Abstract
Like many American mathematicians of his generation, Edward V. Huntington (1874-1952) began his mathematical studies in the United States, but completed his doctoral work in Germany. With others of his generation, he went on to help create a mathematics research community within the United States. Huntington is often remembered today for his efforts to build the infrastructure necessary to support such a community, including the founding of new American professional organizations like the Mathematical Association of America (MAA). Of equal importance to the new community were his contributions to the body of mathematical research produced in the United States, and especially his work in an entirely new field known today as “American Postulate Theory.” In this paper, we discuss Huntington’s 1904 paper Sets of Independent Postulates for the Algebra of Logic as an exemplar of the research agenda of the American Postulate Theorists. We further consider the influence that this body of research had on the development of both mathematical logic and algebra, and its importance in gaining international recognition for the developing mathematical research community in the United States.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
What constitutes “mathematical activity” may be broadly defined, and could include both research and educational interests.
- 2.
As Ackerberg-Hastings notes, it is possible to respond to the work and ideas of people who are long dead, but this does not constitute communication of the type necessary for “interaction” since the other party cannot respond back.
- 3.
Ackerberg-Hastings herself remains undecided on this question.
- 4.
This changed with subsequent generations, as the American mathematical research community became more fully established.
- 5.
Huntington’s method is a revised version of a method first developed by American statistician Joseph Adna Hill (1860–1938); it is thus known today as the Huntington-Hill Method of Apportionment, but is also called the Method of Equal Proportion.
- 6.
Huntington’s footnote: A class is determined by stating some condition which every entity in the universe must either satisfy or not satisfy; every entity which satisfies the condition is said to belong to the class. (If the condition is such that no entity can satisfy it, the class is called a “null” class.) Every entity which belongs to the class in question is called an element (cf. H.Weber, Algebra, vol. 2 (1899), p.3).
- 7.
Huntington’s footnote: A rule of combination ∘, in the given class, is a convention according to which every two elements a and b (whether a = b or a ≠ b) in a definite order determine uniquely an entity a ∘ b (read “a with b”), which is, however, not necessarily an element of the class. In the class of quantities or numbers, familiar examples of rules of combination are +, −, ×, ÷, etc.
- 8.
Huntington’s footnote: A dyadic relation, R, in the given class, is determined when, if any two elements a and b are given in a definite order, we can decide whether a stands in the relation R to b or not; if it does, we write aRb … In the class of quantities or numbers, familiar examples of dyadic relations are = , < , > , ≤ , etc. Relations among human beings furnish other examples.
- 9.
In fact, Huntington’s presentation of the axiomatization for boolean algebra structure in this paper is well suited for use in undergraduate mathematics courses as an introduction to abstract boolean algebra. The guided reading student project (Barnett 2013) offers one approach to doing so.
- 10.
Although Huntington omitted the (universal) quantifiers in stating these properties, his readers would have understood that these are general properties that hold for all elements a, b ∈ K.
- 11.
The second set of postulates began with relation ○ < as the sole undefined symbol. All three sets of postulates defined the three special elements \(\bigvee\), \(\bigwedge\), and \(\overline{a}\) via properties specified in the postulates.
- 12.
In the continuation of this discussion, Huntington again stressed the theme of freedom illustrated by the three equivalent postulate sets for boolean algebra studied in this paper:
In selecting a set of consistent, independent postulates for any particular algebra, one has usually a considerable freedom of choice; several different sets of independent postulates (on a given set of fundamental concepts) may serve as the basis of the same algebra the only logical requirement is that every such set of postulates must be deducible from every other.
- 13.
This special two-valued boolean algebra was first studied in (Boole 1854).
- 14.
The symbol x in models for Ia and Ib represents some element that lies outside of K.
- 15.
The term “boolean algebra” for what had previously been called the “algebra of logic” was first introduced by Scheffer.
- 16.
- 17.
Both papers are discussed in (Awodey and Reck 2002).
- 18.
Huntington first cited (Cayley 1854) for the first description of an abstract group, and (Kronecker 1870) and (Weber 1882) for the earliest explicit definitions of an abstract group in terms of a postulate set. He also gave an overview of how the advantages and disadvantages of each of the nine preceding papers in the series.
- 19.
In addition to these, Huntington also cited a number of researchers in the “algebra of logic” (e.g., Boole, Schröder, and Peirce) and in algebra (including his doctoral advisor Weber).
- 20.
Wiener’s doctoral dissertation compared the treatment of the algebra of relatives given by Schroeder with that given by Whitehead and Russell; he also studied with Russell at Cambridge. Huntington is reported to have sent Wiener a “…set of postulates” as a wedding gift.
- 21.
See Aspray (1991) for further details.
