I receive many emails asking for advice about graduate school applications. I try to answer them all, but I do not always succeed. In the interest of saving some of the time I spend typing out what is essentially the same information over and over--and in the interest of being as helpful as I can be to those who don't email but do stalk my website--here are my answers to some of the questions I often receive. If you have lingering questions about the Harvard philosophy PhD program (or the Cambridge HPS MPhil program) in particular, feel free to reach out via email, and I will do my best to get back to you (but no promises!).
1. Caveats: tl;dr, do not trust me because I do not know that much about this and I don't really understand why people ask me about it so often
I should note at the outset that I am very skeptical that I am the one you should be asking about your graduate school applications. I was not the one reading my application, and I can only speculate about why it was successful. The best advice I have is that you should consult a supervisor, and, if possible, a supervisor who has served on a graduate admissions committee. Insofar as I know anything about graduate school admissions, it is because I trust the people who advised me when I was applying. There are many guides to philosophy graduate school admissions available for free online, and many of them are written by people who have served on graduate admissions committees and are much more knowledgable than I am about grad school admissions. Here are two I recommend: this one and especially this one, which is super comprehensive. These are more helpful than I could possibly be.
Finally, what I say is only relevant to American philosophy graduate school admissions. I also applied to--and was accepted by--comparative literature PhD programs, but I have even less of a sense of what made those applications successful. I have absolutely no idea what makes, e.g., an application to an art history PhD program successful. I have never applied to a PhD program in the UK, so I don't have any idea how to do that either.
2. Do I need to know/say a lot about my specific dissertation project?
No, not if you are applying to an American philosophy PhD program. (I imagine things are different in the UK, but again, I don't know much about that.) American philosophy PhD programs are a minimum of five years, two-ish of which involve coursework. When you apply, you are expected to have some idea what you will work on and what kinds of topics you want to explore, but you are not expected to have a detailed sense of the questions your dissertation will ask, much less how you will answer them. I would imagine that it is in fact kind of a bad idea to talk as if you are certain of what your dissertation will say: a person on an admissions committee has likely seen a lot of students shift gears mid-way through their PhDs. (I thought I wanted to work on epistemology and philosophy of perception when I arrived at Harvard.... Now I work on aesthetics.) My personal statement touched on my areas of interest (at the time, epistemology, philosophy of perception, and Heidegger) and on the broad questions within these areas that intrigued me (e.g., under what conditions is testimony evidentiary?). I did not speculate about my dissertation, nor did I attempt to do philosophy. That is, I didn't walk through any arguments ("I think testimony is evidentiary just in case.."). I just gestured at topics I wanted to explore in the future: I saved the actual philosophizing for my writing sample.
3. Should I contact potential supervisors?
No, not if you are applying to an American PhD program. (Again, I bet it's different in the UK.) Because you cannot really know what your dissertation will be about, you cannot really know who will chair your committee. Moreover, you don't want to make it sound or seem as though your interest in a program hinges wholly on whether you end up working with one particular person. Most of your application should be about what interests you, not about the places you're applying: my personal statement contained only one paragraph about the programs I was applying to, since I assumed admissions committees would grasp why their schools were good fits for me. When you do discuss particular departments, you should explain in more general terms why the relevant programs can support you, given your range of interests (e.g., Harvard is strong in continental philosophy, so even if I had become less interested in Heidegger and more interested in some other continental figure, it would've been a nice place for me to be.) Of course, you can name a few people you'd be interested in working with--I did. Also, if you are accepted to a PhD program, and you have good reason to believe you would want to work primarily with one person, it is a good idea to ensure you get along with the person (as well as to talk to the person's current and former students to make sure the person is a good supervisor). But at the application stage, you do not need to worry about any of this yet!
Hope this helps!