Weighing the Evidence

When discussing the difference between physics and mathematics, 
people sometimes acknowledge that there are paradigm shifts in 
mathematics, but they maintain that these shifts do not invalidate 
previous results, whereas in physics (they claim) a paradigm shift 
DOES invalidate previous results.  As an example, they suggest that 
we no longer believe everything is composed of earth, wind, fire and 
water, nor do we believe that general relativity shows that the 
proof of the Pythagorean theorem in Euclid is wrong.

Let's take these one at a time.  First, as a classification scheme 
for the possible states of matter, the categories of solid, gas, 
liquid, and plasma (earth, wind, water, fire) are not too bad.  Then 
again, it's not clear that it's even possible to "invalidate" a 
classification scheme.  Can we invalidate the classification of 
Europe and Asia as separate continents?  People have reasons for 
classifying things as they do.  We may decide to classify things 
differently someday (although those four categories have been 
remarkably enduring), but that wouldn't really "invalidate" other 
schemes.

Second, does general relativity prove the ancient Greek geometers were 
"wrong"?  There's a sense in which it actually does, insofar as they 
were engaged in an activity that today would be called physics, i.e., 
they were trying to find out facts about spatial relations.  For 
example, we know from a letter written by Archimedes to a friend that 
he actually discovered many of his marvelous theorems by cutting out 
curved shapes (sections of parabolas and so on), and then physically 
weighing them to compare their areas, and dunking spheres and cylinders 
in water and measuring their displacements to compare their volumes.  
Admittedly he then went on to devise synthetic proofs of his theorems, 
but he clearly regarded his results as descriptive of actual spatial 
relations.  In this he was mistaken.

How does this bear on the issue at hand?  In one sense it supports
the idea that physical theories have been invalidated whereas math
theories have not, if we accept that Euclidean geometry has been 
invalidated as a physical theory but not as a mathematical theory.  
However, in two ways, one technical and one philosophical, it also 
shows how mathematical ideas have been invalidated and rejected.

The technical point is that if we remove the empirical justification
for Euclid's geometry (for example), and try to evaluate it as
a purely abstract mathematical creation, it doesn't stand up all
that well to modern scrutiny.  In a way this is inevitable because
Euclid wasn't TRYING to create an arbitrary axiomatic system in
the modern sense.  Nevertheless, if mathematicians want to claim
The Elements as a work of pure math they can't escape the fact that
it fails strictly on those terms.  For example, there are no axioms 
of "betweeness" or "continuity" or many other things that would be 
necessary for a coherent axiomatic system.  To compensate for these 
deficiencies, Euclid relies heavily on physical intuition and visual 
evidence.  So the direct answer to the question is yes, the modern 
judgement of Euclid's proof of Pythagoras's theorem is that it is 
incorrect, i.e., it does not follow by strict deductive reasoning 
from the axioms and postulates and common notions as presented.

The second point is philosophical.  People like Euclid and Archimedes 
considered themselves to be mathematicians and believed what they were 
doing - finding out the TRUE ideal forms - was mathematics.  As a 
mathematical idea this view has been largely invalidated.  Today most 
mathematicians believe that math is the business of working out the 
implications of a set of premises - ANY set of premises that strikes 
their fancy.  In this sense modern mathematics has evolved a new 
understanding of what math is, and has largely rejected and discarded 
the view of mathematics held by most ancient (and some not so ancient) 
mathematicians.