After reading about Kuhn's work, this question still keeps me wondering:

Have there really been paradigm shifts, like Kuhn states?

The examples of paradigm shifts often include the switch from geocentrism to heliocentrism, and of course Newtonian mechanics to the theory of relativity.

However, I don't find these examples convincing, because both a geocentrist and a heliocentrist can give the same (mathematical) description of the orbit of the planets. And secondly, Newtonian mechanics are just a special case of the theory of relativity; all Newtons formulas follow directly from Einstein's.

So, neither of these two cases are satisfactory examples. What other examples are there, which strictly include a scientific revolution and incommensurability between the new and the old theory?

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    My problem with Kuhn is that what he calls a "paradigm shift" a scientist calls "doing good work". The quality of the science is measured by the degree to which it kicks the paradigms around. Kuhnian shifts are common, he just focuses on the most famous ones.
    – Ron Maimon
    Commented Apr 16, 2012 at 15:40
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    @Maimon: But that trivialises the work of many good scientists. A lot of science is incremental advances & minor unifications. And all quite necessary. Commented Dec 6, 2012 at 3:41
  • Kuhn's work is descriptive not proscriptive. He describes the history of science through the Wittgensteinian idea of aspect-seeing. See W's Philosophy of Psychology (used to be called "Part II" of PI), which interestingly enough replaced his Remarks on the Foundations of Mathematics as the "second part" to the Investigations. But, his idea of "paradigm shifts" seems to come directly from W's On Certainty: see secs 97 ("river-bed" shifts) and 341-4 ("hinges" of the door). Kuhn, then, describes the history of science through the lens of aspect seeing to account for the shifts in...
    – Jon
    Commented Jul 29, 2013 at 21:47
  • ...the bed-rock, river-beds, and hinges on the door, so to speak. But the SSR was one of the first works to utilize Wittgensteinian concepts to describe specific historical events. Kuhn's reliance upon Wittgenstein cannot be overstated!
    – Jon
    Commented Jul 29, 2013 at 21:55

5 Answers 5


You seem to be puzzled by the fact that Newtonian mechanics was retained in a successive theory and that this somehow precludes incommensurability. It seems that you would think of incommensurable theories as "not relating to each other" somehow and that retaining a former theory is incompatible with this requirement.

If this is your view, the point to be clarified here is that the possibility of including a former theory within a successive theory does not preclude incommensurability between those theories.

Both Kuhn and Feyerabend - who put forward the concept of incommensurability between scientific paradigms (Kuhn) viz. universal theories (Feyerabend) - were trained physicists, so they knew very well that you can derive Newtonian mechanics as a special case of STR.

So, what was Kuhn's point? He writes:

… the physical referents of these Einsteinian concepts are by no means identical with those of the Newtonian concepts that bear the same name. (Newtonian mass is conserved; Einsteinian is convertible with energy. Only at low relative velocities may the two be measured in the same way, and even then they must not be conceived to be the same.) (SSR, p. 102)

His point is that key concepts in both theories, while retaining the same name - like "mass" - have not only different meaning, but that the meanings of Newtonian mass and Einsteinian mass exclude each other. According to Kuhn, there is no concept of mass, in this case, which can consistently unite both meanings - thus they are "incommensurable" concepts, i.e. concepts "without a common measure". (How this gets us to incommensurable theories and paradigms is a lot more complicated, but you get the picture.)

Kuhn uses this point against what is known today as convergent realism, the view that science shows improving approximation to the truth. Why? Because, according to Kuhn, the concept of mass as devised by Einstein does not extend, but replaces, the concept of Newtonian mass. So, if we want to understand in which way science progresses through these shifts, Kuhn argues, the model of successive extension or piecemeal revision of a concept is not a good candidate.

Did Kuhn argue from this - as it is sometimes assumed - that one couldn't derive Newtonian mechanics as a special case of STR? Certainly not. He argued, however, that the special case derived is not actually Newtonian mechanics, but a substitute of it, a numerical approximation within STR. This qualification is of no interest to the working physicist, but it is of interest to someone - philosopher or physicist alike - who wants to argue that the progress of science consists in (old-style) convergent realism.

