What is the most difficult text on mathematics?

[Demystifier]

There isn't anything that is too rigorous or too abstract, the more the better.

I knew you will say that.

I am curious how do you feel about category theory, which some mathematicians call "abstract nonsense"?

But there are texts that are too difficult for me even if I like the style.

Yes, that's what I want here. Please, can you give some examples?

I am curious now how do physicist (and anyone else) like text like these http://www.ams.org/bookstore-getitem/item=qft-1-2-s

I don't know, I am not one of them.

But how about this one?
https://www.amazon.com/dp/0387968903/?tag=pfamazon01-20
I suspect that this could be one of rare books respected by both mathematicians and physicists.

 

[Demystifier]

That's not to say I like anything that is abstract and rigorous, it depends on the topic.

Which topics in pure mathematics do you not like?

 

[OCR]

I am curious how do you feel about category theory, which some mathematicians call "abstract nonsense"?

You're right ... some mathematicians do call it "abstract nonsense" ...

http://en.wikipedia.org/wiki/Category_theory

 

[martinbn]

I knew you will say that.

I am curious how do you feel about category theory, which some mathematicians call "abstract nonsense"?

Yes, togather with homological algebra.

Yes, that's what I want here. Please, can you give some examples?

The style is not Bourbaki, but everything that Harish-Chandra and Langlands write I find extremely difficult.

But how about this one?
https://www.amazon.com/dp/0387968903/?tag=pfamazon01-20
I suspect that this could be one of rare books respected by both mathematicians and physicists.

:) Well, the fisrt time I tried to read it I found it as difficult as any physics book. And thought that the style is repulsive, the exact oposite of Bourbaki (which Arnold critisies a lot). But if you are already familiar with the topic the book is exelent. But I couldn't learn from it before I had learn enough geometry from other sources.

 

[martinbn]

Which topics in pure mathematics do you not like?

Not like is probably too strong. Its more like they aren't as likebale as others. Generally the more algebraic the better, the more analytic the worse. Does probability count as pure math? Not a big fan.

 

[Demystifier]

Bourbaki (which Arnold critisies a lot)

"It is almost impossible for me to read contemporary mathematicians who, instead of saying, ‘Petya washed his hands’, write ‘There is a t1 < 0 such that the image of t1 under the natural mapping t1 -> Petya(t1) belongs to the set of dirty hands, and a t2, t1 < t2≤0, such that the image of t2 under the above-mentioned mappings belongs to the complement of the set defined in the preceding sentence."
V. I. Arnol’d

 

[Demystifier]

The style is not Bourbaki, but everything that Harish-Chandra and Langlands write I find extremely difficult.

Did you know (see http://en.wikipedia.org/wiki/Robert_Langlands ) that Langlands occupies Albert Einstein's office at the Institute for Advanced Studies in Princeton? As Einstein was a physicist, it is not surprising that you find Langlands difficult.

Harish-Chandra (http://en.wikipedia.org/wiki/Harish-Chandra) was both mathematician and physicist, and his doctoral adviser was the physicist Paul Dirac. No further comment is needed.

 

[Demystifier]

Concerning mathematicians and their problems with physics, I remember my high-school teacher of mathematics. She was really great as a teacher of mathematics, but when she taught us about trigonometric functions she said: "By the way, at the University they told us that trigonometric functions have something to do with physics, something about shaking, but I never understood that. "

 

[phion]

I find anything written in mathematics difficult.

 

[Demystifier]

I find anything written in mathematics difficult.

Really, anything?
How about 1+1=2?
Or to be less trivial, solving for x in 7-x=4?

 

[phion]

Really, anything?
How about 1+1=2?
Or to be less trivial, solving for x in 7-x=4?

Don't be obtuse. I'm sure if I wasn't already familiar with the basic rules of arithmetic or algebra, then it would be rather confusing.

 

[Demystifier]

I'm sure if I wasn't already familiar with the basic rules of arithmetic or algebra, then it would be rather confusing.

Ah, I think I also have some similar mathematical experiences. For example, adding natural numbers in basis 10 is easy for me, but adding them in any other basis is not so easy. (For instance, I must think a little before seeing that 1+1=10 in the binary basis.) Despite the fact that, objectively, no basis is really easier than any other.

 

[martinbn]

Did you know (see http://en.wikipedia.org/wiki/Robert_Langlands ) that Langlands occupies Albert Einstein's office at the Institute for Advanced Studies in Princeton? As Einstein was a physicist, it is not surprising that you find Langlands difficult.

Harish-Chandra (http://en.wikipedia.org/wiki/Harish-Chandra) was both mathematician and physicist, and his doctoral adviser was the physicist Paul Dirac. No further comment is needed.

I know, but I don't think their styles are physics like (by the way I find Einstein pleasant to read). There is an ancdote that Harish-Chandra and Dyson met and talked. Harish-Chandra said something along the lines that theoretical physics is such a mess that he is leaving it for mathematics. To which Dyson replys that he is switching to physics for the exact same reason.

