On 24.05.2012 09:52 Bruno Marchal said the following:
On 23 May 2012, at 20:19, Evgenii Rudnyi wrote:
...
nominalism that they are just notation and do not exist as such
independently from the mind.
But that distinction is usually made in the aristotelian context,
where some concrete physical universe is postulated. With comp we
know this is not possible. You can restate it by saying that the
natural numbers are concrete, but that a property like 'being prime"
is abstract. Then mathematicians are mostly realist, because they
believe that "being prime" is an independent property of natural
numbers. for a mechanical generable set, like the set of prime
numbers, you can come back to nominalism through Gödel numbering, and
through the identification of the concept of primes with the number
(machine) which generates all and only the prime numbers. But this
leads to difficulties for the non mechanically generable sets of
numbers, which *do* play a role in the machine/numbers points of
view.
To me this difference "realism vs. nominalism" seems to be related
to the question whether mathematical objects are mental or not.
But with comp, mental is a number's attributes. And eventually
"physical" is a collection of number attribute. If you make
mathematical object mental, and *only* mental, you have to tell me
what you assume at the start in the theory. If you chose something
physical, then you have to abandon comp, and you have to tell how you
relate mental and physical, by using provably non Turing emulable
components. You will lose also the explanation of why something
physical exist, and why it hurts.
In my view, it would be nicer to treat such a question historically.
Your position based on your theorem, after all, is one of possible
positions. In your paper to express your position you employ a normal
human language. Hence I believe that that the question about general
terms in the human language is the same as about the natural numbers.
Again, the ideal world of Plato was not designed for natural numbers only.
Evgenii
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