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> Second, and more importantly, the system of proof itself is falsifiable. If a theorem says that an equation p(x,y,z) = 0 has no solutions, and a proof without

by rntz 8y ago

> Second, and more importantly, the system of proof itself is falsifiable. If a theorem says that an equation p(x,y,z) = 0 has no solutions, and a proof without mistakes is found, then it better have no solutions. If somebody comes along and demonstrates that x=43,y=23,z=809 is a solution, then the system is in trouble. This ensures that math is not a formal game of arbitrary axioms that is internally consistent but ultimately meaningless. I don't see how what you've said is anything other than the criterion of formal consistency - that a thing and its negation are not both provable. And a system may certainly be consistent, yet not particularly useful. It should be noted that mathematics has not always been as rigorous as people assume. Attempts to ground math in formal logic didn't really get off the ground until the late 1800s. And this isn't a trivial point; the "rules of the game" were up for debate, and often debated, before then (and, to a lesser extent, since). People disagreed about which numbers exist; people disagree about which proof methods are acceptable; and for example, calculus - a highly practical branch of mathematics - was invented before its formal justifications were found.

jules 8y ago
It is more than formal consistency because to show a whole number solution to a^n + b^n = c^n and n > 2, one doesn't need to use a high powered formal framework like ZFC. Thus Wiles' proof of Fermat's last theorem, which uses all kinds of infinite sets and other features of ZFC, can be falsified by a simple manipulation of finite numbers. You just need the numbers a,b,c,n of the counterexample and a calculator. Mathematics thus makes predictions about the world (in this case about the behaviour of calculators, human or electronic) in a somewhat similar way that physics makes predictions about the world. Or as Arnold famously said: "Mathematics is the part of physics where experiments are cheap". You wouldn't have that if math was an arbitrary symbol pushing game with arbitrary axioms, with the only requirement that they be consistent. [If a counterexample to Fermat's last theorem is found it probably isn't because ZFC is inconsistent, but because Wiles' proof has a mistake, which would show that the checking of the proof was inadequate and that mathematics needs to use a higher standard for what is considered a valid proof. So the system of proof is falsifiable in a very broad sense: it can expose a bad proof checking culture within mathematics as well as inconsistencies in the formal system. I don't think that metaphysics has anything equivalent. If somebody comes up with a counterexample a,b,c,n then Wiles would immediately admit that his proof has a mistake, even though he wouldn't even know where in the proof the mistake is. Try convincing a metaphysicist that their argument is wrong or meaningless... That said, there are certainly also stubborn mathematicians who insist that their proof is right even though other mathematicians have pointed out that a particular step isn't clear. However, if the mathematician who came up with the proof cannot clarify that step, ultimately down to the axioms of ZFC, then the proof isn't accepted.]
- rntz 8y ago
  > It is more than formal consistency because to show a whole number solution to a^n + b^n = c^n and n > 2, one doesn't need to use a high powered formal framework like ZFC. Thus Wiles' proof of Fermat's last theorem, which uses all kinds of infinite sets and other features of ZFC, can be falsified by a simple manipulation of finite numbers. This follows from consistency (and is one reason why consistency is important). If there were some whole number solution to a^n+b^n=c^n for n>2, then this would be provable in Peano Arithmetic (a much simpler system than ZFC, that axiomatizes natural number arithmetic). ZFC extends PA (every theorem of PA is a theorem of ZFC, interpreted correctly; this is relatively easy to show); thus, anything ZFC proves is not falsifiable in PA, otherwise we'd have a contradiction provable in ZFC (& it wouldn't be consistent).
  - jules 8y ago
    It doesn't follow from consistency in itself. It is possible to imagine a formal system that is consistent but nevertheless doesn't predict the behaviour of calculators. If you presume that ZFC correctly models the behaviour of calculators, or that PA models calculators and inconsistency of PA implies inconsistency of ZFC, then sure it follows from consistency of ZFC, but that was the point, namely that there is a difference between a purely formal system that only exposes itself to inconsistency tests, and a formal system that exposes itself to external tests. Numbers aren't the only way that math does that. Geometry makes predictions about the behaviour of rulers and compasses, for instance. Contrast this with a discipline in which there is jargon and arguments about that jargon. It might be the case that the jargon and the basic ways of arguing about the jargon forms a consistent system in some sense, even though the whole enterprise is meaningless.

by rntz 8y ago

jules 8y ago

It is more than formal consistency because to show a whole number solution to a^n + b^n = c^n and n > 2, one doesn't need to use a high powered formal framework like ZFC. Thus Wiles' proof of Fermat's last theorem, which uses all kinds of infinite sets and other features of ZFC, can be falsified by a simple manipulation of finite numbers. You just need the numbers a,b,c,n of the counterexample and a calculator. Mathematics thus makes predictions about the world (in this case about the behaviour of calculators, human or electronic) in a somewhat similar way that physics makes predictions about the world. Or as Arnold famously said: "Mathematics is the part of physics where experiments are cheap". You wouldn't have that if math was an arbitrary symbol pushing game with arbitrary axioms, with the only requirement that they be consistent. [If a counterexample to Fermat's last theorem is found it probably isn't because ZFC is inconsistent, but because Wiles' proof has a mistake, which would show that the checking of the proof was inadequate and that mathematics needs to use a higher standard for what is considered a valid proof. So the system of proof is falsifiable in a very broad sense: it can expose a bad proof checking culture within mathematics as well as inconsistencies in the formal system. I don't think that metaphysics has anything equivalent. If somebody comes up with a counterexample a,b,c,n then Wiles would immediately admit that his proof has a mistake, even though he wouldn't even know where in the proof the mistake is. Try convincing a metaphysicist that their argument is wrong or meaningless... That said, there are certainly also stubborn mathematicians who insist that their proof is right even though other mathematicians have pointed out that a particular step isn't clear. However, if the mathematician who came up with the proof cannot clarify that step, ultimately down to the axioms of ZFC, then the proof isn't accepted.]

rntz 8y ago
> It is more than formal consistency because to show a whole number solution to a^n + b^n = c^n and n > 2, one doesn't need to use a high powered formal framework like ZFC. Thus Wiles' proof of Fermat's last theorem, which uses all kinds of infinite sets and other features of ZFC, can be falsified by a simple manipulation of finite numbers. This follows from consistency (and is one reason why consistency is important). If there were some whole number solution to a^n+b^n=c^n for n>2, then this would be provable in Peano Arithmetic (a much simpler system than ZFC, that axiomatizes natural number arithmetic). ZFC extends PA (every theorem of PA is a theorem of ZFC, interpreted correctly; this is relatively easy to show); thus, anything ZFC proves is not falsifiable in PA, otherwise we'd have a contradiction provable in ZFC (& it wouldn't be consistent).
- jules 8y ago
  It doesn't follow from consistency in itself. It is possible to imagine a formal system that is consistent but nevertheless doesn't predict the behaviour of calculators. If you presume that ZFC correctly models the behaviour of calculators, or that PA models calculators and inconsistency of PA implies inconsistency of ZFC, then sure it follows from consistency of ZFC, but that was the point, namely that there is a difference between a purely formal system that only exposes itself to inconsistency tests, and a formal system that exposes itself to external tests. Numbers aren't the only way that math does that. Geometry makes predictions about the behaviour of rulers and compasses, for instance. Contrast this with a discipline in which there is jargon and arguments about that jargon. It might be the case that the jargon and the basic ways of arguing about the jargon forms a consistent system in some sense, even though the whole enterprise is meaningless.