What is the most difficult text on mathematics?

[MathematicalPhysicist]

I am not sure that a Bourbaki approach to numerical analysis is the right way.

 

[MathematicalPhysicist]

this is completely subjective, in that it can only mean what do i myself find difficult. i found EGA a very difficult math book to read (too long, too abstract), and I find Russell Whitehead to be a book of logic not mathematics. Euclid is very easy and clear, although old. I like Riemann's works, although many people have found them impenetrable for decades. I like Dieudonne's Foundations of modern analysis, and spivak's calculus on mNIFOLDS, although not all do. baby Rudin is easy to read but hard for me to get any benefit from. all physics books are hard for me to read for the reason given by David Kazhdan(?) "physics has wonderful theorems, unfortunately there are no definitions".

you also need to define what you mean by difficult. does that mean which text is harder to plow through 10 pages of in a certain amount of time? or which is harder to learn something from? I once spent 3 hours struggling with a few pages of a research paper by Zariski and was very discouraged at my rate of progress in terms of number of pages. however, when i returned to class the next day i answered literally every question on that topic from my profesor until he told me to be quiet since i "obviously know the subject cold." so that research paper was much easier to read in the sense of how much insight can one gain per hour say than baby rudin.

From the books in physics I read there are definitions but they aren't declared as such; For example broken symmetry is defined in Srednicki (I can't remember where but it's defined). It's not like math or logic books which use notation like definition 1.1.1 or other such notation; most definitions are spread over the text.

Remember that physics is an ongoing enterprise that expands according to experiment, obviously it's not as rigorous as math or logic.

 

[Hornbein]

In your opinion, what is the most difficult written text (e.g. a book or a paper) on mathematics?

My candidate: Principia Mathematica by Whitehead and Russell

There are so many texts that I can't read at all that I see no criteria for discernment.

 

[lavinia]

Remember that physics is an ongoing enterprise that expands according to experiment, obviously it's not as rigorous as math or logic.

To me learning Physics is just as rigorous as learning Math if not more so. One starts with a physical law and derives conclusions from it using mathematical deduction.

Mathematics often requires non-deductive techniques to arrive a new mathematical ideas (as opposed to new Physics ideas) and this process while ultimately expressed rigorously, is itself more intuitive than deductive.

Finding new physical laws seems to me to be more like discovering new mathematical ideas.

 

[martinbn]

I am not sure that a Bourbaki approach to numerical analysis is the right way.

Bourbaki approach is the right way for anything.

 

[lavinia]

Bourbaki approach is the right way for anything.

Why?

 

[martinbn]

Why?

That was a half joke. I, personaly, find the approach better than any other.

 

[lavinia]

That was a half joke. I, personaly, find the approach better than any other.

I heard that Bourbaki was a fictitious person who was invented so that mathematicians who were banned by the Nazis could still publish. Is that true?

 

[PAllen]

I heard that Bourbaki was a fictitious person who was invented so that mathematicians who were banned by the Nazis could still publish. Is that true?

Hmm. I heard it was self chosen group of mathemeticians who wanted to revisit foundations. Nothing about a Nazi connection.

 

[Samy_A]

They started the project in 1934, so indeed no direct connection with the second world war.

 

[lavinia]

They started the project in 1934, so indeed no direct connection with the second world war.

I think Jews were banned from Academic positions in Germany during the 1930's . Artists and musicians as well. Many left Germany.

 

[Samy_A]

I think Jews were banned from Academic positions in Germany during the 1930's . Artists and musicians as well. Many left Germany.

Yes, but this was at the beginning a French project. The French Wikipedia page on Bourbaki has some interesting facts about the history of Bourbaki. I think all the founding members were French, or at least lived in France.

EDIT: All were French apparently. Mandelbrojt was born in Poland, but was naturalized French in 1926.

EDIT2: But you may have a point, lavinia. Some of the founding members were Jews, so maybe the pseudonym helped them to continue their collaboration during the war.

 

[PAllen]

Though formed at the same time as the beginning of academic restriction on Jews in Germany, I find no claimed connection at all with fairly extensive internet searching. The motivation for the founding of the group (in France) and the use of secret synonym appear to have no connection to the concurrent German events. I could not find even a hint of a claim that one motivated the other. There were no non-french members until later.

 

[lavinia]

Though formed at the same time as the beginning of academic restriction on Jews in Germany, I find no claimed connection at all with fairly extensive internet searching. The motivation for the founding of the group (in France) and the use of secret synonym appear to have no connection to the concurrent German events. I could not find even a hint of a claim that one motivated the other. There were no non-french members until later.

OK. I don't remember who told me that. I guess it was wrong.

 

[Demystifier]

After rereading this thread, I think I can finally tell the difference between mathematical and physical type of writing. Both use a combination of formal and informal talk, a mixture of rigorous and intuitive arguments. The difference is that in the mathematical text one more clearly indicates which is which.

 

[mathwonk]

my problem with physics books was the lack of completeness and precision in their assumptions. once when trying to learn relativity i read works by some greats like einstein and others. in one book maybe by Pauli, he asserted that some principle was purely derivable from another, so as I do when learning math i closed the book and tried to derive it myself. I did not succeed but when i opened the book and read his derivation he began by saying "Since space is homogeneous...". Now he had never asserted anywhere this assumption before, so I had not used it. This marked my whole experience in freshman physics, namely almost every problem needed some reasonable but previously unstated assumption, to be solved. Either I lacked this reasonable man's intuition or was just too conservative to use things not stated. Once when doing homework I spent a long time making precise some plausible assumption that helped in the solution. The grader handed it back with the remark "You are the first person in over a hundred papers to make clear just what you are doing here." I appreciated the compliment but despaired of having enough time to make precise everything going on in that, for me logically muddied, class. So I quit.

 

[martinbn]

On the subject of mathematicians reading physics, I find this spot on.

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

 

[Demystifier]

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

"...most of us would dismiss the assertion that (1, 3) ∩ (3, 1) = {1, 3} as nonsense, although it is quite correct according to the standard definition of an ordered pair: (a, b) = {{a}, {a, b}}."


That's why physicists don't always appreciate mathematical rigor.

Let me also mention that I have a similar feeling about topological spaces. The formal definition of topological space
https://en.wikipedia.org/wiki/Topological_space#Open_set_definition
simply does not feel to be the same thing as it is intuitively supposed to be.

 

[MathematicalPhysicist]

"...most of us would dismiss the assertion that (1, 3) ∩ (3, 1) = {1, 3} as nonsense, although it is quite correct according to the standard definition of an ordered pair: (a, b) = {{a}, {a, b}}."


That's why physicists don't always appreciate mathematical rigor.

Let me also mention that I have a similar feeling about topological spaces. The formal definition of topological space
https://en.wikipedia.org/wiki/Topological_space#Open_set_definition
simply does not feel to be the same thing as it is intuitively supposed to be.

I thinki it should be {{1,3}} and not as stated {1,3}, since {1,3} is contained in both sets, we use epsilon inclusion and not subset inclusion.
But yes sometimes it's not rigorous enough neither in maths.

 

[jonjacson]

"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

I can't stop laughing! hahaha

I feel the same. A whole bunch of math books are dang like that.