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,...

In reply to a recent question about whether aesthetic judgments are reliable Stephen Maitzen wrote "(1) We often seem to make objective aesthetic judgments, such as the judgments concerning Bach and Rihanna that you mentioned in your question; why not take those judgments at face value? Why think we have to interpret those judgments as non-objective?" Often we (or some of us) feel that the aesthetic value of a work derives from an ontological sense that the music represents, expresses or even manifests a higher reality. We don't take Rhianna very seriously as a great artist because her music doesn't seem to convey anything of profound importance. We can feel that way even if we happen to enjoy her music a lot. If we listen to Suite Number 3 in D Major by Bach we might feel that the music conveys something grand but we can't say for certain what. It's that lack of certainty about what is conveyed by the music that I think makes people question the validity of...

Thanks for your reply. As I did in my previous answer , let me emphasize that aesthetics isn't my specialty, so I hope specialists will come forward to answer your questions. I'm not sure what to say about the idea that a musical work "conveys something grand" or "manifests a higher reality" than what's manifested by another musical work. So I'll leave that to others to address. But we might just compare Bach and Rihanna in terms of the harmonic and rhythmic complexity of their music; their inventiveness in developing a theme during the course of a piece; their skill in writing for various instruments; whether they incorporate enough surprise in a piece to maintain our interest yet not so much that the piece lacks integrity; and so on. Pop music almost always strikes me as very simple music -- it's often more "ear candy" than something having subtle flavors -- which may explain its mass appeal. Now, it's probably unfair to compare Rihanna to Bach, because by definition Bach's music has stood the...

Are mathematical truths such as 2+2 =4 arguable exceptions to the correspondence theory of truth? I mean is 2+2=4 a truth that corresponds to "the world"?

I don't think mathematical truths pose a special problem for the correspondence theory of truth (see this link for more about the theory). The correspondence theorist can interpret "the world" broadly enough to include abstract objects, aspects of mathematical reality, and so on. In other words, "the world" needn't be restricted to the physical universe.

I am really fascinated with Hume's discovery that an "ought" cannot be derived from an "is." However, I've also read that the argument of Hume is a failure. My question then is, what can be the most reasonable response to this accusation of Hume? Is he right or wrong on the matter?

I prefer to think of it as Hume's claim rather than Hume's discovery, since "discovery" implies the truth of what's discovered, and I think Hume was wrong, at least on what seems to me the most natural interpretation of what he says in the Treatise of Human Nature . But the interpretation is part of the problem; scholars disagree on what Hume meant. There's a magazine article on this topic, written by one of Hume's defenders, at this link . There's also a recent collection of essays, Hume on Is and Ought (Palgrave Macmillan, 2010), that goes into minute detail on the interpretation and evaluation of Hume's claim.

Can a thing being distinct from something else be considered a property of that thing? (If my mind is distinct from my body can "being distinct from my body" be considered a property of my mind. It seems to me that if something is distinct from something else it is separate from it and therefore cannot somehow be considered a property of it. But I have a feeling I am missing something. Thank you Samantha R.

Thanks, Samantha, for your question. You wrote, "It seems to me that if something is distinct from something else it is separate from it and therefore considered a property of it." But notice that in the typical case -- and certainly in all concrete cases -- an object is distinct from each of its properties. Any red ball is distinct from the properties being red , being a ball (etc.) that the ball instantiates: the ball is a material object, but its properties are abstract objects rather than material objects, so they must be distinct from the ball. So if being distinct from your body is a property of your mind, it will be distinct from your mind. As I see it, the properties of an object are never parts of the object, so they can be (as you say) separate from the object while still being properties of the object.