Mathematical Symbols

[Laurence Horn]

>In a message dated 02/19/2003 4:14:10 PM Eastern Standard Time,
>bjv6xc at UMR.EDU writes:
>
>>  Some mathematical symbols are similar to letters of the alphabet, but are
>not
>>  exact representations.  For example, the existential quantifier is
>>  represented as a backwards capital E, and the universal quantifier is an
>>  upside-down capital A.  Can anyone help me with the origin of these
>symbols,
>>  and do they have any connection with the letters to which they are similar?
>
>See Jeff Miller's Web site "Earliest Uses of Various Mathematical Symbols"
>     URL http://members.aol.com/jeff570/mathsym.html
>
>
><begin quote>
>Existence. Peano used <backwards E> in volume II, number 1, of his Formulaire
>de mathematiqués, which was published in 1897 (Cajori vol. 2, page 300).
>
>Membership. Peano used <epsilon> in the introduction to volume I of his
>Formulaire de mathematiqués, which was published in Turin in 1895, although
>the introduction itself is dated 1894 (Cajori vol. 2, page 300).
>
>The website at the University of St. Andrews states that Peano introduced the
>symbol in 1889 and that it comes from the first letter if the Greek word
>meaning "is."
>
>Peano's symbol for membership was an ordinary epsilon ; the stylized epsilon
>now used was adopted by Bertrand Russell in Principles of Mathematics in 1903
>.
>
>For all. According to M. J. Cresswell and Irving H. Anellis, the upside-down
>A originated in Gerhard Gentzen, "Untersuchungen ueber das logische
>Schliessen," Math. Z. 39 (1934), p, 178. In footnote 4 on that page, Gentzen
>explains how he came to use the sign. It is the "All-Zeichen," an analogy
>with  for the existential quantifier which Gentzen says that he borrowed from
>Russell.
>
>Cajori, however, shows that Peano used  before Russell and Whitehead (whose
>backwards E had serifs, unlike Peano's).
><end quote>
>
>
>"Cajori" is Cajori, Florian. _A History of Mathematical Notations_. 2
>volumes. Lasalle, Illinois: The Open Court Publishing Co., 1928-1929.
>
>         - James A. Landau

Sounds good.  The standard folk etymology (the one I'd always
accepted) for the upside-down E is that it designates "Exists", a
better translation than "Some" for the (appropriately named)
existential quantifier.  (I think of it as an upside-down rather than
backward E since only the former story provides a parallel with the
universal.)  And the upside-down A for the universal quantifier
stands for "All", as Jim notes.  The fact that the latter involves an
English word ("All") supports the idea that the former should as well
("Exists"), but evidently such is not the case.

Larry