In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history.Lubos Motl liked the essay.
Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths.
But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency.
His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring.
This description of Goedel's theorems is common, but I doubt that Goedel would agree with it.
In spite of the above, ZFC set theory is a perfectly good foundation for mathematics. It is logical, consistent, and good enough to prove all the theorems in your favorite math textbooks.
Yes, ZFC is incomplete in the sense that you cannot use it to program a computer to answer any mathematical question. Life is not so simple.
This reminds me a little bit of Einstein's complaints that quantum mechanics is incomplete. Somehow the theory is good enough to explain the physics underlying about a trillions dollars worth of global production, but somehow it does not answer every question in the way that Einstein thought should be answered.
Supposedly Hilbert was the one who wanted an axiom system that was consistent and complete. I am not sure he did. He did say he wanted an axiom system that was demonstrably consistent, but I don't think he ever said that the consistency proof should be within the axiom system.
If he did, then he made a minor misstatement of what was possible. But the larger goal of axiomatizing mathematics has been very successful, and Hilbert and Goedel played roles in that.
All these articles saying Mathematics and Physics have faulty foundations are wrong.
1. Re. Einstein:ReplyDelete
>> "...Einstein's complaints that quantum mechanics is incomplete."
The truth here is a bit tricky.
QM *is* demonstrably incomplete because (i) QM is a linear theory, and (ii) a linear theory cannot accommodate any mechanism to even address the measurement problem (let alone solve it).
Any system governed by a linear set of laws, and involving waves, is just going to go on oscillating back and forth forever. So, the really pertinent point isn't just that Schrodinger's cat would be in a superposition of being dead and alive at the same time, before measurement. The crucial point is: the cat would remain in such a superposition even *during* and *after* any measurement, so long as the QM formalism is applied to it. Even a momentarily dead cat would turn back alive right the next moment, only to die a second time, with the whole cycle repeating ad infinitum! And yes, it would be absolutely the same cat with the same body [which is unlike the Indian idea of reincarnation wherein only the soul stays the same, but the reincarnated body changes].
It isn't just that we cannot *explain* irreversibility using a linear theory. The point is: We *know* that an irreversible change cannot even *occur* in a linear system.
But the MSQM is linear. That's why, of course, QM is incomplete.
At the same time, Einstein's *reasons* as to why QM must be regarded as incomplete, are in themselves *wrong*! Plain wrong!
Yes, Einstein reached the right conclusion, but for an entirely wrong set of reasons! (In science, it doesn't count for anything.)
Einstein's persistent (even stubborn) position was that the ideas of classical physics should be regarded as providing the ultimate and eternally immutable standard for every theory of physics---past, present and future. He never fully grasped (1) that the ideas of classical physics themselves were based on a certain kind of an implicit ontology (certain assumptions regarding the ultimate nature of objects and actions), and this ontology in turn was based on the a limited set of observations and phenomena which it subsumed, and (2) that as the nature of the subsumed phenomena changed, a corresponding change in the ontology was called for.
Einstein, like all others, always kept the ontology implicit. That's why, their debates would go on forever.
In fact, I think, Einstein's error wasn't even as profound as that. He de facto made aether superfluous (that is, when he was not outright rejecting it). Which implies that, let alone QM, he didn't even have a very firm handle on basic theoretical nature of EM either---certainly not the way Lorentz did. And, the poor fellow didn't even realize it!
2. Re. Hilbert:
>> "He did say he wanted an axiom system that was demonstrably consistent, but I don't think he ever said that the consistency proof should be within the axiom system."
You've absolutely hit the nail right on the head here!
Thank You for refusing to make Imagined Perfection the enemy of Very Good ZFC.ReplyDelete