Some people do argue that there is no arithmetical property independent of us because there is no thing on which they would apply independentkly of us. What we would call their arithmetical properties is simply a set of tautologies that do come with them when they are considered but exist no more than them when they are not considered.
But then what would be an undecidable proposition?
You know, about arithmetic, and about machines btw, a lot of people defends idea which are just no more plausible since Godel has proved its incompleteness theorems. Arithmetical proposition are just not tautologies. This is how Russell's and Whitehead logicism has break down. There is a ladder of arithmetical propositions which ask for more and more ingenuity to be proved. Actually arithmetical truth extend far beyond the reach of any consistent machine (and consistent human with comp). There is an infinity of surprise in there. I guess you know that there is no natural number p and q such that (p/q)(p/q) is equal to 2. If mathematical truth were conventionnal, why did the pythagoreans *hide* this fact for so long? So those propositions are neither tautologies, nor conventions. David Deutsch, following Johnson's criteria of reality, would say that such propositions kick back.
> Giving that I hope getting some understanding of the complex human > from something simpler (number property) the approach of those > people will never work, for me.
And certainly vice versa. Though it is difficult to have them saying it explicitely I have the feeling that the reason why they do not want the natural numbers to be out there and even as not possibly being considered as out there is that they do not accept that the complex human be understood from something simpler (number property). They do not even accept the idea being considered, were it as a mere conjecture or working hypothesis. Their more official argument is that such a view would prevent the foundation of human dignity.
Damned!!! If there is one thing which could prevent the foundation of human dignity, it is certainly that totalitarian idea following which some ideas can not even be considered as an hypothesis or conjecture. But that happens all the time. There has been days you could be burned even just because you ask yourself if by chance it was not the sun but the earth which was moving. Are you defending those guys? Are you asking me how to reply to those guy? My suggestion: if many people things like that around you, just leave them. Like Valery said, those who are not willing to use logic with you (that is to argument) are in war with you. Run or kill them! It is not enough they have good intention, if they does not want arguments, they are dangerous for all humans. I like to insist, in Valery spirit, that logic is not a question of truth, but of politeness.
I have not met any of them physically but I had discussion with some of them via Internet. There might not be so many of them but there are. You will find, at least in the US, a lot of people considering the views of evolution and/or of the big-bang as evil.
Then what? If they disagree with dialog and argumentation, *I* will consider them as evil. (btw I think there are much more people like that in France and in Belgium, especially in Belgium, but that's another story).
If they finally have to abandon these positions due to the amount of evidence in favor of it, the last line of defence for their conception of a personal God and for a significant role for Him could be at the level of artihmetical realism. Artihmetical realism by itself (not from a distinct personal God) is therefore seen as evil by them. As I mentionned, they usually do not put it that way. Rather they argue that such a view would prevent the foundation of human dignity and the like.
They make probably the same confusion of those who believe that determinism is in contradiction with free will. Actually I tend to think that Godel's and other incompleteness result makes comp a sort of vaccine against reductionist view of self and reality (and arithmetic). You know "reason" works only through doubt, and through the ability to listen to different opinions. Now with Godel we can say more, which is that "good faith" never fears "reason" and rationality. Sincere Faith can only extend Ratio, and is always open to dialog.