Internalism, externalism, and beyond: 19th-century British algebra

This paper discovers the roots of symbolical algebra in a three-quarters-of-a-century British discussion of sound reasoning, general terms, and signs—a discussion in which mathematical and philosophical elements were freely and perhaps inseparably intermingled. It establishes, in particular, a link between early-19th-century symbolical algebra and nominalism. Opening with a review of recent attacks on the rigid internal/external dichotomy, which underlies much of the modern history of science, the paper also serves as an example of the fertility of suspending the dichotomy in pursuit of earlier mathematical subcultures.     

Early criticism of the symbolical approach to algebra

For a long time, historians have believed that in the 1830s Sir William Rowan Hamilton was a lone critic of symbolical algebra. Using published and unpublished documents, this article shows that symbolical algebra was a considerably controversial subject among British mathematicians of the 1830s and 1840s. Special attention is paid to William Frend's and Osborne Reynolds' criticism of symbolical algebra. The article ends with a brief discussion of reservations concerning symbolical algebra expressed by Augustus De Morgan, William Whewell, and Philip Kelland.     

Robert Leslie Ellis,William Whewell and Kant: the role of Rev H F C Logan

Reverend H F C Logan is put forward as the formerly unidentified figure to which Robert Leslie Ellis referred in a journal entry of 1840 in which he wrote that it was due to his influence that William Whewell came to uphold particular Kantian views on time and space. The historical evidence of Ellis’s early familiarity with, and later commitment to Kant is noteworthy for at least two reasons. Firstly, it puts into doubt the accepted view of the second generation of reformers of British algebra as non-philosophical, practice-oriented mathematicians. Secondly, in so far as Logan was the correspondent of William Rowan Hamilton, it re-emphasizes that the role of Kantianism in the transition from ‘symbolical’ to ‘abstract’ algebra in nineteenth-century British algebra requires closer scrutiny.     

Babbage,peacock and modern algebra

George Peacock's A Treatise on Algebra of 1830, contained in its preface the first published recognition that Algebra need not necessarily be always associated with Arithmetic, and that non-arithmetical Algebras were possible. The work also contains the first statement of Peacock's principle of the permanence of equivalent forms. It is shown in this paper that Charles Babbage had all these ideas in almost identical or superior form in an unpublished work The Philosophy of Analysis written in 1821. Peacock certainly had access to his friend Babbage's writings, and the suggestion is made of unconscious assimilation rather than deliberate plagiarism.     

Woodhouse, babbage, peacock, and modern Algebra

In a recent article, J. M. Dubbey [Historia Mathematica 4 (1977), 295–302] showed that George Peacock's A Treatise on Algebra (1830) was similar to an unpublished work written by Charles Babbage in 1821. Evidently perplexed about the absence of a dispute over priority, Dubbey concluded that Peacock had unconsciously assimilated Babbage's ideas, and that Babbage was too busy with other activities to be concerned. The thesis of this article is that the innovative aspects of the work of both Babbage and Peacock are extensions of ideas put forth in 1803 by Robert Woodhouse, and that probably neither Babbage nor Peacock was overly concerned with acknowledgments because their approach to algebra was not unique at Cambridge.     

Descartes and the geometrization of thought: The methodological background of Descartes' géométrie

After the relationship between geometry and algebra exemplified in his distinction between geometrical and mechanical lines is examined, the basis for Descartes' limited approach to analytical geometry is discussed in connection with his reflections on method. It is argued that his epistemology, which required that conceptual thinking be accompanied by a construction supplied by the imagination, in conjunction with the significant role he attributed to mnemonic devices, helps to clarify the methodological background for Descartes' distinctive approach to geometry.     

Maestro benedetto da firenze e la storia dell'algebra

This article presents a careful analysis of the chapters devoted to algebra in the Trattato di praticha d'arismetrica written by M^oBenedetto of Florence in 1463 (ms. L.IV.21 Municipal Library, Siena). This analysis permits us to reconstruct partially the history of algebra in Tuscany from Leonardo Pisano to the middle of the 15th century, and shows that the elements of algebra expounded in Luca Pacioli's Summa de arithmetica [1494] were already contained in Benedetto's treatise.     

Buhstaber's work on two-valued formal groups

The paper which follows may be regarded as the best substitute available for the lecture which V.M. Buhstaber would have delivered to the International Congress of Mathematicians, Vancouver 1974, if he had been present. (We would like to say how sorry we are that he was not able to be there.) In fact, we originally agreed to prepare it for submission to the Proceedings of the Congress. The text is in the form of a report on Buhstaber's work by J.F. Adams and A. Liulevicius, and these two authors accept entire responsibility for it. Of course, our primary source is the account of Buhstaber's work which we heard at the Congress from A.T. Fomenko, and we would like to thank him for all his help. But we have also tried to improve our understanding by consulting the papers which Buhstaber has published in Russian.We assume that the reader is aware of the connection between complex cobordism and the theory of formal groups [2, 5]; this work is generally respected. The topic of two-valued formal groups represents an extension of this theory. It is conceived partly as a contribution to pure algebra, but it is inspired by an application to algebraic topology; this application lies in the theory of characteristic classes of symplectic bundles, and in the study of symplectic cobordism.     

On measures of fuzziness and their representations

This research investigates the development of probability calculi for fuzzy sets (abbreviated FS's). Two FS algebras are employed, each as a basis for a FS probability calculus. One employs minima for FS intersections and consequently maxima for unions. The other uses a product rule for intersections and a standard probability-like addition rule for unions. The two derived FS probability theories are not identical and neither is identical with standard probability theory. Finally, ε-fuzzy sets, or almost standard sets, are defined and investigated. The main result of the last section is that while FS's do not satisfy all the laws of Boolean algebra fuzzy sets that are almost standard almost satisfy these laws. Moreover, the extent to which such laws are satisfied decreases as their complexity increases.     

