[ExI] [Extropolis] Gödel and physical reality
Giulio Prisco
giulio at gmail.com
Wed Nov 3 16:08:03 UTC 2021
On Wed, Nov 3, 2021 at 4:59 PM John Clark <johnkclark at gmail.com> wrote:
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> On Wed, Nov 3, 2021 at 10:38 AM Giulio Prisco <giulio at gmail.com> wrote:
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>> > Gödel's result implies an independence of mathematical truth from any human theory of that truth, and I would argue, even makes mathematical truth independent of the physical universe.
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> Mathematics is the language of physics but like any language it can be used to write both fiction and nonfiction. English is also a language and it can be used to write a book about Quantum Mechanics but it can also be used to write a book about Harry Potter, and both books could be equally grammatically correct. I sometimes have the feeling that some of the more abstruse areas of modern mathematics might be the equivalent of mathematical Harry Potter novels.
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>> > But it can also be argued that it makes mathematical truth *strongly
>> dependent* on the physical universe.
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> That's why I would argue physics is more fundamental than mathematics, and that's why a book on advanced computational theory sitting on a shelf cannot add 2+2, the ideas in the book have to be implemented in physics before it can actually do anything. If there is an even number that is not the sum of two prime numbers but the number is so huge that it cannot be calculated by physics even in theory then I think it would be meaningless to say they're really "is" such a number.
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I agree, physics is more fundamental than mathematics.
>> > our universe lacks the computational and memory resources to even list all the properties of the number "3". In this sense, the number 3 is a larger, and more complex object than our physical universe, for there are an infinite number of true statements concerning the number 3,
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> That depends on exactly how you define "complexity", but you wouldn't need to list everything the number 3 can do to uniquely defined it, but if you define the complexity of something as the minimum number of characters needed to define it then the first billion digits of π are more complex than all of the digits of π since all the digits of π are uniquely defined by just 24 characters, 4*(1-1/3+1/5-1/7+1/9 -...), but you'd need 1 billion characters to uniquely define just the first billion characters of π.
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>> > The finiteness of our universe also means there are axiomatic systems which contain more axioms than there are atoms in our universe, so we could never conceive of them.
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> A logical system is only as strong as its weakest axiom, that's why axioms need to be simple and intuitively obvious. We could just add the Goldbach Conjecture as an axiom but it's truth is not intuitively obvious and it would be very embarrassing if after doing this a computer crunching through huge numbers happen to find an even number that was not the sum of two prime numbers, it would mean all the work done in mathematics since the axiom had been added was meaningless.
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> John K Clark
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