I am interested in how mathematical propositions relate to objects in the world; that is, how math and its concepts somehow correspond to the physical world. I have thought a bit about the issue, and realize that what happens, say, with numbers when we do some kind of mathematical operation with them may be the same as when we deduce one proposition in logic from another (If there is a number 2 and an operation "+", and an operation "=", then the result of using 2 + 2 = 4); but my question is this: does the proposition 2 + 2 = 4 mean the same thing as taking two objects and placing two more objects alongside of them, and then counting that there are four objects?

Philosophers continue to debate the relationship of mathematics to the physical world, including why mathematics is so effective at describing the physical world. The SEP entry on "Explanation in Mathematics," available at this link , contains much useful discussion as well as many references to further reading. At least one of the articles cited in the bibliography is available online: The Miracle of Applied Mathematics , by Mark Colyvan. I hope these prove helpful. Strictly speaking, the proposition that 2 + 2 = 4 can't mean the same thing as the process of taking two objects, placing two more objects alongside them, and then counting that there are four objects in total. Propositions and processes belong to different categories. Moreover, one might doubt that the proposition that 2 + 2 = 4 even entails that whenever you take two physical objects and place two more physical objects alongside them, there will be four physical objects to count up. Why?...

Does the fact that our perceptions can be represented geometrically and that geometry consists of eternal truths independent of the mind prove that an external reality underlies our perceptions?

I don't think that such an argument would rationally compel external-world skeptics (who say that no one can know that there's an external world) to abandon their view. External-world skeptics think that no one can know that solipsism is false, where solipsism is the claim that nothing external to oneself and one's mind exists. The solipsist won't grant that geometry consists of truths that are independent of his own mind, because he thinks nothing is. The solipsist could admit that his perceptions have a geometric character to them without having to attribute that character to something external. So I don't think solipsism can be disproven in the way you suggest. All of this assumes that solipsism is otherwise intelligible. But one might argue that solipsism is unintelligible because it relies on the incoherent idea of a 'private language', an idea explored in detail in this SEP article .

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...

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.

Are 3 and √9 the same mathematical object (in light of the fact that they have the same numerical value), or are they distinct mathematical objects? In other words, are the expressions '3' and '√9' co-referential names (both referring to the number 3), or do they refer to different objects?

If "√9" refers to the positive square root of 9 (I'm not sure what the convention is concerning the square-root symbol), then I'd say that 3 and √9 are the same object, just as Mark Twain and Samuel Clemens are the same object. (Indeed, the plural verb "are" in each case is a bit of loose talk.) Leibniz's Law (the Indiscernibility of Identicals) therefore implies that everything true of 3 is true of √9, and everything true of Twain is true of Clemens, which seems right to me.

Hi, I was hoping for some clarification from Professor Maitzen about his comments on infinite sets (on March 7). The fact that every natural number has a successor is only true for the natural numbers so far encountered (and imagined, I suppose). Granted, I can't conceive of how it could be that we couldn't just add 1 to any natural number to get another one, but that doesn't mean it's impossible. It seems quite strange, but there are some professional mathematicians who claim that the existence of a largest natural number (probably so large that we would never come close to dealing with it) is much less strange and problematic than many of the conclusions that result from the acceptance of infinities. If we want to define natural numbers such that each natural number by definition has a successor, then mathematical induction tells us there are infinitely many of them. But mathematical induction itself only proves things given certain mathematical definitions. Whether those definitions indeed...

Thanks for sending a follow-up question. Prof. Heck, who knows this territory better than I do, provided helpful corrections and amplifications in his answer to Question 5068 . I recommend taking another look there. You wrote, "The fact that every natural number has a successor is only true for the natural numbers so far encountered (and imagined, I suppose)." The claim that every natural number has a unique successor, like the claim that 1 isn't the successor of any natural number, is an axiom -- a starting point -- rather than a conclusion drawn from examining or imagining data. Your fifth sentence suggests that you know this already. You're quite right that math induction proves its results only given its starting points, but of course that's true of all proofs: all proofs (even proofs that have no premises) rely on essential assumptions. You say that some professional mathematicians would rather accept the existence of a largest natural number than accept the paradoxical features of...