It seems to me that there are two kinds of numbers: the kind that the concept of which we can grasp by imagining a case that instantiates the concept, and the kind that we cannot imagine. For example, we can grasp the concept of 1 by imagining one object. The same goes for 2, 3, 0.5 or 0, and pretty much all the most common numbers. But there is this second kind that we cannot imagine. For example, i (square root of -1) or '532,740,029'. It seems to me that nobody can really imagine what 532,740,029 objects or i object(you see, I don't even know whether I should put 'object' or 'objects' here or not because I don't know whether i is single or plural; I don't know what i is) are like. So, Q1) if I cannot imagine a case that instantiates concepts like '532,740,029', do I really know the concept, and if so, how do I know the concept? Q2) is there a fundamental difference between numbers whose instances I can imagine and those I cannot? (I lead towards there is no difference, but I don't know how to account...

If you had a child to make yourself happy, as most people do, would that violate the Kantian imperative to avoid treating people as means?

I the Koran subject to interpretation or to be taken literally?

Does one have to be aware that one is exercising one's free will, in order to have free will?

Is it morally acceptable to hate a crime but not the criminal?

When I read most discussions about free will, it seems that there is an implicit unspoken assumption that might not be accurate once it is brought forward and addressed explicitly. We know from research (and for me, from some personal experiences) that we make decisions before we are consciously aware that we have made that decision. The discussions about free will all seem to assume that one of the necessary conditions of free will is that we be aware that we are exercising it, in order to have it. (sorry if I did not phrase that very well). In other words, if we are not consciously aware that we are exercising free will in the moment that we are making a decision, then it is assumed that we do not have free will, merely because of that absence of conscious awareness. Suppose we do have free will, and we exercise it without being consciously aware that we are doing so at that particular moment. That might merely be an artifact that either we are using our awareness to do something that requires...

Lately, I have been hearing many arguments of the form: A is better than B, therefore A should be more like B. This is despite B being considered the less desirable option (often by the one posing the argument). For example: The poor in our country have plenty of food and places to live. In other countries, the poor go hungry and have little to no shelter. It is then implied that the poor in our country should go hungry and have little to no shelter. I was thinking this was a fallacy of suppressed correlative, but that doesn't quite seem to fit. What is the error or fallacy in this form of argument? How might one refute such an argument?

A lot of philosophy seems to be "philosophy of x" -- philosophy of science, philosophy of language, philosophy of mathematics, etc. Given this, should philosophy, institutionally speaking, be treated as a separate discipline at all? I mean, why couldn't the various philosophies of x be absorbed into the various types of x?

Why did all the ancient philosophers seem so fascinated by astronomy? Their interest in math and "physics" is understandable, as math can be seen as very similar to certain branches of philosophy in that it is not the study of a particular existence, but, rather, the study of "existence," and physics is the study of the seemingly occult laws that govern everything, which is also very similar to philosophy in a sense, but astronomy is just the extrapolation of those two fields on "arbitrarily chosen" pieces of mass. Math, and even physics to a large extent, are "implicit" (for lack of better term) to existence, while astronomy is wholly explicit.

Humans can apparently commit to beliefs that are ultimately contradictory or incompatible. For instance, the one person, unless they're shown a reason to think otherwise, could believe that both quantum mechanics and relativity correspond to reality. What I wanted to ask is -- the ability to hold contradictory beliefs might sometimes be an advantage; for instance, both lines of inquiry could be pursued simultaneously. Is this an advantage that only organic brains have? Is there any good reason a computer couldn't be designed to hold, and act on, contradictory beliefs?