The latest issue of the Post-Autistic Economics Review is now out, availableÂ here.Â Â It has an interesting article by philosopher Donald Gillies arguing against theÂ centrally-organized reviews of university research activities which British academics have had to endure these last 20 years, and which now look likely to be adopted in Australia, NZÂ and elsewhere.Â One argument he makes is that one’s peers are usually quite bad at judging the long-run impact and quality of one’s research, especially when the research is innovative, and Gillies gives the example of Frege’s Begriffsschrift, the first axiomatic treatment of propositional and predicate calculus.Â When this was published in 1879, it was slammed by Frege’s contemporaries, and it was only recognized for the seminal work it is two decades later.Â If Frege had been working in a British University a hundred years later, both he and his department may have faced termination by his university administration, given the hostility that his own peers felt towards his work; lots of departments have been closed, and academics made unemployed, as a result of the peer assessments of the British Research Assessment Exercise (RAE).
A longer version of Gillies’ paper is availableÂ on his web-site, here.Â Â
An easy-to-readÂ article on mathematics in biology.
Ronnie Brown, of the University of Wales at Bangor, UK, recently posted some very interesting remarks on mathematical speculation to the categories list. With his permission, I am reposting them here:
“The situation is more complicated in that what could be classed as speculation may get published as theorem and proof. For example, in algebraic topology, sometimes proofs of continuity are omitted as if this was an exercise for the reader, yet the formulation of why the maps are continuous (if they are necessarily so) may contain a key aspect of what should be a complete proof. This difficulty was pointed out to me years ago by Eldon Dyer in relation to results on local fibration implies global fibration (for paracompact spaces) where he and Eilenberg felt Dold’s paper on this contained the first complete proof. I have been unable to complete the proof in Spanier’s book, even the second edition. (I sent a correction to Spanier as the key function in the first edition was not well defined, after Spanier had replied `Isn’t it continuous?’) Eldon speculated (!) that perhaps 50% of published algebraic topology was seriously wrong!
van Kampen’s original 1935 `proof’ of what is called his theorem is incomprehensible today, and maybe was then also.
Efforts to give full details of a major result, i.e. to give a proof, are sometimes derided. Of course credit should be given to the originator of the major steps towards a proof.
Grothendieck’s efforts to develop structures and language which would reduce proofs to a sequence of tautologies are notable here. Colin McLarty’s excellent article on `The rising sea: Grothendieck on simplicity and generality ‘ is relevant.Some scientists snear at the mathematical notion of rigour and of proof. On the other hand many are attracted to math because it can give explanations of why something is true. But `explanations’ need a higher level of structural language than for what might be called proofs.
I can’t resist mentioning that one student questionaire on my first year analysis wrote `Professor Brown puts in too many proofs.’ So I determined to rectify the situation, and next year there were no theorems, and no proofs. However there were lots of statements labelled `FACT’ followed by several paragraphs labelled `EXPLANATION’. This did modify the course because something labelled `explanation’ ought really to explain something! I leave you all to puzzle this out!
In homotopy theory, many matters, such as the homotopy addition lemma, had clear proofs only years after they were well used.Surely much early algebraic topology is speculative, in that the language has not yet been developed to express concepts with rigour so that a clear proof can be written down. It would be a curious ahistorical assumption that there is not at this date another future level of concepts which require a similar speculative approach to reach towards them.”
(Ronnie Brown, posted 2006-03-14 to the categories list).
The Enigma machine was a mechanical encoding/decoding device used by the German military in WWII and which the British were able to successfully crack on occasion. It is now possible to purchase a Do-it-yourself Enigma machine and even a paper-version of one.
The mathematicians Ron Brown and Tim Porter, at the University of Wales in Bangor, UK, have long been at the forefront of mathematics popularization in Britain. (This is in addition to their very valuable contributions to algebraic topology, category theory and theoretical computer science!) Their web-pages include a page of articles and links about popular mathematics, reasons for studying math, the teaching of math, etc.
More logic can be found in John Baez’s latest This Week’s Finds in Mathematical Physics (no. 227), along with pictures of Martian sand-dunes.
Epistemic modal logic was invented by Finnish philosopher Jaako Hintikka to represent knowledge and belief (in a book published in 1962), and is now used by computer scientists to model and design systems of autonomous software agents. It uses modal operators to indicate which propositions are known to which agents.
A common modal system for beliefs is C. I. Lewis‘ system S5, which (among other axioms) assumes that agents know what it is they know (positive introspection) and know what it is that they don’t know (negative introspection). (In other words, if an agent does not know whether or not some proposition is true, then the agent knows that he does not know whether or not that proposition is true). These are quite strong assumptions, and have been criticized as being unrealistic. Two computer scientists, Joseph Halpern and Leandro Chaves Rego, have now identified negative introspection as the axiom which makes the satisfiability problem for S5 NP-complete.
As an aside, discussion of positive and negative introspection by epistemic logicians meant that they fully understood Donald Rumsfeld’s statements about known unknowns vs. unknown unknowns.
Issue 38 of Plus is now out!
Finding those math theorems or conjectures just too hard to prove? Then, you need to attend this workshop.
Understanding of geometry is innate, and cannot be learnt. Discuss.