How Algebra became abstract: George Peacock & the birth of modern algebra (England, 1830)

In this article we look at the ideas of George Peacock whose 700-page opus A Treatise on Algebra (1830) transformed classical algebra into its modern form as an abstract symbolic science, free from the physical interpretation of quantity that had previously restricted it.

Peacock’s Treatise laid the logical foundation for the new science (ch1-7) and rebuilt on this foundation the entire mathematical corpus of the 17th & 18th centuries (ch8-17) including combinatorics, probability, binomial theorem, polynomial roots, primitive roots of unity, complex numbers, trigonometry, the algebraization of geometry, logarithms, and simultaneous equations.

The context for Peacock’s work was the vigorous debates then ongoing at Cambridge on the logical soundness of negative and imaginary numbers, and serious calls to abolish negative numbers from the teaching of algebra in the university, an outcome which for obvious reasons would seriously cripple the ability for English students to advance. Peacock’s motivation was to remove the reason for the debate, by separating universal arithmetic from symbolical algebra, as different subjects entirely

Peacock’s emphasis on uninterpreted operations as the foundation for algebra was highly influential to a generation of British logicians (Boole, De Morgan, Jevons, Venn)and algebraists (Cayley, Hamilton, Sylvester). These would go on to develop symbolic logic, group theory, quaternions, and matrix analysis. Over the next fifteen years Peacock rewrote and re-issued his Treatise in two separate volumes (1842/1845) which then formed the basis of English algebra till the first quarter of the 1900s.

1. Historical context

Algebra in England prior to the 1800s had been an extension of Arithmetic with operations that had physical interpretations. While symbols could be substituted for specific quantities, they were required to be checked at each step in a derivation to ensure they remained valid (e.g. respecting that a greater cannot be subtracted from a lesser, etc.). This was a continuation of the physical interpretation of arithmetic and algebra from its first emergence in Babylonian algebra consisting of cut-and-paste geometric operations with physical meaning (c.2500 BCE) (Hoyrup, 2013, pp.13-20) and Egyptian algebra of approx. the same period Rhind Papyrus where x is denoted by the sign for heap, through to the Muslim mathematician al-Khawarismi (c.830 CE) who designated the subject al-jebr wa al-muqabala, or restoring and balancing quantities, or “al-jebr” (algebra), through to its arrival into Europe through Latin translations of al-Khawarismi in the 1200s (e.g. Fibonacci’s Liber Abaci of 1202 AD). Yet the requirement for physical interpretation of quantity persisted, with negative roots of equations were rejected as fictitious (Cardan 1500s), false (Descartes 1600s), meaningless (Newton 1600s), false (Carnot 1700s), and inconsistent or vague (De Morgan 1800s). [Mpantes]

Algebraic considerations began to broaden during the 1500s based on the discoveries of the Italian wranglers Tartaglia, Cardano, Ferrari who found the cubic and quartic formulas, using imaginary numbers, and on Bombelli (1572) who first interpreted them. From here, Algebra developed a symbolic virtuoso stream in the 1600-1700s that led to the theory of series (Gregory, Taylor, MacLaurin), the differential calculus (Barrow, Newton, Leibniz), and then to the prolific Euler in the mid-1700s whose use of uninterpreted algebraic symbols led him to far-reaching discoveries in every aspect of mathematics, in particular the algebraization of geometry through the remarkable formula , unifying trigonometry through infinite series and symbolic substitution of .

2. The rise of the symbolic approach and “the principle of permanence of form”

The uninterpreted symbolic approach based upon the “principle of permanence” began to appear in the mathematics of the German/Swiss school beginning with Leibniz (who called it the “law of continuity“) and continuing through his mathematical descendants (Leibniz, the three Bernoullis including Johann, Euler, Lagrange, and Laplace).

But this “mathematics by suggestion, by inspiration” generated concerns when preparing the subject for exposition or teaching. Indeed, Leibniz’s own exposition of his infinitesimal calculus, a symbolic approach, caused considerable difficulty to the Bernoulli brothers Jakub and Johann who were its first serious students.

