In science, people have made discoveries and suggested theories that have been widely accepted or completely accepted by the rest of the world only to be proven wrong once new technology or a new method of experimentation has come along. I wonder, could the same thing occur in Math? Could some new way of calculations or something else cause all we know to be proven wrong? Has anyone in history ever proved a theory or formula, etc. that has been completely and utterly accepted as true, only to be proven wrong later? What I am asking is anything ever ABSOLUTELY beyond ALL possibilities proven? Also, on another note, does undefined=undefined? I mean, does 5/0=5/0 and does 2/0=9/0? Thank you, I know you all are beyond intelligent, please give me some insight.
No.
The reason that hard sciences can have these revolutions is that the observations of the physical world on which they are based improve. If you try to measure the diameter of the earth using cubits (the distance from your elbow to the tip of your outstretched middle finger), you get one fairly approximate answer. If you later use satellites with lasers and radar, you will get a different, and more precise answer.
But with math, everything is built logically from unchanging axioms. The only revolutions possible are when someone proves someone else’s proof to be incorrect (ie. not following logically from the axioms), or when someone decides to work out a new system based on different axioms. But neither of those invalidate the previous correct work.
I suppose some would call it a revolution when someone starts a new branch of mathematics (such as the calculus of Newton), but that is just discovering other things the axioms imply. Again, it doesn’t change the work already done.
As far as the undefined value of division by zero, that is one of the axioms of mathematics. Basically, that value lies outside of {the set of values you can do math with}. So it can’t be used in a meaningful way in an equation. On the bright side, that’s what calculus is for. You can do operations with values arbitrarily close to that undefined value. But not with the undefined value itself.
In theory, math is not dependent on evidence the way science is. So once something is rigorously proved, no evidence can really disprove it.
Still, published mathematical proofs have flaws found in them all the time. Historically, I think this has usually happened within a year or two of the proof being published, although I haven’t read a lot of mathematical journals. I understand that today, mathematics is highly fragmented, with only a few people capable of checking any particular proof in most areas — I would expect this would lead to higher levels of fallibility.
There’s another question, though; if you’re reasoning from contradictory axioms, you can prove contradictions: both that X is true and that X is not true. There’s no way to tell whether a set of axioms is contradictory. In particular, there’s no way to tell whether the basic axioms of, say, arithmetic are contradictory, although you’d think we’d have noticed by now if they were.
Undefined does not equal undefined; the expression 5/0=5/0 is meaningless because 5/0 is undefined.
On division by zero being undefined: that is not an axiom. It is just that nobody has found a way to assign any meaningful value to the result of division by zero based on the real axioms of arithmetic (for which see e.g. Principia Mathematica by Russell and Whitehead.)
I disagree, saltire. True, mathematics are consistent, but only as long as the axia are unchanged. When you change axia, you change the mathematics. As you say, the axia are based on the physical world, but that doesn’t mean that the same physical laws hold everywhere. Euclidian geometry works like a champ on earth, but near an even horizon, triangles can (in theory, of course) have more than 180 degrees.
Obviously, this is a nitpick. To answer the first part of the OP, proven mathematical formulae and equations cannot be disproven by their own axia.
When learning “the scientific process” in a psych. class… one of the things my teacher said to us is that nothing, NOTHING can ever be proven. Any theory can be disproven, but never proven… even if ALL evidence points to the theory, it still can not be proven.
As she explained it… suppose I have this theory, “There are no purple penguins, all penguins are black.” I can show a bazillion penguins, but never prove my theory… all this does is support my theory. But all it takes is one purple penguin to shoot my whole theory to heck.
The point is, anytime I see anything in the news that is reported by “science to have been proven…” I immediately doubt the research. (Or at least the reporter). I suppose even math can be disproven… (though I’m certainly not gonna try to do it. 
)
Screeme
That would be an event horizon. As in a black hole.
Even with the preview feature, I still cann’ttypw
Math is an extension of pure logic, using methods to develop truths that follow from axioms. At one level, math is just moving symbols around on paper. Any symbols can do. You can develop a math based around chickenscratches and as long as the symbols are consistent, it’s math. At a conceptual level, math is moving symbols around in your head. The concepts behind math (oneness, twoness, the concept of addition, etc.) are based upon the truths of the physical world, but they are infinitely accurate concepts. The real world isn’t infinitely accurate. It’s based around quanta, the ‘pixels’ all the Universe is made of (There’s a quanta of distance, a quanta of time, and, for all I know, a quanta of mass-energy. The concept of a quanta is that any unit less than the quanta is meaningless. Two things less than a quanta seperated in space-time can be said to be existing in the same space at the same time.) Therefore, math is always going to be more accurate than the real world. New rulers can improve our knowledge of the real world. Nothing can improve our knowledge of math. (BTW, there is a great book on our whole concept of logic and math out there called Gödel, Escher, Bach: the Eternal Golden Braid by Douglas Hofstadter. It’s at http://www.amazon.com and it explains the whole set of concepts around formal logic and Gödel’s Incompleteness Theorem, among other things, extremely well. It also gives insight into art and how we think and exist as rational, thinking beings. Very good, and very readable. I’ve read it a few times myself.)
Certainly anything can be disproven. Who knows… in ten years, they might find that 2 + 2 = 5. Now that may sound like I’m making fun, but what I mean is this: Anything can be disproven, but not everything can be REASONABLY disproven. A lot of things are only found untrue in abstract philosophical viewpoints or extreme circumstances (like the aforementioned black hole). But on a practical, day-to-day level, a lot of things won’t be disproven any time soon.