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    (The case of heliocentric vs. geocentric model is more difficult. Kuhn did think of it as an example of incommensurability, while Feyerabend did reject it, as he thought there was no case of incommensurability to be made. If my current answer isn't enough, I could expand this point.)
    – DBK
    Commented Apr 16, 2012 at 17:45
  • Thanks, this really helps me understanding Kuhn. However, I still wonder whether Einsteinian and Newtonian mass, for instance, differ so fundamentally. We still use the same instruments to measure it, and, according to Wikipedia, the conservation of Newtonian mass still holds, but just slightly different: "The theory of relativity allows particles which have rest [Einsteinan] mass to be converted to other forms of mass which require motion, such as kinetic energy, heat, or light. However, the system mass remains"
    – Harmen
    Commented Apr 16, 2012 at 18:06
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    I agree with your characterization, but I feel compelled to protest that Kuhn's idea of incommensurablility is almost entirely useless to anyone working in physics because the reduction of a newer theory to an older one in a limiting case is just an extension. Indeed, the reduction of relativistic equations to Newtonian ones in the limit of slow speeds (v << c) and modest gravitational fields in fact means that the concept of mass is the same in those cases that we always used it before; just in more esoteric situations, we have to use a more sophistication model. It's still convergent.
    – Rex Kerr
    Commented Apr 16, 2012 at 23:36
  • @Harmen Premise: The problem is that there is no "Einsteinian mass" proper, but relativistic mass and invariant (or rest) mass. When physicists today speak of "mass" (e.g. of a particle), they always mean invariant mass. ("Relativistic mass" is considered a misnomer and one simply speaks of "energy (divided by c-squared)".) …
    – DBK
    Commented Apr 20, 2012 at 11:59
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    @RexKerr Well I don't think (realist) scientists take their best theories to be definitively true, but most certainly approximately true. And that's the general tenet of scientific realism in current phil. of science. There is nothing mystical about the concept of truth here. What does "good description" mean other than "approximately true description"?
    – DBK
    Commented Apr 21, 2012 at 1:11

Both of your examples are actually very valid paradigm shifts. I have tried to summarize them below.

The mathematical explanations of geocentric orbits and heliocentric orbits are actually decidedly different. The geocentric orbit tries to explain retrograde motion of planets by hypothesizing that while all bodies orbit the Earth in a circular motion, they also move in "epicycles" or smaller circular orbits around themselves.

This is very different from the heliocentric model, which in itself explains retrograde motion without these epicycles. Kepler's elliptical model further refines the orbits, and they certainly are not described the same as in the geocentric model. This means that there was a huge mathematical and physical paradigm shift between these two models, both in different mathematical explanations for orbits and in a different physical perspective entirely.

As for Newton's Laws, making them a special case of relativity is quite the paradigm shift; instead of looking at the world as governed by Newton's Laws, we have come to see relativity as the ruling model (or paradigm), where Newton's Laws are a sub-paradigm for certain situations.

Thus, I think both of your examples do actually show a convincing paradigm shift.

As for other (more philosophical) examples, there is the Renaissance, during which the paradigm of man being ruled by God shifted for many to man being ruled by himself. There is also the rise of existentialism in the past century or two; people have stopped thinking "what purpose does the world want me to fulfill" and are now asking "what purpose do I want to fulfill."

  • Your description of the Renneissance is misleading--- the actual change was to deny that any human organization had the power to speak authoritatively for God, and people should freely investigate different aspects for themselves to produce individual contributions which enrich a gradual understanding. The individuals are still just as ruled by the collectives they find themselves in as any other time, and are still just as impotent. It would have been flat-out impossible for Newton to discover quantum mechanics, maybe barely possible for Hamilton--- the difference is the collective.
    – Ron Maimon
    Commented Apr 18, 2012 at 15:49

I think you are misunderstanding Kuhn.

