This post is a set of notes from a talk I gave on December 5th for the discrete mathematics seminar at Marshall University. I want to argue that logical analysis can reveal the “internal combinatorics” of theorems, using some recent results of Dorais, Dzafarov, Hirst, Mileti, and Shafer [DDHMS 2012] as a particular example of the process.
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I have been required to complete a “responsible conduct of research” training module by the research office at my school. The reason I am commenting is that I was asked to answer the following question “true” or “false”. This is not meant to be an opinion, it is from a quiz on which I am supposed to maximize my score.
True or false: “A good alternative to the current peer review process would be web logs (BLOGS) where papers would be posted and reviewed by those who have an interest in the work.”
I thought that would be interesting to people who follow Boole’s rings. The correct answer, of course, was “false”. Here is the rationale they gave:
“Although the peer review process is evolving, the described system would probably not work very well. It is likely that the peer review process will evolve to minimize bias and conflicts of interest. It is, in the best interest of everyone involved in the research enterprise that the scientific review process be fair and rigorous.”
Just throwing it out there….
Here are several interesting results from computable analysis:
Theorem 1. If $f$ is a computable function from $\mathbb{R}$ to $\mathbb{R}$ and $\alpha$ is an isolated root of $f$, then $\alpha$ is computable.
Corollary 2. If $p(x)$ is a polynomial over $\mathbb{R}$ with computable coefficients, then every root of $p(x)$ is computable.
Theorem 3. There is a effective closed subset of $\mathbb{R}$ which is nonempty (in fact, uncountable) but which has no computable point.
Theorem 4. There is a computable function from $\mathbb{R}$ to $\mathbb{R}$ which has uncountably many roots but no computable roots.
My first paper was “An incompleteness theorem for $\beta_n$ models” with Stephen Simpson [1]. It’s a short paper, but the idea is very pretty. We know that the incompleteness theorem implies there are strange models of arithmetic, but these models often seem mysterious, and it’s hard to see what useful properties they can have. But now suppose that a theory of the form $A+B$ meets the hypotheses of the incompleteness theorem, and moreover this theory proves its own consistency, so that $A+B$ is inconsistent. It follows that if $A$ is true (that is, true in the standard model) then $B$ must be false. In this way, we can use the incompleteness theorem to prove facts about the standard model rather than about nonstandard ones. The idea is originally due to Harvey Friedman in his thesis, I believe.
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I arrived home yesterday from the 2012 Joint Mathematics Meetings in Boston, where I was a co-organizer of the AMS/ASL Special Session on the Life and Legacy of Alan Turing. The talks were wonderful, and the session went very smoothly, for which I can thank my co-organizers Jeff Hirst and Damir Dzhafarov. Continue reading
This post is about a research idea I have been thinking about which is quite different from my usual research. It’s an example of a project with an “old fashioned” feel to it, as if it could have been studied 50 years ago. It’s almost a toy problem, so I haven’t spent too long digging through references yet. For all I know it was solved 50 years ago. Continue reading
For a while I have been thinking about migrating to a content management system for my web page. Previously I had a script I wrote that was essentially a simply wiki to let me edit pages from any web browser (which is important to me). This worked, but it was somewhat primitive.