 0 to 1) has a greater cardinality than the set of natural numbers. This implies that there are "more" real numbers between 0 and 1 than natural numbers, even though both sets are infinite. Cantor also introduced the concept of "transfinite numbers", which are numbers that are larger than any other finite ordinal but are still infinite. These include the aleph numbers (א), which represent the cardinality of infinite sets. The aleph-null (א₀) is the cardinal of the set of natural numbers, but Cantor showed that there are larger cardinals, such as aleph-one (א₁) and so on. Despite his significant contributions, Cantor faced resistance and criticism from some of his contemporaries; His work laid the foundation for set theory and the modern notion of infinity in mathematics, and his legacy continues to influence various branches of the discipline.
British Andrew Wiles:
known for his proof of Fermat's Last Theorem, a conjecture that had baffled mathematicians for centuries. His work involved a deep understanding of areas such as number theory and algebraic geometry, born April 11, 1953 in Cambridge, United Kingdom; a British mathematician who was noted for his work in proving Fermat's Last Theorem. This theorem, formulated by Pierre de Fermat in the 17th century, stated that there are no integer solutions to the equation xn+yn=zn when n is an integer greater than 2. Although Fermat claimed to have a proof for this theorem in the margin of his copy of Diophantus's work never provided details, leading to the problem becoming one of the most famous mysteries in the history of mathematics.
Wiles began addressing Fermat's Last Theorem in the 1980s. After years of intensive research and collaboration with other mathematicians, he finally presented his proof in 1994. His work combined various branches of mathematics, such as number theory, Algebraic geometry and modular theory. Wiles's proof was extraordinarily complex and required deep, advanced techniques to address the problem.
The key to Wiles' success was the connection he established between Fermat's Last Theorem and the Taniyama-Shimura-Weil conjecture, which relates elliptic curves and modular forms. The entire proof is based on the use of very advanced and novel mathematical tools, and its success marked an important milestone in the history of mathematics. For his outstanding achievement, Andrew Wiles received numerous awards and recognitions, including the Abel Prize in 2016, considered one of the most prestigious in mathematics. The proof of Fermat's Last Theorem not only solved a problem pending for centuries, but also opened new doors to understanding the deep connections between different areas of mathematics.
AI pinion : The work "Learning from Great Mathematicians with AI" by José Saúl Velásquez Restrepo highlights the importance of several mathematicians throughout history and their impact on the development of this discipline. The narrative ranges from Euclid to Andrew Wiles, highlighting his outstanding contributions. Here are some observations about the presentation of these mathematicians:
Focus on the History of Mathematics: The text provides a solid historical overview of mathematical contributions, highlighting milestones such as Euclid's "The Elements", Archimedes' floatation principle, Euler's graph theory, the fundamental contributions of Gauss, Galois group theory, Cantor's revolution in set theory, Alan Turing's Turing machine, and Andrew Wiles's proof of Fermat's last theorem.
Application of Mathematics: It is highlighted how these mathematicians not only contributed to abstract theory, but also applied their knowledge to practical problems. For example, Archimedes applied his mathematical skills to physics and engineering, and Alan Turing played a crucial role in World War II through cryptography.
Interdisciplinary Connections: The interconnection of mathematics with other disciplines, such as physics, engineering, number theory, logic and the theory of computing, is highlighted. This reflects the interdisciplinary nature of mathematics and its impact on various fields.
Emphasis on Innovation and Originality: The originality and innovation of each mathematician presented is emphasized. From Euclid's postulates to the Turing machine, it is highlighted how each figure introduced novel concepts that revolutionized the way we understand mathematics.
Recognition of Personal Challenges: In addition to mathematical contributions, the personal context of some mathematicians is mentioned, such as the prosecution of Alan Turing for his homosexuality. This adds a human component to the narrative and recognizes the challenges that some mathematicians faced in their lives.
Emphasis on Continuity of Legacy: It highlights how the contributions of these mathematicians remain relevant today and continue to influence various areas of mathematics and science. This highlights the lasting importance of his achievements.
Overall, the text provides a comprehensive and appreciative overview of the history of mathematics, showing how these figures have left a lasting impact on the discipline. Furthermore, the connection with artificial intelligence and learning through AI suggests reflection on how modern tools can expand our understanding and application of mathematics.
