What a wonderful question! It would be great to launch a social experiment in which this question was addressed, e.g. in certain parts of the world large sums could be made available for students to go on to do philosophy life-long and compare regions where there is less money in the offing. I suspect that if there was more money in philosophy, more people would practice the discipline and some people with native good philosophical skills who have chosen other fields due to monetary reasons might stick to philosophy. I believe Bertrand Russell observed that in his day many of the brightest, most promising philosophical students chose non-philosophy fields due to money and politics. More recently, John Searle remarked that the key to a movement in philosophy was youth and funding. That said, many of us in the field of philosophy are not in it for the money. I haven't met a philosopher (yet) who claimed they were in it for the money, but I don't think I have met many philosophers who would complain if they were paid more On a related point, I am not sure that lots of money can help some philosophies. Imagine there is good reason to think utilitarianism is not sound in moral theory. Offering lots of money in the way of grants might get lots of philosophers to try to work out a good defense of utilitarianism, but (given that belief is involuntary) the money alone will not suffice to get philosophers to believe in the adequacy of utilitarianism.
Much depends on the country you're in. You have better prospects in the US, I think, than elsewhere because the US has no mandatory retirement age, holds people responsible for their own retirement savings, and has rules against age discrimination. (In many other countries, when a university hires someone at 45, it gets burdened with the obligation to pay her/him a pension after age 65 or so.) I think that, if you go on the job market at 45 or so with a really good dissertation and perhaps two published articles, you stand a fair chance of landing a job, especially if you also have a track record as a good teacher or teaching assistant. At least your prospects won't be substantially inferior to those of younger, equally qualified candidates.
There are some technical issues with your formula. You need to decide whether you want to understand L as a constant (80 years), thus assigning equal value to each year of human life, or whether you want to understand L as a variable that varies from person to person, thus assigning greater value to life years of persons whose lives are short. There are plausible arguments on both sides.
You might also rethink your reliance on percentage increase in the goodness of a person's life/time. Do you conceive of goodness as being always a positive number? And do you assume there to be some upper bound such that goodness can vary, say, between zero and one? Assuming all this, your formula is prioritarian: you give greater weight to those who are worse off. Thus, according to your formula, getting someone from 0.01 to o.11 is 50 times more valuable than getting someone else from 0.5 to 0.6 (+1000 percent versus +20 percent). Prioritarians often use a different formula, measuring down from the top or from some sufficiency threshold and then tagging on some exponent greater than 1. If the exponent is 2, then this formula yields a value of 0.99^2-0.89^2=0.188 for the first improvement and a value of 0.5^2-0.4^2=0.11 for the second improvement. Here the first improvement is not even twice as valuable as the first.
Leaving these technicalities aside, it seems plainly false to me to assume that "anything really important will eventually be achieved". There are, to use your own example, billions of ways of organizing education in the United States. Do you really think the best of these will eventually be achieved? In many ways, the last thirty years have been the opposite of progress: there have been more chronically undernourished people in the last two years than ever before in human history, for example, and similarly for environmental degradation, resource depletion, and the rapidity of climate change. So I think there are real opportunities not merely to accelerate progress but also to stave off retrogression or to turn retrogression into progress.
The fact that there are so many possible futures (billions of ways of organizing education in the United States, etc.) also saddles you with a baseline problem. If you devote your life to improving the US education system, how will you know how this system would have evolved without your efforts? I believe that such impacts of persons are next to impossible to ascertain with any precision, even long after the fact. For example, how would the world have evolved if Immanuel Kant had died in his crib? I doubt anyone can provide anything like a solidly grounded answer to this question.
This thought of Kant suggests two further points. First, even if you devote your life to the improvement of the US education system, the effects of your life will reverberate throughout the entire human world (witness the "non-identity problem" as one example of such reverberation -- how you decide to live will likely affect who (in terms of DNA) will be born in the future). Second, the effects of your life will last into the distant future, quite possibly making much more of a difference in the fourth millennium than in our 21st century.
The upshot of these last three paragraphs is that, even if you're well on your way toward a reasonable formula, it will be next to impossible to obtain the empirical data and -- especially -- the predictions that the formula requires. In the end, you'll have to make an intelligent guess about how you can sustainably add the most to the collective effort to make this world better. (Here "sustainably" alludes to yet another uncertainty: concerning your adherence to your original plan. It makes sense to choose a path that you will find interesting and challenging so that you will be happy and productive in its pursuit and will want to stick with it rather than switch out before you can really achieve much.)
