Of course, there is plenty of possible answers to this one, and you are welcome to share yours in the comments. However as far as “classical” (basically more or less up to the early XXth century level) mathematics goes, the Oxford User’s Guide to Mathematics appears to provide, at least for me, a reasonable, if not quite perfect,  enhancement for PCM.

OUGM has many omissions of its own and certainly could use more editing and proofreading — in particular, in order to make it somewhat more self-contained, but nevertheless this book provides a fairly broad and reasonably deep (for the beginner) panorama  of the “classical” mathematics as defined above. For instance, it does not cover category theory and related stuff. However, by and large, OUGM does a quite decent job in helping the beginner to advance her/his understanding of a great number of mathematical disciplines from abstract algebra to probability theory, and I certainly recommend to have a serious look into this book if you really want to deepen your knowledge of the “classical” subjects beyond the PCM level.

P.S. I just cannot miss this opportunity to wish merry Christmas and happy New year to the readers of this blog

Stumble!

Update: it can be found here. Owing to a misunderstanding, it was posted before I had any input into it, but in any case, the full proof is here, even if the version that is submitted for publication will have some changes.

The Notices of the AMS have published five back-to-back reviews of the Princeton Companion to Mathematics. They are by Bryan Birch, Simon Donaldson, Gil Kalai, Richard Kenyon and Angus Macintyre.

From Quomodocumque I learned of a new website, Math Overflow, where you can ask and answer mathematical questions. It seems to be very active, with a lot of users, rating systems for comments and commenters, and the like. So in principle it could be another mechanism for pooling the resources of mathematicians with the help of the internet. For example, if you need a certain statement to be true and do not know whether it is known, then my guess is that you could find out pretty quickly if you post a question there. For more discussion, see a post over at the Secret Blogging Seminar.

During my vacation in Virginia, I read 1984. I figured that I should get a few classics and well-known pieces under my belt. I really enjoyed 1984 despite its slow pace in a few places. Overall, a good read and a good way to get your mind thinking critically. Oddly, after I read it the book kept popping up everywhere, even in the news for that Amazon ebook fiasco!

Next, I was on to The Calculus Wars by Jason Socrates Bardi. As a math fanatic, I picked up this book on a mall trip (post getting asked out by some random guy for coffee) back in May but did not get far into it till July. History was never my strong point, but I would like to think I am getting better at it. The first few chapters were rough but I ended up thoroughly enjoying it! I hope my history of math class can be supplemented with the information I gleaned from the book. Anything that will help me get through an after-lunch history class will be worth it.

Current reading: The Lovely Bones by Alice Sebold. As graphic as it is, I am surprised I am six chapters into it! The first couple ones are the worst if you have a weak stomach. Despite its possible grusome factor, it is very emotional and touching. I will update further on this one once I finish it. It should take too long, I am already a third of the way through with only a few hours of reading!

I am also very excited to say that I now own The Princeton Companion to Mathematics! My parents gave me the money to get it since my friend could get me a discount from her amazing job at Barnes & Noble! Most probably think I am far too excited for a book that my brother dubs “The Devil’s Bible.” He didn’t even open it though, one look and he ran. Granted, he was already halfway out the door anyhow. I haven’t had a chance to seriously dive in just yet. I’ll post on any really interesting articles as I go. It has articles on basically everything in mathematics, and I mean EVERYTHING. I love that I have it now!

Resident Assistant training starts back up in less than 20 days and I am psyched to see everyone and to make some friends with the new staff as well! I miss being in the dorms with the staff and yes, even the residents. I got the chance to see a couple of my previous residents while out for my birthday during Harborfest. It was interesting to say the least. Though now I know they thought I was a great RA for them. =D I want to get back to classes as well. I am going to try to keep up with my free reading during the semester while doing all the classwork. We shall see how that goes!

Once again, many thanks to those who have notified me of errors. Particular thanks this time round go to Axel Boldt, who sent a very long list. He was to PCM errata II as Joseph Myers was to PCM errata I. But others too made observations that will lead to significant improvements when the reprint comes out. It has been disconcerting to see just how many mistakes remained after the effort we spent on eliminating them, but I suppose in a book this size it is inevitable, and I imagine that there are several more lurking there. So do please continue to let me know of any errors that you find. At some point in the not too distant future I’ll try to merge the second list with the first to make it easier for people to see which mistakes have already been spotted. For now, after the break, here is a copy of the instructions I have sent to PUP.

