Notices of the AMS Volume 59, Number 2, DOI: http://dx.doi.org/10.1090/noti793

by Garret Sobczyk, who is professor of mathematics at Universidad de Las Américas-Puebla.

His email address is: garret_sobczyk_at_yahoo.com.

Download full paper: http://www.ams.org/notices/201202/rtx120200264p.pdf

In 1878 William Kingdon Clifford wrote down

the rules for his geometric algebra, also

known as Clifford algebra. We argue in this

paper that in doing so he laid down the

groundwork that is profoundly altering the

language used by the mathematical community to

express geometrical ideas. In the real estate business

everyone knows that what is most important

is location. We demonstrate here that in the business

of mathematics what is most important to

the clear and concise expression of geometrical

ideas is notation. In the words of Bertrand Russell,

*…A good notation has a subtlety*

*and suggestiveness which at times*

*make it seem almost like a live*

*teacher.*

Heinrich Hertz expressed much the same thought

when he said,

*One cannot escape the feeling*

*that these mathematical formulae*

*have an independent existence and*

*an intelligence of their own, that*

*they are wiser than we are, wiser*

*even than their discoverers, that*

*we get more out of them than we*

*originally put into them.*

The development of the real and complex number

systems represents a hard-won milestone in

the robust history of mathematics over many centuries

and many different civilizations [5], [29].

Without it mathematics could progress only haltingly,

as is evident fromthe history ofmathematics

and even the terminology that we use today. Negative

numbers were referred to by Rene Descartes

(1596–1650) as “fictitious”, and “imaginary” numbers

were held up to even greater ridicule, though

they were first conceived as early as Heron of

Fig. Herman Gunther Grassmann (1809–1877) was

a high school teacher. His far-reaching

Ausdehnungslehre, “Theory of extension”, laid

the groundwork for the development of the

exterior or outer product of vectors. William

Rowan Hamilton (1805–1865) was an Irish

physicist, astronomer, and mathematician. His

invention of the quaternions as the natural

generalization of the complex numbers of the

plane to three-dimensional space, together

with the ideas of Grassmann, set the stage for

William Kingdon Clifford’s definition of

geometric algebra. William Kingdon Clifford

(1845–1879) was a professor of mathematics

and mechanics at the University College of

London. Tragically, he died at the early age of

34 before he could explore his profound ideas.

Alexandria, the illustrious inventor of the windmill

and steam engine during the first century AD

[18]. What most mathematicians fail to see even

today is that geometric algebra represents the

grand culmination of that process with the completion

of the real number system to include the

concept of direction. Geometric algebra combines

the two silver currents of mathematics, geometry

and algebra, into a single coherent language. As

David Hestenes has eloquently stated,

*Algebra without geometry is blind,*

*geometry without algebra is dumb.*

*Source:* pp. 264 and 265 of http://www.ams.org/notices/201202/rtx120200264p.pdf