On the basis of your remarks, it seems that you aren't especially interested in 'theoretical' philosophy (roughly, metaphysics and epistemology), but that you are interested in 'practical' philosophy (ethics) and aesthetics. You might want further to investigate just what kinds of work are being done by professional philosophers who focus on ethics--including 'applied ethics', such as bioethics and business ethics--and aesthetics, in order to get a better sense of what sorts of issues are currently 'live' in professional philosophy. Depending on the nature of your interest in "public perceptions of philosophy and debates about science, and in general about different attitudes and values in society," you may be able to explore the questions that interest you in a philosophy department; the more empirical your interests, however--that is, the more you are interested in determining just what those interests are, as opposed to assessing the basis for those interests--the less likely that a philosophy department will be hospitable to those interests. (Although I should say that given the current explosion of interest in 'experimental philosophy', it may even be possible for you to explore issues that have a significant empirical component while pursuing philosophy.) In general, in order to determine whether pursuing further study in philosophy is the best intellectual path for you, I recommend that you look more closely at what kinds of work is being done in the areas of interest to you, and also that you try to determine just what kinds of questions, and what approach to those questions, you find most congenial. As you get clearer about your own interests and the kinds of questions that interest professional philosophers, you'll be in a better position to determine whether you can pursue your interests within the context of professional philosophy. I wish you all the best as you explore your interests, and hope that you find a congenial academic home in which to do so!!
First, you should get letters of recommendation from the professors who taught the philosophy courses that you took. If you didn't take classes with enough distinct professors to have the sufficient number of letters--which I take to be an implication of your question--then you should get letters from professors with whom you had a close relationship and who can attest to your general intellectual ability and suitability for graduate work. This won't be seen as a problem, since dedicated Masters programs--such as those at Tufts and Wisconsin-Milwaukee and Virginia Tech, and Northern Illinois, to take four programs with which I am familiar and whose students are generally placed at very good Ph.D. programs (although there are of course other good programs, and you should consult Brian Leiter's Philosophical Gourmet rankings of MA programs for further information--are in the business of preparing students who lack the necessary undergraduate coursework to apply directly to Ph.D. programs to apply to such programs. I might add that the fact that you were a math major will make you a very attractive candidate to Masters programs, since you obviously already have the analytic skills that are essential to philosophical work.
It depends in part on what field you want to go into but apart from studying for the GRE etc make sure to develop strong mentoring relationships with a professor or two. You will need strong and detailed references. Also, work diligently on your writing- which entails reading you work aloud with someone.
My own view is that all this is incredibly wrong-headed. Singularly missing from this list is the project of immersing yourself in philosophical texts, thinking and talking to people about those texts and the issues they raise, and developing a deeper and subtler understanding of philosophical issues. I would like to think that we still live in a world in which, if you were to do this, then the rest will sort itself out appropriately. I might be wrong - but in that case, I personally would be less interested in pursuing such a career.
Great question! You are in a great position if you have the skills to do either pure or applied science. I am not sure about classifying black-hole research as "a very expensive hobby," but I think the answer to your question(s) depend on the urgency of the problems facing your community, family or nation. If you are in a political community that is facing urgent needs involving energy use, and there are few if any people as skilled as you in designing a badly needed solar cell, then I think you would have a prima facie obligation to pursue the relevant applied science. But assuming there are other well qualified scientists that can or are addressing urgent problems in technology, medicine, security and the like, then it seems that there is no such obligation. Besides some of what you might think of as "pure scientific research" may lead to some fruitful, important results in applied science.
Assuming by "philosophy paper" you mean student essay, then what you need to be doing is evaluating arguments, as carefully and as honestly and as rigorously as you can. You must aim for maximum explicitness, maximum clarity, maximum organization of your thoughts. But you are writing ordered English prose, not lab notes or a mathematical proof.
"Personal style" will look after itself, and shouldn't be your conscious concern. (It is always a pleasant surprise to me, e.g. when I have to mark a stack of undergraduate dissertations, how -- despite the fact that students have gone through the same teaching treadmill and drilled by weekly one-to-one essay tutorials -- distinct voices will always come through.)
These are two very different questions-- First, I would not regard the Facebook checking prof as cool. Going on to your computer while someone is giving a seminar or talk is just disrespectful. I doubt our Facebook checking prof would take kindly to someone doing the same to her as she delivered a talk that she had been working on for months.
As for your other monumental question, I would suggest that while there are some acts that are clearly always wrong, there are many that might be wrong in one context and not so in another. Lying is generally wrong but if you are doing it to save an innocent life or lives? The term "boundary" is of course a metaphor that would need to be unpacked to answer your question but on the face of it (another metaphor) I don't think there is an absolute boundary between right and wrong such that we can say of every act- this one is on the righteous side- this evil. Thanks