Is Mathematics always correct?

It seems Mathematical theories/Laws/Formulas are the least questioned in all of the sciences. Is mathematics that good at being closest to the laws of universe, or is it just a logical tool of our own perception of the universe (that being the reason it always works)?

I'll elaborate second part of question above, I thought after reading some answers:

Our natural sense of telling one from many, larger from smaller, numerous from scarce, bright from dark, close from distant, familiar from strange, similar from different and so on, does not needs mathematical axioms and derivation. Even Animals are known to be capable of this. This most primitive logic, the seed of perception is hardwired.So, If Mathematics, itself is based on this sense, starts from here forward, Can we throw "observable phenomenon" and "Physically verifiable" out of the window, when talking about Mathematics ? Isn't this unquestioned nature of our hardwired sense, makes its logical treatment undisputed too ?

Finally,

Perhaps this matter is already in debate.There's a strange, incredible certainty to Mathematics, not found in other sciences. It's still a mystery to me.Though I still think that all sciences are parts of the same universal thread , we got hold of, at different points.

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Subject of much debate over the years/decades/centuries.... Unlikely to be settled here. –  Gerry Myerson Sep 21 '12 at 5:54
The 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." –  Arthur Fischer Sep 21 '12 at 6:57
I'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.\ –  mixedmath Sep 21 '12 at 15:06
@fischer: great quote from einstein. –  Mozibur Ullah Sep 22 '12 at 12:33
@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 relative to 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. –  Jayesh Badwaik Oct 5 '12 at 9:00

Physical sciences rely upon thinking of hypotheses and testing them with experiments. The conclusions from physical sciences are always scrutinized because it is the way of the scientific method. In order for a scientific theory to become better, first a deficiency in the theory is discovered, followed by an altered hypotheses, followed by re-testing.

Some people unfortunately see this method as evidence that science is often wrong and unreliable. Science is however a methodology that involves constant refinements of hypotheses to get a clearer and clearer picture of the truth. Therefore science isn't wrong, but the hypotheses science produces are never 100% right either. It is the nature of the game.

Mathematics however is a completely different game. Mathematics works from axioms upwards. Therefore mathematics doesn't have to constantly refine itself as science does. Mathematics is based on foundations, known as axioms, from which the rest of the subject is built from. Unlike in science, the axioms of mathematics are unchanging.

Science can be seen as working in the opposite direction as mathematics. That is, determining the principles from the results, which is much harder than determining the results from the principles (mathematics).

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No, the beauty of mathematics is axioms are assumptions. So all mathematical conclusions are of the form, assuming A then B. So the conditional nature of maths means it can never be wrong. The axioms can be wrong and don't have to match the universe for the conditional mathematical conclusion to be true. –  Mew Sep 21 '12 at 6:18
No, I tried to emphasize that axioms are not "the most primitive form of our perception of things around us". I can develop a mathematical theory on false axioms, and the mathematical theory would still be true, because of its conditional nature. Mathematics doesn't depend upon the real world around us, that was my main point. –  Mew Sep 21 '12 at 6:35
The difference between mathematics and physics, is in maths, we are free to choose the axioms to be whatever we want. In physics, we must deduce the "axioms" or "principles" from what we observe. Since what we observe is always changing as we do more experiments, our axioms in physics are always changing. This change isn't necessary in maths since axioms don't have to match the real world. –  Mew Sep 21 '12 at 6:37
"Mathematics works from axioms upwards." A great many people have done a great deal of Mathematics, pure and applied, without any reference whatsoever to any axioms. "The axioms of Mathematics are unchanging." History does not support this statement. –  Gerry Myerson Sep 21 '12 at 7:06
By "axioms of mathematics are unchanging" I meant for a particular mathematical theory. A mathematical theory can be built on whatever axioms it likes, and that is ok. The axioms for that theory don't have to change because of experiments. Of course, new theories built on new axioms are being developed all the time. –  Mew Sep 21 '12 at 7:09

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).

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I think the answer to this question lies in the distinction: Science deals with observable phenomena where mathematics deals with abstract notions such as numbers, sets or the nature of computability.

Where science strives to be able to express the true state of the universe, mathematics strives to create consistent systems of thinking. When one speaks of a scientific theory they mean a developed and tested explanation of the natural world which can produce falsifiable predictions. When one speaks about a mathematical theory they mean the current state of exploration into one of these abstract notions. A scientist advances his field by testing hypotheses. A mathematician advances his field by proving theorems.

Mathematics does not claim to be the law of the universe, mathematics doesn't claim to be any one thing at all. It happens that science uses mathematics with the hope that the universe is a system that can be expressed with consistency because if not, how would we?

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There are contradicting assumptions in mathematics, which cannot be resolved, and it is fine! Euclidean and hyperbolic geometry are based on different sets of axioms, which cannot be true simultaneously. However, both geometries are meaningful and have real-world applications.

