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Philosophy

Why do I feel stupid when confronted with questions about Philosophy and yet, I'm strangely attracted to it? Am I a masochist or someone who doesn't know better? Cheers! Victor

Of course I can't say what's up with you, Victor, since I don't know you, but I can report that the questions in philosophy often have that dual effect on people. On the one hand, they are utterly seductive and mesmerizing. On the other hand, their elusiveness and apparent intractability can be painful at times. Some of the greatest of philosophers have felt the stupidity before philosophical problems that you report; for instance, the Austrian philosopher Ludwig Wittgenstein felt this very strongly and would frequently castigate himself — often before his dumbfounded students — for his stupidity. Perhaps you're right in your suggestion that these traits might be connected: for some people are most attracted by precisely that which remains out of their reach.
Mathematics

As a teacher of high school mathematics and a former student of philosophy, I try to merge the two to engage my students in meaningful conversations about the significance of some mathematical properties. Recently, however, I could not adequately defend the statement "a=a" as being necessary for our study of geometry when one student challenged "When is a never NOT equal to a?" What would you tell them? (One student did offer the defense that "Well, if we said a=2 and a=5 then a=a would be false, causing problems.")

I'm not sure whether you're asking (1) What role does the reflexivityof identity (i.e., every object is equal to itself) play in geometry?,or (2) What justification can be offered for the reflexivity ofidentity? As regards (1), I assume that in an explicit axiomatizationof geometry, there would be axioms dealing with identity. As Richardpoints out above, in such an axiomatic system we will want to derivetheorems in which "=" figures; so we had better have some axioms thattell us under what conditions identity statements can be proved. Asregards (2), I'm not sure what to advise you if a student is unwillingto grant that an object is identical to itself. I would infer that s/hedoesn't understand what "identical to" means and would treat the matter as a case of miscommunication.
Mind

It seems that most of my thoughts are expressed as reflections of familiar stimuli received through the agreed-upon 'five senses' (this includes spoken and written language). Is there any appropriate way to speculate on what form the thoughts of a hypothetical person born without access to sight, sound, smell, touch, or taste might take? I guess what I mean is: "please speculate!"

I take it that you mean to ask about the thoughts of an individual whohas been utterly deprived of sensory experience? Sadly, we likely knowthe answer to that question, for we know something about the mentallives of individuals who have been reared in very sensorily deprivedcircumstances. They are cognitively damaged, usually beyond repair. Itseems humans require (and require early enough in their development)rich experiences to feed the development of their mental lives. Absentthese, thoughts, and indeed anything like our familiar mental life,fail to develop.
Mathematics
Mind

How do we resolve the fact that our finite brains can conceive of mental spaces far more vast than the known physical universe and more numerous than all of the atoms? For example, the total possible state-space of a game of chess is well defined, finite, but much larger than the number of atoms in the universe (http://en.wikipedia.org/wiki/Shannon_number). Obviously, all of these states "exist" in some nebulous sense insofar as the rules of chess describe the boundaries of the possible space, and any particular instance within that space we conceive of is instantly manifest as soon as we think of it. But what is the nature of this existence, since it is equally obvious that the entire state-space can never actually be manifest simultaneously in our universe, as even the idea of a board position requires more than one atom to manifest that mental event? Yet through abstraction, we can casually refer to many such hyper-huge spaces. We can talk of infinite number ranges like the integers, and "bigger"...

To add a word or two to Dan's great response: there is no questionthatmathematics deals with infinite collections, but what those are, whatwe mean when we make claims about them, which claims are correct —these have been hotly disputed issues for thousands of years. (Inthe history of mathematics, concern for these foundational questionshas waxed and waned. There have been times, for instance in theearly part of the twentieth century, when disputes over these issues,were very heated and split the mathematical community. There have beenother times, for instance now, when mathematicians have been lessinterested in these issues — although of course there are alwaysexceptions, like Dan.) The basic question — what does it mean to call aset "infinite"? — is so fundamental that it's simply astounding that wedon't know how to answer it. Onone way of looking at the matter, what Dan called "platonism", to saythat a set is infinite is simply to have given a measure of its size.To say that a set is infinite is...
Biology
Ethics

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?

Perhaps it's also worth noting that beliefs aren't like reflexes.Evolution shaped us (I assume) to blink when an object rapidlyapproaches our eyes. No amount of reasoning, thought, or imagination isgoing to stop you from blinking. Beliefs aren't like that. We developthem, hold them, let them go, etc. — often on the basis of arguments orconsiderations that people offer us or that we offer ourselves. So evenif evolution inclined us initially toward certain moral beliefs, onemight still think that they are not hermetically sealed off fromreflection.
Philosophy

If you were sent back 100 years in time and met a fellow philosopher, what advances in the field since his or her time would you tell him or her of? Would you be able to convince him or her of what you said?

To my mind, the formulation, discussion, appreciation, and absorption of the work of Gottlob Frege, Ludwig Wittgenstein, W.V. Quine, Donald Davidson, and Saul Kripke have allowed for far deeper, sharper, and more sophisticated discussions in the philosophy of language than have ever been possible before. Could I convince someone of this after traveling back in time, you ask. Could I convince someone of that now ?
Knowledge
Mind

Are we directly aware of reality, or is what we "sense" merely a representation of reality?

This is a perennial and extremely vexing question about which there continues to be great debate. You might find this essay in the Stanford Encyclopedia of Philosophy to be of value.

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