From FISHERGM@jmu.edu Fri Sep 25 11:13:43 1998
Date: Tue, 21 May 1996 23:18:13 -0500 (EST)
From: FISHERGM@jmu.edu
To: athena-discuss@info.harpercollins.com
Subject: *Right* msg from Thomas to Fisher, fwdd to list

Subject: Re: Concerning "logismos" and axiomatic systems
To: FISHERGM@jmu.edu
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FISHERGM@jmu.edu wrote:

> I remember years ago reading numerous articles concerning the
> juxtaposition of "arithmos" and "logismos" (in the accusative)
> in the passage from Plato's *Phaedrus* we've been discussing.
> My own view is that "logismos" would be better translated
> here as "mathematical reasoning".  For example, in the problem
> from the Rhind papyrus I discussed a while ago, concerning
> the "skd" or "seked" method in connection with building
> pyramids, there is some arithmetic involved consisting of
> use of a division algorithm.  On the other hand, the problem
> and its solution as a whole is an admirable example of
> mathematical reasoning.

It is perhaps a manifestation of my training as an engineer,
but the direct and immediate it would seem to me to be far
preferable, as hypotheses go, to the indirect and far-fetched.
Although I'm not a Greek scholar, it does not escape me that
the Greek word logos means logic, and is the root word that, used
as a suffix, gives us all the "ologies", eg. psychology, epistemology,
etc., denoting "logic of".  Now we have another Greek word, 
"logismos", with the same root, but with as "ismos" thrown in.
That too I'm quite familiar with, the suffix "-ism" having long
ago been imported into the English language.  It gives us words
like capital-ism, commun-ism, centr-ism, etc. where the
connotative burden carried by the "-ism" is to imply a "distinctive 
doctrine, cause or theory" (Webster's Collegiate) in the root
word being qualified.  Webster's also confirms what is
obvious, namely that the English suffix "-ism" derives from
the Greek "-ismos" which is what we have here in the word
being considered.  Therefore, we
have a literal translation of "logismos" as being "logic-ism",
or a theory or calculus of
logic, one translator having already used the term
"calculus" as a translation for "logismos", though not specifying
what kind of calculus.  It was precisely that term
which led to 
the query from one poster as to what "calculus" could have been
meant, the modern differential and integral
calculus being anachronistic in context.  But the 
translator's conflation of
"logismos" and "calculus" brings to mind the term "propositional
calculus", which is precisely the calculus of the logic of
propositions which is a direct lineal descendant of the 
axiomatic logic attributed to the Greeks.  The translator
could not have meant the arithmetic calculus or simple calculation,
because the word "logismos" comes in a list right after 
"arithmos", and therefore meant something different.  It is 
indeed a calculus, it is the propositional or logical calculus.
And it is here being attributed by Plato/Socrates to the
Egyptians!

The suggested translation of "mathematical reasoning" is 
a dodge, not at all convincing, asserted only to avoid giving
credit to the Egyptians for the invention of axiomatic
reasoning.

Invocation of the Rhind papyrus and the damning-with-faint-
praise to which it is subjected, is a red herring.  The absence
of axioms in the Rhind papyrus hardly help us to translate
the Greek word "logismos" in the Phaedrus. 


> I believe it would be putting too
> much weight on "logismos" as it occurs in the *Phaedrus*
> to conclude that the reference is to axiomatic mathematics.

How about simply translating the word and following where it leads...
True science consists of sitting humbly before facts, as
an old Chemistry teacher once taught me.  It does not consist
in distorting facts to fit cherished preconceptions.

> There are some references, or at least one somewhere, to
> *Elements* of geometry "before Euclid".  I forget where they are,
> or it is.  As I recall, there was no attribution given in
> the reference(s), which may have been relatively late.
> 
> Of course, it's always a possibility that the kind of thing
> Euclid did was done earlier in Egypt, before the Hellenistic
> era.  However, there is no evidence yet found to suggest this,
> that I know of.  No earlier *Elements* have survived.  I
> wish we had the lost history of mathematics by Eudemus, which
> is referred to in several places (e.g., by Pappus of Alexandria,
> if I remember correctly).  I once considered going to Istanbul
> to look for a copy, or at least fragments, but I never did,
> and undoubtedly never will.

The totality of the evidence suggests to me quite convincingly
that the Egyptian science had moved beyond the merely empirical,
as various Greeks themselves seem to have admitted.  Some sort
of syllogistic reasoning process must have been used by the Egyptians
to arrive at the results implemented in general algorithmic form
in various of the papyri.  Ie, there was an axiomatization,
necessarily, implicit or explicit.  Now here we have reference
precisely to a literal "logical calculus" being attributed to the
Egyptians, by the Greeks, and still it's not enough...

> Gordon Fisher    fishergm@jmu.edu

Regards,

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