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

In summary: I don't know, maybe they just thought it was an important topic? It's probably not, but I guess the author... I don't know, maybe they just thought it was an important topic?
  • #1
AndreasC
Gold Member
545
304
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?
 
Last edited:
  • Like
Likes Demystifier
Physics news on Phys.org
  • #2
AndreasC said:
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,
 
  • Like
Likes Demystifier
  • #3
George Jones said:
"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.
 
  • #4
AndreasC said:
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.
 
  • #5
George Jones said:
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.
 
  • #6
What's wrong with those theorems?
 
  • #7
martinbn said:
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.
 
  • #8
martinbn said:
What's wrong with those theorems?
The von Neumann's theorem is mathematically correct, but physically irrelevant because it rests on an unreasonable assumption.
 
  • Like
Likes AndreasC
  • #9
Demystifier said:
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?
 
  • Like
Likes dextercioby
  • #10
martinbn said:
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.
 
  • #11
martinbn said:
Why is this relevant for a logic book for mathematicians?
It's probably not, but I guess the author disagrees.
 
  • Like
Likes AndreasC
  • #12
Demystifier said:
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!
 
  • Like
Likes dextercioby and jim mcnamara
  • #13
martinbn said:
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.
 
  • Like
Likes weirdoguy and AndreasC
  • #14
martinbn said:
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."
 
  • Like
Likes vanhees71 and AndreasC

1. What is Yuri Manin's A Course in Mathematical Logic?

Yuri Manin's A Course in Mathematical Logic is a textbook on mathematical logic written by the mathematician Yuri Manin. It covers topics such as propositional logic, first-order logic, and axiomatic set theory.

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

There is some debate among mathematicians about the soundness of Manin's textbook. Some argue that it contains errors or misleading explanations, while others defend its validity. Ultimately, whether or not the textbook is considered unsound may depend on the individual reader's interpretation and understanding of the material.

3. What are some common criticisms of Yuri Manin's A Course in Mathematical Logic?

Some common criticisms of Manin's textbook include its lack of clarity and organization, as well as potential errors or oversimplifications in the explanations of certain concepts. Additionally, some argue that the textbook may not cover all necessary topics in sufficient depth.

4. Are there any alternative textbooks on mathematical logic?

Yes, there are many alternative textbooks on mathematical logic written by different authors. Some popular alternatives include Introduction to Mathematical Logic by Elliott Mendelson and Mathematical Logic by Stephen Cole Kleene.

5. Is it necessary to use Yuri Manin's A Course in Mathematical Logic to study mathematical logic?

No, it is not necessary to use Manin's textbook specifically to study mathematical logic. As mentioned, there are many alternative textbooks available. Additionally, there are numerous online resources and courses that cover the same material. It is ultimately up to the individual to choose the resource that best suits their learning style and needs.

Similar threads

  • Science and Math Textbooks
Replies
6
Views
2K
  • Science and Math Textbooks
Replies
28
Views
1K
  • STEM Academic Advising
Replies
6
Views
129
  • Science and Math Textbooks
Replies
4
Views
1K
  • Science and Math Textbooks
Replies
10
Views
2K
  • Science and Math Textbooks
Replies
2
Views
8K
  • Science and Math Textbooks
Replies
0
Views
670
Replies
7
Views
837
  • Science and Math Textbooks
Replies
7
Views
3K
  • Science and Math Textbooks
2
Replies
46
Views
3K
Back
Top