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Richard Jernigan -> RE: Mathematics: created or discovered? (Apr. 21 2015 3:36:30)
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quote:
ORIGINAL: Ricardo
And I finally finished watching your NOVA program, and by no coincidence it WAS the same physicist in both programs (Mario Livio). At one point the math guy does admit he is sort of exagerating his position as a way to sort of "represent" what he thinks Gødel would have thought....but at one point the philosopher and he disagree about what the Gødel proof actually is SAYING...which of the two would you agree with? (1:13:15)
Ricardo
I don't agree with either one of them. Chaitin says that Gödel showed that mathematics can't provide us with certainty because it's not a formal system.
The philosopher Goldstein says she believes Gödel would say that mathematics provides us with certainty despite not being a formal system.
What Gödel would have said is beyond me. What he proved was that you could not demonstrate the consistency of a formal system rich enough to produce arithmetic, within the system itself. That is, within such a system, following the rules of grammar, their is always a recipe for constructing contradictory propositions whose validity is undecidable by the rules of the system.
I emphasize that this is a proof about formal systems, which are not the same thing as all of mathematics. "Validity" in formal systems was meant to behave like "truth". But they are not the same thing. It's sort of like calling a place on a Monopoly board a piece of real estate. Park Place may have many of the characteristics of real estate, but you can't actually check into a hotel on the Monopoly board.
There's a little book about Godel's proofs by Ernest Nagel and James Newman that gives a clear and detailed exposition of these ideas. The latest edition has slight revisions and a lively foreword by Douglas R. Hofstadter, the author of "Gödel, Escher, Bach, the Eternal Golden Braid." Hofstadter knew the Nagels when he was a boy, and was strongly influenced by his association with Ernest.
Perhaps in his later philosophical essays Gödel may have proposed that mathematics could provide certainty, but I haven't read those essays.
I don't believe mathematics by itself can provide certainty, except in the thoroughly abstract sense of, "If those axioms are true, then these theorems follow." Most of the widely received body of mathematics now in use provides this sort of certainty, though I would bet a few invalid proofs have managed to slide by
The mismatches among Newtonian, Einsteinian and quantum mechanical physics are well known. Each is a fully developed mathematical system. Newtonian and Einsteinian physics fit together neatly at the limit of speeds much slower than light. Relativity and quantum mechanics demonstrate jarring incompatibilities at small distance scales. Both are fully mathematical theories. Their incompatibility is at the mathematical level.
Many working physicists bleep right over the "measurement problem" at the heart of quantum mechanics. When do you apply a linear operator to "collapse" the wave function? And just as importantly, why does it work?
Anyone who has spent serious time working in engineering will be deeply conscious of the fact that all of our measurements and essentially all of our calculated results are approximate. Leaving out the significant probability of simply doing something wrong, almost all of our calculating work potentially suffers the problem of the weather forecasting system. Eventually the weather forecast diverges from observations, despite the validity of the scientific model--valid as far as we know.
Quantum electrodynamics is a marvelously capable and precise theory. Its predictions agree with experimental measurements within one part in a trillion. As far as we know, quantum electrodynamics is right. As far as we know.
That's about as certain as we have managed to be about physical reality. But a lot of people argue over exactly what is physical reality, when you get down to the sub-subatomic level.
Heisenberg finessed this question in the first successful formulation of quantum mechanics. He predicted later measurements from earlier ones, without saying anything about what happened in between. Schrödinger proved that Heisenberg's formulation is mathematically equivalent to the most popular form in use nowadays, with the wave function, etc....
RNJ
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