On 28 Feb 2013, at 14:58, Stephen P. King wrote:

On 2/28/2013 7:46 AM, Bruno Marchal wrote:

On 28 Feb 2013, at 05:04, meekerdb wrote:
You are assuming that justification comes from logic; and indeed it is too much to expect from such a weak source. I look for such "justification" as can be found from experience, which you demoted to mere "motivation".

Hi Bruno and Brent,

Where did I say "motivation"? I use the term "intuition", and I demote nothing, as it correspond to to the first person (the hero of comp, the inner God, the third hypostase, Bp & p; S4Grz1, etc.).

   ISTM that 'motivation' is a 3p view of 'intuition'!

I don't see this at all. Motivation is somehow even more a psychological notion than 'intuition', which admit logical specification.

But justification for me invokes "proof", formal or informal.

   Justification requires a model and/or implementation,no?

Not necessarily, in case of formal justification, or in first order logic: we need only formulas and sequences of formulas, at the meta- level.

[BM] I don't think that what you ask is possible, even if I am pretty sure that x + 0 = x, x + s(y) = s(x + y), etc.

I'm not at all sure that there is successor for every x.

Then you adopt ultrafinitism, and indeed comp does not make sense with such hypothesis, and UDA1-7 suggests that ultrafinitism might save physicalism, but step 8 put a doubt on this. The axiom that all natural numbers have a successor is used in basically all scientific paper though. You need it, or equivalent, to define "machine", "formal systems", "programs", "Church's thesis", "string theory", "eigenvector", "trigonometry", etc.

I need to be sure that I understand this: Numbers are prior to computations. Is that correct?

Once you agree on the axioms and rules of elementary arithmetic, numbers and computations coexist, like even numbers and prime numbers. You can' have one without the other.

If so, then ultrafinitism fails, but if computations are prior to numbers, ultrafinitism (of some kind) seems inevitable. I have always balked at step 8 in that is seems a bridge too far... Why does the doubt have to be taken so far?

It is a conclusion. We will come back on step 8 on the FOAR list, soon. In your neutral monism, primary matter (and thus time and space) also does not exist. I don't see why you have a problem with this non- existence at the ontological level, given that those have to be explained at some other level.

My intuition doesn't reach to infinity. It seems like an hypothesis of convenience.

I propose a theory, that's all. You don't need to believe in infinity, unlike in set theory (yet also used by many). You need just to believe (assume) that 0 ≠ s(x), and that x ≠ y entails s(x) ≠ s(y). Notion like provability and computability are based on this.


I still don't understand how we cannot assume some implicit set with even arithmetic realism. How are integers not a set?

You can assume the numbers, without assuming sets. That means that set will not be "first order citizen" in the reality that you assume, a bit like classes in ZF set theory. Set appears at the metalevel, when you assume only numbers. They can appear also as "mental" objects in the mind of the relative numbers, but they are not existing objects, you can't prove ExP(x) with x denoting them, unless you represent a set by a numbers, which can be done for the RE sets, but not for any set. But yes, some set will exist, even explicitly, through some possible representations. But those sets are not assume, then, they are proven to exist.



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