I love Dorking.
Epistemological Holism and The Incompleteness Theorem
- Thread starter ItWillNeverWork
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I love Dorking.
I don't. It's not something I know much about and it's made me think.
Field theory isn't a big deal in it's own right. It is basically an abstract form for ordinary arithmetic and also clock arithmetic (although I'm going to be pedantic and say that there can be lots of clock arithmetics going on simultaneously and the clocks all need to have a prime number of elements otherwise its not a field its integer domain if I remember rightly).
What you need to absorb is my point that algebraicist just make up their own rules for what ever reason they fancy. There is no profound truth to 1+1=2 to an algebraist. It's just the way you're playing the game. Admittedly its bad form to use the symbol "+" for a binary operation that's not very +like.
This talk of logic and knowledge reminds me of a paradox.
There's prisoner waiting to be executed and the guard says, "you'll be executed within the week, but on the day you won't know that that's the day you'll be executed."
The prisoner reasons: It can't be the last day of the week because then I'll know that I'll be executed on that day. It can be the day before that because I know that I won't be executed on the last and so I'd know that today would have to be the day of my execution... and so on. Eventually the prisoner realises that guard must be lying he can't possibly be executed and not know that today is the day. Surprisingly enough for the prisoner he was executed on Wednesday.
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I think the moral of this paradox is: Don't treat "know" as a definite logical quantity like "provable" or "true".
There's prisoner waiting to be executed and the guard says, "you'll be executed within the week, but on the day you won't know that that's the day you'll be executed."
The prisoner reasons: It can't be the last day of the week because then I'll know that I'll be executed on that day. It can be the day before that because I know that I won't be executed on the last and so I'd know that today would have to be the day of my execution... and so on. Eventually the prisoner realises that guard must be lying he can't possibly be executed and not know that today is the day. Surprisingly enough for the prisoner he was executed on Wednesday.
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I think the moral of this paradox is: Don't treat "know" as a definite logical quantity like "provable" or "true".
I've heard that paradox expressed previously in terms of knowing that there will be a test in class.
It resolves because induction doesn't work forwards, only backwards. Like all apparent but actually false paradoxes, the solution is a lot more boring than the problem.
I do think this way of resolving it is interesting. It means that knowing that you don't know today is distinct from just not knowing that it's today. Knowing what you know and don't know is an odd thing and can invoke back-to-front induction.
It's like the Russell paradox which relies on the conceptual definition of a set.
A little bit more on field theory. I've got a neat way to explain the central idea.
This is well expressed. I'll sumarise your assumption mathematically:
Quantity x and quantity y need to be related by "<" and ">" (is less than and is greater than). The mathematical rule is the "<" and ">" are consistent with addition or "+" so that:
If x<y then x+z<y+z for any z.
Informally, a field is an algebraic structure where addition and multiplication function just like normal but there need not be any concept of "greater than" or "less than". So adding need not be like stacking things up. A very common field would be the set of complex numbers, that is numbers which involve quantities of the square root of negative one. There is no obvious way to decide whether 2i < 2.
This is well expressed. I'll sumarise your assumption mathematically:
Quantity x and quantity y need to be related by "<" and ">" (is less than and is greater than). The mathematical rule is the "<" and ">" are consistent with addition or "+" so that:
If x<y then x+z<y+z for any z.
Informally, a field is an algebraic structure where addition and multiplication function just like normal but there need not be any concept of "greater than" or "less than". So adding need not be like stacking things up. A very common field would be the set of complex numbers, that is numbers which involve quantities of the square root of negative one. There is no obvious way to decide whether 2i < 2.