The art and the science of British algebra: A study in the perception of mathematical truth

This paper investigates the origins of the concept of mathematical truth by focusing on the development of algebra in England in the early 19th century. In particular, it investigates the reasons why the English, despite their attention to the elements of abstract algebra, never produced a system comparable to modern algebra. Special consideration is given to the works of George Peacock, Augustus DeMorgan, William Whewell, and John Herschel. It is argued that what separated the early development of English algebra from modern algebra is a fundamental difference between 19th- and 20th-century views of truth.     

Symbolical Algebra as a Foundation for Calculus: D. F. Gregory's Contribution

Duncan Farquharson Gregory (1813–1844) was a mathematician and founder of the Cambridge Mathematical Journal. His 1841 Examples of the Processes of the Differential and Integral Calculus was an extensive revision of Peacock's 1820 textbook of a similar title. Among the new material in Gregory's version is an exposition of symbolical algebra, prominently featuring the method of “separation of symbols.” We examine Gregory's career; the mathematicians he influenced; and Servois and Murphy, who influenced him. We consider Gregory's use of separation of symbols in the Examples and consider whether he believed these techniques to be capable of providing an adequate foundation for calculus. © 2002 Elsevier Science (USA).Le mathématicien Duncan Farquharson Gregory (1813–1844) fut également fondateur du Cambridge Mathematical Journal. Son ouvrage publié en 1841, Examples of the Processes of the Differential and Integral Calculus, constitue une révision substantielle du livre de Peacock au titre similaire. L'un des apports du livre de Gregory est un exposé de l'algèbre symbolique, et en particulier de la méthode de séparation des symboles. Dans cet article nous passons en revue la carrière de Gregory et nous parlons des mathématiciens qu'il a influencés, ainsi que de l'influence que Servois et Murphy ont eu sur ses travaux. Nous considérons son usage de la séparation des symboles dans ses Examples, et nous examinons si Gregory croyait que ses techniques pouvaient constituer un fondement pour le calcul différentiel. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 01A55, 01A70, 01A50, 26-03, 26A03, 34-03, 34A30.     

Translation-invariant integrals,and Fourier analysis on Clifford and Grassmann algebras

The generalization of Berezin's Grassmann algebra integral to a Clifford algebra is shown to be translation-invariant in a certain sense. This enables the construction of analogs of twisted convolutions of Grassmann algebra elements and of the Fourier-Weyl transformation, which is an isomorphism from a Clifford algebra to the Grassmann algebra over the dual space, equipped with a twisted convolution product. As an application a noncommutative central limit theorem for states of a Clifford algebra is proved.     

A proof of Lyndon's finite basis theorem

A new proof is given of the theorem, originally proved by R.C. Lyndon, that any two element algebra of finite similarity type has a finite basis for its equations. We also provide a new proof of a result of W. Taylor that any equational class generated by a two element algebraic system contains only a finite number of subdirectly irreducible members, each of which is finite. The original proofs of these two theorems relied on E.L. Post's classification of the two element algebraic systems. Our paper uses instead some recent results from universal algebra.     

On an optimal solution for maximizing the probability of retention in PPS sampling

A direct alternative proof of Keyfitz's optimal solution to the problem of maximizing the probability of retention in sampling on a second occasion is given, using techniques of elementary linear algebra. The proof and comments help one to better understand Keyfitz's solution, and they clearly demonstrate that the closed form solution of Keyfitz is one of a possible infinity of solutions offered by a linear programming approach. We also give one of those other solutions offered by linear programming, which is easy to obtain by hand calculations using only the operation of subtraction.     

Lexicographic matching in Boolean algebras

Hall's theorem on systems of distinct representatives implies the existence of matchings between two consecutive levels of a Boolean algebra. In this paper a very simple construction of such matchings is given whose induced chains are in addition symmetric.     

Iterative algorithms for large partitioned linear systems,with applications to image reconstruction

We present a unifying framework for a wide class of iterative methods in numerical linear algebra. In particular, the class of algorithms contains Kaczmarz's and Richardson's methods for the regularized weighted least squares problem with weighted norm. The convergence theory for this class of algorithms yields as corollaries the usual convergence conditions for Kaczmarz's and Richardson's methods. The algorithms in the class may be characterized as being group-iterative, and incorporate relaxation matrices, as opposed to a single relaxation parameter. We show that some well-known iterative methods of image reconstruction fall into the class of algorithms under consideration, and are thus covered by the convergence theory. We also describe a novel application to truly three-dimensional image reconstruction.     

Stochastic calculus of variations

A theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variations to stochastic processes. Generalizations of the Euler equation and Noether's theorem are obtained and several conservation laws are discussed. An application to Nelson's probabilistic framework of quantum mechanics is also given.     

Approximate distributive laws and finite equational bases for finite algebras in congruence-distributive varieties

For a congruence-distributive variety, Maltsev’s construction of principal congruence relations is shown to lead to approximate distributive laws in the lattice of equivalence relations on each member. As an application, in the case of a variety generated by a finite algebra, these approximate laws yield two known results: the boundedness of the complexity of unary polynomials needed in Maltsev’s construction and the finite equational basis theorem for such a variety of finite type. An algorithmic version of the construction is included. Received November 27, 1996; accepted in final form December 16, 2004.