*How wonderful that we have met with a paradox. Now we have some hope of making progress.*(Niels Bohr)

Science appears to be filled with paradoxes, which it itself is a paradox, because true science should be free of paradox. For a true scientist a scientific paradox is thus something unbearable, which requires immediate action, because one paradox is enough to kill a whole theory.

There are logical paradoxes as contradiction between words and there are physical paradoxes as contradictions between theory and observation.

One logical paradox is enough to kill a mathematical theory. Thus Russell's paradox killed set theory as the foundation of mathematics in the early 20th century.

Zeno's paradox (still unresolved) of the arrow which is moving although it is not moving at every instant, triggered the development of Calculus, but with a delay of 2000 years!

One physical paradox is enough to kill a physical theory as a mathematical theory about phenomena of physics, all according to the famous physicist Feynman:

A paradox may be thus devastating to existing theory, while leading to new better theory by focussing on weak points.

Yet, the list of physical paradoxes has remained through the development of modern physics and in fact have multiplied since modern physics is loaded with many more paradoxes than classical rational physics, as if modern physics is irrational. Thus the pillars of modern physics in the form of relativity theory and quantum mechanics are both filled with paradoxes, which have remained unresolved for 100 years. This has formed the deep trauma of modern physics with no escape from ever more paradoxes.

Niels Bohr was a master of handling the many paradoxes of quantum mechanics lifting sophistry to a new level with his "complementarity principle" addressing the wave-particle contradiction with murky statements like:

The first defence line for a classical physicist is to simply deny the existence of a paradox formulated by some renegades. The next is to accept that there is indeed a paradox and then come up with an ad hoc explanation for the contradiction between theory and reality, showing that the contradiction is in fact only apparent, but not really real. If the ad hoc explanation is refuted, a new ad hoc explanation is presented and so on.

For a modern physicist, a paradox thus poses no real problem, but of course it is some kind of nuisance and so occasionally may get some attention. Like the Twin Paradox of special relativity discussed in earlier posts, unresolved since 100 years.

The prime paradox of fluid mechanics is d'Alembert's paradox comparing the prediction of zero resistance to motion through a fluid from potential flow solutions to Euler's equations of slightly viscous flow like air and water, with the observation of heavy resistance increasing quadratically with velocity.

The paradox was formulated by d'Alembert in 1755 but nobody was able to come up with a resolution until the young German fluid mechanician Ludwig Prandtl in 1904 came up with the ad hoc solution to discriminate the zero drag potential solution of Euler's equations because potential flow does not satisfy a no-slip boundary condition coming with a thin boundary layer. With the potential solution thus eliminated form the discussion, the paradox simply disappeared. But the act of discrimination of solutions of the Euler equations of course was not so glorious. Discrimination of prefect exact solutions on formal grounds carries the same weakness as discrimination of good citizens on purely formal grounds.

In 2008 we gave a different resolution of d'Alembert's paradox than Prandtl's, based on the fact that potential solutions of the Euler equations are unstable and thus turn into turbulent solutions with substantial pressure drag. This was not discrimination on formal grounds, but on real grounds; an unstable solution does not persist over time. This opened to a revolution in computational fluid dynamics freed from a perceived necessity to computationally resolve unresolvable thin boundary layers.

You find on this web site, if you are interested and make a search, resolutions of the following paradoxes:

There are logical paradoxes as contradiction between words and there are physical paradoxes as contradictions between theory and observation.

One logical paradox is enough to kill a mathematical theory. Thus Russell's paradox killed set theory as the foundation of mathematics in the early 20th century.

Zeno's paradox (still unresolved) of the arrow which is moving although it is not moving at every instant, triggered the development of Calculus, but with a delay of 2000 years!

One physical paradox is enough to kill a physical theory as a mathematical theory about phenomena of physics, all according to the famous physicist Feynman:

*It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong.*

A paradox may be thus devastating to existing theory, while leading to new better theory by focussing on weak points.

Yet, the list of physical paradoxes has remained through the development of modern physics and in fact have multiplied since modern physics is loaded with many more paradoxes than classical rational physics, as if modern physics is irrational. Thus the pillars of modern physics in the form of relativity theory and quantum mechanics are both filled with paradoxes, which have remained unresolved for 100 years. This has formed the deep trauma of modern physics with no escape from ever more paradoxes.

Niels Bohr was a master of handling the many paradoxes of quantum mechanics lifting sophistry to a new level with his "complementarity principle" addressing the wave-particle contradiction with murky statements like:

*The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth.*

The first defence line for a classical physicist is to simply deny the existence of a paradox formulated by some renegades. The next is to accept that there is indeed a paradox and then come up with an ad hoc explanation for the contradiction between theory and reality, showing that the contradiction is in fact only apparent, but not really real. If the ad hoc explanation is refuted, a new ad hoc explanation is presented and so on.

For a modern physicist, a paradox thus poses no real problem, but of course it is some kind of nuisance and so occasionally may get some attention. Like the Twin Paradox of special relativity discussed in earlier posts, unresolved since 100 years.

The prime paradox of fluid mechanics is d'Alembert's paradox comparing the prediction of zero resistance to motion through a fluid from potential flow solutions to Euler's equations of slightly viscous flow like air and water, with the observation of heavy resistance increasing quadratically with velocity.

The paradox was formulated by d'Alembert in 1755 but nobody was able to come up with a resolution until the young German fluid mechanician Ludwig Prandtl in 1904 came up with the ad hoc solution to discriminate the zero drag potential solution of Euler's equations because potential flow does not satisfy a no-slip boundary condition coming with a thin boundary layer. With the potential solution thus eliminated form the discussion, the paradox simply disappeared. But the act of discrimination of solutions of the Euler equations of course was not so glorious. Discrimination of prefect exact solutions on formal grounds carries the same weakness as discrimination of good citizens on purely formal grounds.

In 2008 we gave a different resolution of d'Alembert's paradox than Prandtl's, based on the fact that potential solutions of the Euler equations are unstable and thus turn into turbulent solutions with substantial pressure drag. This was not discrimination on formal grounds, but on real grounds; an unstable solution does not persist over time. This opened to a revolution in computational fluid dynamics freed from a perceived necessity to computationally resolve unresolvable thin boundary layers.

You find on this web site, if you are interested and make a search, resolutions of the following paradoxes:

- D'Alembert's paradox and other paradoxes of fluid mechanics.
- The Reversibility paradox of classical and quantum mechanics (Loschmidt's paradox)
- Paradoxes of special relativity including the Twin paradox.
- Paradoxes of wave-particle and collapse of the wave function of quantum mechanics.

Yes, it is wonderful to discover a paradox and even more wonderful to resolve it!