What is the most difficult text on mathematics?

[fourier jr]

& taking things out of context

 

[aleazk]

The ones that I find hard to read are those written by physicists or physics minded mathematicians. Those that I find easier are the Bourbaki or Bourbaki style.

In moments like that, keep in mind this:

"Physics is not so mysterious as many mathematicians seem to consider it. It is rather that physicists have different values and a different viewpoint, and this leads them to explain things in a manner uncongenial to mathematicians. If one works at it, it is possible to translate practically all of physics into well-defined mathematics. Moreover, when one does so, one finds a beautifully coherent scheme, which can be rather briefly summarized" - George Mackey.

Is there a hope for me to become a pure mathematician?

You probably are and always have been at heart.

Indeed! because we all know in the deeps of our hearts that:

"Mathematical proofs really aren't there to convince you that something is true—they're there to show you why it is true" - Andrew Gleason.

-Gleason & Mackey.

 

[atyy]

Can mathematicians even define the so called "natural numbers"?

 

[ShayanJ]

I'm in deep love with abstract algebra so in this thread, I'm with mathematicians.

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!)

 

[lavinia]

another one which I thought was hard is lang's algebra. there's a lot of stuff incorporating other subjects so it helps to know a bit about them, and a lot of the problems are good too

A former student of Lang told me that some students in his calculus class complained to him about his book - and Lang told them to throw the book out.

 

[lavinia]

Can mathematicians even define the so called "natural numbers"?

Can you clarify the question? How about an infinite cyclic group on one generator?

 

[atyy]

Can you clarify the question? How about an infinite cyclic group on one generator?

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

 

[PAllen]

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

No, it just says Natural numbers as defined by the Peano axioms cannot be proved consistent within this system itself. Several proofs of consistency for the Peano axioms have been achieved using elements outside them. Of course, then there are other statements whose truth cannot be decided within that system. However it seems a big stretch to me to call any this 'inability to axiomatically define the natural numbers'.

 

[atyy]

No, it just says Natural numbers as defined by the Peano axioms cannot be proved consistent within this system itself. Several proofs of consistency for the Peano axioms have been achieved using elements outside them. Of course, then there are other statements whose truth cannot be decided within that system. However it seems a big stretch to me to call any this 'inability to axiomatically define the natural numbers'.

There is the syntactic version and the semantic version. The semantic version does say that the natural numbers cannot be axiomatically defined, because it says that there is a statement that is true of the natural numbers but that every consistent extension of the Peano axioms neither proves nor disproves.

The version you are thinking about is the syntactic version, proven by Rosser, using key insights from Goedel's work. It is theorem 4.17 in these notes by Victoria Gitman: http://boolesrings.org/victoriagitman/files/2013/05/logicnotespartial.pdf.

The semantic version is theorem 4.13.

There is also the very interesting discussion on p14-16 of http://www.columbia.edu/~hg17/nonstandard-02-16-04-cls.pdf about what we mean by the "standard model" of arithemetic.

 

[PAllen]

[

There is the syntactic version and the semantic version. The semantic version does say that the natural numbers cannot be axiomatically defined, because it says that there is a statement that is true of the natural numbers but that every consistent extension of the Peano axioms neither proves nor disproves.

The version you are thinking about is the syntactic version, proven by Rosser, using key insights from Goedel's work. It is theorem 4.17 in these notes by Victoria Gitman: http://boolesrings.org/victoriagitman/files/2013/05/logicnotespartial.pdf.

The semantic version is theorem 4.13.

I still don't see this as saying you can't axiomatically define natural numbers. It just says for any such axiomatic definition, there will be statements whose truth or falsity cannot be determined. Incompleteness is in no way the same as absence of definition.

 

[atyy]

I still don't see this as saying you can't axiomatically define natural numbers. It just says for any such axiomatic definition, there will be statements whose truth or falsity cannot be determined. Incompleteness is in no way the same as absence of definition.

The important point is that the undecidable statements can be shown to be true, contrary to your assertion.

 

[PAllen]

The important point is that the undecidable statements can be shown to be true, contrary to your assertion.

