What does it mean in mathematics for two things to be equal, or for two things to have the same "identity"? For example, because anything divided by zero is "undefined", can we say that 1/0 = 2/0? What about the relational database concept of "null" which is supposed to stand for "unknown"? In relational algebra, they say NULL is not equal to NULL, but doesn't that violate the law of identity that everything is equal to itself?

I will begin by acknowledging that neither the philosophy of mathematics nor the metaphysics of identity are my specialties. But if you'll take what I say with a grain of salt, perhaps I might make a helpful observation nonetheless. Keep in mind that "being equal" mathematically is not exactly the same thing as "being identical," mathematically or otherwise. "Being mathematically equal" means, one might say, having the same mathematical value, in the sense of amounting to the same thing. Identity on the other hand means being the same thing, in the sense of having all the same properties. This difference can get confusing because commonly in symbolic logic the equal sign, "=", is used to express identity. But, if you follow me, then "2+2" equals "4" but is not identical to "4." Why? Well, while they both have the same mathematical value, they don't both have the same properties. "2+2", for example, is a formula composed of two Arabic numbers and a symbol for the mathematical relation of addition....

I am a baseball coach/manager. In my stepson's baseball league, another team has a child (these are pony league players - 13 & 14) who has some arm problems. I know he has had an MRI (know the MRI tech) and also that his doctor instructed him never to pitch again. The coach and parents are aware of this too - yet the coach still pitches him in games. Other parents discuss this problem, yet no one seems willing to step up and do something about this. Since I know the story, would it be ethical if I anonymously informed the league? There may be a potential liability issue at stake here too. This kid is going to ruin his arm before he gets to high school. I am also trying to balance the confidentiality of the medical relationship vs. the kid's welfare. Should I even be considering this?

I agree with Thomas Pogge's remarks, but I also have a couple of cents to add. First, consider very seriously and act in light of the fact that your information comes to you second hand (from a lab tech and not the child's physician or parent)--unless, of course you are the MRI technician. Second hand reports are notoriously inaccurate, and so I suggest proceeding with caution and when you act qualifying your comments with the acknowledgment that your information may be inaccurate. The tech may be exaggerating the physician's instructions or otherwise distorting them. Secondly, it's worth pointing up front that (again unless you are the technician) that it was probably unethical for the technician to have given you medical information about the boy. Medical information is by law, custom, and moral principle extremely private material. The technician's poor conduct in providing you with medical information about the child further calls his or her credibility into question. Thirdly,...

Is it ethical for surgeons to use economic considerations when setting their fees? For example, is it ethical for a surgeon who is known to have better results for a certain operation to charge more than a surgeon who has worse results? Likewise is it ethical for a surgeon who has a scarce skill in a region to charge exorbitant fees for that skill simply because it would be unaffordable for most patients to travel to another region to attend another surgeon?

This is a fascinating question because medical care is not a commodity like many others--for example televisions or ice cream. It is a service related to the most pround of human needs. For that reason, I answer your first question with a "no"--but with qualification. It really depends upon what you mean by "economic considerations." I think it would be wrong to use simple supply-and-demand considerations where the supplier (the surgeon) charged the highest price the market will bear. Why? Because higher prices will exclude those with less money from the service, and I don't think it morally defensible to distribute essential medical services on the basis of wealth. Moreover, people suffering from illness are not in a position to bargain for fees with medical providers in the absence of coercion. (Think of how little a surgeon would charge if he or she were at risk of dying if a prospective patient decided to seek care elsewhere.) For this reason, I find the American medical system on the whole...

Hi- I got this question from Harvard Econ. Prof. Greg Mankiw's blog. He got it from Richard Rorty. Here it is: "Aliens from another planet, with vastly superior intelligence to humans, land on earth in order to consume humans as food. What argument could you make to convince the aliens not to eat us that would not also apply to our consumption of beef?" What's the answer!?!?! Thanks!

