Is Yuri Manin's A Course in Mathematical Logic generally unsound?

[AndreasC]

A book that has really caught my attention recently is Yuri Manin's A Course in Mathematical Logic for Mathematicians. I am very interested in the foundations of mathematics and mathematical logic, plus I noticed that it had some chapters on quantum logic, so I started skimming through it. However I realized that the chapter on quantum logic uncritically accepts the "impossibility proofs" about hidden variables in quantum mechanics, and presents the one by Kochen and Specker.

However I know from reading J.S. Bell that these proofs were proven to be incorrect because they relied on some unreasonable assumptions and ruled out only specific hidden variable theories, as opposed to hidden variable theories in general. I do not KNOW why exactly it is wrong, but I know that it was demonstrated to be so before the first edition of the book was published, and long before the second edition.

This did make me skeptical of the book. Why was an entire chapter written on something that was already demonstrated to be incorrect, without mentioning anything about it being incorrectly applied, even if mathematically it is sound?

I guess the concern here is, does anyone have experience with the book, can it be relied on in general, are these chapters actually incorrect or am I mistaken, and are there any books on the subject you consider to be better?

 

[George Jones]

However I realized that the chapter on quantum logic uncritically accepts the "impossibility proofs" about hidden variables in quantum mechanics, and presents the one by Kochen and Specker.

However I know from reading J.S. Bell that these proofs were proven to be incorrect because they relied on some unreasonable assumptions.

"Kochen-Specker Theorem" is an example of Stigler's Law. Not only did Bell not show that the proof of Kochen-Specker is correct, Bell formulated and proved the Bell-Kochen-Specker theorem before Kochen and Specker,

 

[AndreasC]

"Kochen-Specker Theorem" is an example of Stigler's Law. Not only did Bell not show that the proof of Kochen-Specker is correct, Bell formulated and proved the Bell-Kochen-Specker theorem before Kochen and Specker,

Sorry, I got confused. However the point about von Neumann's impossibility proof still stands, since this is also presented in the book in the same vain. I mistakenly thought Kochen-Specker was a different formulation of von Neumann's proposition that hidden variable theories are impossible, thanks for pointing this out.

As far as I understand, von Neumann's idea wasn't exactly incorrect, however the conclusion people drew from it was incorrect, in that it simply ruled out a class of hidden variable interpretations, instead of hidden variable interpretations as a whole. However this doesn't seem to be referenced in Manin's book, which caused my skepticism.

 

[George Jones]

Sorry, I got confused. However the point about von Neumann's impossibility proof still stands, since this is also presented in the book in the same vain.

On what page(s) of the second edition is von Neumann's impossibility proof discussed.

 

[AndreasC]

On what page(s) of the second edition is von Neumann's impossibility proof discussed.

Page 78. Though now that I think about it it doesn't seem as bad as I first thought. But check it out and tell me what you think.

 

[martinbn]

What's wrong with those theorems?

 

[AndreasC]

What's wrong with those theorems?

From what I've read (and I may very well be mistaken), the issue with them is that while they are purported to show the impossibility of hidden variable theories, they actually only rule out a very specific class of hidden variable theories, but I didn't find mention of that in the book and it seemed like a major oversight when an entire section has been devoted to this subject. This made me a little bit concerned about the reliability of the rest of the book too, although the concern may be entirely unwarranted. I know Manin is a great mathematician, however I know that even great scientists sometimes kinda rush their books, and it's a bit hard to tell how valid it is overall when you know nothing on the subject. That is why I decided to ask people who know better.

 

[Demystifier]

What's wrong with those theorems?

The von Neumann's theorem is mathematically correct, but physically irrelevant because it rests on an unreasonable assumption.

 

[martinbn]

The von Neumann's theorem is mathematically correct, but physically irrelevant because it rests on an unreasonable assumption.

Why is this relevant for a logic book for mathematicians?

 

[AndreasC]

Why is this relevant for a logic book for mathematicians?

Well the book does seem to make the assertion that it does rule out hidden variable theories, and the authors decided it was important enough to devote an entire section on it.

Anyways judging from the rest the book is pretty great so I think I'll keep reading it and just bear in mind that this section is not 100% right.

 

[Demystifier]

Why is this relevant for a logic book for mathematicians?

It's probably not, but I guess the author disagrees.

 

[martinbn]

It's probably not, but I guess the author disagrees.

Why is it releveant if the assuptions are physically reasonable or not, if it is mathematically sound and the book is about logic!

 

[Demystifier]

Why is it releveant if the assuptions are physically reasonable or not, if it is mathematically sound and the book is about logic!

Because in a mathematical logic textbook, one should not just put random theorems which are logically correct. Unless it is an exercise for a student (which in this case it isn't), one should select theorems which are somehow important and significantly contribute to a general understanding of a whole field.

 

[Demystifier]

Why is it releveant if the assuptions are physically reasonable or not, if it is mathematically sound and the book is about logic!

A quote from https://iep.utm.edu/val-snd/ :

"A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false."

"A deductive argument is sound if and only if it is both valid, and all of its premises are actually true."