If you look carefully, you will see that I haven’t acted on every single comment, because I’m supposed to confine myself to easily implemented corrections that will have a minimal effect on page and line breaks. But I’ve taken account of almost all of them, and when I have not written about one, it is with regrets.

16.2 line 13

It’s been pointed out to me that I use the subset symbol without having
said what it is. That’s not ideal, but I think it’s not enough of a problem to
be worth the hassle it would take to fix it.

43.1 line 20

It would be better to say “a point on the line at infinity of $P’$” rather than
just “the point at infinity of $P’$”.

44.2 line 14

“ball” would be a better word than “sphere” here. In fact, this is an
essential change because of the wording earlier in the sentence.

126.1 Weierstrass quotation

Someone points out that it looks odd to date this 1988, but I think it’s
our convention that we refer to the date of the edition that is being
quoted and not the original book. Is that right? If so, then no need for
any action here.

149.2 line 17

Not sure how, but a “with with” slipped through the net.

163.2 line -3

Where it says SO(2m) it should say SO(m).

164.2 line 10

“motion through space-time” would be better replaced by “motion
in space-time”

180.2 lines 24-26

There are a couple of lower case phis that should be upper case.
(All phis round here should be upper case.)

186.9 line 9

In this heading, it should say $C_c^\infty[-1,1]^*$ rather than just
$C^\infty[-1,1]^*$. The font of the suffix c should be as it is in the
ensuing paragraph.

208.2 line -3

It should say “equal angles, just like the more familiar Euclidean rotations.”

218.2 line 6

In the definition of the ellipticity condition it should say \sum_{i,j}a_{ij}\xi^i\xi^j
In other words, the sum over i and j is missing.

265.1 line -6 and 265.2 line 2

In both places it should say \binom nk p^k(1-p)^{n-k}

That is, the power of 1-p should be n-k rather than just k.

287.1 line 13

The final psi in this definition of the Laplacian shouldn’t be there.
In other words, it should end with $\sum_{j=1}^n\frac{\partial^2\psi}{\partial q_j^2}$.

293.1 line 12

It should say \cos^{n-2r}\theta(1-\cos^2\theta)^r at the end. (In other words,
there is a theta missing.)

299.1 line -5

The words “a diffeomorphism” should be replaced by the single word
“symplectic”

303.2 line 19

The words “with a_r=b_r=0 for r>\frac 12 n” should be omitted.

313.2 line 9

Ideally, after the sentence that ends “vector space” I’d like to add
the following sentence:

Moreover, for every sufficiently small region $U$ of $X$, the inverse
image of $U$ is homeomorphic to $\R^n\times U$ (this property is
called \textit{local triviality}).

347.1 line -10

Where it says (b,q)=1 it should say (b,p)=1.

378.1 line -22

Am waiting for an email from the author before I know what to do here.

457.1 line -16

“derivatives of $f$” should be “derivatives of $u$”

470.1 line -3

It should say $B_{a+ct}(x_0)$ where it currently says $B_{a+t}(x_0)$.

478.2 line 11

The following rewriting is needed to remove an unfortunate ambiguity.

From this we wish to dedcue that the derivative of $S$ at $\phi$, in
some appropriate sense, is zero, and hence …

479.2 line 7

The sentence in parentheses needs to be rewritten as follows:

(For instance, for energy the one-parameter group of …

479.2 line 19

The displayed formula is missing two factors of \frac 12, which
should appear in the places where they appear in formula (58).
I think it would also be appropriate to have a dx at the end of
the integral, for consistency with (58).

490.1 line -14

\nabla_i A_0 should be \partial_i A_0

490.2 line 2

\partial_{x^i}^2 should be \partial_i^2

512.1 line 22 It would be slightly better to write

… a \textit{unitary} operator is an invertible …

but if this causes problems with line or page breaks then it is not essential.

520.1 line -16

The Z inside C^*(Z) should be blackboard bold, as it is four lines earlier.

521.2 middle of page

Where it says $T^*T-T^*T$ it should say $T^*T-TT^*$. (A mistake
that it doesn’t require any mathematical knowledge to spot …)

524.1 line -7

The word “period” should be replaced by the phrase “angular frequency”

527.2 line -4

There is a factor \frac 12 missing before the x^2.

