[FOM] 605: Integer and Real Functions

[George McNulty]

John,

         I was interested in the integer/reals  discussion on FOM.

         Having myself been a student of Euclid since 1971 when I took a 
foundations of geometry course from Lezsek Sczcerba when he was a 
post-doc in Berkeley having just finished his PhD with Wanda Szmielew in 
Warsaw, I thought you might help me put my own thought here straight.  I 
have very small Latin and no Greek and cannot even begin to consider 
myself a scholar of mathematical history of these ancient times.  But I 
have read my way through what remains to us of Euclid and Acrchimedes 
(in English!) a number of times, as well as both the first and second 
editions of Hilbert's book (in can make out the German...), and the very 
nice book of Borsuk and Szmielew.

         So here is what I gather.

        0.  Euclid, and probably other Greek mathematicians several 
centuries preceding Euclid, had a firm grip on addition and 
multiplication of natural numbers (apart from 0).  While lacking a good 
notational scheme,  the theorems in the number theoretic books of the 
Elements, and even the proofs often enough, are at home in undergraduate 
number theory courses today.

       1. Euclid makes a careful distinction between numbers used to 
count with and numbers used to measure with.  Heath puts into Euclid's 
mouth the word ``multitude'' in reference to counting, while he uses 
``magnitude'' in reference to measuring.  I would guess that Euclid 
takes a kind a care here that to people in our time seems over fussy, 
since the Pythagorean discovery of the incommensurability of the side of 
a square with its diagonal was still, after several generations, still 
fresh.  The resolution of this issue, credited to Eudoxus, came a couple 
of generations after Pythagoras and a couple before Euclid and 
represented the greatest achievement (my judgment) enshrined in the 
Elements.

       2.  Euclid also is careful to distinguish the kind of magnitudes 
used to measure line segments and those used to measure figures. (I 
remember my physics professor harping heavily on units of measure---he 
was a Euclidean!)

       3.  I could not find anywhere in the Elements where Euclid 
actually multiplies magnitudes, although Book VI might have something 
tucked away.  Archimedes, I seem to recall, is less hesitant.

       4.  What Euclid actually does is deal with ratios or 
proportions.  In expounding Eudoxus' theory to Book V (the toughest to 
read, I found) and then developing its consequences in Books VI, and X 
through XIII.  It is, in more modern terms, as if he chose to develop 
the theory of division rather than that of multiplication. At any rate, 
the theory of similar figures gives a way to understanding the scaling 
of figures---in more modern terms, scalar multiplication.  I would not 
hesitate the say that the theory of similar figures houses a theory of 
multiplication of magnitudes.  As Euclid has also developed a theory of 
adding line segments (and also of areas, cf the Pythagorean Theorem), I 
would say that the Elements has in fact, some version of addition and 
multiplication of magnitudes.  Moving to Archimedes I think this is even 
more readily apparent.

       5.  I recall my surprise, on first reading Euclid, that none of 
the familiar formulas of geometry I learned in high school, were to be 
found.  Euclid thought that the Pythagorean Theorem was about drawing 
actual squares and showing how to match up areas.  NO  a^2 + b^ 2= c^2.

       6.  The great champion of the algebraic rendition of all this 
stuff was not Descartes but Richard Dedekind.  After his docent years in 
G\"ottingen, Dedekind took at appointment at ETH in Zurich and found 
himself teaching calculus for the first time in 1858. While he realized 
that he had to do a lot of hand waving (just like the rest of us!) he 
set himself the task of figuring out why it all really worked.  He said 
he finally succeeded on 24 November 1858. He wrote it all down in a 
paper call ``Continuity and Irrational Numbers'' that was eventually 
published in 1872.  This is the paper with Dedekind cuts.  Dedekind say 
quite explicitly in this paper that he wants to completely banish any 
dependence on geometry and do everything algebraically.  In a later 
paper published in 1887 ``Was sind und was sollen die Zahlen'' (where 
Dedekind develops Peano arithmetic and introduces Dedekind finite sets)  
Dedekind observes that the notion of a Dedekind cut is closely linked 
with Eudoxus's theory from Book V of the Elements, but with a very 
important difference.  Where Eudoxus addressed the task of reconciling 
two notions already in hand, namely counting numbers (that is positive 
rationals) and magnitudes,  Dedekind wanted instead to build the real 
numbers (the magnitudes) from the rationals.  It the process he was even 
able to frame a geometric completeness axiom:

          ``If all points of the straight line fall into two classes 
such that every point of the first class lies to the left of every point 
of the second class, then there exists one and only one point with 
produces this division of all points into  two classes, this severing of 
the straight line into two portions.''

This is from the 1872 paper.   Of course, Euclid used some principle 
like this in lots of places.  It was also used by HIlbert in the second 
edition of his book (the first edition has a completeness axiom only a 
model theorist could love).

        7.  Finally, if I recall Hilbert's book correctly, he uses the 
theory of similar triangles to define multiplication, when he goes about 
the business of coordinatizing his plane.  I don't recall area being 
used to get at multiplication.  Hilbert had to deduce enough geometry 
from his axioms to proved that the addition and multiplication he 
defined for points on a line to impose on it a complete ordered field.  
I remember from Szczerba's course that things like the Pappus-Pascal 
Theorem and Desargue's Theorem has important roles, as did a lot of 
stuff from Euclid, especially the theory of parallelograms and of 
similar triangles.

       8.  I guess my bottom line is that the ordered field of real 
numbers and the Euclidean plane are interchangeable mathematical 
objects.  So it seems to me that Eudoxus, Euclid, and Archimedes and 
their followers knew a lot about the real numbers.


How much of that corresponds to your understanding of this stuff?

Best Regards,

George