FOM: degrees and foundations
Raatikainen Panu A K
Praatikainen at elo.helsinki.fi
Fri Oct 29 05:41:36 EDT 1999
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 !
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