Problems with a dictionary definiton of "vector"

[James A. Landau]

The following was posted on the Historia Matematica mailing list:

<begin quote>
from my CD-ROM version of a Merriam-Webster unabridged
dictionary:  There are two entries under the word _vector_ .  The first reads

Main Entry: vector
Pronunciation: *vekt*(r) sometimes -*t*(*)r or -*(*)
Function: noun
Inflected Form: -s
Etymology: New Latin, from Latin, carrier, from vectus
(past participle of vehere to carry) + -or * more at WAY
1 a : RADIUS VECTOR 2  b or    vector quantity : a quantity
that requires for its complete specification a magnitude,
direction, and sense and that is commonly represented by
a line segment the length of which designates the magnitude
of the vector, the orientation of which designates the direction
of the vector, and the sense of which is designated by an
arrowhead at one end of the segment : a quantity having both
magnitude and direction  c : a course or compass direction
especially of an airplane
2 : an agent capable of transmitting a pathogen from one
organism to another either mechanically as carrier (as
houseflies that transport typhoid bacteria) or biologically by
playing a specific role in the life cycle of the pathogen
(as mosquitoes in relation to the malaria parasite) *fleas
are vectors of plague* *aphids are vectors of plant viruses*
3 : a behavioral field of force toward or away from the
performance of various acts;  broadly   : DRIVE

The second reads:

Main Entry: vector*
Function: noun
  : an element of a vector space

A certain circularity arises if one then looks at this dictionary's
definition of _vector space_ (what are the _vectors_ in the system
that got generalized?):

Main Entry: vector space
Function: noun
  : a set representing a generalization of a system of vectors and consisting
of elements which comprise a commutative group under addition, each of
which is left unchanged under multiplication by the multiplicative identity
of a field, and for which multiplication under the multiplicative operation of
the field is commutative, closed, distributive such that both
c(A + B) = cA + cB and (c + d)A = cA + dA, and associative such that
(cd)A = c(dA) where A, B are elements of the set and c, d are elements
of the field

I note also that from the meaning 2 in the first definition above, one
may deduce that some bugs are vectors, and that presumably
where they are moving around can more or less legitimately be
called a vector space.  Right?

Gordon Fisher     gfisher at shentel.net
<end quote>

Problem 1 is that the two definitions are more or less circular
Problem 2 is that the entry for "vector" gives the geometric definition,
whereas the entry for "vector space" uses a much more abstract definition.
Some mathematicians get a little perturbed on the subject and claim the
abstract definition is the only correct one, with the geometric definition
being merely ONE visualization of a vector.  Physicists and engineers prefer
the geometric definition because they (in most cases) find it much more
useful than the abstract definition.

(Applied mathematicians, I might add, point out that the two definitions are
equivalent and go ahead, use whichever you prefer, you get the same results).

        - Jim Landau