5

I was reading Gödel's incompleteness theorem and Heisenbers's uncertainty principle. I found some similarities although one is based on a physical phenomenon and the other is mathematical.

Q: Are they interrelated? Can one interpret the incompleteness and inconsistency of Gödel as uncertainty as well?

Seyhmus Güngören
  • 7,848

1 Answers1

10

Heisenberg's uncertainty principle can be given a precise mathematical formulation that, as far as I can tell, has nothing to do (mathematically) with the incompleteness theorem.

One version is a statement about how spread out a function on $\mathbb{R}$ can be relative to how spread out its Fourier transform can be; roughly speaking, a function and its Fourier transform cannot simultaneously be localized (physically the function can be interpreted as describing the position of some particle and its Fourier transform can be interpreted as describing its momentum, but the mathematical statement is independent of this interpretation). A more general version is a statement about the variances of noncommuting random variables. In this form it is essentially an application of the Cauchy-Schwarz inequality.

If there are similarities to the incompleteness theorem, they are philosophical, not mathematical.

Qiaochu Yuan
  • 419,620
  • I am asking from the locical point of view. In both of them when you optimize one of them you make the other imperfect. In Gödel if you have completeness then you dont have consistency whereas in Heisenberg we have more perfection at momentum for example but then we dont have at the location. This is what I mean. – Seyhmus Güngören Sep 06 '12 at 18:09
  • 2
    Yeah, but Heisenberg allows you to have "partial" results for both values, whereas you can't really have different levels of consistency and completeness. You either have consistency or not, you either have completeness or not. – Thomas Andrews Sep 06 '12 at 18:50
  • 3
    Also, one way to think of Heisenberg is to realize that it doesn't mean there are actual numbers that you can know. There is no accurate pair of location and momentum that we can't determine, rather, that intuition that such values exist is wrong. We don't have an intuition about logic the Gödel breaks. Sure, we'd like completeness and consistency, but I don't feel like it is intuitive that it should be true. – Thomas Andrews Sep 06 '12 at 18:56
  • @Thomas yes you re right. Thank you very much for your comment. – Seyhmus Güngören Sep 06 '12 at 19:10
  • 1
    @SeyhmusGüngören "In Gödel if you have completeness then you don't have consistency". Not exactly. Gödel's theorem doesn't rule out complete consistent theories even of arithmetic (the set of all true statements in the language of first-order arithmetic constitutes a consistent complete theory). What Gödel's theorem rules out is complete, consistent and effectively axiomatized theories containing a modicum of arithmetic. – Peter Smith Sep 06 '12 at 21:00
  • @PeterSmith what is effectively axiomatized theories containing a modicum (I heart this word in my life for the first time here) of arithmetic? – Seyhmus Güngören Sep 06 '12 at 21:56
  • 1
    @SeyhmusGüngören: "effectively axiomatized" means that it is possible to decide, using a definitely specific mechanical process, whether or not something is an axiom of the theory. (The point of this restriction is to prevent some wiseguy from declaring that the axioms of his theory are exactly every true statement about $\mathbb N$ -- which would make it both both complete and consistent). – hmakholm left over Monica Sep 06 '12 at 23:55
  • @SeyhmusGüngören And the modicum (small amount) of arithmetic could be, e.g., Robinson Arithmetic (often called simply $\mathsf{Q}$). See http://en.wikipedia.org/wiki/Robinson_arithmetic – Peter Smith Sep 07 '12 at 10:17
  • @HenningMakholm thank you very much for the explanation. – Seyhmus Güngören Sep 07 '12 at 10:37
  • @PeterSmith thank you very much for the explanation. I just realized that you have a book on that matter. I will try to read it. Thanks again. – Seyhmus Güngören Sep 07 '12 at 10:38
  • have a look at the latest answer. – Seyhmus Güngören Mar 14 '15 at 23:10