What is the most difficult text on mathematics?

[PAllen]

He has to do it for interacting fields and mention the Osterwalder-Schrader conditions (or an equivalent thing). For free fields, all the physics texts are essentially rigourous.

Major universities continue to use Peskin and Schroeder, and it appears to use what you describe as 'obsolete' and 'not what any physicist uses'.

Also Weinberg's books are not based on your 'unique correct approach'. I guess Weinberg is not a physicist.

 

[atyy]

Major universities continue to use Peskin and Schroeder, and it appears to use what you describe as 'obsolete' and 'not what any physicist uses'.

Also Weinberg's books are not based on your 'unique correct approach'. I guess Weinberg is not a physicist.

The other approach is the Wilsonian effective field approach. Both Peskin and Schroeder and Weinberg mention it. Also, one should distinguish between use and understand. The usual method that is used is not understandable. The method that is understandable is impractical to use. As far as I can tell, Folland only presents the method that can be used but is not understandable.

Overall, the Wilsonian effective field approach is the most important conceptual advance in QFT, and I never understand why the standard texts present it only in the later chapters, and in a way that is still quite hard to understand. If one knows what one is looking for, the relevant ideas are in Srednicki's chapter 29 http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf, where the key equations are Eq (29.9 -29.11) and the conclusion on p193 "The final results, at an energy scale E well below the initial cutoff 0, are the same as we would predict via renormalized perturbation theory, up to small corrections"

 

[atyy]

Nonsense. The statement that any formalization of the natural numbers does not encompass all true statements about them does not mean natural numbers are not formalized let alone not defined. Limitations or incompleteness of a formalization does not mean the formalization doesn't exist, or is useless, or doesn't serve to define anything. These are wild overstatements, IMO.

Basically the reason I am right is that what I mean is exactly "any formalization of the natural numbers does not encompass all true statements about them".

 

[PAllen]

Basically the reason I am right is that what I mean is exactly "any formalization of the natural numbers does not encompass all true statements about them".

It comes down to definition. Including definition of definition. You are claiming a formalization that is incomplete is not a formalization or a definition. I claim it is still both a formalization and defintion despite incompleteness. This is a matter of definition. So far as I know, my definition is much more popular among experts than yours. And there really is no debating definitions, thus we keep going in circles.

 

[atyy]

It comes down to definition. Including definition of definition. You are claiming a formalization that is incomplete is not a formalization or a definition. I claim it is still both a formalization and defintion despite incompleteness. This is a matter of definition. So far as I know, my definition is much more popular among experts than yours. And there really is no debating definitions, thus we keep going in circles.

You are misreading my claim. I don't disagree that there are incomplete formalizations. But the point remains that no formalization of the natural numbers can encompass all true statements about them. And this does not hinge just on "incompleteness". Incompleteness only means that if you have a formalization, then there is an undecidable statement. The important additional point is that one cannot say that since either statement is consistent with the axioms, I will just choose one and add it. If one does that, the formal system will not have as a model the standard natural numbers. So the point is beyond "incompleteness", and hinges on the "true natural numbers".

The incompleteness you mention is a syntactic point. The failure I am referring to is a semantic point.

 

[PAllen]

You are misreading my claim. I don't disagree that there are incomplete formalizations. But the point remains that no formalization of the natural numbers can encompass all true statements about them. And this does not hinge just on "incompleteness". Incompleteness only means that if you have a formalization, then there is an undecidable statement. The important additional point is that one cannot say that since either statement is consistent with the axioms, I will just choose one and add it. If one does that, the formal system will not have as a model the standard natural numbers. So the point is beyond "incompleteness", and hinges on the "true natural numbers".

The incompleteness you mention is a syntactic point. The failure I am referring to is a semantic point.

I am not missing that point since I described it. Per my definition of definition and formalization it remains interesting but not limiting. I still have (several) possible formalizations that can serve as definitions of natural numbers. Their failure to encompass all true statements doesn't change that. We disagree on even on the definition incompleteness. To me, both the feature of true but unprovable statements, or undecidable statements that can be added as either the statement or its contradiction (consistently) , are different flavors of incompleteness, and neither is more problematic to me. In fact the 'true but unprovable flavor' is the first that I studied.

 

[suremarc]

You are misreading my claim. I don't disagree that there are incomplete formalizations.

That's not what you said. Look over your posts:

Well, doesn't the Goedel incompleteness theorem basically say that the natural numbers cannot be axiomatically defined?