They can be shown to be true outside of that axiomatic system. I meant inside the given system. I have never seen this, or any discussion of Godel's theorem as saying you can't axiomatically define Natural numbers. Just that the resulting system has limitations not previously recognized or expected.

 

[atyy]

They can be shown to be true outside of that axiomatic system. I meant inside the given system. I have never seen this, or any discussion of Godel's theorem as saying you can't axiomatically define Natural numbers. Just that the resulting system has limitations not previously recognized or expected.

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.

 

[PAllen]

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.

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.

 

[atyy]

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.

"The statement that any formalization of the natural numbers does not encompass all true statements about them"

Well, that means that you have an intuitive sense of the natural numbers that cannot be formalized.

Either that, or you formalize the natural numbers in ZFC. But that means that you do not acknowledge that Goedel's incompleteness theorem applies to ZFC, which is unorthodox but fine. But then that means the metalanguage used to define ZFC, when using notions like "finite" is really about steps that a human mathematician acting as a robot, or that a computer as a physical machine can take.

So if one has the intuitive natural nunbers, that is basically a lack of rigour. If one does not have the intuitive natural numbers, then ZFC is defined by physics.

 

[PAllen]

Incomplete does not mean non-rigorous. In fact, the finding of incompleteness is the result of using rigor. Thus, as I see, rigor requires acceptance that meaningful axiomatic systems are not complete - not that they don't exist, or don't act as definitions, etc.

 

[atyy]

Incomplete does not mean non-rigorous. In fact, the finding of incompleteness is the result of using rigor. Thus, as I see, rigor requires acceptance that meaningful axiomatic systems are not complete - not that they don't exist, or don't act as definitions, etc.

But incomplete does mean that one used the "intuitive natural numbers".

 

[PAllen]

But incomplete does mean that one used the "intuitive natural numbers".

I disagree. I have a formally defined axiomatic system. It's consistency can be shown by going outside itself. Rigorous methods show it is incomplete in a specific sense, as are all substantive formal systems. We will probably never agree, but I will say I have never seen your expansive interpretation of the consequences of Godel in literature I've read. In particular, I have never seen anyone besides you suggest it means there is no formal definition of natural numbers.

 

[atyy]

I disagree. I have a formally defined axiomatic system. It's consistency can be shown by going outside itself. Rigorous methods show it is incomplete in a specific sense, as are all substantive formal systems. We will probably never agree, but I will say I have never seen your expansive interpretation of the consequences of Godel in literature I've read. In particular, I have never seen anyone besides you suggest it means there is no formal definition of natural numbers.

In the (usual) proof of Goedel's theorem, the notion "natural number" is used without definition. It is assumed intuitively.

 

[PAllen]

In the (usual) proof of Goedel's theorem, the notion "natural number" is used without definition. It is assumed intuitively.

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.

 

[atyy]

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".

 

[lavinia]

I'm in deep love with abstract algebra so in this thread, I'm with mathematicians.

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.

 

[atyy]

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.

 

[Demystifier]

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.

 

[atyy]

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.

 

[lavinia]

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.

 

[martinbn]

http://www.ams.org/notices/201009/rtx100901121p.pdf

 

[atyy]

http://www.ams.org/notices/201009/rtx100901121p.pdf

Is Folland's work really rigourous? He writes like a physicist (I've looked at his QFT text).

 

[aleazk]

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."

 

[atyy]

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.

 

[atyy]

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.

 

[martinbn]

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.

 

[atyy]

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?

 

[PAllen]

Does he mention either idea in post #101?

Hmm. How about a whole section with that title:

https://books.google.com/books?id=u...#v=onepage&q=Weinberg Wightman axioms&f=false

 

[atyy]

Hmm. How about a whole section with that title:

https://books.google.com/books?id=ucaUAAAAQBAJ&pg=PA121&lpg=PA121&dq=Weinberg+Wightman+axioms&source=bl&ots=T1gtwrIM1j&sig=oU0idHcnBsainHNjFOrsxf24RaQ&hl=en&sa=X&ei=HmBaVdOLAYW1oQSvr4GgCw&ved=0CEcQ6AEwBw#v=onepage&q=Weinberg Wightman axioms&f=false

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.