It's a fine question, isn't it. Short, sweet, and deeply provocative. In the interests of full disclosure, however, I should, at the outset, let you know that I don't think we should eat beef--in part because of the sort of reasons this question elicits. That being said, I don't think that the claim the question seems to advance is by itself decisive--namely that it's human's superior intelligence that provides grounds for eating beef. After all, if minimal intelligence itself justified eating an organism, then humans with minimal intelligence (including the aged, those with brain injuries, infants and fetuses, the mentallly retarded, public officials, etc.) would be candidates for consumption, and various computers would have moral standing. But establishing moral standing isn't simply a matter of determining intelligence. Rather, I'd say that what principally (not exclusively) marks an entity as one not to be consumed is its sharing or its capacity to share (or have shared) in certain projects and...

Why are performance-enhancing drugs seen negatively for athletes, but no problem for musicians? Why do we worship The Beatles (big-time drug takers and their creativity amplified substantially through drug use) and attack Ben Johnson?

I think this is a fascinating question, one which will probably bounce around in my mind for a while. I can well imagine music companies, for example, writing recording contracts only for musicians who pass drug tests. But I do think there are a couple of relevant differences between musicians and athletes concerning performance enhancing drugs. (1) The nature of the competition in music is not as exclusive. And (2) the extent to which drugs enhance rather than undermine performance is clearer in sports than in music. You see in a running race or playing a match, there can be only one winner. The victory of one implies the defeat of another. In music, by contrast, many musicians can be successful, and it's not clear that the success of one prevents the success of others. Many records can go gold. Now, I'll grant you, in music sometimes success is exclusive. Only one person can be first violin of the New York Philharmonic. Only one performer can win the Grammy in a given year. It's in cases like...

1.) Would you label free market/vanilla capitalism (however you choose to answer) as moral or immoral? Now, is it more or less moral in comparison to its alternatives, such as socialism and all its variations? 2.) Is it moral/immoral to infringe upon property rights in the name of the "common good"? Thank you for your time. I am a student with a strong interest in governmental philosophy and appreciate the concise answers your website provides. Sincerely, Alexander C. R.

Okay, here's a concise answer: Immoral, unless certain predictable consequences are mitigated or corrected. Why? Well, it depends precisely what you mean by capitalism. If one takes a pure form of market capitalism, I'd say it's immoral because it makes no evaluative judgments that take into account the distribution of goods and services or costs and benefits, the suffering of various actors, or considerations of flourishing. Says the radical capitalist: whatever outcome the market produces is okay. I disagree. I think that because of (1) the profound importance in human life of the issues and (2) the intimate connection between (a) economic matters like distribution and (b) moral matters like need, fairness, flourishing, justice etc. the operations of a capitalist economy must be regulated, guided, and directed to produce morally desirable outcomes. In particular moral considerations should guide economic activity to produce outcomes that are fair, moderate and equitable, outcomes that promote...

How can anybody, including myself, be sure that what is seen is real? My right eye was scratched, and I can see this scratch-mark before "reality", as one would see their right hand before their left if they arranged the two that way. I wonder if this proves the external to be an actual place within something (the universe?), like it has an absolute position within my (a sentient being) perception. This brings me to my final question: How can I prove the distance between my two hands? When I look at my right hand in front of my left hand, I see them as two objects apart from each other, but I sometimes see a flat picture, like a movie screen: it is manifestly flat but produces 3-dimensional pictures. Does this mean that my eyes create reality to be other than what it is, like how they create depth to be where it really is not? Or does this mean that my eyes are perceiving reality as it should be perceived? Ugh! And the thought that those who cannot "see things" in ink-blots on white paper have learning...

Yeah, these are the kinds of questions that lead many of us to "Argggh!" They're also the kind of questions that I approach with a great deal of trepidation because they are knottier than knotty. So, please understand that what I say here by the nature of this kind of exposition will be very rough and overly simplistic. You'll also probably find more than a few of my colleauges to disagree. But anyway, let's barrel right on with it. I don't think you can be absolutely sure that what you see is what's "real"--though you really ought to take some time to parse out what you mean by that word because it's LOADED. I take it that you mean by "real" something like what's out there independently of us. In a sense, actually, my best shot is that what you or we see isn't exactly real in that sense. Do remember that old Aristotelian question, "When a tree falls in the forest and there's no one around to listen, does it make a sound?" Well . . . get ready . . . almost yes, but no. No in the way that...

