How should I respond to a terrible copyediting job? I just received the galley proofs for an article which has been accepted to a well-regarded math journal.

The copyediting is, quite frankly, terrible. The copyeditor has introduced dozens of mathematical, typographical, and bibliographic mistakes. The proofs were accompanied by a list of changes and questions, and some of the newly introduced mistakes were described in this list, but not all. 

A few representative examples:


The editor has, in many places, replaced the mathematical notation &lt;&lt; with "is much less than". It is universal within my subfield of mathematics that &lt;&lt; means something very different from "is much less than".
The editor has typeset fractions in different ways, so that what was formerly in the denominator is now in the numerator.
The editor has monkeyed unacceptably with the bibliography. I used a software package, for which I looked up on their FAQ how to cite it. The editor replaced my correct bibliography entry with something different.
The tables now appear in wrong places. A cursory reading of the edited text reveals that it now makes no sense, and this would be obvious even to someone without specialized knowledge of my field.


I could go on and on, but I don't want to just rant. Instead I have two closely related questions:


First of all, is this typical? I find it difficult to imagine that a publisher who did this kind of shoddy work on a regular basis could stay in business. I wonder if I just had the bad luck to get the new employee this time around.
More importantly, is a strong response warranted? I am inclined to write to the journal staff, tell them that their work is unacceptably sloppy, explain in detail why, and ask them to start over from scratch, and to furnish a list of all changes made, no matter how minor.

This is not just because I want to pick an argument. I have gone through my paper line-by-line several times in the past, making very sure that everything I said was correct. With this level of copyediting work, I am back to square one and I suspect I might accidentally miss several errors introduced by the copyeditors.

Would such an e-mail be likely to produce the kind of results I'm looking for?


Thank you very much.

Update: Thanks to everyone who replied. I wrote a strong, but I hope polite, e-mail to my contact at the journal, listing several of the mistakes, and asking them to start over and to send me a complete list of changes. His first reply was a little bit ambiguous, appearing to perhaps misunderstand what I was asking for -- but he has since apologized and agreed to my requests.

One point of departure from Anonymous Mathematician's advice: I haven't said anything to the editorial board and the publisher, or discussed this issue (other than here, anonymously) with anyone but my coauthor and my contact at the journal -- happily, it looks like there won't be any reason to.

Update 2: As I requested, my contact at the journal started again from scratch, did a much more conservative job of copyediting, and provided me a copy of my file which was marked up in red and blue with every change they made. Needless to say this made my job quite easy and I thanked them for their good work.
That does sound terrible.  I haven't run into anything so bad myself, but I know people who have.  It seems to vary a lot by publisher.  My impression is that it happens occasionally when they try to save money on copyediting by outsourcing to a new, cheap copyediting company.  If the results are terrible, then the publisher will switch to the next-cheapest company, and this will continue until the publisher finds an acceptable level of quality.  Unfortunately, it leads to bad results for authors who get caught in the middle of this process.  Hopefully the publisher will learn from this incident and the problem won't continue.


  More importantly, is a strong response warranted?


Yes, certainly.  


  ask them to start over from scratch, and to furnish a list of all changes made, no matter how minor


I'd guess that getting them to start over from scratch will be difficult, but you should certainly insist on seeing a second set of proofs.

Asking for a list of all changes made could be useful, but it's not clear whether you can trust that they will compile a complete list (they may be sloppy about that too).  Many copyeditors mark up a paper copy before changing the file, so you may have some luck in getting a photocopy/scan of the marked up copy.

I'd also recommend letting the editorial board know, and perhaps the publisher too.  If you just resolve the issue directly with the copyeditors, it's possible that nobody else will find out what happened.