What is the most difficult text on mathematics?

[atyy]

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.

I replied to this in post #140, and as I said my main point in bringing up PRA had nothing to do with the incompleteness theorem. However, it does seem that PRA is also incomplete http://www.personal.psu.edu/t20/notes/logic.pdf (Theorem 6.6.4, p122).

 

[atyy]

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

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.

In fact, Bourbaki also says that the language of mathematics rests on the informal language of physics, biology and psychology. For example, they say in the Introduction to their Theory of Sets that one needs to assume that we know what is meant by a letter of the algebra being "the same" in two different places on a page.

In the same Introduction they also say "The verification of a formalized text is a more or less mechanical process". Again that is physics, implicit in the word "mechanical".

They also say that it is impractical to carry out all mathematics in the formalized way, and they will therefore use informal arguments in which the existence of the intuitive natural numbers will be assumed before any formal arithemetic is defined.

As far as I can tell, my views are very Bourbakist

 

[lavinia]

In fact, Bourbaki also says that the language of mathematics rests on the informal language of physics, biology and psychology. For example, they say in the Introduction to their Theory of Sets that one needs to assume that we know what is meant by a letter of the algebra being "the same" in two different places on a page.

In the same Introduction they also say "The verification of a formalized text is a more or less mechanical process". Again that is physics, implicit in the word "mechanical".

They also say that it is impractical to carry out all mathematics in the formalized way, and they will therefore use informal arguments in which the existence of the intuitive natural numbers will be assumed before any formal arithemetic is defined.

As far as I can tell, my views are very Bourbakist

In practice mathematics is done through intuition and insight. Formalism is always an after thought - part of the process of verification - but not the source of ideas.

 

[atyy]

While I can't follow the details, I have always found Perelman's papers to seem well written. There is plenty of description of the idea to be established, how it fits with other idea, and how it will be used. On the other hand, experts in the field are pretty unanimous that the logical 'step size' is way above average. This is the aspect that made it so hard, and meant every verification of it was 10 times the size of the original. On the other hand, my first point about a clear game plan led, in my recollection, to experts 'believing the program' way before they completed detailed verification.

One of the interesting things about Perelman's http://arxiv.org/abs/math/0211159 is that he throws a bone to physicists, saying that the Ricci flow may be linked to the renormalization flow. I've always wondered about that, and googling brings up this piece by Urs Schreiber which gives the whole strategy in such a simple way that even a physicist can understand it (of course, the details are still far off)!

http://ncatlab.org/nlab/show/Ricci+flow:
"[Ricci flow] is the renormalization group flow of the bosonic string sigma-model for background fields containing gravity and dilaton (reviewed e.g. in Woolgra 07, Carfora 10, see also the introduction of Tseytlin 06). In (Perelman 02) Ricci flow for dilaton gravity in 3d was shown to enjoy sufficient monotonicity properties such as to complete Richard Hamilton’s proof of the Poincaré conjecture."

On the subject of difficult texts, I've found Hairer's paper on the KPZ equation inpenetrable http://arxiv.org/abs/1109.6811. It should be very interesting for physicists, because the KPZ equation is a standard equation in statistical physics, and physicists have known for years that it is meaningless because it multiplies distributions, yet instinctively it must mean something, so they've worked around its non-existence for 30 years. Apparently Hairer has now found a way to make sense of it in a way that makes the physics way of treating it correct, as well as giving it meaning beyond what physicists have known.

Incidentally, here is a video of Kardar (the "K" of KPZ) saying at 1:11:00 in a different context "It's good that you know that these equations can do all kinds of strange things. But when you take a particular physical system, you have to beat on them until they behave properly."



If one is thinking physically, then it would seem that Hairer and Perelman's works are linked, since the KPZ is motivated by renormalization group thinking.

 

[mathwonk]

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.

 

[disregardthat]

Definitely have to second Perelman's "paper" on the Poincare conjecture (apparently he wrote the proof in his blog). It took a long time until people had filled in the steps, and as far as I remember a Chinese research group took up the task and filled in the details in a large paper.

As far as popular grad student books, I'd say Harthshorne's Algebraic Geometry is far up the list of books considered pretty difficult. EGA is considered easier in comparison (at least according to those who I have talked to who reads it), due to the way it generously fleshes out proofs. I haven't read much of it, though (french is difficult).

