FOM: degrees and foundations

[Raatikainen Panu A K]

Reply to Friedman 10:11  28 Oct 99

To begin with, I would like to thank Friedman for his illuminating 
comments on my private thoughts. The reason why I did put them 
out at all was that I hoped to get some feedback from the experts. 
This was much more than I expected.

Now to the details:
 
> I would state your conclusion more cautiously as follows. "I don't see 
> any relevance of the theory of r.e. degress (etc.) for the foundations of 
> mathematics and the philosophy of mathematics, and the burden of proof 
> is on those who claim that there is such relevance."  

Yes, this is certainly much better. (Can I conclude that, with 
certain qualifications (above), you actually agree with my basic 
point ?)
 
> > (but please note that my claims do not, as such, imply anything  
> concerning the mathematical interest of the field) 
> 
> This is a very common hedge that people make in lots of contexts. The 
> general feeling is that "mathematical interest" is somehow a clearer 
> concept or is more easily understood and agreed upon than such 
> concepts as "foundational interest", "philosophical interest", "general 
> intellectual interest." But I don't agree.

I did not mean to give any such  impression. As a philosopher,  I 
merely wanted to express my incompetence in judging such 
issues (i.e. "purely mathematical interest" - whatever that may be)


> In fact, I find  
> 
> "mathematical interest" or "mathematical importance"  
> 
> is *less* understandable, and *more* ambigious.  
> 
> Of course, to some people it might just mean  
> 
> "complicated, challenging, and intriguing" 
> 
> in which case I would say that it is more understandable and less 
> ambiguous. 
> 
> But normally, it is intended to mean something much more subtle than 
> that. And that's where it becomes quite unclear just what it means. 
> Because it seems to depend on some discriminating view of what (good) 
> mathematics is - and that is something we just don't have. The core 
> mathematicians have not been good at - or even cared very much about - 
> clarifying this in generally understandable and/or useful terms.  

As I said, I did not mean to claim the contrary. But I am happy this 
come out, for getting your opinions on this issue are, I think, very 
interesting. Actually I agree with you completely, but my subjective 
opinion on this question is hardly reliable. 

> >(1)  The most natural decidable theories are the successor arithmetic 
> and Presburger arithmetic (of successor and addition).   
> 
> Well, I'm not quite sure what you mean by "natural" here. The axioms of 
> elementary algebra and of elementary geometry (say as given by Tarski) 
> are very natural. Incidentally, they can be given by finitely many 
> schemes in the sense of postings #65 asnd #66 - I think Tarski presented 
> them this way, among other ways. 
> 
> I have the feeling that the best way to think about what you are getting 
> at is in terms of simplicity. 

Yes, you are right. I was only thinking of theories of natural 
numbers. (By the way, I have found the issues you have dealt with 
in your last few postings extremely interesting. I have thought 
related issues for a couple of years...)

> But there may be foundationally interest purposes to recursion theory 
> and r.e. sets/degrees that are not related to formal systems at all. Note 
> the Conjectures in posting #68 that do not involve formal systems. 

This, again, is certainly a very good qualification. 

*** 

This was, at least for me, an extremely interesting and clarifying.  
Thank you very much !