Now, mathematicians also deals with definitions, and there are of course different ways to define the same thing. Now, it took a while to actually define things like limits, groups etc. and they have looked slightly different throughout history. Some things become considerably nicer with a "better" definition. Some prefer to use $2\pi=\tau$ as THE circle constant which everything is based on, and a lot of formulas becomes simpler using $\tau$ instead of $2\pi.$

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The Socratic method: ask the questioner what he means with his words:

• What do you mean with "laws"? There are thorough (mathematical) definitions of what a "theory" is and what a "formula" is, but what is a "law"? Can you tell the difference between a "law", an "axiom" and - say - a "definition"?

• What do you mean with "questioned"? How and why are other-than-mathematical "theories/laws/formulas" more questioned than mathematical ones.

If you happen to give at least partial answers to these questions, it seems worthwhile to continue the talk.

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In fact, "What is Counting ?" can spawn a study of its own. –  user2411 Sep 22 '12 at 6:41

Mathematics is often taken as a kind of path to true. But its methods are not as simple as is popularly made out.

Although mathematical systems are often described axiomatically, this is not how these systems are born. Its often their final form, or rather the form that they are expressed in to bring out their most important properties and to make it look as though they are almost inevitable. Though this is as much psychological for a certain kind of mind.

An example is calculus: Archimedes investigated integration synthetically but could not put it on a formal axiomatic system ala euclid. Its development stalled until Newton/Leibniz utilised the coordinitisation of geometry to begin to fully realise its capability. It was of course noticed that these 'fluzions' were not fully rigorous, and Berkelys criticism of 'ghosts of departed quantities' stung. It wasn't until Cauchy developed the idea of a limit that the foundations of calculus began to be put on a rigourous basis. Now there are a plethora of different axiomatics for the calculus: Synthetic Differential Geometry, Nonstandard Analysis, Diffeological Spaces. Which one of these is the one true & correct axiomatic framework?

Similarly with the more well-known story for Euclidean geometry. The fabric of space-time is much better modelled by Lorentzian geometry.

One could argue, that the axioms are derived empirically, by understanding what important questions can be cast into this kind of language, but surely logic remains a priori.

Again, this is not so simple. We have classical logic from the time of Aristotle which affirmed the law of the excluded middle, (but he noted that this didn't hold for future events), this was eventually formalised as boolean logic, but Brouwer advocated intuitionistic logic that doesn't (his supervisor advised him to establish his reputation in some traditional area before advocating such startling views). People are now researching logics where the law of non-contradiction doesn't hold, where time and modality is taken into account, and so on.

The nature of mathematical truth is not simple. Nor has it shown to be always true. There is a great deal of truth to what social constructivists maintain, that mathematical truth is socially constructed, but that doesn't mean to say that it is solely that, and that it doe not have some sophisticated relation to reality too.

This is what Felix Klien had to say (he was a mathematician famous for formulating the Erlangen programme amongst others):

'Quite often you may hear non-mathematicians, especially philosophers, say that mathematics need only draw conclusions from clearly given premisses and that it is irrelevant whether those premisses are true or false – provided they don’t contradict themselves. Anybody who works productively in mathematics, however, will talk in a completely different manner, In fact, those people base their judgements on the crystallized form in which mathematical theories are presented once they’ve been worked out. The research scientist, like any other scientist, does not work in a strictly deductive way but essentially makes use of his imagination and moves forward inductively with the help of heuristic aids.'

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I have Klein's "Development of Mathematics in the Nineteenth Century" on my bookshelve. Going to read it. You could find my answer below interesting I guess. –  Riga Feb 10 '13 at 21:59

The most credible answer I know is given by Henri Poincaré in his "Science and Hypothesis":

He writes about reasoning by recurrence as an example of a true scientific value that is different from tautology. Then he compares mathematics with physics in this aspect:

It cannot escape our notice that here is a striking analogy with the usual processes of induction. But an essential difference exists. Induction applied to the physical sciences is always uncertain, because it is based on the belief in a general order of the universe, an order which is external to us. Mathematical induction — i.e., proof by recurrence — is, on the contrary, necessarily imposed on us, because it is only the affirmation of a property of the mind itself.

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Mathematics can only answer limited questions. All mathematics uses deterministic equations, there is no non-deterministic mathematics. We can only solve for 1 variable while holding other variables constant. This is not how the real world operates. The classical three body problem in physics is an example of this. Other examples are flow dynamics and chaos.

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Mathematics certainly can be wrong in that a mathematician presents a faulty theorem with an error in its proof, and it passes the scrutiny of peers and is commonly accepted as true.

Of course after a time the error will be found and the necessary corrections made. Any theorem that follows the rules up from the axiom is correct. It may be totally unrelated to physics or workings of our universe, or it may be related and very similar but with important deficiencies, still, within its own framework it's correct as long as no (stupid) mistakes have been made along the way.

Now an interesting point is some branches of mathematics use theorems without proofs. Famous mathematicians offer a hypothesis with a faulty proof, with known fault - the proof covers a large part of cases but some remain unproven. Now the mathematics follows by building upon that theorem, always with a little disclaimer "Assuming X's theorem is correct", and meanwhile there's a race between enthusiasts to produce a full proof, or alternatively disprove the dubious theorem. In these cases mathematics can be wrong, but only within range of the disclaimer.

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