[FOM] Euclid's axioms

[William Tait]

A number of postings to the list in the fairly recent past have 
concerned the absence of the axiom of continuity in Euclid's 
_Elements_. With some blushing, let me post a somewhat fanciful 
response to this, adapted from an equally fanciful course of lectures 
on Greek mathematics I once gave.

Bill Tait

.................

Euclid is often criticized for not postulating the existence of the 
points of intersection of
curves. For example, let c be a circle and l a curve with points in the 
interior and points in the exterior of
c. The criticism is that Euclid does not postulate that l has points on 
c, but assumes in proofs that it does. For example, Book I, Theorem 1 
already assumes that there are such points: Euclid takes the two points
of intersection of two circles of radius equal to the distance between 
their centers. But I think
that there is some evidence that this criticism is open to question.

The cirumference of a circle is a line of a certain sort (I. Def.15). 
The definition of a line has
two parts, Def 2 and 3. Def. 2 says that a line is length without 
breadth. Def. 3 says that the
perata  of a line are points. De.f 1 has already stated that a point is 
that which has no part.
So, presumably, to say that a line has length is to say that it has 
parts. Suppose that we cut it
into two parts. To say that it has no breadth is to say that what is 
common to those parts, their
boundary, has no parts. The term 'perata ' is usually translated as 
'extremities'. But this
doesn't make sense: a circumference is a line and has no extremities. 
The term is translated
  in other contexts to mean, for example, 'limit' or 'determinant'. It 
is reasonable to think that
what Euclid has in mind here by perata  is what constitues the boundary 
of any two parts. Indeed,
Def 13 states that a boundary is that which is the peras  of anything. 
Thus, on this reading,
Euclid is realizing that to say that the boundary has no parts is not 
enough: the boundary might
be null. He therefore supplements Def.2 by asserting that there is a 
boundary---which consists of
points. In modern terms, he is requiring lines (i.e. curves) to be 
connected. Note that there is a similar
phenomenon in connection with his Def. 5 and 6 concerning the notion of 
a surface: a surface is
length and breadth without depth and the perata  of a surface are 
lines. In other words, if we cut
a surface into two parts, its boundary has no breadth, i.e. at most 
consists of lines. This is
supplemented by requiring that the boundary indeed consist of lines.

One should compare these definitions to Poincare's informal definition 
of the dimension of a
continuum in Revue de metaphysique et de moral(1912),p.486, quoted in 
Hurewicz-Wallman's _Dimension
Theory_, p.3. "... if to divide  a continuum it suffices to consider as 
cuts a certain number of
elements all distinguishable from one another, we say that this 
continuum is of one dimension ;
if, on the contrary, to divide a continuum it is necessary to consider 
as cuts a system of
elements themselves forming one or several continua, we shall say that 
this continuum is of
several dimensions . "If to divide a continuum C, cuts which form one 
or several continua of one
dimension suffice, we shall say that C is a continuum of two dimensions 
; if cuts which form one
or several continua of at most two dimensions suffice, we shall say 
that C is a continuum of three
dimensions ; and so on."

It is, or so the argument that I may or may not quite believe runs, 
this inductive character of
the notion of dimension that Euclid is anticipating in his definitions. 
The main difference, aside
from considering only continua of dimension <4, is that he is 
considering a less general class of
continua than Poincare was: he requires that all cuts of the n+1 
dimensional continuum determine one
or more continua, all of dimension n. For example, Euclid would not 
consider a line segment with a
point missing as a line (and perhaps could make no sense of the idea). 
Likewise, he would not consider two tangent disks
as forming a surface---a ``figure''.

In any case, if this reading of Euclid is correct, he needs no separate 
postulation of a point of
intersection of a circle c and a curve l: the circle cuts the curve 
into the part interior and the
one or two parts exterior to the circle. (Of course, he never makes 
this precise.) This
yields the one or two points of intersection.

One problem with this reading is that, as I understand it, there is no 
knowing what definitions
were actually in the original text. David Fowler recently pointed out 
that the earliest text that
we have is closer to our times than to Euclid's. Fowler himself (in 
other contexts) likes to take this as an argument
that one likely (or perhaps unlikely) story is as good as another. 
Whether or not that is so, I like my story, which
makes Euclid and his colleagues smarter than anyone else was for a long 
time after.

......

Now let me send this off before I lose my nerve.  Bill