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This is more like a comment to the question in Mathematics that starts with: "If you have a line, and it goes on forever, and you choose a random point on that line, is that point the center of that line? And if you ..." The answer provided by the panelist, as well as the initial question, assume that one can distinguish between points at infinity. As far as Math goes however, one cannot do that, and this is the reason the limit for cos(phi) does not exist, as phi goes to infinity. Revisiting the argumentation provided by the panelist, the error starts with the 'definition' of the distance between a fixed point and infinity - this distance cannot be defined, and therefore it cannot be compared (at least, as math goes). A somewhat similar problem can be stated, without the pitfalls of the infinity concept, for a point on a circle, or any closed curve.

It seems to me that you are reading things into the original question, and my answer to it, that were not there. I do not see, either in the original question or in my answer, any reference to "points at infinity". The orignal question talks about a line going on forever, and my answer talks about the line extending infinitely far in either direction from some point P on the line. But this just means that for every number x, there are points on the line more than x units away from P in either direction, not that there are points that are infinitely far away from P. I claimed that the parts of the line on either side of P are congruent, and you can see this by observing that if you rotate the line 180 degrees around P, each side gets moved so that it coincides with the other side.

My previous answer was based on a particular definition of "center". There is another, slightly different definition of "center" that could lead to the sorts of worries that you raise. Suppose we define the center point to be the point that is equidistant from the endpoints. This works fine for a finite line segment, and leads to exactly the same center as the definition I originally proposed. But for an infinite line, if you tried to apply this definition then you would, indeed, find yourself looking for endpoints of the line--points at infinity--and you would find yourself trying to compute the distances from those points at infinity to other points. So this definition of "center" would lead to the sorts of worries that you raise, but it is not the definition I was using in my previous post.

For what reason should beliefs of others be honored or respected? That is to say, if something I say makes another uncomfortable because of their belief, what reason do I have to not say it? I have heard many times people say, "Don't say that, it will insult people because of their beliefs." Given this reason, if there were a person who was deeply insulted by the word "is" in any conjugation I would have to really tip toe around any speech! I suppose I am talking mostly about religious/superstitious belief. There doesn't seem to be any reason to respect beliefs in this regard when the belief may or may not be true. My second question: Did I just answer my own question?

Do people's beliefs deserve our respect? I'm not sure what this would mean. I think that often what people who offer this sort of advice mean is that one should be respectful to other people whose beliefs are different from one's own. But I don't think that a respectful attitude toward others requires us to pretend that we don't disagree or to refrain from saying anything that might lead them to question their beliefs. It's true that I can disrespectfully disagree. I can be condescending, abrasive, ordismissive. But equally, I think, I could disrespectfully agree (or at least not voice my disagreement). I might think that the other person is so irrational that neither of us could profit from a discussion of the basis for our disagreement.

This is a follow up on http://www.amherst.edu/askphilosophers/question/51, whether the mind can understand how the mind works. In Alexander George's response, he said, "it 'follows' from Gödel's result that there is some basic fact about our minds that we cannot ever know, that we could not in principle access." But is that fact necessarily about the how the mind works, or could it be about some other aspect of mind? As a second question, if we were told what it was, we might not be able to prove it for ourselves, but what would keep us from understanding it in its stated form?

Very loosely and given all the assumptions of my original response, the "basic fact about our minds" in question is the fact that the rules that constitute our minds do not produce conflicting results. Is that a fact "about how the mind works" or "about some other aspect of mind"? That's too vague a question to answer, I think. The fact in question is (given all the assumptions, etc.) a basic property that our minds possess but one that we could not know that they possess. We could "understand" this property, in the sense that we could formulate the claim that says that our minds possess that property. But our minds would not have the means to establish that the claim is true.

Is it possible to deify an object, perhaps a penguin? If so, what qualities and/or properties would make it godlike? D.D.

In Chapter XII of Leviathan, Hobbes says that "there is almost nothing that has a name that has not been esteemed...in one place or another, a god or a devil....Men, women, a bird, a crocodile, a calf, a dog, a snake, an onion, a leek, [were] deified." Hobbes would probably say that somewhere, already, penguins have been deified. So it is certainly possible to deify a penguin.

The question is whether one would be justified in deifying a penguin. Hobbes--and most Christians--would say no, because only a being with all the attributes of the Christian God (omniscience, omnipotence, etc.) is justly worshipped, but no finite being has such attributes, and consequently, no finite being ought to be worshipped.

Perhaps other religious traditions would allow one to deify, and hence worship, a penguin. But I'm not familiar enough with other religious traditions to say.

Why are there no bad color combinations in nature? Colors in nature never clash. Why not?

Is this so? I don't know (I'm color blind). But assuming it is, shouldwe reach for evolution to explain this (I realize you didn't suggest this)? Somehow, one might hold,evolutionary pressures shaped our aesthetic sensibilities (for I takeit that that's what judgments about color clashes amount to) in justsuch a way that we find all color arrangements in nature pleasing. Butwhat possible evolutionary advantage would there be to that? In fact,you might have expected that there would be an evolutionary advantageto finding certain color combinations repellent: e.g., thatit would promote our survival if we were to judge that the colors ofthat dangerous carnivore's coat "just don't work together". Also,aesthetic judgments are, well, judgments. And as such, theyare sensitive to the rational give and take of reflection, experience,imagination, and so on. The judgment that these colors clash is not abrute, immutable reaction, but rather a judgment that is sensitive tomany considerations and hence beyond the reach of evolutionary shaping.Does this last thought point to a more fruitful direction ofexplanation? Might it be that our aesthetic judgments are shaped inpart by our experiences and hence it's no wonder that we judge thecolor combinations found in nature to be harmonious?

