[FOM] Are Proofs in mathematics based on sufficient evidence?

Vaughan Pratt pratt at cs.stanford.edu
Thu Jul 15 02:21:51 EDT 2010

On 7/13/2010 8:46 PM, Monroe Eskew wrote:
> On the other hand, one can point to certain
> arguments in Euclid's Elements which are not valid as ordinary
> mathematical arguments (i.e. not valid when charitably formalized into
> classical first order logic).  Therefore since he singled out Euclid
> he probably meant to refer to these defects rather than the lack of
> explicit deduction rules.

It's hard to separate these.  Ignoring the complaints about Proposition 
1, which I've never understood, and the awful but technically correct 
proof of Proposition 2 (there is a *much* simpler proof), to my thinking 
the problem happened at Proposition 16, "In any triangle, if one of the 
sides is produced, then the exterior angle is greater than either of the 
interior and opposite angles."  The relevant figure and argument can be 
seen at


The problem comes at the step "join FC".  Postulate 5 ruled out 
hyperbolic geometry but not elliptical.  On the sphere, if the distance 
from F to C is more than half the circumference the segment FC will be 
the complete line FC less the segment drawn by Euclid in the diagram. 
This screws up the angles in the rest of the argument.

People over the last half millennium or so have excused this on the 
ground that Postulate 2, that you could extend a line segment 
indefinitely, meant that it would never meet itself if you did.  Suppose 
that was what Euclid in fact intended.  How does Proposition 16 follow 
under that assumption?  It just doesn't make any sense.

If Euclid had had explicit and sound deduction rules, and stuck to them 
like a leech, this problem would have come to light.  Had they been 
unsound the unsoundness would have come to light.  Without formality one 
is thrown back on common sense, which is what the argument above 
depended on.  As such it is open to attack, which I'm happy to field.

Vaughan Pratt

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