What are the major open questions of mathematical philosophy? Of these, which are mathematically significant, if any? By "mathematically significant," I mean "would affect the way mathematicians work." For example, the question of whether mathematics is created or discovered has no impact on working mathematicians. On the other hand, studies into the foundations of Math were certainly mathematically significant, and although one could argue that that was more Math than Phil, we can give Phil some credit. But that question is now closed, as far as mathematicians are concerned.

You write that "the question of whether mathematics is created or discovered has no impact on working mathematicians", but this doesn't seem so to me. If that question is a vivid way of asking whether intuitionistic logic or rather classical logic is correct, then the answer to the question has great consequences for how mathematicians work. For instance, if intuitionism captures the inferences that are really sound, then mathematicians will have to curtail the use of reductio ad absurdum arguments. (For more on this form of reasoning, see Question 121 .) Classical mathematicians do not hesitate to infer "P" from the derivation of a contradiction from the assumption "not-P". But intuitonists believe that this inference is not in general correct and so should be avoided. The German mathematician David Hilbert thought this had such great "impact" that it was like depriving the boxer of the use of his fists! (For some more on intuitionism, see Question 168 .)

Is mathematics independent of science? And, vice versa.

Mathematics investigates number systems some of whose properties arecritical for measurement. And without measurement, we wouldn't be ableto provide our scientific theories with sharp reality checks.Furthermore, those theories are themselves shot through with extremelysophisticated mathematics; most central claims of the advanced sciencescannot even be stated without a generous helping of mathematics. Manyhave been amazed that mathematics, often developed independently ofempirical research, turns out to be so useful, indeed necessary, forscience. But whatever one's explanation, it's a fact that it has. Mathematics,by contrast, appears independent of science in important respects. It's true thatmathematical inquiry was often initially prompted by scientific inquiryinto the natural world. But what inspires mathematics is one thing, andwhat it owes its justification to is something else entirely. Mostmathematicians will tell you that the only ground for accepting amathematical claim is that someone has...

Can 2+2 equal ANY other answer than 4?

It's hard to imagine how I could be convinced to doubt that 2+2 = 4.Whatever argument you gave me, there would surely be some assumption orstep of reasoning in it that was less obvious to me than that 2+2 = 4.Faced with the decision of denying that 2+2 = 4 or of rejecting somestep in your argument, it seems it would always be more rational for meto do the latter. Do you now want to object: "I'll grant you itwould always be more rational for me to believe that 2+2 = 4 — butstill, couldn't it be false!?"

Isn't everything relative? For example, mathematics was invented by man — did it exist before man invented it?

You would have thought that we would have worked out by now whethermathematics is a human invention or not. We haven't. There is still aheated debate between those who believe that mathematics describes arealm of entities (numbers, sets, functions, etc.) that exist quiteindependently of us and those who believe that the mathematical worldis in some sense constructed as a result of human activity. This is thecontrast between platonism and constructivism that has been touched on elsewhere here.

Is the underlying mathematics of string theory both complete and consistent? If it is, then apparently Gödel was wrong; if it is not, then how can it be a theory of everything? Would not an endless string of metatheories be needed for sufficiency? If not, what did Gödel, Tarski, etc. miss. Dave

I don't know anything about string theory, but I assume that itemploys rich enough mathematics that, were we to articulate thatmathematics in a formal system, Gödel's 1931 Incompleteness Theoremwould apply to it to yield the result that, if the system isconsistent, then it is incomplete, that is, then there is somemathematical statement in the language of the system that is neitherprovable nor disprovable in that system. You ask whether theconsistency and (hence) incompleteness of the system would conflictwith the claim that string theory is "a theory of everything". Itdepends on what "a theory of everything" means. If it means that thetheory can answer all questions about physical phenomena ,then there need be no conflict: the undecidable statement of the formalsystem (the statement that can neither be proved nor disproved if thesystem is consistent) is one in the language of mathematics. It is notmaking a claim about the physical world. If, on the other hand, by "atheory of everything" one...

As a teacher of high school mathematics and a former student of philosophy, I try to merge the two to engage my students in meaningful conversations about the significance of some mathematical properties. Recently, however, I could not adequately defend the statement "a=a" as being necessary for our study of geometry when one student challenged "When is a never NOT equal to a?" What would you tell them? (One student did offer the defense that "Well, if we said a=2 and a=5 then a=a would be false, causing problems.")

I'm not sure whether you're asking (1) What role does the reflexivityof identity (i.e., every object is equal to itself) play in geometry?,or (2) What justification can be offered for the reflexivity ofidentity? As regards (1), I assume that in an explicit axiomatizationof geometry, there would be axioms dealing with identity. As Richardpoints out above, in such an axiomatic system we will want to derivetheorems in which "=" figures; so we had better have some axioms thattell us under what conditions identity statements can be proved. Asregards (2), I'm not sure what to advise you if a student is unwillingto grant that an object is identical to itself. I would infer that s/hedoesn't understand what "identical to" means and would treat the matter as a case of miscommunication.

How do we resolve the fact that our finite brains can conceive of mental spaces far more vast than the known physical universe and more numerous than all of the atoms? For example, the total possible state-space of a game of chess is well defined, finite, but much larger than the number of atoms in the universe (http://en.wikipedia.org/wiki/Shannon_number). Obviously, all of these states "exist" in some nebulous sense insofar as the rules of chess describe the boundaries of the possible space, and any particular instance within that space we conceive of is instantly manifest as soon as we think of it. But what is the nature of this existence, since it is equally obvious that the entire state-space can never actually be manifest simultaneously in our universe, as even the idea of a board position requires more than one atom to manifest that mental event? Yet through abstraction, we can casually refer to many such hyper-huge spaces. We can talk of infinite number ranges like the integers, and "bigger"...

To add a word or two to Dan's great response: there is no questionthatmathematics deals with infinite collections, but what those are, whatwe mean when we make claims about them, which claims are correct —these have been hotly disputed issues for thousands of years. (Inthe history of mathematics, concern for these foundational questionshas waxed and waned. There have been times, for instance in theearly part of the twentieth century, when disputes over these issues,were very heated and split the mathematical community. There have beenother times, for instance now, when mathematicians have been lessinterested in these issues — although of course there are alwaysexceptions, like Dan.) The basic question — what does it mean to call aset "infinite"? — is so fundamental that it's simply astounding that wedon't know how to answer it. Onone way of looking at the matter, what Dan called "platonism", to saythat a set is infinite is simply to have given a measure of its size.To say that a set is infinite is...