Everything needs a cause, right, or it couldn't happen, right? But, if everything needs a cause, how could anything happen? Because the thing that would cause it to happen would also need a cause. So does that means the universe can't happen/could never get to now? Or is time a cause in and of itself? And "drags" things as time goes forward, like a replay in a video game? But then time would need a cause too, right?

Many quantum physicists say that lots of events occur without being caused to occur. But let's assume that they're wrong and that every event needs a cause. One way to answer your challenge is to allow for an infinite regress of contingent events: a series of events stretching back endlessly in which no member is logically or metaphysically required to happen. I don't see what's wrong, in principle, with an infinite regress of events. One might reject such a regress on the grounds that "time couldn't stretch back forever," but I see no good reason to say that it couldn't. But even if time couldn't stretch back forever, you can still squeeze infinitely many events into a finite time if they "telescope" so that the time between them decreases geometrically as you go back. We needn't treat time itself as a cause in any of this. Indeed, if (as almost all philosophers have held) some events are contingent, and if every event has a sufficient explanation why it occurred rather than not, then an...

Can paradoxes actually happen?

Yes! But bear in mind that a paradox is an apparent contradiction, an apparent inconsistency, that we're tasked with trying to resolve in a consistent way. For example, a particular argument implies that the Liar sentence ("This sentence is false") is both true and false, and a similar argument implies that the Strengthened Liar sentence ("This sentence is not true") is both true and not true. Usually it's our conviction that those arguments can't be sound that impels us to seek out the flaw in each argument. So too for other famous paradoxes, such as the Paradox of the Heap. Paradoxes abound! But that doesn't mean that contradictory situations do. Now, some philosophers, such as Graham Priest, say it's a mistake to demand a consistent solution to every paradox. Priest says that the Liar Paradox has an inconsistent solution, i.e., the Liar sentence is both true and false: it's both true and a contradiction. So Priest would say that not only do paradoxes actually occur but inconsistent...

Is it rational to believe that some of my beliefs are false? This seems like a reasonable claim. After all, most people have some false beliefs, and I know that I've had plenty of beliefs in the past which I later learned were false. On the other hand, I obviously believe that each of my beliefs is true (otherwise, they wouldn't be my beliefs). So how could I also believe that some unspecified beliefs among them are false?

It certainly looks like the height of rationality for you to believe that at least some of your beliefs are false. Yet, as you point out, there's no particular belief of yours that you regard as false. Any given belief of yours you regard as true; otherwise, it wouldn't be a belief of yours. This pair of attitudes gives rise to what's usually called the "Paradox of the Preface." One place to start looking is the SEP entry on "Epistemic Paradoxes," available here , which contains both discussion and references.

Would it be logically coherent to have a world in which everything that happened was bad or have a world in which everything that happen was good? Can good and evil exist independently of each other? Do we need one to define or contrast the other? Can each of them be definable in their own right? Is there any arguments that can be put forth to show that good and evil are not polar concepts?

You've raised a large and complex set of issues. I'll address just one part of one of your questions. It seems to me that the burden of proof rests with whoever claims that good can't possibly exist without evil. For one thing, the claim implies that the monotheistic God is impossible, since God is supposed to be perfectly good and independent of anything distinct from himself (or at any rate independent of evil). Moreover, it's not as if every property is instantiated only if its complement is instantiated. The property of being self-identical is instantiated by everything, but necessarily its complement, the property of being self-distinct, is instantiated by nothing. The property of being physical is instantiated by many things, but it's at least controversial whether the property of being nonphysical is instantiated at all.

In mathematics numbers are abstract notions. But when we divide number say we do 1 divided by 2 i.e. ½ does this mean abstract notions are divisible. It gives me a feeling like abstract notions have magnitude but then it comes to my mind that abstract has no magnitude.1=1/2 + 1/2 can we say the abstract notion 1 is equal to the sum of two equal half abstract notions? How should I conceptualize the division? The other part related to abstract notion is that how is the abstract notion of number 1 different from the unit cm? how can we say that the unit cm is abstract when we consider it a definite length. How is the unit apple different from unit cm if I count apples and measure length respectively? I am in a fix kindly help me to sort out this. I will be highly- highly grateful to you.