The fact that a geocentrist and a heliocentrist can both give the same description of the orbit of the planets is precisely the point-- they are two different explanatory mechanisms for interpreting the same data, yet the "incommensurability between the new and the old theory" is clear.

Similarly, Einsteinians recognize that Newtonian mechanics are just a special case of the theory of relativity, but Newton didn't see that himself.

Re-read Kuhn, and ponder more.

  • Actually, the underdetermination of theory by evidence is not what Kuhn addresses when speaking about incommensurability. "Re-read Kuhn, and ponder more." :)
    – DBK
    Commented Apr 16, 2012 at 16:40
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    Perhaps I wasn't clear enough; I wasn't speaking merely about the underdetermination of theory by evidence, but rather, the fact the incommensurability that arises from the differing center-points of the the formulas. Similarly, although Newtonian mechanics can be framed in terms of relativity, the reverse is not the case-- they are speaking a different language. See plato.stanford.edu/entries/incommensurability Commented Apr 16, 2012 at 19:44
  • From another angle, GR could be seen as a culmination of Newtons project: Newton taught that a particle moves in a straight line at constant velocity when no forces are acting. Einstein modified that to all particles move in a straight line on a curved 4-d space at exactly the same constant velocity. This is why physicists generally think of GR as a classical theory. Commented Apr 17, 2012 at 3:52
  • Along these lines, we teach that the "geocentric universe is wrong," which is an interesting oddity, if they are merely mathematical transforms of each other.
    – Cort Ammon
    Commented Dec 13, 2015 at 16:57

I also would disagree with Kuhn on whether many of his supposed examples of "paradigm shifts" would actually qualify as such. However, I think there are better and clear examples of paradigm shifts in the history of science, which if nothing else serve as proof of concept. In fact, I was reminded of what is probably the best example by the questioner's accidental word choice here:

What other examples are there, which strictly include a scientific revolution and incommensurability between the new and the old theory?

Probably the most important incommensurability in the history of science was none other than the discovery of incommensurability of course! Sometime around the 5th century BC a Pythagorean (conjectured to be Hippasus) discovered and logically proved that some numbers like √2 can't be expressed as a ratio of whole numbers. This brought Pythagorean mathematics, founded on the basic assumption that all of mathematics and everything in the universe can be reduced to whole numbers and their ratios, to a screeching halt. Greek mathematics at this stage underwent a complete philosophical reboot, and geometry supplanted number as providing the foundations of the rest of mathematics.

Had the doctrine of number not failed, we'd probably refer to Pythagorean Number Theory and the Pythagorean Method like we do to the geometry of Euclid and the axiomatic-deductive method as the beginning of the history of formal mathematics.

One might wonder whether this mathematical paradigm shift also counts as a scientific one, and the answer is yes. The distinction between physical geometry and mathematical geometry wasn't really understood until centuries later. For the ancients, geometry could be considered the science of space. The unification of method geometry offered across diverse fields such as statics, astronomy, navigation, surveying, and geodesy (a science invented by the Greeks) only served to cement the transition from number to space.


I think the Copernican turn was important for not just the switch from geocentric to heliocentric universe, but because it was the beginning of a new intellectual current. After all this discovery had been made 2 millenia ago, and given the circulation of Archimedes texts, astronomers would have known of this hypothesis by Aristarchus. Instead, it marks the ascendency of science as intellectual arbiter and the eclipse of the Church and the slow abandoning of the public sphere by Christianity.

The metaphor for switching geo to helio became a kind of metaphor for intellectual introspection and critical inquiry. A similar move can be detected this century with the word 'quantum' playing the same role. Examine the phrase a 'quantum leap', and when one understand that in the physical sense this just means an extremely tiny discrete jump, it doesn't seem important enough to carry the weight of meaning that it does, but thats because its second metaphorical meaning is a large conceptual change.

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