FOM: An historical Perspective for the term ``arbitrary''

[Insall]

Dear FOMers,
     In some offlist discussion, professor Kanovei suggested to me that an
approach one might take to help settle the discussion about the mathematical
meaning of the term ``arbitrary'' would be to take a look at some older
writings.  He specifically mentioned Euler.  I have not read Euler's
writings, for I never studied Latin.  (I thought he may have written some in
German, and I am almost as weak in German as I am in Latin.)  I am also a
member of another discussion list, ``Historia Matematica'' (HM), and I
thought someone on that list might be able to help us clarify how some of
the older mathematical writings use the term ``arbitrary''.  Thus I posted
the following message to that discussion group.  Ed Sandifer replied, and I
have asked his permission to quote his response in its entirety to the FOM
list.  While I await his approval (which I presume he will give), you may
read the message I posted to the HM list.

Matt Insall

-------------------Posting to HM------------------------------

I am not sure that the members of HM want to participate in this
discussion from another list (fom at math.psu.edu), which, I (and professor
V. Kanovei, in some offlist discussions) believe needs some historical
input and perspective.  In particular, the question that is currently
being discussed is "What, in Mathematical terms, does the word 'arbitrary'
really mean?"  Various philosophical views are being posited on this topic,
and I personally feel that a reasonable attack on this question may involve
some investigation in the writings of some of the great mathematicians of
all time.  It seems to me that Euler fits the bill, and professor Kanovei
even suggested that answers to the questions currently being posed in the
FOM forum might be obtained, at least in part, by consulting the writings
of Euler in particular.  Since Ed Sandifer has mentioned Euler's articles,
and I expect that most of them (all of them, I expect, actually) are not
written in English, I am not the right person to comment on the context
in which Euler used the German (or Latin??) equivalent of the English word
"arbitrary" in his mathematical writings.  Is someone on this list familiar
enough with Euler's writings, or with commentaries about them, to answer
such questions as the following?

Did Euler use the (equivalent of the) word "arbitrary" frequently, and did
it seem to carry the same semantic content in each use?

When Euler used the term "arbitrary" (or its equivalent), was his use what
one now frequently refers to as "redundant"?  For example, if I asked a
reader to consider an "arbitrary group", I may as well begin with "Let G
be a group...", instead of "Let G be an arbitrary group...".  Is this
really the only way Euler seemed to use the term "arbitrary", or did some
other, less "redundant" uses appear in his writings?

Did Euler appear to attempt to avoid what he seemed to consider redundant
in general, but still use the term "arbitrary" frequently, as if he might
have been uncertain whether the word "arbitrary" was redundant, or does
the evidence support the hypothesis that Euler considered the use of the
term "arbitrary" redundant in the same way I discussed above, but used the
word only for emphasis, rather than technical accuracy, precision and
efficiency in his communication of his mathematical discoveries?

Matt Insall