The notion of "free will" implies an agent can make its own choice independent of the deterministic laws of nature. However, within a causally closed system this is impossible. Why then would evolution endow agents with the feeling of control? Would it not be more efficient (and more expected) for evolution to produce automata without subjective (and superfluous) mental phenomena?

One way to respond to this question is to reconceive the notion of control at issue. Rather than accepting that the control that agents feel they have requires that they be able to make choices independent of the laws of nature, one might argue that all the control that agents need in order to be responsible for their choices is for their choices to be sensitive to their reasoning. On this conception of freedom, an agent would be responsible for her choices, and have control of them, because she chose for a reason. Rational control of this sort seems eminently compatible with determinism.

On such an account, the feeling of control that one has is not taken to indicate that one is exempt from the laws of nature, but rather reflects the fact that one's choices are up to one because they reflect one's reasons.

Critical thinking: We are bombarded with information all the time so I think it's very important to use "critical thinking" but it's not easy. So my question is: what are the basics in critical thinking?

I think it is also useful to think about the separate skills that are necessary for applying the concepts and techniques that Joseph described to complex real-life situations.

Alas, we often have the most need for critical thinking when confronting the situations where this is the hardest to do: situations that are really complex, that matter a lot to our lives, which involve complex emotional dynamics or serious interpersonal conflict, and so on.

So, to best use critical thinking in our own lives we need to be able to handle "messy" situations like those. For example, it is useful to understand your own and others' agendas and motivations, the emotional dynamics of a situation, how to act in ways that have the best chance of making a difficult discussion more rational and more constructive.

When I teach critical thinking, I prefer to teach the reasoning skills that Joseph describes together with serious reflection on the "messy real life" issues that I sketch out above. The best critical thinking text I've seen that does this is Douglas Walton's Informal Logic, which adopts an interesting dialogical approach (Amazon.Com link is here). Walton’s text is difficult for my students at first, but most come to love it and over the years many have told me that they have purchased copies for spouses, children, or parents.

I am married to a man who earns a considerable amount of money doing a job he enjoys. It is possible for me to earn a similar amount of money, but I now feel considerable discomfort in the profession that over the years has allowed me to do so. My preferred work (writing fiction) will bring in no money in the short term and has little chance of a lot of money in the end. Is it ethical to choose to earn less, and yet to share the rewards of my husband's salary - the big house, nice car, holidays and so on?

I would have thought that this sort of decision was one to be made by you and your husband jointly. How your household earns its income is for the two of you to decide, and it is also for the two of you to decide how that income will be used. I can well understand that there might be emotional costs to writing full-time and so not making much (if any) money doing it. The concerns you express already reveal some of those costs. But they can be mitigated in various ways. If your husband were to be truly supportive of your writing and believed in what you were doing, that would presumably go a long way. You could think of yourselves as investing the income you could otherwise earn in your writing. Perhaps that will pay off financially and perhaps it will not. But perhaps there are more important things than money, and you are really investing in them.

ID theorists and creationists like to say that the Theory of Evolution is "just a theory." Is that true? What does that mean? What's the difference between "truth" and "theory"?

Theories are descriptions, and they come in two flavors: true and false. So the Theory of Evolution can be both a theory and true, which is just what a great number of scientists believe. When evolution by natural selection is called a theory, however, this is sometimes intended to emphasise that there is no proof that it is true. Now if by 'proof' we mean what pure mathematicians produce, then this is correct. There is no proof of the Theory of Evolution, and there is no proof of any other empirical theory either. Proof in this sense is not an option in science, because all theories go beyond the evidence upon which they are based. There can similarly be no proof that the sun will rise tomorrow. But the sense in which it is true that there is no proof of evolution is compatible with the claim that there is overwhelming evidence that it is true, which is again what a great number of scientists believe.

If science (i.e. evolutionary psychology) can explain why I have the morality I do, does that mean morality is subjective? If what I believe about morality is just a product of my evolution and my upbringing, can I still expect other people to live up to my principles even though they may have had a different upbringing? What about myself? Can I still hold myself to my own standards or am I being deceived by my evolution into thinking it would be wrong to do so?

It might be helpful to follow a strand of British empiricism and to think about 'morality' as a social phenomenon, involving various 'sanctions' such as blame, guilt, shame, and so on. (So in that respect it is rather like law, though the sanctions there are somewhat different.) Your worry is that some moral principle you accept -- that it's wrong to cause serious suffering merely for fun, say -- has emerged only because of the evolutionary advantage conferred on groups which accept something like that principle. So it seems quite contingent which principles we come to believe -- as you imply, in different circumstances we might accept different principles. But, to pick up Alex's point, we have the capacity to stand back from our 'morality' and assess whether we have independent reason to accept its principles. In which case, if you believe there is a reason not to cause suffering for fun, you may think that this justifies the moral principle which forbids it (as it would also justify a law forbidding it).

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