543.1 formula (2)

There is a left square bracket missing at the beginning of the first line.

548.1 line 18

A valid-sounding objection has been raised to this formula. I’m not
sure what to do about it and may consult the author. But I think we’ll
have to leave it for the time being, unless I get back in touch about it.

639.2 line -10

It would be good, but not essential if it causes problems, to change
“all the true statements” to “all the true first-order statements”

645.1 line -1

Where it says R(x_i,z) it should say (x_i\sim z) (if \sim gives the
tilde symbol used just above). And where it says \neg R(y_i,z)
it should say \neg(y_i\sim z)

681.1 line -1

It should say “is at most $6x$” rather than “is $6x$”.

714.1 First sentence of V.25

Apparently I misstated the conjecture, which is embarrassing. Here
is an alternative first sentence.

The Poincar\’e conjecture is the statement that a COMPACT [III.9]
smooth $n$-dimensional manifold that is HOMOTOPY EQUIVALENT
[IV.6.2] to the $n$-sphere $S^n$ must in fact be homeomorphic to $S^n$.

This requires a further change at the end of the paragraph. Where it says,
“Thus, the Poincar\’e conjecture asks” it should be changed to “In three
dimensions, the Poincar\’e conjecture asks”

852.1 line -20

It should say

… which takes the value 1 between 0 and $\frac 12$ and $-1$ between
$\frac 12$ and 1.

The secondary purpose of this post is to suggest that you should wait a bit if you are thinking of buying the book. I don’t want to hit sales too hard, but I’m guessing that not all PCM buyers are avid readers of this blog so I might as well reward those who are. Of course, you could take the attitude that the error-strewn version is a collector’s item: if so, hurry while stocks last.

The errors that have so far been spotted.

I will use the notation A.B to stand for page A column B.

An erratum with three asterisks in front of it denotes one that is problematic because it is more than just a minor change, or will affect pagebreaks.

Back flap: I suggest “illustrating a logarithmic spiral” instead of the phrase about the Fibonacci sequence.

14.1 Displayed statement (4′) should read

(4′) For all , lifelong happiness is at least as good as .

(I hope that still fits on one line.)

14.2 line -17

The reference to sentence (6) should be to sentence (9).

24.2 First displayed equation should read

27.2 line 18

… it is a simple exercise to show that every homomorphism that is not identically zero is automatically an isomorphism …

30.1 line 7

… such that is proportional to …

The v here should be bold face.

33.2 line -5

It should say “rate of change of ” rather than just “rate of change “.

40.1 Sixth line of section 6.4

It should be n-dimensional rather than d-dimensional.

42.2 line -12

It should say ““.

Two lines later it should say ““.

In line -2 it should again say ““.

43.1 line 13

Change “with a “point at infinity”" to “with a “line at infinity”"

***43.1 line -6

Ideally, we would omit everything from “(This is quite hard to imagine” to “but a projective plane.)” since what I wrote in those brackets is unfortunately wrong. But omitting it will affect many pagebreaks. I can’t immediately think of any text that could replace what we have at present. We may need to discuss this.

43.2 line -1

The very last equation of the column should be ““.

57.2 Middle of column, it should say

… natural properties (for instance for all functions , , and ) …

62.1 In the paragraph headed (i), it should say

… which leads to the formula .

line -6

replace “” by ““

63.2 At the end of section 6.2 it should read

… was proved by Huxley in 2003 (the latest in a long line of successive improvements), is that is at most for some constant .

64.2 At the very end of section 6.3 it should say

… as many as .

In other words, ms need to be ns.

75.1 line -21

The number 181 needs to be added to the list of numbers here (in the appropriate place).

121.1 line 4

This displayed equation should read

124.1 line -7

Replace “in every bounded infinite set” by “for every bounded infinite set”.

131.1 line -13

It should be DBCF rather than DBFC here.

169.1 Middle of column.

… if one maps the real number to the point , then …

That is, the final “squared” comes inside the bracket.

171.1 End of penultimate paragraph.

This list is missing some numbers. It should go

and then continue as before. If it’s a problem to add a line here, then we could end the list with “4, -4, …” instead of ending at “5, -5, …”

***172.1 line -20

A new half-sentence is needed here. It should say

… for any elements and ; this involution is required to satisfy the -identity .

