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An American Postulate Theorist: Edward V. Huntington

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Research in History and Philosophy of Mathematics

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.

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Notes

  1. 1.

    What constitutes “mathematical activity” may be broadly defined, and could include both research and educational interests.

  2. 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. 3.

    Ackerberg-Hastings herself remains undecided on this question.

  4. 4.

    This changed with subsequent generations, as the American mathematical research community became more fully established.

  5. 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. 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. 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. 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. 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. 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. 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. 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. 13.

    This special two-valued boolean algebra was first studied in (Boole 1854).

  14. 14.

    The symbol x in models for Ia and Ib represents some element that lies outside of K.

  15. 15.

    The term “boolean algebra” for what had previously been called the “algebra of logic” was first introduced by Scheffer.

  16. 16.

    These were (Huntington 1902a) and (Huntington 1905), respectively. See also (Franci 1992) for a description of the entire collection of papers.

  17. 17.

    Both papers are discussed in (Awodey and Reck 2002).

  18. 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. 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. 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. 21.

    See Aspray (1991) for further details.

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Authors and Affiliations

  1. Colorado State University - Pueblo, Pueblo, CO, USA

    Janet Heine Barnett

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  1. Janet Heine Barnett
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Correspondence to Janet Heine Barnett .

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Editors and Affiliations

  1. Mathematical Information and Computer Sciences, Point Loma Nazarene University, San Diego, California, USA

    Maria Zack

  2. Department of Philosophy, University of California, Davis, Davis, California, USA

    Elaine Landry

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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

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