[FOM] Fwd: Forward of moderated message

Sat Sep 12 09:45:02 EDT 2015

George,

I think we agree on almost everything. The point is 7; Euclid uses area to
the sides of similar triangles are proportional and Archimedes to
accomodate incommensurables.

Hilbert defines, multiplication and then area.  Thus he has no need of
Archimedes.

One technicality.  Hilbert (in my opinion ) misreads Euclid as requiring
areas of polygons to be preserved only by
literal decompostions.  Hilbert calls the more general notion where you are
allowed to both add and subtract content.

He notes that computing area  in the narrow sense requires Archimedes but
computing content does not.  I think it clear from common notion 3 and 4
that euclid means by area what HIlbert calls content.

--------

Archimedes and Dedekind's axioms are add on's (In Dedekind's case
literally) for Hilbert to make the Euclidean field complete.

I argue that the greeks knew a lot about particular real numbers but no
thought of all them.

See the following papers on my website.

most of the argument is in the first.

 *Axiomatizing Changing Conceptions of the geometric continuum I:
Euclid-Hilbert*  http://homepages.math.uic.edu/~jbaldwin/pub/axconIsub.pdf

http://homepages.math.uic.edu/~jbaldwin/pub/axconIIfin.pdf

http://homepages.math.uic.edu/~jbaldwin/pub/pisub.pdf

John Baldwin

John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607

On Wed, Sep 9, 2015 at 9:41 PM, George McNulty <mcnulty at math.sc.edu> wrote:

> 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
>
>
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