176.2 Middle of column.

Where it says “where and .” it should say “where and .”

180.2 line 5

Instead of “” it should say ““.

Earlier in the same line it would be better to write “” instead of ““.

183.2 line 16

… written either as 0.1 or as 0.02222…

(instead of 0.22222…)

184.2 lines 12-14

It should read

… sense: if is less than , then the -dimensional Hausdorff measure of is infinite, while if is greater than , then it is 0.

187.2 line 7

It should be “octahedron” rather than “octagon”.

194.2 line -8

The final should be (so that it matches the in the exponent).

***202.2 End of column

I would like to add the following sentence if possible.

… coefficients quickly. A method for doing this was discovered by Cooley and Tukey in 1965 (though it turned out that Gauss had anticipated them over 150 years earlier).

223.1 line -16

In the display here there is a stray comma in the last expression, which should be . (NB the “e” should still be a Roman e — it’s just the comma that needs removing.)

228.1 line 3

It should end with . I.e., it’s half the square root of 3 and not the square root of half of three.

232.1 bottom of column

It should read

… ,

***238.1 line 1

The factor 6 here is not explained. Unfortunately, it may take more serious rewriting to clarify this. (It’s not exactly wrong, but comes from a slightly different form of the original equation.) So I think we’ll have to leave it for now.

However, it should say

… equation . If we add …

240.2 line 8

It should say

… is the matrix defined by .

249.2 line 11

It should be

I.e., there’s a missing q in the bracket.

266.1 line -21

It should say “,” instead
of “,”.

272.2 line 4

It should say “generated by the polynomial .”

277.1 line -4

Reword as follows.

… a system of numbers where we introduce not just one square root of -1 but three, called i, j, and k (together with their negatives). Once one knows …

***278.2 Middle of column, item (ii).

Ideally we should extend the sentence as follows:

… is a real number, and similarly for multiplication on the right.

If the extra line causes problems, then we can perhaps live without this, but it will bother some readers.

297.1 line 21

It should say

… spherical harmonics are eigenvectors of the Laplacian, but they …

310.2 Second paragraph of section 2.1

This should read

Suppose, then, that we are given two points P and Q in the plane. We take as our class of admissible functions all smooth, real-valued functions , defined on some interval , such that and . The length …

314.2 line -5

It should say “when one does include the”

323.2 display in line -9

Where it says it should say

324.2 middle

It should say “published in 1952″ rather than “published in 1934″.

325.1 line -11

It should say “It is easy to deduce that is itself a prime in the …”

NB the “i” should be in the same font as the other ones round here when this correction is made.

327.1 lines -8, -6, -5

The three s that appear here should be s.

339.1 line 3

It should say “arithmetic progression” rather than merely “arithmetic” here.

353.2 line 12

“Pollardand” should be “Pollard and”.

355.1 line -16

“Pollardfor” should be “Pollard for” (!)

424.2 line 18

Change “the first three” to “three of the”

425.2 Middle of column.

Change to (it comes at the end of a line).

***444.1 line 18

Change “non-trivially” to “”properly discontinuously”", so it should read

… if a group acts “properly discontinuously” as a set of isometries on …

A better change, but one that would affect linebreaks and pagination, would be

… if a group acts properly discontinuously (this means that for any compact set there are only finitely many such that and intersect) as a set of isometries on …

445.1 line 19

Replace “smooth compact manifold” by “complete Riemannian manifold”.

line 20

Replace “Given any closed path” by “Given any contractible closed path”.

460.1 line 4

It should say

… matrices that satisfy …

496.1 line -19

A few small changes needed here. It should say

However, if we apply ten times to and , then their respective eleventh digits have shifted leftwards and become the first digits of and . These two …

523.2 line -21

Replace “slide rules” by “log tables”.

531.2 line 9

It should say image( ker() rather than
the other way round.

555.1 line -7 it should say

… is by constructing a (preferably nice) bijection …

558.1 line 22

It should read “When there is exactly … “

568.1 line -13

It should read

… set of integers using …

575.2 line -5

It should say “Two very obvious conditions” here.

595.1 lines 16-17

.. at least . Therefore, the probability … is at most , which is …

645.1 last line

The right-hand side should read

This one will need careful formatting.

