3 ms·

It often seems to me that Godel's incompleteness is the same phenomenon as Heisenberg's uncertainty, just in different domains.

eli_gottlieb 10y ago
https://arxiv.org/abs/quant-ph/0402197 https://arxiv.org/abs/quant-ph/0402197 >In 1927 Heisenberg discovered that the ``more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa''. Four years later G\"odel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the {\it converse} implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a ``formal uncertainty principle'' which implies Chaitin's information-theoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. In fact, the formal uncertainty principle applies to {\it all} systems governed by the wave equation, not just quantum waves. This fact supports the conjecture that uncertainty implies randomness not only in mathematics, but also in physics.
danbruc 10y ago
Absolutely not. The uncertainty in Heisenberg's uncertainty principle stems from the fact that you can not have a signal simultaneously well localized in both time and frequency. If you want a signal that is very well localized in time, then you need a short signal, a single spike. But the frequency of a single spike is not well defined, if you want a signal with a well defined frequency, you need something like a sine wave. And to make the frequency of a sine wave well defined, you need a long piece of it which of course means that the signal is no longer well localized in time. That is the heart of Heisenberg's uncertainty principle, you can not have signals that are simultaneously well localized in time and frequency. Gödel's incompleteness theorems have in some sense much deeper reasons, they are based on the logical consistency of the entire construction. Maybe you can look at it in a similar way, a theory is an object like a signal above and the properties of being consistent and complete can not be realized at the same time. But I have a hard time imagining that this could really be similar to signals where you can trade localization in time for localization in frequency and vice versa, but how would you trade a bit of consistency for a bit of completeness? EDIT: To be a bit more concrete, in classical mechanics you have to specify position and momentum (velocity) of a particle to specify its state, those are two independent properties that can have specific and independent values. That is not true in quantum mechanics, there position or momentum alone fully specify the state of the system. The wave function (in position space) tells you where the particle is with what probability, the frequencies of the wave function tell you what the momenta are with what probability. And from here it is the same as above, if you force a particle into a very well localized position, i.e. make the wave function a narrow spike at some place, then the frequencies and therefore the momenta are no longer well defined. If, on the other hand, you make the wave function of the particle like a sine wave, then you get a well defined frequency and therefore momentum but the wave function becomes spread out across space and the position is therefore no longer well localized.

eli_gottlieb 10y ago

https://arxiv.org/abs/quant-ph/0402197 https://arxiv.org/abs/quant-ph/0402197 >In 1927 Heisenberg discovered that the ``more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa''. Four years later G\"odel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the {\it converse} implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a ``formal uncertainty principle'' which implies Chaitin's information-theoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. In fact, the formal uncertainty principle applies to {\it all} systems governed by the wave equation, not just quantum waves. This fact supports the conjecture that uncertainty implies randomness not only in mathematics, but also in physics.

danbruc 10y ago

Absolutely not. The uncertainty in Heisenberg's uncertainty principle stems from the fact that you can not have a signal simultaneously well localized in both time and frequency. If you want a signal that is very well localized in time, then you need a short signal, a single spike. But the frequency of a single spike is not well defined, if you want a signal with a well defined frequency, you need something like a sine wave. And to make the frequency of a sine wave well defined, you need a long piece of it which of course means that the signal is no longer well localized in time. That is the heart of Heisenberg's uncertainty principle, you can not have signals that are simultaneously well localized in time and frequency. Gödel's incompleteness theorems have in some sense much deeper reasons, they are based on the logical consistency of the entire construction. Maybe you can look at it in a similar way, a theory is an object like a signal above and the properties of being consistent and complete can not be realized at the same time. But I have a hard time imagining that this could really be similar to signals where you can trade localization in time for localization in frequency and vice versa, but how would you trade a bit of consistency for a bit of completeness? EDIT: To be a bit more concrete, in classical mechanics you have to specify position and momentum (velocity) of a particle to specify its state, those are two independent properties that can have specific and independent values. That is not true in quantum mechanics, there position or momentum alone fully specify the state of the system. The wave function (in position space) tells you where the particle is with what probability, the frequencies of the wave function tell you what the momenta are with what probability. And from here it is the same as above, if you force a particle into a very well localized position, i.e. make the wave function a narrow spike at some place, then the frequencies and therefore the momenta are no longer well defined. If, on the other hand, you make the wave function of the particle like a sine wave, then you get a well defined frequency and therefore momentum but the wave function becomes spread out across space and the position is therefore no longer well localized.