You’re misunderstanding the implications of both the halting problem and Gödel’s first incompleteness theorem.
What Turing and Gödel independently proved is that a human observer can (theoretically) always have insights about mathematics and programming that are incomputable. That is, you cannot program or axiomatize or formalize or digitize everything that a mind can do. Period.
Analog computers are sufficiently different from digital systems to potentially emulate brain activity. But digital (discrete) methods are probably too constrained.
What Turing and Gödel independently proved is that a human observer can (theoretically) always have insights about mathematics and programming that are incomputable. That is, you cannot program or axiomatize or formalize or digitize everything that a mind can do. Period.
that is not what either of them proved. like… at all
I study this stuff. You will find what I said in any philosophy of mathematics textbook dealing with the subject. In fact, I am paraphrasing the Oxford logician Joel David Hamkins.
You’re welcome to also read Shapiro’s famous paper for a rephrasing. These results have been well understood for half a century, although because the implications are ultimately metaphysical and not mathematical, we can’t be sure of the wider consequences, if any.
Going through some of the related paper abstracts, including speculative comments by Gödel: this is pure philosophy. Nothing that is set in stone. Which now points me back to my initial statement, where we can discuss all we want, but in the end it’s philosophy. Not “hard” (“provable”) science
There can be no enumerable list of axioms for the true statements of mathematics. No computational procedure could exist to determine whether propositions are valid, provable, or even equivalent. And no matter how you formulate the number-theoretic axioms, a mathematician would always have insights (for instance, about whether a Diophantine equation has a solution) that were both clearly “true” and obviously unprovable. This holds true for all digital systems.
Here is what we don’t know for sure:
Anything else.
Your distinction between science and philosophy is incorrect. Science is inductive and abductive. It doesn’t “prove” things. It’s not deductive. Mathematics and philosophy can “prove” things.
You’re misunderstanding the implications of both the halting problem and Gödel’s first incompleteness theorem.
What Turing and Gödel independently proved is that a human observer can (theoretically) always have insights about mathematics and programming that are incomputable. That is, you cannot program or axiomatize or formalize or digitize everything that a mind can do. Period.
Analog computers are sufficiently different from digital systems to potentially emulate brain activity. But digital (discrete) methods are probably too constrained.
that is not what either of them proved. like… at all
I study this stuff. You will find what I said in any philosophy of mathematics textbook dealing with the subject. In fact, I am paraphrasing the Oxford logician Joel David Hamkins.
You’re welcome to also read Shapiro’s famous paper for a rephrasing. These results have been well understood for half a century, although because the implications are ultimately metaphysical and not mathematical, we can’t be sure of the wider consequences, if any.
ah, now we’re getting somewhere.
Going through some of the related paper abstracts, including speculative comments by Gödel: this is pure philosophy. Nothing that is set in stone. Which now points me back to my initial statement, where we can discuss all we want, but in the end it’s philosophy. Not “hard” (“provable”) science
Here is what we know for sure:
There can be no enumerable list of axioms for the true statements of mathematics. No computational procedure could exist to determine whether propositions are valid, provable, or even equivalent. And no matter how you formulate the number-theoretic axioms, a mathematician would always have insights (for instance, about whether a Diophantine equation has a solution) that were both clearly “true” and obviously unprovable. This holds true for all digital systems.
Here is what we don’t know for sure:
Anything else.
Your distinction between science and philosophy is incorrect. Science is inductive and abductive. It doesn’t “prove” things. It’s not deductive. Mathematics and philosophy can “prove” things.
agreed
and here I disagree