This is a question about the pertinence and legitimacy of the approach towards contemporary philosophy. Increasingly it seems that philosophy has become divorced from common culture, which is sad as the subject has offered so much insight on, and for the sake of, society throughout the ages. Since the advent of the 'new realism' philosophers do not, as I understand, attempt to build systems of philosophy but rather try to answer small and well-defined questions with consistency and through giving a justification for their own notions. However, there seems to be several problems with this approach which I will present: (a) One can be consistently false. In particular, if one focuses on small questions, chances are one is just not including anything within the remit which will challenge one's argument. (b) If one begins from the standpoint of one's own intuitive notions, this is effectively reinforcing one's own opinion and bias. If two people give an argument justifying their opinion, this will not...

I have often found the appeal to intuitions, unsatisfying and sloppy. But I'm not sure it's always so, especially in cases where the intuition is widely shared, or anyway shared by the audience or readership. In that case, it is true that the intuition itself lacks scrutiny, but I have my doubts that we can ever get to the sort of bedrock Absolutes you describe. In fact, even with an appeal to the Absolute, I don't see any way one can be sure one isn't consistently supporting a delusion--since one's appeal to the Absolute might be erroneous. In fact, the many different Absolutes that people have promote today demonstrates that most and perhaps all appeals to the Absolute comprise systems, perhaps even consistent systems, of falsehood. And in any case, is there any difference between calling "p" an Absolute Truth from saying simply, "I believe p very strongly"? I'm inclined to say that appeals to Absolutes and articles of faith impede philosophical discussion by placing certain ideas beyond...

Today in English class we were shown a list of "moral developments" that seemed to progress linearly - how people determine what is moral when they are 5, and how they determine this when they are 40. At lunch, my friend said, "I think it is silly to say there are developments of morality". I replied, "No, they were not developments of morality, but developments how we DETERMINE what is moral. You cannot develop morality because there IS only one true answer to what is moral and what isn't. The list was just showing how people differ in the way they DETERMINE whether something is moral or not." My friend replied that there is NOT only one true answer to what is moral and what is not - that everyone has "his/her own" set of moral values, and there is not any set that is more correct than another, that I was just biased for thinking so. (In other words, she claims that although murder might seem immoral to me, this does not mean that is IS immoral, only that is is immoral by my moral standards. Nothing...

You and your friend have articulated extremely well a philosophical problem that's been debated for thousands of years. Some of my favorite ancient places to think about the question are Plato's Republic and Gorgias and Cicero's De Finibus . I'm afraid I must tell you, however, that the matter really isn't settled--though I do think we're a bit more sophisticated today in working through the alternatives. Some philosophers think there is an objective truth to morality--that morals is somehow grounded in elements of the world independent of our subjective feelings. There are various candidates for this kind of grounding--the divine, nature, language, even the idea that 'goodness' is an objective property of conduct. Others think that the essential element, or at least a necessary condition, of morality is subjective. A couple of things you might consider of the debate you're having with your friend: First, the idea that morality is grounded in something subjective, entirely or just in...

Do we "see" black objects in the same sense that we see objects of other colors? Black objects being those which reflect no light, how is looking at a black object different than closing your eyes (it seems absurd to say that we see anything with our eyes closed); in either case, no light reflects from the object to our eyes. If I have a white piece of paper with a black spot on it, do I "see" the spot, or do I infer it?

This question reminds me of an experience I had going to rent a tuxedo. I told the clerk I wanted a black tuxedo, and he responded with the question, "What shade of black?" I suppose the answer here depends upon what you mean by "see" and by "black." I'm inclined to think that all "seeing" of objects involves a kind of inference or judgment. That is one judges what one sees to be an object. Does it really matter whether the physical cause of what one sees and hence the basis of judgment is light or the absence of light? I'm not sure I see why. What my haberdasher taught me, however, should also be said. That when we see some "black" object, we really don't see utter colorlessness. We see all kinds of shades of gray, etc., so much so that one might argue that we never really see a purely black object (unless one is staring a black hole, I suppose, which perhaps isn't really an object, anyway). We also see boundaries with other hues, as well as shadows and patterns of motion and interaction...