Is Knowledge Cumulative? 

Shut up the words, and seal the book, even to the end of time. Many shall run to and fro, and knowledge shall be increased. 
Daniel 12:4 

I gave up my heart to know wisdom, and to know maddness and folly. I perceived also that this is vexation of spirit. For in much wisdom is much grief, and he that increaseth knowledge increaseth sorrow. 
Ecclesiastes, 1:17 

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 the ancient belief that everything is composed of earth, wind, fire and water has been falsified by modern physics, whereas general relativity has not falsified the proof of the Pythagorean theorem found in Euclid. 

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 judgment 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. 

In the popular imagination, Isaac Newton (one of the founders of modern physics) is often regarded as espousing a somewhat naive mechanistic and deterministic worldview, but in fairness it should be said that Newton was (in public) admirably circumspect about the underlying structure of reality. He even claimed that "I make no hypothesis". In fact he was severely criticized on precisely these grounds, i.e., he insisted on simply describing things, and declined to offer explanations of things. He knew the difference between a scientific theory and an interpretation. When people talk about scientific theories being overthrown, they usually mean 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 disproved  it still works today as well as it ever did. It was simply replaced by a different interpretation that proved to be more comprehensive, elegant, and powerful. Of course, what people have in mind when they say Ptolemy’s theory was rejected is the change in interpretation. 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 superceded 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 noncommutative multiplications and so on, represented a real change in the interpretation of the subject matter. The old "permanence 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 observables 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 throughout history much like the theories and interpretations in other branches of knowledge. The view that mathematical knowledge is uniquely enduring is based on an underestimation 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 reinterpreted). Of course, the classical counterargument 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? 
