What is the most difficult text on mathematics?

[Demystifier]

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

 

[jedishrfu]

What about the notebooks of Ramanujan?

 

[Demystifier]

What about the notebooks of Ramanujan?

At least I understand the notation in it (which cannot be said for PM).

 

[lavinia]

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

why?

 

[Demystifier]

why?

Have you tried to read it?

 

[Greg Bernhardt]

Have you tried to read it?

Difficult math or difficult writing?

 

[Demystifier]

Difficult math or difficult writing?

In the case of PM, it is definitely difficult writing.
First, they use a rather unfamiliar notation (at least to modern mathematicians, including logicians).
Second, while other books on logic have a human friendly combination of formal and informal talk, PM is almost entirely formal.
Third, it's really big, probably much bigger than necessary to explain all what really needs to be explained.

 

[micromass]

In the case of PM, it is definitely difficult writing.
First, they use a rather unfamiliar notation (at least to modern mathematicians, including logicians).
Second, while other books on logic have a human friendly combination of formal and informal talk, PM is almost entirely formal.
Third, it's really big, probably much bigger than necessary to explain all what really needs to be explained.

And fourth, the kind of logic they are using makes things much more complicated than using a more modern logic. The entire endaveour was to eliminate Russel's paradox by introducing type theory. This goes on to make an extremely complicated kind of mathematics. The more "modern" elimination of Russel's paradox (by the ZFC axioms) is much easier and intuitive.

 

[micromass]

Anyway, to answer your question: I find about any old mathematics text very difficult to read. The older the text is, the more difficult in general. This is mainly because they have a certain kind of philosophy of mathematics that is not common anymore. For example, Newton's Principia or ancient Greek texts are difficult to read because they wanted to do everything geometrically (again: to eliminate certain paradoxes that we have solved much more adequately). The modern texts have a more balanced view of geometry/algebra. Other difficult texts (to me) are the ones written by Galois (which proves that introducing some abstraction is certainly beneficial).

Just open any mathematical history book and try to read the old statements of mathematical results. You will almost always find the modern statement to be more comprehensible. And this (I think) mainly because we are more used to the modern philosophy of mathematics.

 

[wabbit]

I don't know if it's the most difficult, but http://arxiv.org/abs/math/0211159 isn't easy : )

 

[lavinia]

The things that make math hard - other than bad writing - are non-intuitive arguments,formalism, and high levels of abstraction. Every area has some of this.
Most mathematicians find Spanier's Algebraic Topology unreadable. Try Rational Homotopy theory by Halperin and Felix. Or have fun with grown-up Rudin.

I sat in on a course in Algebraic Topology given by one of the immortals, and he shunned abstraction and formalism. In fact if a student tried a formal demonstration he would say, "That's not a proof" To him the proof is the idea not the demonstration. I believe that all mathematics can be seen as ideas but sadly many books have neither the time or space for it. The only way to sunlight is to talk with others and to concentrate until the ideas come through.

 

[lavinia]

Another thought.

Mathematics in 2015 is a larger and more elaborate field than it was in the early 20'th century. Whole new fields have come into existence and older fields have achieved a new sophistication. The sheer volume of theory that arose on the 20'th century dwarfs all mathematical knowledge of prior centuries. Because of this mathematics is much harder today than it was back then. To read a book or a paper nowadays requires knowing a lot of machinery. From this point of view, the hardest books are probably be written now.

 

[pwsnafu]

The things that make math hard - other than bad writing - are non-intuitive arguments,formalism, and high levels of abstraction. Every area has some of this.

An example from integration theory would be texts written by Henstock.

 

[Demystifier]

And fourth, the kind of logic they are using makes things much more complicated than using a more modern logic. The entire endaveour was to eliminate Russel's paradox by introducing type theory. This goes on to make an extremely complicated kind of mathematics. The more "modern" elimination of Russel's paradox (by the ZFC axioms) is much easier and intuitive.

The type theory may me complicated when studied in all details, but the idea of type theory (which I read about from other books, not directly from PM) is quite intuitive to me.