‘Proving’ and ‘Disproving’ mean using logic to argue from axioms. You can no more disprove a valid chain of reasoning than you can prove an invalid one. In the real world, everything is probabilites. At the level of abstract math, all quantities are exact and all processes are known.
But what if my theory says “There are some penguins containing the color black.”? I could clearly prove this just by showing you one black penguin.
I guess at this point your teacher launched into a discussion of falsifiability (is that how it’s spelled?). My statement is not falsifiable, it could never be proved false, so it doesn’t count as a valid theory. So just by the way we define what a theory is, one can never be proven.
Actually, the above-mentioned Gödel Incompleteness Theorem provides an example of something close to what the OP is talking about. Gödel’s conclusion is that there are some mathematical statements which are true but which cannot be proven. He showed this by considering a particular, very specialized statement, and proving that it could not be proven, and then proving that it was true. How did he do this? The statement could not be proven in the system in which it was formulated. He then used a different, and slightly more advanced, system to prove that it was true. Furthermore, in any system of mathematics, no matter how advanced, there exists such a statement. The also above-mentioned book Gödel, Escher, Bach does an excellent job of explaining this.
And Kragen, anyone who refers innocent people to Russel and Whiteheads Principia has a mean, sadistic streak a mile wide. 
Division is defined in terms of multiplication. a/b is that number x which satisfies xb = a. From the definition, you can see that if b=0 and a is not zero, there is no number x such that x0 = a. If a is also zero (and division by zero is allowed at all) then any x will satisfy x*0 = 0.
Very good observation. Godel’s second theorem says: If there exists a proof of consistency of a system, then the system is not consistent. Therefore to prove a system consistent, we need to show that there does not exist a proof of consistency. The problem with this is that an inconsistent system can prove anything.
Derleth recommended Gödel, Escher, Bach: an Eternal Golden Braid by Douglas Hofstadter. I love that book. Hofstadter gives an excellent explanation of Gödel’s theorems (among a lot of other topics). I recommend the book highly.
Axia? OK, I think I know what you mean. Mathematical theorems derived from axioms can be disproven by logic from the same axioms. That’s the meaning of inconsistency. Some sets of axioms are inconsistent. And we don’t know if our usual axioms (those of arithmetic) are consistent.
This is true for systems at least as advanced as arithmetic, but not for simpler ones like Euclidean geometry, which Gödel showed could not generate undecidable propositions.
picmr
One problem with the above-mentioned penguin example is that no two penguins are exactly alike… every 2 is always a 2, they’re always the same, 2 always equals 2, no matter what happens (in our perceived universe, anyway). I once had a geometry teacher who began the course with “Can you give me a pile of seven?” (the WHOLE speech went on for about an hour). His whole point was that numbers themselves are merely ideas… ideas that are attached to, basically, a series of input-output formula. Disproving numbers is like trying to prove if I’m thinking of chocolate cake right now… well, sort of. But I AM thinking of chocolate cake.
Now, higher mathematical concepts (physics, calculus), on the other hand, are different, since they’re based on widely diverse probabilities and “s’posed ta’s”… that is, every theory has been right so far, and it’d take a new theory that marks a dark spot on the current theory to make a revisement considered (oi, that’s a tough sentence). To wit, you need something new to disprove the old… the old can’t disprove itself.
To dumb it down a bit, math is all subjective, so, no, it can’t be proven wrong. “7+6=13” is true because the system we created says it is true (converely, a base 8 system would give a different answer . . . 14?). Gravity is true because my hamburger falls on the floor – best thing for me, really, what with my diet and all.
Not much new here, but just putting it in different terms.
VarlosZ
I disagree that math is subjective. I think you might mean that it depends upon the axioms. It is objective in the sense that if we agree upon the axioms we obtain the same results.
Screeme
I think you are confusing mathematical theorems and scientific theory. It is scientific theory that cannot be proven, just supported by the evidence or disproven. A mathematical theorem can be proven. (Subject to valid (consistent) logic and consistent axioms.)
If you demonstrate that some (real) triangles have more than 180 degrees, you have not disproven the mathematical theorem that (in euclidian geometry) all triangles have 180 degrees, you disproven the scientific theory that reality behaves like euclidian geometry.
SPOOFE Bo Diddly wrote:
Huh?
If physics is mathematics then accounting is too. Why are you conflating a physical science with a purely abstract science?
Theorems are not theories. If you believe that calculus sports ‘theories’ you weren’t paying attention in class.
Look, a lot of mathematics revolves around taking a well understood system of axioms, say Euclidian geometry, and altering the axioms and investigating the properties of the new system. Thus non-Euclidian geometry was born. Does non-Euclidian geometry disprove plain old geometry? Is 2 + 2 = 4 contradicted by the fact that 2 + 2 = 1 mod 3? Of course not, you are trying to compare two logically independent systems without a common objective reality. Are both Euclidean and non-Euclidian geometries useful in physical sciences? Yep, but that’s got nothing to with their logical soundness.
Do mathematicians disagree on what is mathematically ‘true?’ Yep, but it is mostly a disagreement over what basic axioms are appropriate and useful.
Andrew Warinner
MAth can not be proven false because it is not “true” in any kind of meaningful way. Math is purely subjective and based upon a set of logical rules everyone agrees on. If you were to change the rules (say to a base 8 system rather than base 10) all the mathematics change. MAthematics is a purely human creation and is not founded in real world phenomenon.
theories based upon mathematics can be proven false however. Consider Ptolemy’s theory that the sun revolved around the EArth. This was actually based on some unflawed mathematics…but the conclusions were, of course, false.