The only thing you have to assume to be unconditionally true in Mathematics is some minimal logic (and yes, that's despite having axiomatic systems for logic; you still have to use some form of logic to actually *define* those axiomatic systems). But logic is assumed to be true in any science (because without it, you cannot draw any conclusions).

But apart from logic, all statements in mathematics are ultimately conditional statements on the chosen axioms. For example, take the statement "there are infinitely many prime numbers." How can we know this to be really true? Well, we have a definition of the natural numbers through a set of axioms, and we have a definition of what it means to be a prime number. From those axioms we can *logically* derive that there are infinitely many primes. But that statement is implicitly conditioned on the axioms: We have to *assume* that what we are looking at really fulfils the Peano axioms. If we look at something which doesn't, the claim doesn't hold. However, mathematics doesn't look at a specific system. The statement it derives is not "for this real world object we have infinitely many primes." It says "*Whenever* we have something which fulfils those axioms, we know that we will find infinitely many primes." It also tells you that if we make certain other assumptions (such as that the axioms of set theory hold), we can derive that we'll find something fulfilling those axioms.

This is also why mathematics is so useful in natural sciences: It does *not* tell us what assumptions are true. But it tells us what follows *if* certain assumptions are true (and also, if certain assumptions cannot hold together). So if we have for example a physical phenomenon, we can formulate the hypothesis that it has certain properties. This hypothesis is not part of the real world, but a set of assumptions. Therefore we can now go to mathematics, which tells us what to expect from systems with such assumptions (and also, which additional assumptions we might want to make). Note that this step is completely independent of reality. After we've found what to expect if those assumptions are true, *then* we can go back to the lab and check if our experiments show the behaviour we just have derived from our assumptions. If yes, we've got a confirmation and may be more confident in our hypothesis, otherwise we have falsified our hypothesis and have to modify it (and again, mathematics will tell us what assumptions will be compatible with our new knowledge from the experiment).

Note that there's another type of questioning theories which *is* done in mathematics as well as in natural sciences: Namely the questioning whether your results are actually correct. In mathematics, this means checking that there are no errors in the proof (and in some sense this is similar to the experimental tests of theories in natural sciences: We are confident in a proof if it has been sufficiently looked at and nobody has found an error), in physics it means checking that there's no error in the measurement procedure (that is, we really have measured what we thought we measured) and no error in the application of mathematics (that is, we correctly applied the tools we got from mathematics and made no hidden assumptions, and thus our conclusions about what to expect are correct).

15Subject of much debate over the years/decades/centuries.... Unlikely to be settled here. – None – 2012-09-21T05:54:10.883

Well, math is a priori, while other sciences are a posteriori... – Alex Becker – 2012-09-21T05:55:13.613

Thanks, I'm not into mathematics, I usually come across Maths to solve some problems. But some times I'm left wondering about mathematics itself. Didn't know this is already debated, just posted out of curiosity. Personally , yes I do think that there are always new ways to represent our perception, and would be glad to see a new number-system or so. – None – 2012-09-21T06:03:40.403

31The following quote by Einstein is apropos: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." – None – 2012-09-21T06:57:08.100

3I've moved this from math to philosophy, because although it got a lot of good attention from math, it was closed as off topic. Perhaps it will get a different perspective here.\ – davidlowryduda – 2012-09-21T15:06:22.063

@Alex Becker, that is subject to debate – smartcaveman – 2012-09-21T19:43:07.213

2@fischer: great quote from einstein. – Mozibur Ullah – 2012-09-22T12:33:44.643

1May I add the equally apropos quotation: "Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." -Bertrand Russell – user 726941 – 2012-10-05T08:57:52.557

4@wingman: "Mathematics is not about being correct or wrong, it is about being consistent." A statement inside a theory is correct or incorrect or unknown

relativeto other statements in that theory. If it is proven to be consistent with other statements, then it is correct, if it is proven to be inconsistent, it is wrong, if it is undecided, it is unknown. You can then apply such theories to real life to do probably describe some phenomena, in which case then it is appropriate model for the phenomena, else you try to find some other models, and probably invent a new theory for that. – None – 2012-10-05T09:00:55.443The answer is neither yes nor no, henceforth the question is wrong!! – None – 2012-10-05T09:05:26.390

@JayeshBadwaik Well, my maths teacher DID mark correct and wrong, so maybe I'm used to think like that :D – None – 2012-11-24T18:21:31.707

@wingman correct or wrong was in relation to the certain axioms that have not been formalized at school level but are thought to have been understood intuitively. – None – 2012-11-24T19:28:23.927

Mathematics deals with the measures of quantity, or proportions, which is related to the problem of the one and the many, in my mind. Also, if mathematics bears no relation to the world, then you have a far larger burden to explain what we experience. And I wouldn't say it's a matter of experimentation. We can't experimentally compute 1 + 1 + 1 = 3. It's a matter of meaning. We know what 1 is, what + is, and what three is. – Robert LeChef – 2012-11-26T22:40:18.767

I asked myself the same question. I don't think that mathematics are always true, but I do believe that they are THE path which can lead us to truth. – Bek – 2012-09-21T08:10:05.500