meekerdb wrote:
> On 10/4/2011 1:44 PM, benjayk wrote:
>> Bruno Marchal wrote:
>>>> Bruno Marchal wrote:
>>>>>> But then one 3-thing remains uncomputable, and undefined,
>>>>>> namely the very foundation of computations. We can define
>>>>>> computations in
>>>>>> terms of numbers relations, and we can define number relations in
>>>>>> terms of
>>>>>> +,*,N. But what is N? It is 0 and all it's successors. But what is
>>>>>> 0? What
>>>>>> are successors? They have to remain undefined. If we define 0 as a
>>>>>> natural
>>>>>> number, natural number remains undefined. If we define 0 as having
>>>>>> no
>>>>>> successor, successor remains undefined.
>>>>> All theories are build on unprovable axioms. Just all theories.
>>>>> Most scientific theories assumes the numbers, also.
>>>>> But this makes not them undefinable. 0 can be defined as the least
>>>>> natural numbers, and in all models this defines it precisely.
>>>> But natural *numbers* just make sense relative to 0 and it's
>>>> successors,
>>>> because just these are the *numbers*. If you define 0 in terms of
>>>> natural
>>>> numbers, and "least" (which just makes sense relative to numbers), you
>>>> defined them from something undefined.
>>>> So I ask you: What are natural numbers without presupposing 0 and its
>>>> successors?
>>> This is a bit a technical question, which involves logic. With enough
>>> logic, 0 and s can be defined from the laws of addition and
>>> multiplication. It is not really easy.
>> It is not technical at all. If you can't even explain to me what the
>> fundamental object of your theory is, your whole theory is meaningless to
>> me.
>> I'd be very interested in you attempt to explain addition and
>> multplication
>> without using numbers, though.
> It's easy.  It's the way you explain it to children:  Take those red
> blocks over there and 
> ad them to the green blocks in this box.  That's addition.  Now make all
> possible 
> different pairs of one green block and one red block. That's
> multiplication.
OK. We don't have to use numbers per se, but notions of more and less of
Anyway, we get the same problem in explaining what addition and
multiplication are in the absence of any concrete thing of which there can
be more or less, or measurements that can be compared in terms of more and

meekerdb wrote:
>> Bruno Marchal wrote:
>>> But to get the comp point, you don't need to decide what numbers are,
>>> you need only to agree with or just assume some principle, like 0 is
>>> not a successor of any natural numbers, if x ≠ y then s(x) ≠ s(y),
>>> things like that.
>> I agree that it is sometimes useful to assume this principle, just as it
>> sometimes useful to assume that Harry Potter uses a wand. Just because we
>> can usefully assume some things in some contexts, do not make them
>> universal
>> truth.
>> So if you want it this way, 1+1=2 is not always true, because there might
>> be
>> other definition of natural numbers, were 1+1=&.
> It's always "true" in Platonia, where "true" just means satisfying the
> axioms.  In real 
> life it's not always true because of things like: This business is so
> small we just have 
> one owner and one employee and 1+1=1.
Yeah, but it remains to be shown that platonia is more than just an idea. I
haven't yet seen any evidence of that.
Bruno seems to justify that by reductio ad absurdum of 1+1=2 being dependent
on ourselves, so 1+1=2 has to be true objectively in Platonia. I don't buy
that argument. If our mind (or an equivalent mind, say of another species
with the same intellectual capbilites) isn't there isn't even any meaning to
1+1=2, because there is no way to interpret the meaning in it. It only seems
to us to be true independently because we defined it without explicit
reference to anything outside of it. But this doesn't prove that it is true
independently anymore than the fact that Harry Potter doesn't mention he is
just a creation of the mind makes him exist independently of us eternally in

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