Theories and Interpretations

A somewhat naive mechanistic and deterministic world-view is often 
attributed to Isaac Newton, but in fairness it should be said that 
he was (in public) admirably circumspect about the underlying 
structure of reality.  As he said, "I make no hypothesis".  In fact 
he was severely criticized for precisely that reason, i.e., he 
insisted on simply describing things and obstinately refused to 
EXPLAIN things.  He knew the difference between a scientific theory 
and an interpretation.  When people talk about scientific theories 
being overthrown what that usually mean is an *interpretation* has 
been replaced.  

The classical case is Ptolemy's astronomy, which was used to 
describe and predict events with acceptable accuracy for centuries.  
It was never disproven - it still works today as well as it ever 
did.  It was simply replaced by a different theory that was more 
comprehensive, elegant, and powerful.  Of course, what people have 
in mind when they say Ptolemey's theory was rejected is the change 
in interpretation that occurred at the same time, but we shouldn't
confuse scientific theories with their various interpretations.

Now compare this with the history of mathematics.  As an example,
consider the old Theory of Equations which was studied and developed
for centuries.  Eventually it was superceeded by Galois Theory and
abstract algebra, and the old theory was largely abandoned.  This 
doesn't mean the old theory was wrong - it still works as well as 
it ever did.  It was simply replaced by a more comprehensive, elegant, 
and powerful theory.  Moreover, I would argue that the advent of 
abstract algebra, with its non-commutative multiplications and so 
on, represented a real change in the interpretation of the subject 
matter.  The old "permanance of forms" was overthrown.  Thus, even 
if we go back and look at an old book on the Theory of Equations, 
we will see it in a different light and attribute to it a somewhat 
different meaning than the author had in mind.  We now think of 
algebraS (plural), rather than conceiving of One True Algebra located
eternally at the center of the universe.  All the observeables may be
the same, but our idea of "what is really going on" has changed.

In short, I think mathematical theories and interpretations have 
evolved and changed thoughout history much like the theories and 
interpretations in other branches of knowledge.  The view that
mathematical knowledge is uniquely enduring is based on an under- 
estimation of the extent to which past mathematical ideas have been 
displaced, and an overestimation of how much knowledge in other 
areas has actually been falsified (as opposed to re-interpreted).

Of course, the classical counter-argument to the claim that
mathematical knowledge alone among all branches of knowledge is 
cumulative, is to point out that if this were really the case 
we would expect to have no more knowledge of physics (for example)
today than we did at the beginning of recorded history, and we 
would expect the majority of our knowledge to be mathematical, 
since even if its rate of accumulation is slow, the fact that it 
is cumulative should eventually make it overwhelm every other 
branch of knowledge.  Does our experience support these expectations?  
I would say no.  Since the time of Archimedes, which branch of 
knowledge has seen more cumulative progress, mathematics or physics?