I’ve started struggling through Wolfram Pohler’s recent Proof Theory: The First Step Into Impredicativity. And it is, I’m afraid, a struggle — for a textbook exposition “pitched at undergraduate/graduate level”, it is really quite unnecessarily hard going. For example, I can’t imagine that anyone who hasn’t already encountered the idea would have much hope of cottoning on to what is going on with the Veblen hierarchy from the discussion in Sec. 3.4. And even if you are very familiar with completeness proofs for first-order logic, it’s made ridiculously hard work to see what’s going in the completeness proof in Sec. 4.4.

I’ll certainly keep ploughing on through, given the book’s coverage. But not with relish. Why on earth write like this, without the introductory informal motivating comments and explanations of concept definitions and proof ideas that you’d give in lectures? I’m quite baffled that anyone can think that this is the right way to write a book intended for a student readership.

Anonymous comments: I think this is the usual way of writing a math book. Most, if not all, math books are written this way.

I don’t think that’s entirely true. I’m sitting here surrounded my shelves of relatively recent maths books, both pure and applied maths (dating from when I was writing my chaos book ten years ago, or teaching the philosophy of space-time theories). They do illustrate a whole spectrum of modes of presentation from rather relaxed to take-no-prisoners relentless formality. To be sure, there are too many of the latter kind — but it is possible to do better!

I think this is the usual way of writing a math book. Most, if not all, math books are written this way. I am sure that’s different in philosophy.

I haven’t gotten to reading Pohler’s book yet, so can’t really comment on that.

Picking a random text on my shelf, Sack’s

Higher recursion theory, I find it contains, on most topics, an informative overview, brief historical remarks now and then, motivation, and discussion of the significance of the results established. The book is by no means unique in this.It is, I feel, quite unfair to say most, if not all, books in mathematics are written the way Peter finds frustrating. (Admittedly, there are some truly awful examples of awfulness out there, such as Kleene’s and Vesley’s text on foundations of intuitionism and Vopenka’s

The Theory of Semi-Sets.)a whole spectrum of modes of presentation from rather relaxed to take-no-prisoners relentless formality. To be sure, there are too many of the latter kind — but it is possible to do better!These are not the two available options at all. The optimal is to be strictly formal and readable. These two faculties are independent, i.e., one does not exclude the other.In fact, if I have to choose between a book which is informal, non-self contained, too relaxed, and on the other hand a strictly formal one, I certainly choose the latter. I’m not willing to read a mathematical book, and struggle with the proofs, when I have a slight suspicion about its accuracy.

I’ve read up to the middle of Chapter 5 in Pohlers’s book. Before finding this post, I wondered if the dread I felt at the thought of reading further was due to my own idiosyncrasies. After all, this is the first proof theory book I’ve read; perhaps this is how proof theorists write, I thought.

Then I picked up a copy of “Basic Proof Theory” by Troelstra and Schwichtenberg. The expository style is relatively enticing. However, aside from a brief remark in the preface, the authors seem to eschew ordinal analysis entirely. For me this is unfortunate, since I was hoping to learn about it now rather than later.

So I ask your advice: should a first-year graduate student aiming to specialize in set theory and philosophy of mathematics persevere through Pohlers in hope of acquiring the knowledge he wants when he wants it; or should he consider shelving Pohlers for a few months until he’s read Troelstra and Schwichtenberg? (If it matters, he has read classic first-year graduate texts in set theory, model theory, and computability theory.) Of course, he could also read something else.

Well, I think there’s something to be said still for starting with Takeuti’s classic book, which is relatively accessible.