Abstract
In The Structure of Science, Ernest Nagel finds fault with Werner Heisenberg’s explication of the uncertainty principle. Nagel’s complaint is that this principle does not follow from the impossibility of measuring with precision both the position and the momentum of a particle, as Heisenberg intimates, rather it is the other way around. Recent developments in theoretical physics have shown that Nagel’s argument is more substantial than he could have envisaged. In particular it has become clear that there are in fact two uncertainty principles; as a result, there are four pairs of quantities to examine, whereas Heisenberg considers only one. These findings throw new light on Nagel’s criticism. They enable us to see that his intuition was surprisingly apposite, but also make clear where his argument misses the mark.
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Notes
- 1.
Of course the domain has to be specified, e.g. certain singular situations have to be excluded, but this is to be understood.
- 2.
It is important not to confuse Nagel’s criticism with that of Niels Bohr. In 1927, both at the Volta conference in Como, Italy, and at the Solvay conference in Brussels, Bohr objected to Heisenberg’s claim that the uncertainty relations follow from the gamma ray thought experiment; according to Bohr, that thought experiment is defective (see Sect. 7.3). Bohr was right, but the objection is not the same as Nagel’s, which is that the uncertainty relations do not follow from uncontrollability.
- 3.
The actual formula in Heisenberg’s seminal paper is p1q1 ∼ h, where p1 is “die Genauigkeit, mit der der Wert p bestimmbar ist” and q1 is “etwa der mittlere Fehler von q” (Heisenberg, 1927, p. 175).
- 4.
Camilleri (2009, p. 93) has argued that for Heisenberg the matter is actually not epistemic but semantic: an accurate measurement of the position of an electron renders the concept of its momentum meaningless, rather than meaningful but unknowable. This distinction is interesting, but for our argument not relevant.
- 5.
See Kennard (1927). Heisenberg was formally Privatdozent in Göttingen from 1924 to 1927; but from May 1926 to the end of 1927 he was actually employed in Copenhagen as lecturer and assistant to Niels Bohr.
- 6.
Our translation from the German: “Dieses ist das etwas verallgemeinerte Unbestimmtheitsgesetz von Heisenberg. Es setzt eine untere Grenze für das Produkt der Unbestimmtheitsmaße für jedes Paar kanonischer Variabeln fest.” The inequality that Kennard is talking about is inequality (27) in his text.
- 7.
Another example is the textbook by L.E. Ballentine (1990, p. 166), who first derives in a very elegant way what is actually the preparation uncertainty principle and then ascribes it to Heisenberg. On a personal note, one of us remembers from his time as student a feeling of unease when confronted first with one version and then with another as if they were equivalent. At the time he did not worry too much about the matter but now finds it rather satisfying to learn that his unease was justified after all.
- 8.
This goes not only for an electron, but for all objects and processes in both the micro- and the macroworld.
- 9.
Because of the structure of quantum mechanics, it is possible in special cases for the noise to be zero (a so-called noiseless measurement). According to Heisenberg’s 1927 version of the measurement uncertainty principle, this would mean that the disturbance in the momentum is infinite; but Ozawa’s correction gives merely a finite, and not necessarily large lower bound for this disturbance.
- 10.
Hilgevoord and Uffink (2001/2016), and (1990). In a somewhat similar vein, Uffink and Hilgevoord (1985) earlier criticized the Kennard measure, explaining that for some exceptional probability distributions it gives too ample an estimate of the real spreading of position or momentum. One is more interested in where, say, 95% of the distribution is located than in what the least-squares measure would give. For normal, or near normal distributions with Gaussian tails, the criticism would be of little consequence; but one can think of other cases in which the Kennard measure would give a gross overestimate of the real uncertainty. This is not to say that the measure is wrong; rather for some abnormal distributions it is too crude.
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Acknowledgements
We would like to thank Matthias Neuber and Adam Tamas Tuboly for organizing the conference “Ernest Nagel and the Making of Philosophy of Science a Profession”, and the participants for posing interesting questions. We are also grateful to Jan Hilgevoord, John Norton, Jos Uffink and an anonymous referee for having commented on earlier versions of this paper. Hanoch Ben-Yami we thank for showing us the heights and depths of Budapest’s old and recent history.
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Atkinson, D., Peijnenburg, J. (2022). Putting the Cart Before the Horse: Ernest Nagel and the Uncertainty Principle. In: Neuber, M., Tuboly, A.T. (eds) Ernest Nagel: Philosophy of Science and the Fight for Clarity. Logic, Epistemology, and the Unity of Science, vol 53. Springer, Cham. https://doi.org/10.1007/978-3-030-81010-8_7
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