 

[phion]

Ah, I think I also have some similar mathematical experiences. For example, adding natural numbers in basis 10 is easy for me, but adding them in any other basis is not so easy. (For instance, I must think a little before seeing that 1+1=10 in the binary basis.) Despite the fact that, objectively, no basis is really easier than any other.

Abstract algebra throws me for a loop at the moment. Also, I "enjoy" reading about real analysis currently.

 

[Demystifier]

by the way I find Einstein pleasant to read

That's good news, there is still some hope for you to become a mathematical physicist.

I myself am a theoretical physicist (not even a mathematical physicist) who finds Mac Lane pleasant to read. Is there a hope for me to become a pure mathematician?

By the way, see
https://www.physicsforums.com/threa...didactics-comments.812627/page-6#post-5108561
Is it helpful?

 

[martinbn]

That's good news, there is still some hope for you to become a mathematical physicist.

The more I read physics the more I get used to the style. Now I can read general relativity from physicists without any convulsions.

I myself am a theoretical physicist (not even a mathematical physicist) who finds Mac Lane pleasant to read. Is there a hope for me to become a pure mathematician?

You probably are and always have been at heart.

 

[fourier jr]

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

Most mathematicians find Spanier's Algebraic Topology unreadable.

I remember reading about that one too now that you mention it. it would be good to give that one a go sometime too

 

[aleazk]

One of the problems with Varadarajan's book is that it attracts interest from physicists and physics students who don't have the prerequisite mathematical knowledge. But there are many other issues. The proofs are difficult to follow, and it's difficult to skim through it to get a "big picture" view or an idea about which parts of the book are important. For example, how much projective geometry do you need to know, and do you have to know everything about systems of imprimitivity or measure theory on simply connected locally compact topological groups to understand the later chapters?

In my case, I learned most of the topics from other sources. Armed with this background, I read Varadarajan and found these topics exposed in a much more detailed (but still clear and in most cases motivated) way and ordered in a beautiful and consistent narrative, which makes the foundations of QM to sound and flow like a Mozart piano concerto.

But it was indispensable to know beforehand most of the topics involved and to some degree of precision.

For example, I learned symplectic geometry in a course I took about the topic (and it was preparing a final monograph for this course that I discovered the book in question, since I wanted to talk about the structural analogies between classical mechanics in the symplectic geometry formulation and QM). So, I could easily follow the first chapter and concentrate in the relevant aspects that are needed later (like symmetries of the configuration space and momenta observables, etc.).

For chapters II,III and IV, I learned these topics for the first time from this book. I strongly recommend it, it has a very clear and "only the essential" point of view in the exposition of QM as generalized probability measures on the lattice of projectors (it also contains a complete exposition of all the functional analysis needed for QM; it also has material in quantum symmetries, like projective representations, multipliers, extensions, etc., chapter VII in Varadarajan).

For chapters V and VI, definitely Folland's A Course in Abstract Harmonic Analysis. It's of course a math book for mathematicians, but Folland is very clear. A mathematically minded physics student and with the necessary math background shouldn't have any problem with it.
It contains accessible statements and proofs of the Imprimitivity theorem and the Mackey Machine for semidirect products (also all of the stuff about compact groups, like the Peter-Weyl theorem, etc.)

For chapter VIII, Jauch's Foundations of Quantum Mechanics gives an introduction on the application of the Imprimitivity theorem in QM.

For chapter IX, Folland's Quantum field theory, a tourist guide for mathematicians gives a nice and basic introduction on how to use the Mackey Machine to obtain Wigner's classification (supplemented with material from the Moretti book for the galilei group in order to include non-relativistic QM).

From my experience with all this, I think the method that worked for me was: first, read the books that deal with the physical part (in particular, Jauch and Folland's QFT). From these books I was able to understand which are exactly the relevant points for physics. Then I went to Folland's Harmonic analysis to learn more about the math (not so much how to prove the theorems, but about their precise statement and background material, like, e.g., semidirect products, dual groups, characters, induced representations, etc.). Then I went back to the more physical books and learned how these things are used in physics, i.e., concrete things like how to obtain the Dirac equation from the induced representations, etc. Simultaneously, I was studying from the Moretti the lattice approach and all that.

After all that, I went to Varadarajan and read chapters VIII and IX. Since I already knew some of the topics, I finally was able to understand the big picture (which wasn't completely clear to me from the other books). I found the exposition relatively clear, motivated and also I was able to fill many of the details missing from the other more elementary expositions. But yes, I ended using the book more as a reference. In this post I tried to give a summary of the big picture of these two chapters.

Finally, now that I think I undersatand most of the physical applications and implications, I'm studying in more detail the proofs of the theorems. I'm still studying from the other books rather than Varadarajan. I rely on the latter when details are missing in the others (for example, the Moretti only gives the easy proof about how Gleason's theorem reduces to show that all frame functions are regular, but doesn't prove this, it refers the reader to Varadarajan, but that's fine since the book doesn't want to overwhelm the reader).