References
Ackerberg-Hastings A (2015) Did American professors form a mathematical community in the early 19th century? Contributed talk, special session on ‘Mathematical Communities’. Mathematical Association of American Mathfest, Washington, DC
Aspray W (1991) Oswlad Veblen and the origins of mathematical logic at Princeton. In: Drucker T (ed) Perspectives on the history of mathematical logic. Birkhäuser, Basel/Boston/Berlin
Awodey S, Reck EH (2002) Completeness and categoricity. Part I: nineteenth-century axiomatics to twentieth-century metalogic. Hist Philos Log 23:1–30
Barnett J (2013) Boolean algebra as an abstract structure: Edward V. Huntington and axiomatization. Loci: convergence. doi:10.4169/loci003998
Bell ET (1938) Fifty years of algebra in America, 1888 to 1938. In: Archibald RC (ed) A semicentennial history of the American Mathematical Society 1888–1938, vol 1. American Mathematical Society, New York, pp 1–34
Bernstein BA (1914) A complete set of postulates for the logic of classes expressed in terms of the operation “exception,” and a proof of the independence of a set of postulates due to Del Re. Univ Calif Publ Math 1:87–96
Bernstein BA (1915) A set of four independent postulates for Boolean algebra. Trans Am Math Soc 17:50–51
Bernstein BA (1916) A simplification of the Whitehead-Huntington set of postulates for Boolean algebras. Bull Am Math Soc 22:458–459
Bernstein BA (1926) Sets of postulates for the logic of propositions. Trans Am Math Soc 28:472–478
Boole G (1854) An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities. Walton and Maberly, London
Cayley A (1854) On the theory of groups, as depending on the symbolic equation θ n = 1 - Part I. Philosophical Magazine 7:40–47
Corcoran J (1981) From categoricity to completeness. Hist Philos Log 2:113–119
Corry L (2004) Modern algebra and the rise of mathematical structures, revised 2nd edn. Birkhäuser, Basel/Boston/Berlin
Dines LL (1915) Complete existential theory of Sheffer’s postulates for Boolean algebras. Bull Am Math Soc 21:183–188
Finkel B, Colaw J (1894) Introduction. Am Math Mon 1(1):1–2
Franci, R (1992) On the axiomatization of group theory. In: Demidov, A et al (eds) Amphora: Festschirtf for Hans Wussing on the occasion of his 65th birthday. Birkhäuser, Basel/Boston/Berlin
Henle P (1932) The independence of the postulates of logic. Bull Am Math Soc 38:409–414
Hilbert D (1899) Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen. Leipzig, Teubner
Huntington EV (1902a) Two Definitions of an Abelian group by sets of independent postulates. Trans Am Math Soc 2:27–30
Huntington EV (1902b) A complete set of postulates for the theory of absolute continuous magnitude. Trans Am Math Soc 3:264–279
Huntington EV (1904) Sets of independent postulates for the algebra of logic. Trans Am Math Soc 5(3):288–309
Huntington EV (1905) Note on the definition of abstract groups and fields by sets of independent postulates. Trans Am Math Soc 6:181–197
Huntington EV (1932) A new set of independent postulates for the algebra of logic with special reference to Whitehead and Russell’s Principia Mathematica. Proc Natl Acad Sci 18:179–180
Huntington EV (1933) The method of postulates. Philos Sci 4(4):482–495
Kronecker L (1870) Auseinandersetzung einiger eigenschaft en der klassen-zahl idealer complexer zahlen. Monatsberichte der königlich preusssischen Akademie der Wissenschaften zu Berlin 881–882
MAA (n.d.) Edward Vermilye Huntington: 1918 MAA President, maa.org. http:www.maa.org/about-maa/governance/maa-presidents/edward-vermilye-huntington-1918-maa-president. Accessed 12 Febr 2016
Parshall K, Rowe D (1994) The emergence of the American Mathematical Research community 1876–1900: J. J. Sylvester, Felix Klein, and E. H. Moore. American Mathematical Society and London Mathematical Society, Providence, RI
Scanlan M (1991) Who were the American postulate theorists. J Symb Log 56(3):981–1002
Scanlan M (2003) American postulate theorists and Alfred Tarski. Hist Philos Log 24(4):307–325
Sheffer HM (1913) A set of five independent postulates for Boolean algebra, with application to logical constants. Trans Am Math Soc 14:481–188
Veblen O (1904) A system of axioms for geometry. Trans Am Math Soc 5:343–384
Weber H (1882) Beweis des Satzes, dass jede eigentlieh primitive quadratisehe Form unendlieh viele Primzahlen darzustellen fähig isis. Math Ann 20:302–329
Wiener N (1917) Certain formal invariances in Boolean algebras. Trans Am Math Soc 18:65–72
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Barnett, J.H. (2016). An American Postulate Theorist: Edward V. Huntington. In: Zack, M., Landry, E. (eds) Research in History and Philosophy of Mathematics. Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-46615-6_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-46615-6_16
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-43269-4
Online ISBN: 978-3-319-46615-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)