You can take the undecidable sentence and add it or its negation to the axioms and obtain a consistent system. However, you are not free to add either one if you insist the system models the natural numbers. Therefore the natural numbers cannot be formalized.

You insisted multiple times that the natural numbers cannot be "formalized", complete or incomplete.

This "http://boolesrings.org/victoriagitman/files/2013/05/logicnotespartial.pdf" point you cited (4.13) says that number theory cannot be axiomatized--it does not say that the natural numbers cannot be axiomatized.

 

[atyy]

I am not missing that point since I described it. Per my definition of definition and formalization it remains interesting but not limiting. I still have (several) possible formalizations that can serve as definitions of natural numbers. Their failure to encompass all true statements doesn't change that. We disagree on even on the definition incompleteness. To me, both the feature of true but unprovable statements, or undecidable statements that can be added as either the statement or its contradiction (consistently) , are different flavors of incompleteness, and neither is more problematic to me. In fact the 'true but unprovable flavor' is the first that I studied.

You insisted multiple times that the natural numbers cannot be "formalized", complete or incomplete.

This "http://boolesrings.org/victoriagitman/files/2013/05/logicnotespartial.pdf" point you cited (4.13) says that number theory cannot be axiomatized--it does not say that the natural numbers cannot be axiomatized.

Hmmm, ok, interesting point. Is there a difference between natural numbers and number theory?

For example, http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic:

"The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …."

There's also this interesting passage in http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers:

"A consequence of Kurt Gödel's work on incompleteness is that in any effectively generated axiomatization of number theory (i.e. one containing minimal arithmetic), there will be true statements of number theory which cannot be proven in that system. So trivially it follows that ZFC or any other effectively generated formal system cannot capture entirely what a number is.

Whether this is a problem or not depends on whether you were seeking a formal definition of the concept of number. For people such as Bertrand Russell (who thought number theory, and hence mathematics, was a branch of logic and number was something to be defined in terms of formal logic) it was an insurmountable problem. But if you take the concept of number as an absolutely fundamental and irreducible one, it is to be expected. After all, if any concept is to be left formally undefined in mathematics, it might as well be one which everyone understands.

Poincaré, amongst others (Bernays, Wittgenstein), held that any attempt to define natural number as it is endeavoured to do so above is doomed to failure by circularity. Informally, Gödel's theorem shows that a formal axiomatic definition is impossible (incompleteness), Poincaré claims that no definition, formal or informal, is possible (circularity). As such, they give two separate reasons why purported definitions of number must fail to define number."

 

[ShayanJ]

Basically the reason I am right is that what I mean is exactly "any formalization of the natural numbers does not encompass all true statements about them".

I can't agree with that because you're misusing the word "true". What it means for a statement to be true about natural numbers? I can only imagine two meanings: 1) Axioms imply it. 2) We usually assume it to be true. Either as someone who couldn't care less about axiomatization of natural numbers or as a mathematician who knows about incompleteness and just chooses a statement or its negation to add as a new axiom.
Looking at it this way, it seems to me your statement is meaningless.

 

[PAllen]

It's time to bring in Bill Clinton to discuss what the meaning of "is" is.

 

[Demystifier]

Folland's book is the only book about QFT that I can read. Everything else that I have tried leads to frustration.

It's good to know that, for the case a mathematician asks me to suggest him a book on QFT. Have you also tried the books
by Araki https://www.amazon.com/dp/0199566402/?tag=pfamazon01-20
or Ticciati https://www.amazon.com/dp/0521060257/?tag=pfamazon01-20 ?

 

[martinbn]

Some time ago I tried Ticciati and I couldn't read it. I have seen Araki, but haven't tried it. My guess is that i probably could read it. My tolerance to physics style text has increased and there is a chance that I can actualy read physics text if I tried.

 

[atyy]

It's good to know that, for the case a mathematician asks me to suggest him a book on QFT. Have you also tried the books
by Araki https://www.amazon.com/dp/0199566402/?tag=pfamazon01-20
or Ticciati https://www.amazon.com/dp/0521060257/?tag=pfamazon01-20 ?

Some time ago I tried Ticciati and I couldn't read it. I have seen Araki, but haven't tried it. My guess is that i probably could read it. My tolerance to physics style text has increased and there is a chance that I can actualy read physics text if I tried.

I would not recommend Folland's QFT. As far as I can tell, it is old style QFT which even Dirac and Feynman considered nonsensical, but which we knew was a fragment of something correct because of experiment. This is a case where one should not develop a tolerance to physics style!

