How knowledge grows

Later theories contradict earlier theories

Knowledge does not grow in an incremental fashion as justificationists and inductivists believe; it grows in a revolutionary way. New theories do not extend older ones; they contradict them. This can be illustrated by means of the following diagram:

diagram illustrating how knowledge grows and Feyerabend's principle of proliferation

This diagram represents how theories in mechanics have developed over the years. On the left we have terrestrial mechanics and on the right celestial mechanics; the theories in the middle apply both to terrestrial and celestial phenomena. The earliest theories are those of Aristotle and Ptolemy. Each theory with at least one predecessor is inconsistent with its predecessor or predecessors. There follow some examples of these inconsistencies.

Galileo's theory contradicts Aristotle's

One of the many differences between Aristotle's mechanics and Galileo's is that, according to Aristotle, if you drop two objects from a great height, the heavier one will hit the ground before the lighter one. Galileo famously dropped two objects of unequal weight from the top of the Leaning Tower of Pisa and they hit the ground at the same time. His mechanics could account for this phenomenon.

Newton's mechanics contradicts Galileo's

(1) According to Galileo's mechanics a projectile fired from the surface of the Earth moves in a parabola, but according to Newtonian mechanics it moves in an ellipse. (2) According to Galileo's mechanics a free-falling body (for example, dropped from a high object) has constant acceleration, but according to Newtonian mechanics its acceleration increases slightly as it moves closer to the centre of the Earth.

Copernicus's theory contradicts Ptolemy's

For Ptolemy the planets and the Sun moved in circles around the Earth, whereas for Copernicus, the planets, including the Earth, moved in circles around the Sun.

Kepler's theory contradicts Copernicus's

According to Copernicus, the planets moved in circles around the Sun, whereas, according to Kepler, they moved around the Sun in ellipses.

Newton's theory contradicts Kepler's

In framing his laws of planetary motion Kepler ignores the mutual attraction that exists between heavenly bodies. Thus, according to Kepler's third law a3/T2 = k, where a is the mean distance between two bodies with mass m0 and m1, T is the time it takes for one complete revolution of one about the other and k is a constant. For Newton, however, a3/T2 = m0 + m1.

Einstein's theory contradicts Newton's

There are many differences between Newtonian and Einsteinian mechanics. In Einstein's theory of relativity nothing can travel faster than the speed of light, whereas there is no maximum velocity in Newtonian mechanics. In Einsteinian mechanics two events are only simultaneous relative to a frame of reference, whereas in Newtonian mechanics events are simultaneous or non-simultaneous absolutely.

Currently, Einstein's theory is the best we've got, but that doesn't mean that it is the last word on the matter.

What do we learn from this?

Popper and Feyerabend were very fond of illustrating the way in which knowledge grows by this kind of example. Some of the above examples of contradictions between theories come from chapter 5, "The Aim of Science", of Popper's book Objective Knowledge. Popper goes on to give the following advice (Objective Knowledge, p. 266):

Whenever a theory appears to you as the only possible one, take this as a sign that you have neither understood the theory nor the problem which it was intended to solve.

Some people, who favour the idea that knowledge grows incrementally, on being presented with the above diagram say that Kepler's theory, for example, is a good approximation to Newton's theory. I don't deny that, but my point is that it is inconsistent with Newton's theory and that refutes the idea that science grows incrementally.

Feyerabend is regarded by some as being rather an oddball in the philosophy of science and scorn is heaped on his principle of proliferation. However, reflecting on the what the scientists above, apart from Aristotle and Ptolemy, did we see that, in effect, they were following Feyerabend's advice. No theory in the history of science has been as successful as Newton's; it was highly confirmed and generally accepted and, yet, Einstein invented a theory that contradicted Newton's. Without knowing it he was acting on Feyerabend's principle of proliferation (quoted from Preston's book Feyerabend, p. 138):

Invent, and elaborate, theories which are inconsistent with the accepted point of view, even if the latter should happen to be highly confirmed and generally accepted.

Science only grows if scientists follow Feyerabend's principle of proliferation. Of course, a theory is not accepted because it contradicts existing theories. A theory is accepted by the scientific community because it explains most, if not all, of the same phenomena explained by existing theories and because is also explains some new phenomena not explained by existing theories.


  • Karl Popper, Objective Knowledge: An Evolutionary Approach, London, Oxford University Press, 1972.
  • John Preston, Feyerabend: Philosophy, Science and Society, Cambridge, Polity Press, 1997.

© Antoni Diller (2 April 2014)