On Sunday, December 16, 2018 at 5:53:57 PM UTC-6, Brent wrote:
>
>
>
> On 12/16/2018 1:56 PM, Jason Resch wrote:
>
>
>
> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker <meek...@verizon.net
> <javascript:>> wrote:
>
>>
>>
>> On 12/15/2018 10:24 PM, Jason Resch wrote:
>>
>>
>>
>> On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker <meek...@verizon.net
>> <javascript:>> wrote:
>>
>>>
>>>
>>> On 12/15/2018 6:07 PM, Jason Resch wrote:
>>>
>>>
>>>
>>> On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker <meek...@verizon.net
>>> <javascript:>> wrote:
>>>
>>>>
>>>>
>>>> On 12/15/2018 5:42 PM, Jason Resch wrote:
>>>>
>>>> hh, but diophantine equations only need integers, addition, and
>>>>> multiplication, and can define any computable function. Therefore the
>>>>> question of whether or not some diophantine equation has a solution can
>>>>> be
>>>>> made equivalent to the question of whether some Turing machine halts. So
>>>>> you face this problem of getting at all the truth once you can define
>>>>> integers, addition and multiplication.
>>>>>
>>>>>
>>>>> There's no surprise that you can't get at all true statements about a
>>>>> system that is defined to be infinite.
>>>>>
>>>>
>>>> But you can always prove more true statements with a better system of
>>>> axioms. So clearly the axioms are not the driving force behind truth.
>>>>
>>>>
>>>> And you can prove more false statements with a "better" system of
>>>> axioms...which was my original point. So axioms are not a "force behind
>>>> truth"; they are a force behind what is provable.
>>>>
>>>>
>>> There are objectively better systems which prove nothing false, but
>>> allow you to prove more things than weaker systems of axioms.
>>>
>>>
>>> By that criterion an inconsistent system is the objectively best of all.
>>>
>>>
>> The problem with an inconsistent system is that it does prove things that
>> are false i.e. "not true".
>>
>>
>>> However we can never prove that the system doesn't prove anything false
>>> (within the theory itself).
>>>
>>>
>>> You're confusing mathematically consistency with not proving something
>>> false.
>>>
>>
>> They're related. A system that is inconsistent can prove a statement as
>> well as its converse. Therefore it is proving things that are false.
>>
>>
>> But a system that is consistent can also prove a statement that is false:
>>
>> axiom 1: Trump is a genius.
>> axiom 2: Trump is stable.
>>
>> theorem: Trump is a stable genius.
>>
>
> So how is this different from flawed physical theories?
>
>
> The difference is that mathematicians can't test their theories.
>
> Brent
>
Chaitin ("The Limits of Reason") says math is "quasi-empirical". One aspect
of experimental mathematics
[ https://en.wikipedia.org/wiki/Experimental_mathematics ]: Some proofs may
be so big and complex, the biggest supercomputer can only "test" them by
running for months. To replicate that "experiment", another supercomputer
would need to retest it (because one might not 100% be sure of the first
supercomputer.)
- pt
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