The course will meet Monday, Wednesday, and Friday 1pm, in Smith-Buonanno 201. I will lecture a fair bit, as there will be material in our readings I need to explain, and material not in our readings I'd like to introduce. It is my hope, however, that there will be a good bit of discussion during our meetings, as well. It will be particularly important, therefore, that everyone come to class prepared not just to listen but to participate, every time, and of course that means reading in advance the material we will be discussing. I will talk some the first day about how to read mathematics.
Formally speaking, Philosophy 0540, 1630, or 1880 is a prerequisite for this course, though it is possible successfully to take the course without having satisfied the prerequisite, if one's math background is strong enough. Several students do this each time this course is offered. If you are in that situation, however, you are likely to need to do a bit of 'catch-up'. I've collected some online resources that may be of use for this purpose here.
I will be presuming a familiarity with basic logical notation, with how it can be used to represent the `logical forms' of ordinary English statements and of mathematical claims, and with basic facts about validity, implication, and formal deduction. So you should understand what something like ∀x∃y(Fxy) might mean, and how it would differ from ∃y∀x(Fxy). And you should understand what it means to say that the latter implies the former, but not conversely, and how this could be shown.
Though helpful, Philosophy 1880 is not required. While there is some overlap between the material covered in this course and the material covered in that one, the way we approach the common material will be very different, and I will explain all the concepts from 1880 that we need. That said, it would not be a bad idea to have a copy of the text from that course, Computability and Logic, by Boolos, Burgess, and Jeffrey, to use as a reference for some of what we shall be reading.
What will be most important, though, is that students should have a degree of mathematical sophistication. The course will be very mathematical in content. It is absolutely essential that students have a solid understanding of what it is to prove something mathematically.
The only required book for the course is Undecidable Theories, by Tarski, Mostowski, and Robinson. I did not order it through the bookstore, as it is readily available from the usual outlets, such as ABE Books, Amazon, and Barnes and Noble. You may also wish to purchase a copy of The Undecidable, edited by Martin Davis (ABE Books, Amazon, Barnes and Noble). This has Gödel's paper in it, along with a couple other things we might read (and several other things you should read, such as Turing's original paper on computability, though we will not be reading these in class). If you just want a copy of Gödel's paper, you can buy it too from Amazon, Barnes and Noble, or ABE Books. Note that all three of the books I've just mentioned are Dover editions, so they are quite cheap.
Other readings will be distributed electronically. Some of these are available online through Brown's digital journal holdings; others will be scans of articles, or chapters from books, that are not otherwise available digitally.
To view PDFs, you will of course need a PDF reader. For the DjVu files, you will need a DjVu reader. Free browser plugins for Windows and Mac OSX are available from Caminova; Linux users can likely just install the
djviewlibre package using their distro's package management system. Another option is Okular, which was originally written for Linux's KDE Desktop Environment but which can now be run, experimentally, on Windows and OSX, as well. A list of other DjVu resources is maintained at djvu.org.
The program I use to convert PDFs to DjVu is
a simple Bash script
I wrote myself,
pdf2djvu. It relies upon other programs to do the real work and should run on OSX as well as on Linux.
There will be seven problem sets, connected with our readings, due about every couple weeks. There will also be a final exam during the final exam period, possibly take-home.
The exercises will mostly involve filling in the details of proofs from the readings or proving results that are related or similar to ones we discuss in class. As I will emphasize as we go, however, students should really be doing more of this kind of work than will be assigned. It is impossible to learn mathematics without doing mathematics, and in this case doing very often means: working out details the author leaves unstated, proving results for which the author does not give a proof, and so forth.
Students are encouraged to work on the problems together, though submitted material should be one's own work. That means you should feel free to get together to discuss the problem sets with others, but your answers should be your own. Do not come up with a "group answer" and then each submit it. Indeed, I'd suggest you not come up with a group answer in the first place. If you do, destroy it, and then re-do the problem by yourself. This is what you need to be able to do, anyway.
You are welcome to do your problem sets by hand or on a computer. But if you are going to do the latter, then I would strongly recommend that you not use a traditional 'word processor' to do so. They are simply not optimized for mathematics, and their output is awful. A much better option is LaTeX, and if you want to use LaTeX in an environment that feels a lot like a word processor, then you can use LyX, which can be downloaded for free from http://lyx.org/. (I am one of the lead developers of LyX, so you should feel free to ask me any questions you may have about it.) Especially if you have any intention of ever doing serious technical writing, you should start using LaTeX sooner rather than later. In the sciences, especially, it is the standard tool. Many scientific journals do not accept submissions in any other form.
Final grades will be determined by a variety of factors.
There is a Canvas site for the course, but it will really only be used to record grades. Please do not pay any attention to any 'grade average' that Canvas might report. These are useless.
Problem sets are due in class on the day specified. I will not accept late problem sets. On the other hand, you will find that I am quite prepared to grant extensions, so long as they are requested in advance, that is, at least one day prior to the due-date. Extensions will not be granted after that time except in very unusual and unfortunate circumstancess.
Because I am so reasonable, exploitation of my reasonableness will be taken badly.
You should thus expect your total time commitment for this class to be about 180 hours.
Students may use laptops and the like to take notes in class or to access material we are discussing in class, but all other use of computers, tablets, and mobile devices is prohibited during class. This includes but is not limited to checking email, texting, and searching the web, even if the search is related to the course. I establish this rule not for my benefit, not even for yours, but rather for that of your peers.
In a study entitled "Laptop multitasking hinders classroom learning for both users and nearby peers", Faria Sana, Tina Weston, and Nichola Cepeda showed eactly that. It is not just that students who "multi-task" during class—check e-mail, text, or whatever—received significantly lower grades in the study than students who did not. This is not surprising, since the human brain simply cannot focus on very many things at one time. (If you're skeptical about this, then watch this video or perhaps some of these ones.)
Rather, the surprising conclusion was that students who were sitting near other students who were "mutl-tasking" also received significantly lower grades than students were who not. In fact, they were almost as distracted as the students who were actually doing the multi-tasking. There is thus evidence that "multi-tasking" does not only hurt the person doing it. It also harms the people around them. And that is the basis of my request that students not engage in such activities during class. If someone near you is doing so, you should feel free to ask them to stop.