Logically what is the difference between conceivable and probable or possible?

Sometimes people use the word “possible” to mean something like, “does not entail a contradiction.” This sense of “possible” is narrower than “consistent with the laws of nature.” I gather this is what Thomas Pogge was alluding to when he said that there is a sense of “possible” that coincides with “conceivable.” Some have raised worries about whether conceivability implies possibility in this more narrow sense. Here’s an interesting example from Barry Stroud’s book on Hume (p. 50). Goldbach’s Conjecture states that every even number is the sum of two primes. At this point (as far as I know) no one has offered a proof of Goldbach’s Conjecture, and no one has disproved it either. If Goldbach’s Conjecture is false, it presumably says something that is not possible (in the narrow sense). But one might argue that I can nevertheless conceive of a state of affairs in which someone proves that Goldbach’s Conjecture is true. In that case, it seems, I would have conceived of something impossible. ...

Ernst Mach asserted the the world consists entirely of sensations. Does this make him a solipsist, and how might one refute him?

I'm not an authority on Mach, but (as Peter Lipton suggests) some of the philosophers who believe that “the world consists entirely of sensations” do not think they are thereby committed to solipsism. One obstacle for this view is how to account for the possibility that more than one mind can perceive the same object. When I see a table, on this view, the table is just a collection of sensations in my mind. When you see a table, it’s a collection of sensations in your mind. So how is it possible for us to see the same table?

Why do people participate in meaningless activities such as politics, education, mathematics, philosophy and such when either we are all going to die so it won't matter what we have done, or maybe our existence and/or this world is all an illusion so it doesn't matter what happens because it's not real?

I'm not sure I understand what it is for something to "matter", if it isn't shorthand for mattering to someone or other. As Richard Heck points out, something a person does might matter to other people after her death. And certain things I do matter very much to me right now–or will matter to me later on–even though I know that I will die someday. Perhaps there will come a time when there's no one left in the world—at that point, "nothing will matter" in the sense that there will no longer be anyone to whom anything could matter. But it doesn't follow that the things we do now matter very much (to ourselves or to our descendents). I also don't see how the possibility that "the world is all an illusion" should affect the issue. Perhaps if this possibility were actual, then many of the things that I assume will matter to other people will not really matter, since in that case no other people would exist. (Though there may still be things that matter to me.) For example, one might argue that...

What is the current philosophical viewpoint (from professional academics) regarding the concept of "the embodied mind"? I have just finished rereading "Philosophy in the Flesh" (Lakoff, Johnson); I would like to know the current philosophical standpoint regarding the proposition of the embodied mind. Thanks in advance for all replies!

I wouldn't say that there is a "current philosophical standpoint" about the idea of the "embodied mind," but I do think it's something that many philosophers of mind are sympathetic to these days. Here are two recommendations for further reading: 1) "Being There: Putting Brain, Body and World Together Again," by Andy Clark 2) "Action in Perception," by Alva Noe.

I can see images and hear sounds inside my head at command. How is our mind able to perceive these things without them being real? I can create whatever image I want, and recall sounds, but I don't understand where or how this information is stored in the brain, and how we can see or hear it.

It might help to distinguish between imagination and perception. There's a sense in which what I perceive is not up to me. Although I can control what I see by turning my head, opening or closing my eyes, or taking off my glasses, once I've done those kinds of things, what I see as a result is out of my control. If someone orders me to see an elephant in the middle of my living room, it's not clear how I could carry out that command (aside from calling the zoo and asking if they deliver). On the other hand, I can easily imagine an elephant in my living room. Though as Peter Lipton mentioned, some philosophers (e.g., Locke, Hume) thought that what we can imagine is limited to what we can cobble together from materials acquired through perception.