What is the most difficult text on mathematics?

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The discussion centers on identifying the most challenging mathematical texts, with "Principia Mathematica" by Whitehead and Russell frequently cited for its difficult writing style, unfamiliar notation, and extensive formalism. Participants highlight that older texts, like those by Newton and ancient Greeks, are also hard to read due to their geometric proofs and outdated philosophical approaches to mathematics. The conversation notes that modern mathematics has become more complex, requiring a deeper understanding of advanced concepts and formalism. Additionally, some texts in algebraic topology and mathematical physics are mentioned as particularly difficult. Overall, the consensus is that the difficulty of mathematical texts often stems from both their writing style and the abstract nature of their content.
  • #91
PAllen said:
So what? I don't see that having any bearing at all on whether some axiomatization of natural numbers (that is incomplete) constitutes a rigorous definition.

There are two general routes to proving Goedel's theorem.

1) Assume the intuitive natural numbers. This is the usual route, and leads to the view that ZFC itself is incomplete.

2) Deny the intuitive natural numbers. Define ZFC and define the natural numbers in them, and then prove Goedel's theorem. This route does not prove that ZFC is incomplete, which is fine. But then how does one define ZFC? One is then basically saying something about a human mathematician or a computer as physical objects. The basic point is that the top level is always "intuitive".
 
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  • #92
Shyan said:
I'm in deep love with abstract algebra so in this thread, I'm with mathematicians.:biggrin:

I don't think math is just english or physics. atyy seems to say that because the rigorous ways the mathematicians tend to use can't actually give everything from start, so there should be another thing at the beginning. But from the things I've understood, mathematicians have a sense of seeing that there should be a mathematical concept for something. I mean, they just encounter some calculation and say "oh man...this should have a name on its own! people should work on this...because this is great!". I had such a feeling in its elementary form. I think mathematics is on its own and its beauty is just its own! I just love it. The reason I'm pursuing physics more than mathematics, is that I'm self-studying things and its really hard to self-study rigorous mathematics.(But hey, I love physics too!)

I agree with you. Mathematics exists as a realm of beautiful ideas. The correspondence between the sensed world - what some people call the "real world" - and Mathematics is a wonderful mystery. I love it when physical experiments suggest or even demonstrate theorems.
 
  • #93
Another argument that shows that mathematical rigour depends on intuitive physical statements is that a rigrourous proof is one that is executed step by step by a computer. For example, the proof of the Kepler conjecture is an attempt at rigrourous proof. First, it assumes that we know what a "computer" is, which is already an appeal to physics. Then, it assumes that there was no cosmic ray that struck the computer and unluckily produced an erroneous step. One could run the entire thing multiple times to check that the same answer is given, but that assumes things like the probability of a cosmic ray is low, and assumptions about space and time translation invariance.
 
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  • #94
atyy said:
Another argument that shows that mathematical rigour depends on intuitive physical statements is that a rigrourous proof is one that is executed step by step by a computer. For example, the proof of the Kepler conjecture is an attempt at rigrourous proof. First, it assumes that we know what a "computer" is, which is already an appeal to physics. Then, it assumes that there was no cosmic ray that struck the computer and unluckily produced an erroneous step. One could run the entire thing multiple times to check that the same answer is given, but that assumes things like the probability of a cosmic ray is low, and assumptions about space and time translation invariance.
In the same style, one could also argue that mathematical rigor depends on psychological assumptions, which are even less rigorous than those in physics. Namely, when I perform a mathematical proof based on precisely defined logical rules, I assume that I am not insane, so that I can be confident that I really do follow the rules when I think I do.
 
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  • #95
Demystifier said:
In the same style, one could also argue that mathematical rigor depends on psychological assumptions, which are even less rigorous than those in physics. Namely, when I perform a mathematical proof based on precisely defined logical rules, I assume that I am not insane, so that I can be confident that I really do follow the rules when I think I do.

So what I don't understand is how the measurement problem fits in. Basically, rigourous mathematics always has (at least) two levels, the top level is intuitive and the bottom level is formal. This seems similar to the Heisenberg cut of Copenhagen, with the top level being the classical observer and the bottom part being the quantum system. So it seems mathematics must intrinsically have something like a Heisenberg cut and a measurement problem. Then it seems tempting to say that since mathematics has a cut, physics must have a cut. Yet there seems to be the counterexample of Bohmian Mechanics. Or perhaps it is that Bohmian Mechanics does have a cut which it inherits from mathematics, but the difference is that in Copenhagen some "key" features (like the observer) of the top level are not reflected in the bottom level, whereas in BM those key features of the top level are reflected in the bottom level? The mathematical analogy is that if we let the top level have the intuitive natural numbers, then the bottom level is "faithful" if it captures "enough" of the natural numbers, eg. ZFC (analogous to BM) as the bottom level is believed to be faithful to all known "mathematics", whereas Peano's axioms (analogous to Copenhagen) are not faithful to things like the Paris-Harrington theorem.
 
  • #96
Demystifier said:
In the same style, one could also argue that mathematical rigor depends on psychological assumptions, which are even less rigorous than those in physics. Namely, when I perform a mathematical proof based on precisely defined logical rules, I assume that I am not insane, so that I can be confident that I really do follow the rules when I think I do.

I think it more accurate to say that correctly applying mathematical rigor depends on psychological assumptions. The rigor itself is,in my mind, independent of our fallacies.
 
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  • #99
Maybe you should read the preface again: "This book is an attempt to present the rudiments of QFT (...) as actually practiced by physicists (...) in a way that will be comprehensible for mathematicians. (...) It is, therefore, not an attempt to develop QFT in a mathematically rigorous fashion."
 
  • #100
aleazk said:
Maybe you should read the preface again: "This book is an attempt to present the rudiments of QFT (...) as actually practiced by physicists (...) in a way that will be comprehensible for mathematicians. (...) It is, therefore, not an attempt to develop QFT in a mathematically rigorous fashion."

I'm not sure the book is even comprehensible to physicists, so if mathematicians can understand it, maybe there is something wrong.
 
  • #101
My specific complaints about Folland's QFT text are that there are two ways in which physicists understand QFT.

1) Wightman axioms, and explicit construction via Osterwalder-Schrader axioms

2) Heuristic but physical Wilsonian effective field theory viewpoint.

My quick impression was that Folland mentions neither of these. So what he is writing is incomprehensible old QFT that Dirac and Feynman knew, but did not understand.
 
  • #102
Folland's book is the only book about QFT that I can read. Everything else that I have tried leads to frustration. atyy before complaining read the book.
 
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  • #103
martinbn said:
Folland's book is the only book about QFT that I can read. Everything else that I have tried leads to frustration. atyy before complaining read the book.

Does he mention either idea in post #101?
 
  • #105
  • #106
atyy said:
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.
 
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  • #107
PAllen said:
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"
 
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  • #108
PAllen said:
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".
 
  • #109
atyy said:
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.
 
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  • #110
PAllen said:
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.
 
  • #111
atyy said:
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.
 
  • #112
atyy said:
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.
 
  • #113
PAllen said:
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 said:
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."
 
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  • #114
atyy said:
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.
 
  • #115
It's time to bring in Bill Clinton to discuss what the meaning of "is" is.
 
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  • #117
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.
 
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  • #118
Demystifier said:
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 said:
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.
 
  • #119
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.
 
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  • #120
atyy said:
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."
 
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