Paradigm and paradigm shift are so over used and misused that the world would benefit if they were simply banned. Originally Thomas Kuhn (1922–1996) in his 1962 book, The Structure of Scientific Revolutions, used the word paradigm to refer to the set of practices that define a scientific discipline at any particular period of time. A paradigm shift is when the entire structure of a field changes, not when someone simply uses a different mathematical formulation. Perhaps it is just grandiosity, everyone thinking their latest idea is earth shaking (or paradigm shifting), but the idea has been so debased that almost any change is called a paradigm shift, down to level of changing the color of ones socks.
The archetypal example, and I would suggest the only real example in the natural and physical sciences, is the paradigm shift from Aristotelian to Newtonian physics. This was not just a change in physics from the perfect motion is circular to an object either is at rest or moves at a constant velocity, unless acted upon by an external force but a change in how knowledge is defined and acquired. There is more here than a different description of motion; the very concept of what is important has changed. In Newtonian physics there is no place for perfect motion but only rules to describe how objects actually behave. Newtonian physics was driven by observation. Newton, himself, went further and claimed his results were derived from observation. While Aristotelian physics is broadly consistent with observation it is driven more by abstract concepts like perfection. Aristotle (384 BCE – 322 BCE) would most likely have considered Galileo Galilei’s (1564 – 1642) careful experiments beneath him. Socrates (c. 469 BC – 399 BC) certainly would have. Their epistemology was not based on careful observation.
While there have been major changes in the physical sciences since Newton, they do not reach the threshold needed to call them a paradigm shifts since they are all within the paradigm defined by the scientific method. I would suggest Kuhn was misled by the Aristotle-Newton example where, indeed, the two approaches are incommensurate: What constitutes a reasonable explanation is simply different for the two men. But would the same be true with Michael Faraday (1791 – 1867) and Niels Bohr (1885–1962) who were chronologically on opposite sides of the quantum mechanics cataclysm? One could easily imagine Faraday, transported in time, having a fruitful discussion with Bohr. While the quantum revolution was indeed cataclysmic, changing mankind’s basic understanding of how the universe worked, it was based on the same concept of knowledge as Newtonian physics. You make models based on observations and validate them through testable predictions. The pre-cataclysmic scientists understood the need for change due to failed predictions, even if, like Albert Einstein (1879 – 1955) or Erwin Schrödinger (1887 – 1961), they found quantum mechanics repugnant. The phenomenology was too powerful to ignore.
Sir Karl Popper (1902 – 1994) provided another ingredients missed by Kuhn, the idea that science advances by the bold new hypothesis, not by deducing models from observation. The Bohr model of the atom was a bold hypothesis not a paradigm shift, a bold hypothesis refined by other scientists and tested in the crucible of careful observation. I would also suggest that Kuhn did not understand the role of simplicity in making scientific models unique. It is true that one can always make an old model agree with past observations by making it more complex. This process frequently has the side effect of reducing the old models ability to make predictions. It is to remedy these problems that a bold new hypothesis is needed. But to be successful, the bold new hypothesis should be simpler than the modified version of the original model and more crucially must make testable predictions that are confirmed by observation. But even then, it is not a paradigm shift; just a verified bold new hypothesis.
Despite the nay-saying, Kuhn’s ideas did advance the understanding of the scientific method. In particular, it was a good antidote to the logical positivists who wanted to eliminate the role of the model or what Kuhn called the paradigm altogether. Kuhn made the point that is the framework that gives meaning to observations. Combined with Popper’s insights, Kuhn’s ideas paved the way for a fairly comprehensive understanding of the scientific method.
But back to the overused word paradigm, it would be nice if we could turn back the clock and restrict the term paradigm shift to those changes where the before and after are truly incommensurate; where there is no common ground to decide which is better. Or if you like, the demarcation criteria for a paradigm shift is that the before and after are incommensurate. That would rule out the change of sock color from being a paradigm shift. However, we cannot turn back the clock so I will go back to my first suggestion that the word be banned.
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 This is known as the Duhem-Quine thesis.
 There are probably paradigm shifts, even in the restricted meaning of the word, if we go outside science. The French revolution could be considered a paradigm shift in the relation between the populace and the state.