In the 1800s, across Europe, the logical basis for using the symbolic mathematics that had been so successful in the 1700s, was coming under greater scrutiny. In Paris, Cauchy’s attempts to make the calculus rigorous led to his rejection of what he termed “the generality of algebra” principle, and search for a more rigorous foundation that culminated in his revision of the theory to rebuild it upon the concept of limit.

In Cambridge, the soundness of Algebra was questioned with controversy on whether number should be restricted to positive numbers, and whether negative and imaginary numbers should be abolished altogether. On the side opposed to their use were Robert Simson, Francis Maseres, and William Friend. Fundamentally, the opposition to the idea of negative and imaginary numbers was founded on viewing number as based on the geometric concept of length, i.e. a concept of quantity that can be observed, dating back to classical Greek mathematics. [Mpantes/20xx] Numbers were not accepted until their correspondence with a length could be shown e.g. the construction of the irrational as diagonal of a unit square. On this basis negative numbers had no reality, except by a semantic addition of directionality or debts, and certainly not imaginary numbers. [Phillips/20xx]

On the symbolic side of this debate, several mathematicians in England contributed including Woodhouse, Babbage, and Bromhead. Adrien Buee, a French emigre priest escaping the Revolution, musical compositor, and mathematical enthusiast proposed the interpretation of as a measure of perpendicularity, i.e. taking a geometric approach toward normalizing the meaning of imaginary numbers. John Warren took a similar geometric interpretation of introducing the work of Caspar Wessel and Jean Argand.

But perhaps the most influential of the proponents of symbolic mathematics was George Peacock, himself a Cambridge educated mathematician, who proposed a fundamental shift in algebra toward an independent symbolic foundation with emphasis on the algebraic operations. This was to become the viewpoint of modern abstract algebra, and indeed of modern mathematics generally. Central to his development of algebra was a committed embrace of the “principle of permanence”, which he called “the permanence of equivalent forms”. Peacock went further than his predecessors by presenting the entire corpus of arithmetical and algebraical mathematics of the time in a single text, built fundamentally on the new symbolic/operational foundation that would find final expression in the group, ring, and field axioms of modern abstract algebra at the hands of Cayley (groups) and others.

Among these others was Sir William Rowan Hamilton who although he was vocally opposed to what seemed to be the arbitrariness of the new approach as opposed to the self-evident truths that had guided mathematics since Euclid, ironically he was the first to use the freedoms conferred by uninterpreted symbolical algebra to create in 1843 his quaternions, a new (4-dimensional) algebraic system that extends the complex numbers and can model 3-D space transformations. In 1850, Cayley laid down the mathematical structure for matrices and matrix algebra which is associative but not commutative, and in 1854 formulated the modern notion of abstract groups.

3. A closer look at Peacock’s Treatise on Algebra, 1830

In the Preface his Treatise Peacock attempts to express the ideas that inspired his 730 page work. He seeks to justify and motivate his novel approach to the familiar subject matter of algebra. But, the preface is 30 pages of slightly verbose prose, with Peacock appearing overwhelmed by the significance of this achievement, its novelty, its strangeness, and appears to have still been coming to terms with the right way to make it acceptable to his academic peers as well as accessible to the wider public. I have attempted a compression of Peacock’s preface into two pages , (PDF). The below is the first page. I paraphrased freely but have attempted to remain true to his arguments and his writing style. Where commentary has been added, this is indicated in brackets [ ].

George Peacock, 1830 Preface to Treatise on Algebra (compressed and paraphrased):

[1] ALGEBRA HAS MISTAKENLY BEEN CONSIDERED AS AN UNRESTRICTED SYMBOLICAL ABSTRACTION of Arithmetic in which the operations of arithmetic are transferred to Algebra without any need to re-state either their meaning or application. The symbols of Algebra are assumed to be the general and unlimited representation of all quantity, and the operations of Addition and Subtraction (denoted by signs + and -) able to be freely used in connecting such symbols with each other in any symbolic expression, without any restrictions or modification to their meaning or scope, and similarly for Multiplication and Division. (p.vi)