 

[Fredrik]

The most difficult math (or mathematical physics) book I own is "Geometry of quantum theory" by Varadarajan. The second most difficult is "A course in functional analysis", by Conway.

 

[PAllen]

I don't know if it's the most difficult, but http://arxiv.org/abs/math/0211159 isn't easy : )

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.

 

[wabbit]

I agree there's a lot of motivation - it's really not that the exposition is poor, rather than the content is mathematically very hard - the step size may be a good explanation, its just several sizes above my league :)

 

[cr7einstein]

I don't know if it is actually hard (or am I plain stupid :P), but I find the exercises in Mathematical methods for physicists by Arfken and Weber quite a handful. Rudin and Goldberg's mathematical analysis is also demanding. In case of mathematical physics, I am not good friends with Straumann's GR book.

 

[MathematicalPhysicist]

PM by Russell and Whitehead is indeed quite hard to read. I read the Quine did what they do in shorter amount of pages (something like ~200 pages in comparison to what they do, three volumes of more than 200 pages).

For me alongside this there's also Schwinger's monograph in Particles, Fields and Sources volume I which is hard to read and follow, but that's a physics book, and physics books are notoriously hard to follow.

 

[stabu]

While I can't follow the details, I have always found Perelman's papers to seem well written.

I think the original question overlooks what may be a valid distinction: some thing may be difficult and yet well-written.

 

[Demystifier]

For me alongside this there's also Schwinger's monograph in Particles, Fields and Sources volume I which is hard to read and follow, but that's a physics book, and physics books are notoriously hard to follow.

Schwinger is a special case. For example, S. Schweber in his book "QED and the Men who Made It" said the following:
"Other people publish to show you how to do it, but Julian Schwinger publishes to show you that only he can do it."

 

[martinbn]

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.

 

[RaulTheUCSCSlug]

Anything having to do with Abstract Algebra

 

[rollingstein]

. For example, Newton's Principia or ancient Greek texts are difficult to read because they wanted to do everything geometrically (again: to eliminate certain paradoxes that we have solved much more adequately).

Interesting! Which paradoxes are these that forced the older texts to stick to Geometric proofs.

 

[Demystifier]

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.

How about numerical/computational mathematics? Is there a Bourbaki-style text on numerical/computational mathematics? And if there is, do you find it easier than more common texts on this branch of mathematics which are, as a rule, written in a physics/engineer style?

 

[micromass]

Interesting! Which paradoxes are these that forced the older texts to stick to Geometric proofs.

I recommend to read Kline's: https://www.amazon.com/dp/0195061357/?tag=pfamazon01-20
Basically, the ancient Greeks did not trust algebra because $\sqrt{2}$ was not rational. They had a very hard time accepting this fact. They thought that integers (and thus rational numbers) were intuitive, but irrational numbers were not. This is why they forced their proofs to be geometrical, since these "paradoxes" do not show up in geometry. Hence everything they did was geometric. For example, Euclid's Elements contains a lot of number theory, but it is all stated in a (rather awkward) geometric language.

The Greeks had such an influence on the rest of the history of mathematics, that their example was followed. Hence Newton's famous Principia was geometric too since he wanted to imitate Euclid's Elements. On the other hand, irrational numbers were so convenient that they started being used in mathematics anyway. But nobody knew what they were or why they worked. This (together with the use of infintesimals in calculus) started an era where a lot of math was done nonrigorously. It was only 150 years ago that the concept of an irrational number was clarified (although not everybody agrees with this clarification) and that math was starting to be rigorous again. Whether this is a good thing is a matter of opinion.

In any case, geometry had a very big influence on mathematics. So much that old mathematicians in the 17th and 18th century were often called "geometers". Geometry lost this influence mainly because of the discory of non-Euclidean geometry, which showed that Euclidean geometry was not really the only system that could describe our world. Since that discovery, algebra has been dominant in most mathematics and it still is.