I haven't read Ticciati, but have glanced at Araki, which seems good. For rigourous QFT, I would also recommend
Dimock https://www.amazon.com/dp/1107005094/?tag=pfamazon01-20
Rivasseau http://www.rivasseau.com/resources/book.pdf

However, rigourous QFT is still not able to deal with physically important QFTs like QED. For that, one needs the other great conceptual advance of Wilsonian effective theory that Dirac and Feynman did not know about, and has still not been made rigourous in all cases of interest. However, it is related to rigourous renormalization, and Rivasseau does discuss it. Wilson's ideas came from classical statistical mechanics (and particle physics, as Wilson was a particle physicist who worked on statistical mechanics), and the key physics ideas are usually better described there than in QFT texts. A good non-rigourous text is Kardar https://www.amazon.com/dp/052187341X/?tag=pfamazon01-20.

 

[martinbn]

atyy, you seem very quick to judge textbooks (that you haven't read) and people's understanding (of people you haven't spoken to, and about topics that probably you don't understand)! If you have an opinion about something or someone you need to write it as an opinion, not as god's given truth.

 

[Demystifier]

I would not recommend Folland's QFT. As far as I can tell, it is old style QFT which even Dirac and Feynman considered nonsensical, but which we knew was a fragment of something correct because of experiment. This is a case where one should not develop a tolerance to physics style!

That reminds me of an old joke:

A physicist constructed a new theory and shown it to his friend mathematician to say him if it looks mathematically consistent to him. The mathematician took some time to study it and eventually concluded that the theory doesn't make any sense. But in the meantime, the theory turned out to be in a perfect agreement with experiments, and the physicist earned the Nobel Prize for it. Then the physicist talked to his friend mathematician again: "Look, the theory is in perfect agreement with experiments, so it cannot be totally wrong. Can you take a look at it again?" Then the mathematician studied it again, and after a lot of time he made his final conclusion: "Yes, the theory does make sense, but only in the trivial case when x is real and positive."

 

[martinbn]

That reminds me of an old joke:

A physicist constructed a new theory and shown it to his friend mathematician to say him if it looks mathematically consistent to him. The mathematician took some time to study it and eventually concluded that the theory doesn't make any sense. But in the meantime, the theory turned out to be in a perfect agreement with experiments, and the physicist earned the Nobel Prize for it. Then the physicist talked to his friend mathematician again: "Look, the theory is in perfect agreement with experiments, so it cannot be totally wrong. Can you take a look at it again?" Then the mathematician studied it again, and after a lot of time he made his final conclusion: "Yes, the theory does make sense, but only in the trivial case when x is real and positive."

:) This is one of the most frustrating things about physics. How hard is it to write that x is real and positive!

 

[atyy]

atyy, you seem very quick to judge textbooks (that you haven't read) and people's understanding (of people you haven't spoken to, and about topics that probably you don't understand)! If you have an opinion about something or someone you need to write it as an opinion, not as god's given truth.

I have read large parts of Folland's book. Everything I write is obviously an opinion and not god's given truth.

 

[Demystifier]

:) This is one of the most frustrating things about physics. How hard is it to write that x is real and positive!

I guess that similar problems logicians and set theorists have with "normal" mathematicians. How hard is it to write that you assume consistency and axiom of choice?

Which reminds me of another joke, about a physicist, mathematician and logician he saw a black sheep during a trip ... But you probably know that one.

 

[martinbn]

I guess that similar problems logicians and set theorists have with "normal" mathematicians. How hard is it to write that you assume consistency and axiom of choice?

Haha, may be, but how much 'real' math do they read.

 

[Demystifier]

Haha, may be, but how much 'real' math do they (logicians) read.

And how much physics mathematicians read?
And how much real engineer stuff physicists read?
And how much about real economy engineers read?
And how much about politics economists read?
And finally, to close the circle, how much logic politicians have any idea about?

 

[atyy]

And how much physics mathematicians read?

Do physicists need mathematics? Or do they just need logic? If mathematics is just a short hand for meaningless combinatorial symbols that logicians use, then in the Copenhagen spirit, can I just say that mathematics is the correspondence between mathematical objects and meaningless symbols, while physics is the coorespondence between physical objects and meaningless symbols, so physicists do not need mathematics?

BTW, did you see post #95?

 

[atyy]

And how much physics mathematicians read?
And how much real engineer stuff physicists read?
And how much about real economy engineers read?
And how much about politics economists read?
And finally, to close the circle, how much logic politicians have any idea about?

It's time to bring in Bill Clinton to discuss what the meaning of "is" is.

Apparently they read a lot!

