Mathematical Symbols

[James A. Landau]

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