 

[MathematicalPhysicist]

I recommend to read Kline's: https://www.amazon.com/dp/0195061357/?tag=pfamazon01-20
Basically, the ancient Greeks did not trust algebra because $\sqrt{2}$ was not rational. They had a very hard time accepting this fact. They thought that integers (and thus rational numbers) were intuitive, but irrational numbers were not. This is why they forced their proofs to be geometrical, since these "paradoxes" do not show up in geometry. Hence everything they did was geometric. For example, Euclid's Elements contains a lot of number theory, but it is all stated in a (rather awkward) geometric language.

The Greeks had such an influence on the rest of the history of mathematics, that their example was followed. Hence Newton's famous Principia was geometric too since he wanted to imitate Euclid's Elements. On the other hand, irrational numbers were so convenient that they started being used in mathematics anyway. But nobody knew what they were or why they worked. This (together with the use of infintesimals in calculus) started an era where a lot of math was done nonrigorously. It was only 150 years ago that the concept of an irrational number was clarified (although not everybody agrees with this clarification) and that math was starting to be rigorous again. Whether this is a good thing is a matter of opinion.

In any case, geometry had a very big influence on mathematics. So much that old mathematicians in the 17th and 18th century were often called "geometers". Geometry lost this influence mainly because of the discory of non-Euclidean geometry, which showed that Euclidean geometry was not really the only system that could describe our world. Since that discovery, algebra has been dominant in most mathematics and it still is.

But $\sqrt{2}$ appears geometrically in the digonal of a unit square which is a valid geometric shape, they just closed their eyes?

 

[micromass]

But $\sqrt{2}$ appears geometrically in the digonal of a unit square which is a valid geometric shape, they just closed their eyes?

This is why they preferred to work with this things geometrically, since it made sense to them. The diagonal of the unit square is clear. As a number however, it is not so clear. So to them, all numbers were rational numbers. If anything irrational was needed, then they would do it geometrically.

From Kline:

The limitations of Greek mathematical thought almost automatically imply the problems the Greeks left to later generations. Thc failure to accept the irrational as a number certainly left open thcequestion of whether number could be assigned to incommensurable ratios so that these could be treated arithmetically. With the irrational number, algebra could also be extended. Instead of turning to geometry to solve quadratic and other equations that might have irrational roots, these problems could be treated in terms of number, and algebra could develop from the stage where the Egyptians and Babylonians or where Diophantus, who refused to consider irrationals, left it. Even for whole numbers and ratios of whole numbers, the Greeks gave no logical foundation; they supplied only some rather vague definitions, which Euclid states in Books VII to IX of the Elements. The need for a logical foundation of the number system was aggravated by the Alexandrians' free use of numbers, including irrationals; in this respect they merely continued the empirical traditions of the Egyptians and Babylonians. Thus the Greeks bequeathed two sharply different, unequally developed branches of mathematics. On the one hand, there was the rigorous, deductive, systematic geometry and on the other, the heuristic, empirical arithmetic and its extension to algebra. The failure to build a deductive algebra meant that rigorous mathematics was confined to geometry; indeed, this continued to be the case as
late as the seventeenth and eighteenth centuries, when algebra and the calculus had already become extensive. Even then rigorous mathematics still meant geometry.

 

[Demystifier]

The most difficult math (or mathematical physics) book I own is "Geometry of quantum theory" by Varadarajan.

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.

I guess physicists find Varadarajan difficult precisely because it is not written in a physicist style, despite the fact that it is related to physics. To test this hypothesis, it would be interesting to see what martinbn has to say about this book; is it Bourbaki enough?

 

[martinbn]

How about numerical/computational mathematics? Is there a Bourbaki-style text on numerical/computational mathematics? And if there is, do you find it easier than more common texts on this branch of mathematics which are, as a rule, written in a physics/engineer style?

I don't know.

I guess physicists find Varadarajan difficult precisely because it is not written in a physicist style, despite the fact that it is related to physics. To test this hypothesis, it would be interesting to see what martinbn has to say about this book; is it Bourbaki enough?

I have only looked in the book and have not read it, but my first impression was that, yes, it would be a book that I would enjoy. I do like his other books, so the style is bourbakish enough for me.