You asked, "Does this mean that [these particular] abstract notions are divisible?" I'd say yes . But that doesn't mean they're physically divisible; instead, they're numerically divisible. Abstract objects have no physical magnitude, but that doesn't mean they can't have numerical magnitude. The key is not to insist that all addition, subtraction, division, etc., must be physical. I'd say that the number 1 (an abstract object) is different from the cm (a unit of measure) in that the cm depends for its existence on the existence of a physical metric standard: for example, a metal bar housed in Paris or the distance traveled by light in a particular fraction of a second (where "second" is defined in terms of the radiation of a particular isotope of some element). In a universe with no physical standards, there's no such thing as the cm and nothing has any length in cm. By contrast, the number 1 doesn't depend for its existence on anything physical. Apples are physical, material objects. Units...

How would a philosopher of math describe what happened when ancient mathematicians discovered (?) the number zero?

I think the answer will depend on which philosopher of math you ask. As you seem to recognize, some philosophers of math deny that numbers exist independently of us in such a way that their existence is genuinely discovered by us. Even philosophers of math who think that numbers are discovered might say that your question -- "What happened?" -- is an empirical historical or psychological question rather than a philosophical one. In any case, you'll find relevant material in the SEP entry on "Philosophy of Mathematics" at this link .

Does a point in geometry (cartesian and euclidean) occupy space or have volume (if we consider 3-D geometry)? And is a line segment always perpendicular to its point of origin? Or can we frame this as, is a line perpendicular to each and every point lying on it?

As I understand the theory, an individual point in geometry has no extension and no volume; it's in space but doesn't occupy space in the sense of taking up a nonzero amount of space. Being perpendicular is a relation between lines (or line segments) rather than a relation between a line (or a line segment) and a point. A point can't be perpendicular to anything. At any rate, there's no more reason to say that a line is perpendicular to each point lying on it than to say that it's parallel to each point lying on it. I think it's neither.

What does it mean when a certain axiom is neither provable nor deniable? Does it imply that such axiom is self-evident and can't be doubted? I don't think that "real skeptics"(a skeptic who is so deep in doubt that he doubts his own existence and even his own doubt) like Pyrrho would be happy with that.

Let's consider, for example, what philosopher Hilary Putnam has called "the minimal principle of contradiction": (MPC) Not every contradiction is true. Arguably, MPC is unprovable because whichever premises and inference rules we might use to try to prove MPC are no better-known by us, and no more securely correct, than MPC itself is. But MPC would also appear to be undeniable, since in standard logic to deny MPC is to imply that every contradiction is true, and it's hard (for me, anyway) to make any sense of the notion of denying something in circumstances in which every contradiction is true. So, arguably, MPC is self-evident and can't be doubted: that is, the notion of MPC' s being doubted makes no sense. You suggest that this result would bother...

They say that relativism can not be affirmed without contradiction because to do so would imply that relativism had truth in an absolute sense. Is this simply an oversimplification or a strawman?

I suspect that one can affirm relativism without contradiction provided one is willing to embrace an endless regress . One can affirm the following statements: (R1) No statement is true except relative to some perspective (or worldview, or standard, or set of assumptions, or conceptual scheme). (R2) Statement R1 is true, but only relative to some perspective (or worldview, or standard, or set of assumptions, or conceptual scheme). (R3) Statement R2 is true, but only relative to some perspective (or worldview, or standard, or set of assumptions, or conceptual scheme). ...and so on without end. The endless regress allows one to postpone indefinitely any commitment to a non-relative truth. To be fair, however, one might wonder whether such a position has any cognitive content and, even if it does, whether our finite minds can truly understand such a position. For more, you might consult the detailed SEP entry on relativism available at this link .

Is length an intrinsic property or is it something which is only relative to other lengths? Is an inch an inch? Or is it simply a relation between other (length) phenomena?

Interesting questions. As I understand it, special relativity in physics says that having a particular length isn't intrinsic to an object, because observers in various "inertial frames of reference" can measure different values for the length of an object without any of them being mistaken: the length of an object is always relative to an inertial frame, and no inertial frame is objectively more correct than any other. As for units of length such as an inch, I'm inclined to say that they're always relative to some physical standard, whether the standard is a single physical object such as a platinum bar or, instead, some physical phenomenon like the path traveled by light in a given period of time (with units of time also being physically defined). In a universe containing no physical standard that defines an inch, nothing has any length in inches even if things have lengths in (say) centimeters when a physical standard exists for the centimeter. I hesitate a bit in holding this position,...