 

[Demystifier]

Do physicists need mathematics? Or do they just need logic? If mathematics is just a short hand for meaningless combinatorial symbols that logicians use, then in the Copenhagen spirit, can I just say that mathematics is the correspondence between mathematical objects and meaningless symbols, while physics is the coorespondence between physical objects and meaningless symbols, so physicists do not need mathematics?

What is wrong here is the claim that they are "meaningless". No physicist or mathematician (not even Hilbert) really finds these symbols meaningless. These symbols have a meaning in the heads of physicists and mathematicians, and that's why they find them useful.

BTW, did you see post #95?

Or perhaps it is that Bohmian Mechanics does have a cut which it inherits from mathematics

Yes, I am quite sure this is the case.

 

[atyy]

What is wrong here is the claim that they are "meaningless". No physicist or mathematician (not even Hilbert) really finds these symbols meaningless. These symbols have a meaning in the heads of physicists and mathematicians, and that's why they find them useful.

Yes, but let me see if I can make myself clearer. The symbols alone and the rules alone for manipulating them are meaningless. It is the correspondence between the symbols and "real" objects that gives them meaning. In the case of mathematics, the real objects are mathematical objects, and in the case of physics the real objects are physical objects.

Do physicists need the mathematical objects?

 

[Demystifier]

In the case of mathematics, the real objects are mathematical objects, and in the case of physics the real objects are physical objects.

And the concrete physics objects are modeled by abstract mathematical objects. (For instance, an astrophysicist models a physical planet by a mathematical ball.) So yes, physicists do need mathematics.

 

[atyy]

And the concrete physics objects are modeled by abstract mathematical objects. (For instance, an astrophysicist models a physical planet by a mathematical ball.) So yes, physicists do need mathematics.

Sometimes it is argued that this issue goes away if we assume that physicists use second-order logic, since in some sense second order logic can uniquely specify mathematical objects.

But there seem to be counterarguments (you linked to this very interesting blog, I think in the thread on Lowenheim-Skolem): http://lesswrong.com/lw/g7n/secondorder_logic_the_controversy/

I think many prefer first order logic, because the completeness theorem http://en.wikipedia.org/wiki/Gödel's_completeness_theorem fails for second order logic.

 

[Demystifier]

Sometimes it is argued that this issue goes away if we assume that physicists use second-order logic, since in some sense second order logic can uniquely specify mathematical objects.

But there seem to be counterarguments (you linked to this very interesting blog, I think in the thread on Lowenheim-Skolem): http://lesswrong.com/lw/g7n/secondorder_logic_the_controversy/

I think many prefer first order logic, because the completeness theorem http://en.wikipedia.org/wiki/Gödel's_completeness_theorem fails for second order logic.

Most logicians prefer first-order logic, but all other people ("normal" mathematicians, scientists, lawyers, etc.) use higher-order logic. For example, without second order logic, a biologist could not say that "Cell is an aggregation of molecules and all cells have cytoplasm." (If you wonder why, that's because "all cells" is a quantification over sets, which is not allowed in first-order logic.)

 

[atyy]

Yes, I am quite sure this is the case.

If Bohmian Mechanics also has a cut, then what is the difference between Copenhagen and Bohmian Mechanics? (I made a proposal in #95 also, would you agree?)

 

[atyy]

You insisted multiple times that the natural numbers cannot be "formalized", complete or incomplete.

This "http://boolesrings.org/victoriagitman/files/2013/05/logicnotespartial.pdf" point you cited (4.13) says that number theory cannot be axiomatized--it does not say that the natural numbers cannot be axiomatized.

I also replied earlier to this in post #113. Here is another thought. In the section Question 2.1 on p10, the notes do refer to the standard model of arithmetic as the "natural numbers", which is the same thing referred to in Corollary 4.13 on p45, so my terminology is not entirely unconventional. However, I do agree it could be interesting to distinguish the concepts.

 

[atyy]

I am not missing that point since I described it. Per my definition of definition and formalization it remains interesting but not limiting. I still have (several) possible formalizations that can serve as definitions of natural numbers. Their failure to encompass all true statements doesn't change that. We disagree on even on the definition incompleteness. To me, both the feature of true but unprovable statements, or undecidable statements that can be added as either the statement or its contradiction (consistently) , are different flavors of incompleteness, and neither is more problematic to me. In fact the 'true but unprovable flavor' is the first that I studied.