695.1 line -6

A mathematical mistake here. It should say

We define the index of the fixed point to be the number of turns made by the vector from to , counting this negatively if these turns are in the opposite direction to the way that goes around . (This definition is problematic if for some , but again …

701.2 line 7

It should say “no proof of the formula , where is the …”

706.1 lines 9 and 11

The two capital Fs should be small Fs. That is, we want
“a function ” and we want f(x,y) inside the integral.

708.1 line -16

It should say “tile the plane if and only if the algorithm
fails to halt.”

708.2 line 13

It should say “if and only if the algorithm halted at .”

708.2 line -3

It should say “in the form .”

714.2 line 1

Change to .

***lines 2-5 This parenthetical remark should be removed, though it will change the pagination.

756.1 Middle of column.

For consistency with the rest of the book, we should say “going on to find seven more proofs over the years.” However, we should not make this change unless Jeremy Gray is happy with it.

767.1

It would be better to entitle this article “Ernst Eduard Kummer”

789.1

The first word of the column should be spelt “Theologie.”

813.1 line 5

Omit “closed”, so that it reads “formally real fields”.

832.1 At the end of the first full paragraph, add words so that it reads

… can take any value greater than 1.

***889.2

The description of CBC and OFB modes is oversimplified. The CBC description gives the impression that two successive cleartext blocks are added before encryption (so a 128-bit repeat in cleartext would result in a 64-bit repeat in ciphertext, similar to the ECB problem described in the previous paragraph)
when actually each cleartext block is added to the previous ciphertext block before encryption. In turn this reads like the description given of OFB mode; OFB does not feed the input cleartext through the block cipher at all, but starts with an initialization vector and repeatedly enciphers that to generate
a keystream, that is then added mod 2 to the cleartext to generate a ciphertext.

I think this is going to be too complicated to change this time, but I’ll keep a record of it.

909.1 line 14

“Becky has five siblings” should read “Desta has five siblings”.

919.1

“” (in the second paragraph) should read ““.

920.2

In the display the should be removed inside the “exp”, and the “” in the square root on the bottom should be ““.

Four lines later, there is again a stray , and again the “” on the bottom needs to be a ““. And this time, we also need to change the “” to a ““.

984.1 line -2

“quatre-vingt” should be “quatre-vingts”

997.2

In equation (7), “” should read ““.

1014.2 The remark in brackets that looks as though it explains Robinson’s Non-Standard Analysis has nothing to do with it and should appear after the following entry on the Langlands program.

Incidentally, before I say any more, I want to say that a huge amount of proofreading has gone into the book, so I expect the density of mistakes to be pretty small. But because the book is a big one, I also expect the number of mistakes to be not all that small.

The mistake occurs on page 24 of the book, where I give a formula for the product of two elements of the dihedral group of order 8. I am discussing semidirect products of groups, so I use the notation to stand for the element that would normally be written where is a quarter turn and is a reflection. I point out that and then claim that it follows from that that . Of course, the power of on the right-hand side should in fact be .

How could I write something that jumped out at me as obviously false when I came back to it just now? Probably because I was focusing on the interesting case when . But it was still pretty careless. And the general point I want to make here is that such carelessness is incredibly tempting. Several years ago, David Preiss and I were colleagues at University College London, and he said something that I’ve never forgotten, which was that if you are suspicious of a mathematical argument and want to find the mistake, look for anywhere where it says something like, “An easy argument shows that …” or “It is not hard to prove that …” I dare say he was not the first person to make this observation, but it doesn’t seem to be quite as standard as it might be.

For the purposes of an expository book like the Princeton Companion, it was quite common, and the right thing to do, to miss out details of arguments, and to say things like, “If you do this calculation, you will find that …” I was inclined to trust authors (including myself) when they wrote things like this, but at a late stage I found one or two that were embarrassingly false, and began to think that it would be a good idea to check them systematically. Vicky Neale, a graduate student at Cambridge, very kindly agreed to do a lot of this task (though unfortunately she came in at such a late stage that this final check did not cover the whole book), and she discovered that an extraordinarily high percentage of these not fully justified assertions were wrong. In most cases, most readers wouldn’t notice, because all that really mattered was that the calculation had an answer: it wasn’t so important what the answer was. Unfortunately, the one I’ve just discovered will be genuinely confusing to the reader for whom that particular article was intended. And I’d very much like to eliminate as many as possible of the ones that remain.

