Definition:Halmos Symbol

Definition

The Halmos symbol is the character: $\blacksquare$ used to indicate the end of a proof.

It replaces the old-fashioned and embarrassingly uncool Q.E.D. which muggles sometimes use when pretending to be clever.

Also known as

The Halmos symbol is apparently, according to Halmos himself, also known as the tombstone symbol.

Also defined as

Some sources show this, in some contexts, as a heavy vertical line, something like: $\Rule{3px} {2ex} {0ex}$ or $\Rule{5px} {2ex} {0ex}$, rather than a box, but the intention is the same.

This form is used, for example, by Donald E. Knuth in his The Art of Computer Programming to indicate the point of termination of an algorithm.

Some sources use $\Box$ in preference to $\blacksquare$.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to reserve the former symbol for the end of a subproof within a long proof, as a way of breaking up the flow of thought for clarity.

Source of Name

This entry was named for Paul Richard Halmos.

Historical Note

While the Halmos symbol is indeed named for Paul Halmos, and many attribute it directly to him, he himself did not actually invent it.

However, he is generally credited with introducing it into mathematics, having seen it used in general magazine literature to indicate the end of an article.

In his own words:

The symbol is definitely not my invention — it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like $\Box$, and is used to indicate an end, usually the end of a proof. It is most frequently called the 'tombstone', but at least one generous author referred to it as the 'halmos'.

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Preface
• 1964: William K. Smith: Limits and Continuity ... (next): Preface
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Introduction: Special Symbols
• 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.3$: Functions and mappings. Images and preimages: Theorem $1$ (footnote)
• 1977: Gary Chartrand: Introductory Graph Theory ... (next): Preface
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
• 1985: Paul R. Halmos: I Want To Be a Mathematician: An Automathography
• 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): Notation and terminology
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.1$: Algorithms