# Re: On the ontological status of elementary arithmetic

```On 11/2/2012 12:55 PM, Bruno Marchal wrote:
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On 01 Nov 2012, at 21:42, Stephen P. King wrote:

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```On 11/1/2012 11:39 AM, Bruno Marchal wrote:
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Enumerate the programs computing functions fro N to N, (or the equivalent notion according to your chosen system). let us call those functions: phi_0, phi_1, phi_2, ... (the phi_i)
```Let B be a fixed bijection from N x N to N. So B(x,y) is a number.

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The number u is universal if phi_u(B(x,y)) = phi_x(y). And the equality means really that either both phi_u(B(x,y)) and phi_x(y) are defined (number) and that they are equal, OR they are both undefined.
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In phi_u(B(x,y)) = phi_x(y), x is called the program, and y the data. u is the computer. u i said to emulate the program (machine, ...) x on the input y.
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OK, but this does not answer my question. What is the ontological level mechanism that distinguishes the u and the x and the y from each other?
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The one you have chosen above. But let continue to use elementary arithmetic, as everyone learn it in school. So the answer is: elementary arithmetic.
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```Dear Bruno,'

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If there is no entity to chose the elementary arithmetic, how is it chosen or even defined such that there exist arithmetic statements that can possibly be true or false?
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Nobody needs to do the choice, as the choice is irrelevant for the truth. If someone choose the combinators, the proof of "1+1= 2" will be very long, and a bit awkward, but the proof of KKK = K, will be very short. If someone chose elementary arithmetic, the proof of 1+1=2 will be very short (Liz found it on FOAR), but the proof that KKK = K, will be long and a bit awkward. The fact is that 1+1=2, and KKK=K, are true, independently of the choice of the theory, and indeed independently of the existence of the theories.
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Dear Bruno,

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No, that cannot be the case since statements do not even exist if the framework or theory that defines them does not exist, therefore there is not 'truth' for a non-exitence entity.
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We can assume some special Realm or entity does the work of choosing the consistent set of arithmetical statements or, as I suggest, we can consider the totality of all possible physical worlds
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As long as you make your theory clearer, I can't understand what you mean by "physical world", "possible", "totality", etc.
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I use the same definitions as other people use. I am not claiming a private language and/or set of definitions, even if I have tried to refine the usual definition more sharply than usual.
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Physical world:

http://oxforddictionaries.com/definition/english/physical?q=Physical

1) relating to the body as opposed to the mind:
/a range of physical and mental challenges/
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2) relating to things perceived through the senses as opposed to the mind; tangible or concrete:
```the physical world
3) relating to physics or the operation of natural forces generally:
/physical laws/"

http://en.wikipedia.org/wiki/Possible_world

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"Those theorists who use the concept of possible worlds consider the actual world to be one of the many possible worlds. For each distinct way the world could have been, there is said to be a distinct possible world; the actual world is the one we in fact live in. Among such theorists there is disagreement about the nature of possible worlds; their precise ontological status is disputed, and especially the difference, if any, in ontological status between the actual world and all the other possible worlds."
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Totality: http://www.merriam-webster.com/dictionary/totality
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1:*an aggregate amount*:*sum <http://www.merriam-webster.com/dictionary/sum>,whole <http://www.merriam-webster.com/dictionary/whole>
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/a/*:*the quality or state of beingtotal <http://www.merriam-webster.com/dictionary/total>*:*wholeness <http://www.merriam-webster.com/dictionary/wholeness>
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as the implementers of arithmetic statements and thus their "provers". Possible physical worlds, taken as a single aggregate, is just as timeless and non-located as the Platonic Realm and yet we don't need any special pleading for us to believe in them. ;-)
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?
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I refuse to believe that you cannot make sense of what I wrote. Can you understand that I find your interpretation of Plato's Realm of Ideals to be incorrect? You seem to have read one book or taken one lecture on the subject and not read any more philosophical discussion of the ideas involved. I have asked you repeatedly to merely read Bertrand Russell's small book on philosophy - with is available on-line here http://www.ditext.com/russell/russell.html, but you seem unwilling to do that. Why?
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Bruno

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My thinking here follows the reasoning of Jaakko Hintikka. Are you familiar with it? Game theoretic semantics for Proof theory <http://www.hf.uio.no/ifikk/forskning/publikasjoner/tidsskrifter/njpl/vol4no2/gamesem.pdf>
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How about considering that there are alternatives to your idea of timeless Truths? Jaakko Hintikka does a nice job exploring one of those alternatives!
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Onward!

Stephen

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