 

[PAllen]

Definitely have to second Perelman's "paper" on the Poincare conjecture (apparently he wrote the proof in his blog). It took a long time until people had filled in the steps, and as far as I remember a Chinese research group took up the task and filled in the details in a large paper.

Perelman published his papers to arxiv, not a blog. It is true that he didn't submit them for publication. Instead, he pointed out the papers in e-mails to several mathematicians he knew and who had relevant expertise. He also did one tour of several universities explaining his papers to mathematicians. Finally, for a year or so, he responded to questions posed by experts by e-mail.

There were 3 (later, a fourth, involving Hamilton) independent groups that verified his papers. The Chinese group exaggerated the nature of the gaps, plagiarized the work of a different group (claiming it was accidental from mixing up notes), and had to retract and re-issue their paper (with different title, abstract, and apology).

 

[Wes Tausend]

"atyy said: ↑
In fact, Bourbaki also says that the language of mathematics rests on the informal language of physics, biology and psychology. For example, they say in the Introduction to their Theory of Sets that one needs to assume that we know what is meant by a letter of the algebra being "the same" in two different places on a page.

In the same Introduction they also say "The verification of a formalized text is a more or less mechanical process". Again that is physics, implicit in the word "mechanical".

They also say that it is impractical to carry out all mathematics in the formalized way, and they will therefore use informal arguments in which the existence of the intuitive natural numbers will be assumed before any formal arithemetic is defined.

As far as I can tell, my views are very Bourbakist "

In practice mathematics is done through intuition and insight. Formalism is always an after thought - part of the process of verification - but not the source of ideas.

I am a firm believer in the above statement from lavinia being true.

Math is important, but I once worked out an observation about intuition, oft called insight, being foremost in the ultimate gain of human knowledge. I call it the Race Team principle.

Suppose we observe a successful, typical race team that races cars, a NASCAR team for example. The "win" seems to depend upon the driver having an intuition that most closely approximates real physics. In a nutshell, he, or she, must quickly calculate the best balance between tire adhesion and centrifugal forces. This seems to me to be a rather pure example of intuition. On the other hand, the car cannot win without the skills of a top notch mechanic, no matter the naturally gifted extent of the drivers intuition.

The mechanic uses skills of physics that can be taught, thermodynamics, material selection, a variety of tools which he, or she, knows how to apply quite well... even if it involves some head scratching on occasion. This is not unlike a well trained mathematician and his, or her, tools. The driver wins because he, or she, is an exceptionally well tuned child to Mother Natures laws of motion and friction. So well tuned that the driver intuitively knows where to go when there is no time allocated for head scratching; a quick conjecture in the raw... a sudden eureka of sorts.

This is not to say that the best driver is not an accomplished mechanic, nor the best mechanic an accomplished driver, and the best of both would therefore be a driver/mechanic that surpassed any of either. But in reality almost all gain is still made by teams. And so is it true of the giants of physics whom at least stand on the shoulders of their team-mates.

As examples, many of our scientific giants, our scientific "drivers", could apparently see, could conjecture, Nature's geometry before hashing out the mathematical proof. Copernicus (Heliocentricity), Kepler (eliptical orbit), Newton (the most far-flung falling cannonball), Einstein (rods do get shorter, Equivalence), Feynman (his diagrams, the Lost[/PLAIN] Lecture) and more.

Back to subject, I found a purported partial english copy of Principia Mathematica here: http://www.olimon.org/uan/principia_3.pdf .
By substituting 1 and 2 in the above address, I find earlier sections of the book. However, by substituting 4, I find no more. Is the above address 3, the final end of it all? The last page of the series looks incomplete. A glance at http://www.olimon.org home reveals the main website is in Spanish.

Wes
...

 

[Demystifier]

How about numerical/computational mathematics? Is there a Bourbaki-style text on numerical/computational mathematics?

Perhaps this book is not too far from that:
https://www.amazon.com/dp/1483256375/?tag=pfamazon01-20

 

[MathematicalPhysicist]

Perhaps this book is not too far from that:
https://www.amazon.com/dp/1483256375/?tag=pfamazon01-20

Numerical analysis has a wide scope; I am for one in my thesis am using this book by Gustaffson et al:
https://www.amazon.com/dp/B00QD9SGN0/?tag=pfamazon01-20

And for numerical analysis for a first course there's de Boer's and Conte's book: Elementary Numerical Analysis which is a classic.

 

[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.