But this is not the case, and there are problems that arise with the unrestricted usage of symbols to represent unspecified quantities with arithmetic operations that have precise and restricted meaning. (p.vii) Firstly, + and – need be confined to quantities of the same kind. [For example: it is not correct to add 2 eggs + 1 dozen eggs, or to add lines and areas, or times and velocities.] Secondly, subtraction is valid only when the first number is greater than the second. [Example: 3 – 5 is not valid, as one cannot take away 5 from 3 without making explicit some other concept, such as a debt of 2.] (p.vii)
A third challenge is that of unrestricted substitution. Consider:
c := (a+b), and a-c.
Both expressions look valid. But substituting gives:

which is only valid if a>b. This restriction is not apparent in the 2nd expression. (p.ix)

[2] WHAT WE WILL DO IN THIS TREATISE IS TO FREE THE SYMBOLS + AND – FROM THEIR RESTRICTIONS IN ARITHMETIC and establish them as new symbols with no meaning until we have indicated this, i.e. they must have meaning that is independent of any laws of arithmetic in order to avoid the inconsistencies which have led to the critique of Algebra so far. (p.ix) In this way, this work aims to show that Algebra can be a demonstrative science, and to resolve these recent critiques of the logical imperfections in Algebra as it is currently explained. (p.v-vi)

[3] THE KEY OBSERVATION IS THAT THE CONNECTION OF ALGEBRA TO ARITHMETIC IS *BY CONVENTION*, and not mathematically nor logically necessary, i.e. Algebra has been inspired by suggestion by Arithmetic, not founded upon it. The operations and laws of symbolical combination are assumed, not arbitrarily, but with a general reference to their anticipated interpretation within a subordinate science. An important consequence is that the laws for combination of symbols are completely separated from their interpretation. (pp.xx)

[4] WHEN CONSIDERING THE INTERPRETATION OF OPERATIONS, WE MUST ONLY LOOK AT THE FINAL RESULT, and not look to interpret every intermediate part of the transformation process, for while the connection between successive forms is algebraically necessary, the algebraic laws that govern them are independent of their interpretation. In this way Algebra becomes accommodated to the form and peculiar character of any subordinate science: to Arithmetic which is the most common algebraic system, but also to Geometry [as Vector Algebra] after defining through the very comprehensive sign the relations of line segments to each other with respect to both magnitude (length) and direction (angle); to Mechanics and Dynamics by defining forces as vectors; and similarly to any other branch of precise investigation, which can be reduced to fixed and invariable principles, or laws of operation. (p.xxi) [AE: Indeed it is possible to define the symbols and operations in some other manner provided so long as these definitions do not lead to contradictions, as the work of Boole and Hamilton and others (in the years to come) will show.]

[5] WITHIN ALGEBRA, MISUNDERSTANDINGS ARISE FROM beginning with the meaning of operations and attempting to make the results obtained dependent upon this meaning. This leads constantly to results which are at variance with such interpretations, for example the situation of numbers with negative signs, but even more problematic is the square- and even- roots of a negative quantity which violates the sign rules for multiplication. (p.xxvi)

[6] BY ALLOWING THE SYMBOLIC EXISTENCE of the sign within the essential generality of Algebra as an uninterpreted symbolical language, we can then adduce a meaningful interpretation, as has been done by Adrien Buee (1806) who interpreted as indicating perpendicularity in Geometry. This was followed by John Warren (1828) who gave the geometric interpretation of roots of unity in the context of line segments (vectors) in Geometry. (p.xxvii) This line of inquiry sees its culmination in the observation that provides the continuous two-dimensional representation of a unit vector through the plane, representable through the work of Euler as , the general complex number. Indeed, we are be able to interpret expressions such as and , once only the symbolical laws of the sign () have been determined. (p.xxviii-xxxi) This singular geometric sign make it necessary to incorporate the science of Trigonometry within Algebra, taking Geometry as the underlying analogy, just as previously this was done for Arithmetic. There is a very deep connection between Geometry and Algebra which deserves the minute examination of the definitions and first principles of Geometry so that it may be superseded by Algebra. (p.xxxii-xxxiii)

[7] WE THUS CONCLUDE THAT THE SYMBOLICAL ALGEBRA IS A SUBJECT THAT IS MUCH BROADER than Arithmetic, and not limited nor determined by its laws, but can take these on and in doing so provide much service to the mathematician in solving general arithmetic problems [such as the finding of roots and solving of equations]. (p.viii) [AE: Peacock’s Treatise on Algebra has broad scope including theory of equations, combinatorics, probability, trigonometry, logarithms, polynomials, and imaginary roots.]

