#incompleteness-theorems — Public Fediverse posts

“I love to talk about nothing. It’s the only thing I know anything about.”*…

Try as they might, scientists can’t truly rid a space or an object of its energy. But as George Musser reports, what “zero-point energy” really means is up for interpretation…

Suppose you want to empty a box. Really, truly empty it. You remove all its visible contents, pump out any gases, and — applying some science-fiction technology — evacuate any unseeable material such as dark matter. According to quantum mechanics, what’s left inside?

It sounds like a trick question. And in quantum mechanics, you know to expect a trick answer. Not only is the box still filled with energy, but all your efforts to empty it have barely put a dent in the amount.

This unavoidable residue is known as ground-state energy, or zero-point energy. It comes in two basic forms: The one in the box is associated with fields, such as the electromagnetic field, and the other is associated with discrete objects, such as atoms and molecules. You may dampen a field’s vibrations, but you cannot eliminate every trace of its presence. And atoms and molecules retain energy even if they’re cooled arbitrarily close to absolute zero. In both cases, the underlying physics is the same.

Zero-point energy is characteristic of any material structure or object that is at least partly confined, such as an atom held by electric fields in a molecule. The situation is like that of a ball that has settled at the bottom of a valley. The total energy of the ball consists of its potential energy (related to position) plus its kinetic energy (related to motion). To zero out both components, you would have to give a precise value to both the object’s position and its velocity, something forbidden by the Heisenberg uncertainty principle.

What the existence of zero-point energy tells you at a deeper level depends ultimately on which interpretation of quantum mechanics you adopt. The only noncontentious thing you can say is that, if you situate a bunch of particles in their lowest energy state and measure their positions or velocities, you will observe a spread of values. Despite being drained of energy, the particles will look as if they’ve been jiggling. In some interpretations of quantum mechanics, they really have been. But in others, the appearance of motion is a misleading holdover from classical physics, and there is no intuitive way to picture what’s happening…

More on the development of our understanding of “zero-point energy” and on the questions that remain: “In Quantum Mechanics, Nothingness Is the Potential To Be Anything,” from @georgemusser.com in @quantamagazine.bsky.social.

For the most amusing of musings on nothing, see Percival Everett‘s Dr. No.

* Oscar Wilde

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As we noodle on nought, we might spare a thought for Kurt Gödel; he died on this date in 1978. A  mathematician, logician, and author of Gödel’s proof. He is best known for his proof of Gödel’s Incompleteness Theorems (in 1931). He proved fundamental that in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular, the consistency of the axioms cannot be proved… thus ending a hundred years of attempts to establish axioms to put the whole of mathematics on an axiomatic basis. [See here for a consideration of what his finding might mean for moral philosophy…]

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@avlcharlie
Thanks for the link, would like to read up on #Godel #incompletenessTheorems.
🔗 https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

ゲーデルの不完全性定理によって，ヒルベルトの計画がヒルベルトが思い描いたような形で成就することはあり得ないことが示された，ということは，ヒルベルトの計画自身が無意味であったということを意味するものではないはずですし，ヒルベルト計画が無意味になった，ということも意味するものではないように思えます．

第2節でも既に述べたように，ヒルベルト自身は，ヒルベルトの計画が，彼の思い描いていたような仕方で完結した暁には，数学者たちは，数学の基礎付けの研究を終えて，安心して，従来の数学研究に戻ってゆくことができる，というように考えていたようで，彼の書いた論説の中には，実際にそのような表明も見られます．ゲーデルの不完全性定理が否定したのは，数学者が，この「普通の」数学に戻ってゆくことができる可能性でした．この意味では，数学が健全な科学として発展するためには，未来永劫にわたって，数学の基礎付けや，その研究から派生した数理論理学を，避けて通ることはできない，というのが，この定理の結論のはずなのですが，それを理解できない，あるいは理解することを拒否している数学者があまりに多いことには，驚嘆の念を禁じ得ませんし，本書のような本を書いてしまう人まで出てくることは，開いた口が塞がらない，とでも形容するしかほかないようにも思えます．

https://fuchino.ddo.jp/misc/superlesson.pdf

#HilbertProgram
#IncompletenessTheorems

In 1931, #Gödel published his famous #IncompletenessTheorems:
• A consistent system of logic is incomplete
• Such a system cannot prove its own consistency

I wonder how #programmers might profitably apply these theorems in their next meeting with the #management….

@hanuljeon95 People very often argue "... is not proved scientifically" to mean "... is wrong". Perhaps it is exacty this kind of fallacy. #IncompletenessTheorems

"그런데 쿠르트괴델이 메타수학까지 포함해서 스스로 모순이 없는 체계가 만들어지는 게 불가능하다는 걸 증명해버렸음" https://twitter.com/hyekkim/status/1615005099096674304?s=61&t=Sd1bItLPKig-jZbDEc_yOw
I cannot understand why this type of misunderstanding of the Incompleteness Theorems is so universal. It should be obvious that the lack of a (strictly finitist) consistency proof does not necessarily mean inconsistency. Is this somehow connected to the "talent" of the majority of the people to be able to believe mathematical theorems without understanding their proofs? @hanuljeon95 #IncompletenessTheorems

Is there a connection between Gödel's (https://en.wikipedia.org/wiki/Kurt_Gödel) #IncompletenessTheorems (https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems) and #CategoryTherory (https://en.wikipedia.org/wiki/Category_theory)? If so, what is it?

Reading #Gödel, #Escher, #Bach