OK, so technically we agree. I don't mind your terminology although you object to mine. But how about the larger point that mathematics needs either an idea of the natural numbers before any formalization, or it needs some idea of a physical machine like a computer. Again the argument is:

1) The semantic version of the Goedel incompleteness theorem - it means that we have an intuitive notion of the true arithemetic of the natural numbers before any formalization

2) If one doesn't accept that argument, by saying that the Goedel incompleteness theorem is only proved for natural numbers defined within a formal system like ZFC, then one still has to define ZFC. But for ZFC, or even PRA, the definition already assumes the natural numbers when terms such as "countable" or "recursive" are used (eg. http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic), so again we need the intuitive idea of the natural numbers before any formalization

3) If one rejects the intuitive natural numbers, then one cannot have things like the Turing machine (infinite memory tape), and one is basically saying something like I can make a computer that will verify such and such a theorem in a finite time, which is a physical statement. So one needs an intuitive view of the natural numbers or of physics in order to formalize mathematics.

 

[Demystifier]

If Bohmian Mechanics also has a cut, then what is the difference between Copenhagen and Bohmian Mechanics? (I made a proposal in #95 also, would you agree?)

Yes, I agree.

 

[suremarc]

OK, so technically we agree. I don't mind your terminology although you object to mine. But how about the larger point that mathematics needs either an idea of the natural numbers before any formalization, or it needs some idea of a physical machine like a computer. Again the argument is:

1) The semantic version of the Goedel incompleteness theorem - it means that we have an intuitive notion of the true arithemetic of the natural numbers before any formalization

Does physics need a theory of everything before it can make any predictions? You're saying that the whole of mathematics is invalid unless mathematicians relinquish 100% rigor. By that logic, I assert that the standard model is moot until physicists derive the values of its 26 free parameters analytically.

2) If one doesn't accept that argument, by saying that the Goedel incompleteness theorem is only proved for natural numbers defined within a formal system like ZFC, then one still has to define ZFC. But for ZFC, or even PRA, the definition already assumes the natural numbers when terms such as "countable" or "recursive" are used (eg. http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic), so again we need the intuitive idea of the natural numbers before any formalization

There is no distinction between countable and uncountable in PRA itself. No uncountable ordinal exists in PRA (its proof theoretic ordinal is $\omega^{\omega}$).
Nor does recursion involve quantification over the natural numbers, which is a condition of the incompleteness theorems.

3) If one rejects the intuitive natural numbers, then one cannot have things like the Turing machine (infinite memory tape), and one is basically saying something like I can make a computer that will verify such and such a theorem in a finite time, which is a physical statement. So one needs an intuitive view of the natural numbers or of physics in order to formalize mathematics.

Please explain the reasoning behind your first statement. Turing machine tape does not involve any arithmetic, save for addition and subtraction by 1.

 

[atyy]

Does physics need a theory of everything before it can make any predictions? You're saying that the whole of mathematics is invalid unless mathematicians relinquish 100% rigor. By that logic, I assert that the standard model is moot until physicists derive the values of its 26 free parameters analytically.

Did you mean "You're saying that the whole of mathematics is invalid unless mathematicians are 100% rigorous"?

 

[suremarc]

Did you mean "You're saying that the whole of mathematics is invalid unless mathematicians are 100% rigorous"?

I will say it differently, then: "You're saying that the whole of mathematics is invalid unless mathematicians accept that mathematics is not 100% rigorous."

 

[atyy]

I will say it differently, then: "You're saying that the whole of mathematics is invalid unless mathematicians accept that mathematics is not 100% rigorous."

My point is the exact opposite! What I am saying is that even the most rigourous mathematics makes use of intuitive notions, eg. the intuitive natural numbers. For this reason, I don't believe there is an essential difference between the intuition needed to understand a physics text and a mathematics text. In short, physicists are not worse sinners than mathematicians

There is no distinction between countable and uncountable in PRA itself. No uncountable ordinal exists in PRA (its proof theoretic ordinal is $\omega^{\omega}$).
Nor does recursion involve quantification over the natural numbers, which is a condition of the incompleteness theorems.

Here I am not using the argument that there is a "true arithmetic of the natural numbers" not captured by any formal system. I am using a different argument because I acknowledge it is possible to reject the argument via the incompleteness theorem. This second argument is simply that to even define many formal systems, terms like "countable infinite" or "recursive" are used, which assume the intuitive natural numbers. (Maybe that is not needed for PRA, but most specifications of PRA do use such words, eg. the one on the Wikipedia page.)

Please explain the reasoning behind your first statement. Turing machine tape does not involve any arithmetic, save for addition and subtraction by 1.

The Turing machine tape is a countable infinity, and usually "countable" uses an intuitive understanding of the natural numbers.