II.2 1800 Shift of Focus, Bolyai and Lobachevskii, Acceptance of Non-Euclidean Geometry, Convincing Others, Looking Ahead, II.3 The Development of Abstract Algebra, Old Babylon and Hellenistic Era, The Medival Islamic World, The Latin West, Algebra is Born, The Search for Roots of Algebraic Equations, Exploring the Behavior of Polynomials in n Unknowns, The Quest to Understand the Properties of Numbers, Modern Algebra, II.4 Algorithms, What is an Algorithm?, and Abacists and Algorists, Pages 88 – 105.

Gauss was suspicious of the parallel postulate and thought that there might be another geometry other than Euclidean.  Geometry was divided into the cases were the angles of a triangle summed to greater than two right angles, called G, equal to two right angles, called E, and less than two right angles, called L.  The E case is Euclidean geometry.  The G case was proved to not exist.  The L case loomed.

The fame for discover of non-Euclidean geometry goes to Bolyai (1831) and Lobachevskii (1829).  Analogous to the circle was the L-curve of Bolyai and the horocycle of Lobachevskii.  The arguments were both made in 3 dimensional geometry.  Here Bolyai described the F-surface and Lobachevskii the horosphere.  The question became which geometry is “true”?  During their lifetimes their ideas were mostly neglected.  Gauss did not support the ideas during his lifetime.

When Gauss died in 1855 unpublished papers showed support for the ideas of Bolyai and Lobachevskii.  Gauss’s student Riemann advanced the ideas through his development of manifolds.  Riemann died in 1866 and by this time Eugenio Beltrami had also developed the same ideas.  Minding developed a pseudosphere that had constant negative curvature.  This was rediscovered by Liouville and developed further by Codazzi.  In 1871 Klein developed these ideas further.  Klein became a full professor at age 23.

Klein moved the direction away from the figures of geometry to the rigid body transformations of rotation, translation and reflection.  Poincare reformulated Beltrami’s disk model to make it conformal meaning that angles in non-Euclidean geometry were represented by the same angles in the model.  Augments ensued and it was postulated that one could not determine whether or not light rays were straight or angles did not sum as expected.  Poincare stated that in a particular situation there were two possible conclusions:  light rays are straight and the geometry of space is non-Euclidean or light rays are somehow curved and and space is Euclidean.  It was decided there was no logical way to choose between these positions.  This was called conventionalism.  Federigo Enriques was a prominent critic of conventionalism.  A discussion ensued on whether there was a difference and if it mattered that the law of gravity could not be altered but masses could be moved and it studied.  He argued that a one could decide whether a property was geometrical or physical by seeing whether we had any control over it.

In 1899 Hilbert axiomatized geometry.  In 1915 Einstein proposed his general theory of relativity which is in a large part a geometric theory of gravity.  He used Riemann’s work on manifolds to describe gravity as a kind of curvature in the four-dimensional manifold of spacetime.

Abstract algebra started with  he usual high school variables and constants represented by letters of the alphabet and morphed into the study of groups, rings and fields by the research mathematician.   Abstract algebra includes the abstract structures of groups, rings and fields defined in terms of a few axioms and built up of substructures, such as, subgroups, ideals and subfields.  Also included are maps such as homomorphisms and automorphisms.  In what follows is the relationship between high school algebra which includes the analysis of polynomial equations and modern or abstract algebra.

First and second degree polynomial equations can be found in old Babylonian cuneiform texts of around 2,000 B.C.  The solutions of these texts always followed a geometric interpretation of the equations.  Euclid and Archimedes continued along this geometric solution idea.  Diophantus had a text called Arithmetica.  In this text any solution was regarded as the answer and he did not always follow a geometric interpretation of a solution method.

Al-Khwarizimi in Baghdada advanced many of the old mathematical ideas.  The title of one of his books contained the word for completion “al-jabr” from which the word algebra was passed down to us.  He made explicit the relationships between geometric areas and lines interpreted in terms of multiplications, additions and subtractions.  This suggested a move away from geometric solutions and towards algebraic solutions of problems.

Around 1100 A.D. in a book entitled “al-jabr” Omar Khayyam analyzed a form of the cubic equation.  Al-Karaji synthesized this work and the work of Diophantus.  He and Bragmagupta four hundred years later had techniques for finding integer solutions to Pell’s equations.  Pell’s equations are equations of the form ax2 + b = y2, where a and b are integers and a is not a square.

