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*To*: fom@math.psu.edu*Subject*: FOM: submission*From*: Rod Downey <Rod.Downey@MCS.VUW.AC.NZ>*Date*: Sat, 31 Jul 1999 08:38:38 +1200 (NZST)*Sender*: owner-fom@math.psu.edu

I would be happy to continue this discussion over a beer, but fail to see the point in getting involved in a long discusson. Yes I do think that there is not enough introspection in mathematics, and do agree that there is some differences between philosophically driven and structurally driven investigations. The references to the combinatorial argument are given in the paper definability, computability and algebraic structures, available from my web site. Khisamiev proved that if $X$ is a ce presented torsion free group then it is computably presented. It is interesting that I spoke with Chuck miller about this proof, and when I described it to him he understood why they could not find it, since they were looking for a computable isomorphism, whereas the isomorphism is only delta 2) It is the paper called Hierarchies of Torsion free abelian groups) The Higman argument is in the Higman-Scott monograph. Sorry I had thought that we were talking about computability in mathematics, not priority, although many of the arguments mentioned use priority. This was my mistake. BTW it might be pointed out that that the ideas of most areas are not used in others. Computablity theory is remarkable in the prevalence of its ideas in other areas. And to me very interesting that while some areas only use PM rarely (such as combinatorial group theory) there are others where it is dense. One of the problems is that people know little of techniques from other areas. By the way, when a combinatorialist uses a backtracking argument as in e.g. the methodology for matching in a bipartite graph, or in some competitive online algorithm, or a computer scientist uses e.g. methods like the decidability of S2S which, after Gurevich-Harrington certainly resembles an infinite injury argument, are there 'priority?' I suspect not but the ideas are very similar, as one would expect. I remember when I taucht CS students at cornell spector forcing, they immediately said ''oh that's easy, its just the proof that you cannot solve consensus in a distributed environment' after Fischer-Lynch Patterson. Rod

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