I know this post is a little convoluted, but that was my experience.

From the experience I recounted here, I think Varadarajan was difficult for me because it touches on so many different topics and with a lot of detail. From topics in pure math (at the mathematician's level) to their application in physics. From projective geometry to the representation theory of non-compact groups.

 

[Fredrik]

Great post aleazk. That book by Moretti looks very interesting, and I'm sure I'll find the other parts of your post useful as well.

 

[Hesch]

The proof of Fermat's last theorem by Andrew Wiles 1994.

Aⁿ + Bⁿ = Cⁿ ( A, B, C, n are all positive integers ).

Proof: No solutions for n > 2.

 

[PAllen]

Oh come on, if we're going to do monster proofs, how about:

1) Proof of the 4 color map theorem. No human has digested it.

2) Classification of finite simple groups

 

[martinbn]

Hesh, what you've written is not a proof, it's the statement! And did you try to read Wiles and found it difficult or do you go by its reputation?

 

[Hesch]

Hesh, what you've written is not a proof, it's the statement! And did you try to read Wiles and found it difficult or do you go by its reputation?

I think that what Fermat wrote was a statement. Many mathematicians ( amongst Euler ) tried to prove it with no succes, until Wiles spent 7 years of his life to do it. That was about 357 years after the statement had been written. Wiles got a reward of about 1 million $, that had been offered by some mathematical/physical institute in Berlin ( maybe Max Planck institute? ).

No, I've not read the proof itself, but I've read a book ( "Fermat's last theorem" ) about the history of the theorem, and about the struggle several mathematicians had had to prove it. What amazes me is that the theorem is so very simple, but the proof is not.

Do you fully understand the proof?

 

[aleazk]

Great post aleazk. That book by Moretti looks very interesting, and I'm sure I'll find the other parts of your post useful as well.

Yes, I found it interesting because it gives a clear exposition of the very basics: how to derive all the formalism of QM from the basic set-up of a generalized probability measure on the lattice of projectors on a Hilbert space (and to give a good feeling on the motivations for the concepts).

I think this is the right approach for an introduction. The problem with a lot of the (highly specialized) references on the subject is that they also go (if that's not their primary interest in the first place) into a lot of details about the general theory; for an introduction, it simply becomes very difficult when they start to bombard you with all sort of definitions and results on the abstract/general theory that are not really necessary for the basics. Of course, it's not their fault, since, usually, these references, like Varadarajan, aim to be encyclopedic references on the subject.

 

[atyy]

Is there a rigrorous distinction between mathematics and physics? Physics is essentially English or some intuitive, non-rigourous language like German or Chinese, and so if mathematics is physics, then mathematics is essentially non-rigourous.

Here's an example of how mathematics seems to be physics. For example, if mathematics claims that 7+1 = 8, that is physics, because it is a prediction about a physical action: if I google 7+1, then I will get 8.

 

[martinbn]

Here's an example of how mathematics seems to be physics. For example, if mathematics claims that 7+1 = 8, that is physics, because it is a prediction about a physical action: if I google 7+1, then I will get 8.

Can you do that with the proof of the Riemann hypothesis?

 

[atyy]

Can you do that with the proof of the Riemann hypothesis?

No. But I can argue it in full generality. Mathematics is ZFC. To define ZFC, we need the metalanguage. The metalanguage is essentially English. So all of mathematics is just English.

 

[micromass]

Can you do that with the proof of the Riemann hypothesis?

http://arxiv.org/abs/1101.3116

 

[martinbn]

No. But I can argue it in full generality. Mathematics is ZFC. To define ZFC, we need the metalanguage. The metalanguage is essentially English. So all of mathematics is just English.

You can make even simpler by starting with "mathematics is the multiplication table up to ten" instead of ZFC.

 

[atyy]

http://arxiv.org/abs/1101.3116

Ha, ha, yes most physicists secretly hope that the Riemann hypothesis is just QM (eigenvalues of an operator http://arxiv.org/abs/1105.2342).

 

[martinbn]

http://arxiv.org/abs/1101.3116

Ah, but it doesn't give the prove of the conjecture.

 

[atyy]

You can make even simpler by starting with "mathematics is the multiplication table up to ten" instead of ZFC.

Yes, and obviously I can google the multiplication table :)

Actually, I do need more than that, don't I? Don't I need to know how to use the multiplication table for numbers larger than 10?

 

[martinbn]

Yes, and obviously I can google the multiplication table :)

Deutch has a better take on this. Ah, and Arnold has repeatedly state that maths is that part of physics where experiments are cheap. As a bourbakist a must disagree.

 

[atyy]

Deutch has a better take on this. Ah, and Arnold has repeatedly state that maths is that part of physics where experiments are cheap. As a bourbakist a must disagree.

But wasn't Arnold rigourously right, given that to define ZFC we need the metalanguage (ie. physics)?

 

[martinbn]

In your statements you make so many implicit assumptions, as to what physics is and what maths is and so on, and you do it in a way as if that is the only possible and universally accepted view.