The general moral I would draw from this experience is that in expository writing one should be especially scrupulous about checking calculations, because there tends to be a higher density of assertions that are not fully justified (because they are plausible and their justifications would get in the way of the exposition). To put it in a punchier way: if you haven’t checked it, then it is wrong. That’s particularly true if you worked it out in your head, as I think I did in this instance. 

As for research articles, I think my advice here is pretty hard line: proofs should be written out in full, or, failing that, precise algorithms should be specified for generating such proofs. To explain what I mean with an example: instead of saying, “An argument similar to the proof of Lemma 2.3 shows that …” one should say, “The proof of this fact is similar to the proof of Lemma 2.3; the differences are that in this case we replace by , and …” etc. Of course, another important algorithm for generating part of a proof is something like “Insert here the main result of [23].” I just mean that “It is not hard to show that” should appear only if giving the proof would be an insult to the intelligence not just of experts but even of mathematicians who are not familiar with your area. Doing this not only forces you to deal with the weak parts of your arguments, but it also makes what you write easier to read. After all, if it’s true that “An easy argument shows that …” then you should be able to give the argument reasonably elegantly, and the reader is free to ignore it. If it’s not true, then the reader may well waste time trying to see the obviousness of what you claim.

UPDATE 14TH OCTOBER: here are links to a few blog posts that have discussed the Princeton Companion recently. Terence Tao had a few thoughts on receiving his copy. Brian Hayes did too. Fred Shapiro, editor of The Yale Book of Quotations, has an interest in reference books and commented about the PCM in the New York Times Freakonomics blog. Dmitry Vostokov recommends the PCM for people wanting a broad overview of mathematics. Isallaboutmath points out that the Nautilus shell on the cover does not in fact have anything to do with the golden ratio. And the book has five reviews on Amazon, some more sensible than others.

FURTHER UPDATE (to which I’ll add as the occasion arises): Peter Woit has written a short review on his blog. Ian Stewart has written a review for The Times. Edmund Harriss has reviewed it on his blog. Alexander Bogomolny reviews it here. Does Antonio Cangiano like it? Perhaps you’d better judge for yourself. Having honed your interpretative skills, you’ll be in a better position to appreciate Scott Guthery’s helpful antidote. A different Scott, Scott Aaronson, reviews it on his blog. And Robin Wilson reviews it for the LMS newsletter. A nice review here, in French.

29TH OCTOBER: It’s been a long time coming, but I’ve just noticed that the PCM now appears to be available on Amazon UK, and not just from third-party sellers. According to them, it’s popular amongst geographers. If I knew how, I’d suggest they recategorized it.

As an example of the important issues we now face, here is a question about hyphens: I’m fairly sure there will be a small but passionate minority of mathematicians who care about these, and a question has come up. I am curious to know what other people think, so I’m not going to say what I think: I’ll just try to present the question as neutrally as possible. And here it is.

Consider the following four phrases: “intermediate value theorem”, “travelling salesman problem”, “twin prime conjecture”, “minimal surface equation”. Do they need hyphens? And how about “three body problem”?

Brief argument in favour: the general rule is that if you have a phrase made up out of an adjective, a noun and a noun then you put in a hyphen. (An example is “brute-force search”.)

Brief argument against: the cases above are exceptions because they are a bit like proper names: the first two words are not really functioning as an adjective that describes the main, third, word; rather, the three words form a single phrase. In addition, the lack of hyphen does not lead to any conceivable ambiguity.

If you buy something like the second argument, can you make it more precise?

Also, if you wouldn’t write the hyphen in “intermediate-value theorem”, do you find it positively wrong and strange looking, or merely not necessary?

In other words, should there definitely be hyphens, should there definitely not be hyphens, or would either decision be acceptable? (To judge from Google, standard mathematical practice seems to be not to put hyphens, but that alone is not a sufficient argument.)

I’ve had a more mathematical blog post planned for months, but it’s still going to have to wait. But once again I’d like to make clear that this blog’s still just about alive and will I hope become more active again before too long.

Important biographical notes: Tim has a Fields Medal and a blog.