We define the empty set as the set that contains no elements, but is there more than one empty set? So is there "an" empty set as opposed to "the" empty set? May one be able to receive values, while another is truly empty, etc.? And how is it possible to define the empty set by the absence of members or by emptiness?

The empty set is indeed defined to be that set which contains no elements. Another definition we need is that of identity of sets: we say that set A and set B are identical just in case they contain exactly the same elements, i.e., whatever is in A is also in B, and vice versa . So, with these two definitions in hand, consider empty set E 1 and empty set E 2 . Well, they are equal since any element that is in the one is in the other (for the trivial reason that neither set contains any elements). So there really is only one empty set - which is what licenses our use of the definite article "the" in " the empty set". I'm not sure I understand your last question. In set theory, you've specified a set completely when you've specified its elements. And when we say that the empty set contains nothing, we have indeed specified exactly which elements it contains (namely, none).

So, it's my understanding that Russell and Whitehead's project of logicism in the Principia Mathematica didn't work out. I understand that two reasons for this are (1) that some of their axioms don't seem to be derivable from pure logic and (2) Gödel's incompleteness theorems. However, particularly since symbolic logic and the philosophy of mathematics are not my area, it's hard for me to see how 1 & 2 work and defeat the project.

To follow on some of Richard's observations: I have never found it at all a compelling argument against logicism that it would have the existence of infinitely many natural numbers be a logical truth. That is not an argument against logicism so much as a restatement of the claim that it is incorrect. Richard's discussion of Boolos reminds me of Gödel's own caution with regard to what his Incompleteness Theorems establish with respect to Hilbert's Program (roughly, Hilbert's attempt to show that if a basic, or finitary, proposition of mathematics can be established using the powerful, or infinitary, methods of classical mathematics, then it could already have been established using very basic, or finitary, reasoning). [For more on Hilbert's Program, you might see here or here .] I don't myself think that the phenomenon of incompleteness puts paid to Hilbert's project (as divorced from certain other beliefs that Hilbert may have held, such as the belief that all true mathematical...

Can philosophy of mathematics influence mathematics, or it is just an abstraction of what actually works?

As Peter Smith's examples make clear, sometimes "the philosophy of mathematics" appears in other than philosophy journals and is done by other than people in philosophy departments. Another instance of this is the long current of constructivism in mathematics. The development by mathematicians of constructivist mathematics (most notably, intuitionistic mathematics) is often motivated by their -- for want of a better term -- philosophical reflection on the nature of mathematics.

Mathematics: Does a function in mathematics change anything. For instance: take the function ()+3. If the input is 2 and the output is 5 for this function, then is 5 'derived' from 2 and the function ()+3? Is the input 2 or the funtion ()+3 changed in any way? or is this strictly an assignment, i.e. 2 is assigned 5 3 is assigned 6, etc. Let's take a another example, If I change the color of an object, I really apply a function to the property of that object. For instance, say I have a red ball. I add some yellow and make the color of the ball orange. Have I changed the property of color or have I changed the ball? If I apply the function AddYellow() to the color of the ball, my input is red, AddYellow() is applied, and I get the result Orange. Is this a change or an assignment from red to orange. Specifically, does the value of the ball change or the ball itself because of the function assignment of the value of the ball. How can an assignment change anything?

One says "The value of the function f( x ) = x 2 changes with the value of x ", but nothing is actually changing. Perhaps you can compare it to our saying that the landscape changes as one drives along the road. (One difference though: trees and hills can change over time, but numbers and other mathematical entities cannot.) Mathematicians often look at functions as you suggest: as a collection of ordered pairs of objects <a, b> that satisfies the condition that if <a, b> and <a, c> are both in the collection, then b=c. These pairings are what you've called assignments. Functions are then just particular kinds of sets: sets that contain instructions about what is assigned to what. You could define a function f( t ) = the color of the ball at time t . It too could be viewed as a set of ordered pairs. But what such unchanging functions help us to describe is a physical object, the ball, that is changing. I wonder whether at the root of your question is the thought...

I’ve run into a problem in philosophy recently that I do not completely appreciate. Certain sets are said to be “too big” to be sets. In Lewis’ Modal Realism, the set of all possible worlds is said to be one such set. These are sets whose memberships is composed of infinite individuals of a robust cardinality. I (purportedly) understand that not all infinities are equal. But I don’t quite see why there can be a set of continuum many objects, but not a set of certain larger infinities. Am I misunderstanding what it is to have “too big” a set?

When a set theorist says that such and such collection is "too big" to be a set, what he typically means is that if that collection were taken to be a set a contradiction would arise. The collection of all sets is such a collection. If we assume it's a set then, applying the argument that generates Russell's Paradox, we arrive at a contradiction. And so we conclude that we were wrong to assume that the collection of all sets is a set. As set theorists put it, it's a proper class , not a set. Is there any way of telling whether a collection is "too big", i.e., a proper class, or whether it's a set, besides seeing whether the assumption that it's a set leads to a contradiction? No. So really, "too big" is just a colorful way of saying "leads to a contradiction if assumed to be a set".

Can something be infinite if there is a definitive number of it? Here's an example: I take a number, the largest I can think of, and never stop adding one to it. The number becomes infinite. Now if you take the number of human beings, and never stop adding to it, is the number of human beings infinite? In contrast, dinosaurs cannot be added to therefore they would not be infinite. Does this make sense?

If there are exactly 5 chairs in my apartment, then the number of chairs in my apartment is not infinite. But what if, in terms of those chairs, I could generate a sixth, and in terms of those a seventh, and so on, without end? (Of course, that is false. I'm assuming that's why you chose humans as your example, because you're imagining that given any collection of human beings we could "generate" another -- I don't think that's true as a matter of fact either, but let's not get hung up on the example.) Some mathematicians (they are sometimes called "intuitionists") would say that under those circumstances there would be an infinite number of chairs. It would be true to say that, while at any given time there are actually only a finite number of chairs, still the collection of chairs is infinite. Its infinitude doesn't consist in the fact that there are actually an infinite number of chairs (for there never is actually more than a finite number of chairs), but rather in the fact that we can...