Bruno Marchal wrote:
> Hi Mirek,
> Long and perhaps key post.
Thank you a lot for a prompt and long reply. I am digesting it :-)
Just some quick comments.
> There is no shame in being ignorant. Only in staying ignorant :)
I've ordered the dialogue from a second-hand book shop :-) The Stanford
"Arguably, it is his (Plato) greatest work on anything."
So I'll give it a try :-)
>> judgment, and, finally, knowledge as a true judgment with an account.
> The remarkable thing is that if you accept to modelize "account" by
> "sound machine provability",
This is probably the key problem for me. I know next to nothing about
provability, the logic of provability, PA/ZF provers.
I know that quite often you reference Boolos 1993 - The Logic of
Provability. I took a look at it at Google Books preview but ... there
is something missing in my education. From the beginning I am puzzled
with "Why?, what?". What a headache :-)
> In french students are burned alive if they dare to create new
> adjective, and I thought that in English we have more freedom, but I may
> be wrong. Sorry.
I'd grant this freedom to rational native speakers only :-)
> x divides y if and only if it exists a number z such that y = x*z.
I don't dare to correct your english but "there is/exists a number ..."
is what I would write.
>> Ad 3) If natural numbers and their relations are the only entities which
>> do exist then me, you, everything is a recipe of a Turing-computable
> No. Not at all. Sorry. Gosh, you will be very surprised if you follow
> the UDA-7. On the contrary. Arithmetical truth VASTLY extends the
> computable domain. Most relations between numbers are not Turing emulable.
Aha! Then I really have a wrong mental picture of your work. I
understood to arithmetical realism along the lines of this quotation
from the Stanford article on realism:
"According to a platonist about arithmetic, the truth of the sentence '7
is prime' entails the existence of an abstract object, the number 7.
This object is abstract because it has no spatial or temporal location,
and is causally inert. A platonic realist about arithmetic will say that
the number 7 exists and instantiates the property of being prime
independently of anyone's beliefs, linguistic practices, conceptual
schemes, and so on."
So I thought that you essentially take
a) Numbers and their properties and relations exists.
b) Now, since you don't assume existence of anything else => your body,
your bike and coffee must emerge as patterns in the world of numbers.
c) Taking the Church-Turing thesis, these patterns are Turing-computable.
d) Definitely, the vast majority of all patterns is not Turing-computable.
This is how I have thought about your working framework. Notice, that I
don't talk about what you try to show, argue for, want to end up with etc.
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