After the fall of the Roman Empire in the Latin West, Fibonacci, in Italy around 1202, presented Al-Khwarizmi’s work almost verbatim, the book was called “Liber Abbaci”.  In 1494 the Italian Luca Pacioli published a compendium of all known mathematics.  It is one of the earliest printed mathematical texts.  This book highlighted the question ” Could algorithmic solutions be determined for the various cases of the cubic?”  Cardano and his student Ludovico Ferrari answered this in the affirmative for the cubic and extended it to the quartic.

Algebra was popularized with the translation of Diophantus’s “Arithmetica” in the 1560’s.  Viete in 1591 was concerned with the law of homogeneity.  He was concerned about the units of his numbers.  He also used letters to designate the numbers.

After Viete both Fermat and Descarte removed the law of homogeneity problem.   Descarte understood that there were n roots to the n-th degree polynomial, this is called the fundamental theorem of algebra.  Descarte highlighted the fundamental theorem of algebra and the solution of polynomial equations of degree greater than four.  D’Alembert, Euler and Gauss were all concerned with the fundamental theorem of algebra.  

The search for roots of algebraic equations propelled algebra through the 1700 and 1800’s.  This search advanced the use of radicals and the idea of solubility by radicals.  Quadratics, cubics and Quartics had been solved.  The insolubility of the quintic was shown Abel in the 1820’s.   Modular arithmetic was developed by Gauss who in 1801 showed that the 17th root of unity was constructable.  He also developed cyclic groups and the idea of a primitive element or generator of a group.

Galois around 1830 formulated a theoretical process to determine whether or not an equation was solvable.  He used fields and groups and included automorphisms and the idea of invariance.  In 1893 Heinrich Weber gave abstract definitions for the ideas of a field and a group.

Liebniz was interested in equations with multiple unknowns.  He used the determinant as later call it.  The determinant is associated with an n by n square array of number or matrix.  Cramer, Vandermonde and Laplace also studied determinants.  Sylvester originally developed matrices to linearly transform variables.  Cayley explored these and eventually developed the area of vector spaces.  Sylvester coined the term invariant which was further used by Boole.  This area was further developed by Gotthold Eisenstein, Hermite, Otto Hesse, Paul Gordon and Alfred Clebsch.  Hilbert in the 1880’s and 1890’s advanced their ideas and shifted the emphasis to abstract modern algebra.

It is the work of Hilbert that was used in my PhD thesis to develop multidimensional smoothing splines.

In the 6th centry BC Pythagoreans had defined perfect numbers a positive integer which is the sum of it divisors.  6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7+ 14.  Complex numbers were used by Cardano and Bombelli in the sixteenth century.  In the 17th century Fermat claimed that he could prove that xn + yn = zn for n an integer greater than 2 had no nontrivial solution in the integers.  This is known as Fermat’s last theorem.

In the 1770’s Gauss had used a type of complex number with integer coefficients to prove  Fermat’s last theorem with n = 3.   He extended this to integer coefficient complex numbers in the 1820’s.  These are called Gaussian integers and are closed under addition, subtraction and multiplication.  He also defined notions of unit, prime, and norm in order to prove an analogue of the fundamental theorem of arithmetic.

Quaternions were developed by Hamilton in the 1830’s.  The quaternions are not commutative and represent 4 dimensions.

Further generalizations to n dimensional matrices and vector spaces were developed.

Th classification of finite simple groups inspired set theoretic and axiomatic work of Cantor and Hilbert.  In 1930 the classic textbook “Moderne Algebra” by van der Waerden describes modern algebra that continues to characterize algebraic thought today. 

A precise definition of algorithm is not available but it means rule, technique, procedure or method.  A formal definition was achieved in the 20th century.

Abacists are ones that use an abacus to calculate.  The Chinese counting frame is an abacus.  Our number system today (base 10) is positional rather than additive (such as the Roman numerals).  The arithmetic operations of the decimal number system were labled algoritmus.  Algorists used these algoritmus methods to perform calculations.

The origin of the word algorithm is Arabic and it came from a distortion of the name La-Khwarizmi who is the author of the oldest known work on algebra in the 9th century.  The title translates to: “The compendious book on calculation by completion and balancing”.  This title gave rise to the word “algebra” from the word “al-jabr” from the Arabic title “al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala”.

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