4. The Transition to Modern Abstract, Symbolical Mathematics

Peacock’s work was much discussed in the Cambridge and Royal Society circles, and indeed it led directly to the publication by Boole of the Laws of Thought (1854) in which the algebraic structure of propositional logic was discussed. Hamilton and the quaternions were another instance in which removing the restrictions of arithmetic on algebraic operations bore fruit. Grassmann in Germany did much the same in his algebraization of geometry. Cayley, building equally upon this freedom, developed matrix algebras, and Galois, who took this freedom to the ultimate step and demonstrated categorically using algebra of permutations that there could be no quintic formula, i.e. a fifth or higher degree polynomial cannot be guaranteed to have a solution relying only upon arithmetical operations along with powers and roots, leading to final resolution of the quintic problem (Impossibility of the Quintic) and duplication of the cube (Impossibility of Duplication of the Cube).

The freeing of mathematics from classical interpretations was also seen in Geometry where the parallel postulate of Euclidean geometry was relaxed and found to lead, instead of to contradiction, to a new world of geometries (hyperbolic, elliptical) which would lead to the Riemannian theory of manifolds and to Einstein’s theory of general relativity.

In differential equations, Fourier found solutions to the heat equation using infinite series of trigonometric functions, opening up many controversies. Attempting to understand and resolve these led Cantor to invent the notion of infinite sets and pursue their investigation leading to profound and paradoxical results about the sizes of familiar sets of the number systems necessary for Geometry and Calculus.

The rapid advances across mathematics in the 1700s, in Algebra, Series, Calculus, Differential Equations, Geometry led to reviews and freeing of mathematics from classical interpreted constraints in the 1800s and the move to abstract, symbolical and formal mathematics based on set theory has led to where we are today: the establishment of abstract and formal viewpoint in all mathematical fields. Despite the examination of foundations and the impossibility theorems of Godel in the early 1900s, the formal approach has prevailed in mathematics, with an adventurism and pragmatism that has taken it forward every direction in both pure and applied directions in the 2000s.

References

1. [Peacock/1830] A Treatise on Algebra (ebook), George Peacock, 1830. The 2nd edition was completely rewritten in 2 vols separating vol 1. Arithmetical algebra (1842) from vol 2. Symbolical algebra (1845) (ebook). Reviews were still favourable 100 years later! (1942) — 1830 Preface — two page summary
2. [Pycior/1981] George Peacock and the British Origins of Symbolical Algebra, 1981, Historia Mathematica 8, 23-45
3. [Phillips] George Peacock and the Development of British Algebra 1800-1840, David Phillips, Cambridge University Summer Research Mathematics
4. [Princeton/1892] On the Permanence of Equivalence, Mathematics Dept. Princeton University, Annals of Mathematics, Vol.6, No.4, Jan 1892, pp.81-84
5. [Franci/2010] The History of Algebra in Italy in the 14th and 15th Centuries. Some Remarks on Recent Historiography. Raffaella Franci, 2010, Actes D’Historia de la Ciencia i de la Tecnica, vol 3 (2), p.175-194
6. [Steele/1922] The Earliest Arithmetics in English, Robert Steele, 1922
7. [Buee/1806] Memoir on Imaginary Quantities (FRENCH), Adrien Buee, Philosophical Transactions of the Royal Society of London, Vol.96, 1806. This was one of the first observations of the geometrical interpretation of complex numbers.
8. [Warren/1828] Treatise on Geometrical Representation of Square Roots of Negative Quantities, 1828
9. [Gregory/1838] Foundations of Algebra: On the Real Nature of Symbolical Algebra, 1838, Duncan Gregory, Transactions of the Royal Society of Edinburgh, in The Mathematical Writings of Duncan F. Gregory, William Walton, 1865
10. Grassmann/1844, extending Leibniz concept of a spatial geometry to a symbolical elaboration of vector geometry and vector spaces. He describes how he was led there in his 1844 foreward. Later Universal Algebra
11. [Kleiner/1986] The Evolution of Group Theory: A Brief Survey, Israel Kleiner, 1986, Mathematics Magazine, Vol.59, No.4, October 1986