This post on Scott Aaronson’s weblog is every internet discussion thread in microcosm (except for ours, which are shockingly polite by internet standards). Scott thought he was explaining the notion of non-constructive proof to a dense but argumentative student. If you read the comment thread, though, you’ll discover that the “student” thought Scott was some sort of nut who imagined he’d invented a computer more powerful than a Turing machine.
Back from vacation
I’m back from vacation. Tragically, my laptop, one that I literally took with me around the world, has died. But my commitment to your mathematical pleasure is so high that I plan on breaking into my neighbors’ houses to use their computers to post to the site.
My way or the highway.
I have to admit that I have a somewhat dismissive nature sometimes, and have been known to make critical remarks for non-public consumption; “Programmers cannot do math at ALL” (hi Dale!) Of course, this is more out of shock than a belief in any natural order. I firmly believe that anyone of everyday intelligence can learn math. I am of the opinion that I could teach calculus* to a dead twig if the twig where sufficiently motivated.
Which is why I was happy to see that someone had written a blog entry on learning math being what you make of it. The only thing I would criticize is that he is completely wrong.
Ok..just kidding, but I do have to say that I would not follow his councel on exercises. When I read a GTM on a new subject (papers don’t usually have many exercises :), I don’t really view the problem sets as seperate from the explanitory text – I do every single problem. This is because the author of the text did not view the problem sets as seperate either. It goes beyond “having been shown the idea, cement it in your mind with the excercises”. Most of the time, realizations that the author wants you to have are set up in the problem sets because they would be TOO padantic and verbose in the main text.
*This isn’t restricted to calculus of course.
Enigma Machines
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.
Popular Math
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.
Week 227
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.
Knowing what you don’t know is hard
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.
In Our Time
Over at the BBC, Melvyn Bragg presents a weekly radio programme called In Our Time in which are discussed a wide variety of subjects ranging from art and philosophy to politics and drama. Surprisingly, considering the wide range of human culture covered, Bragg devotes a good proportion of episodes to mathematics. He’s just presented a programme on the Negative Numbers and recently presented one on Prime Numbers. If you dig into the science archives you’ll find recordings of programmes on Zero, Infinity, π, Chaos Theory and Renaissance Mathematics. Some of these are conveniently available as podcasts.
(Note that some of the comments about the mathematical topics on the In Our Time website are a little inaccurate, eg. “a team of researchers…calculated the highest prime number”, but the actual guests on these shows are generally mathematicians or historians of mathematics who know what they are talking about.)
Plus Magazine 38
Issue 38 of Plus is now out!
Vacation, and McCloskey on Mathematics
I’ll be on vacation for the next few days, so I leave you to the tender ministrations of my co-bloggers, and this anecdote (from Deirdre McCloskey’s Secret Sins of Economics):
I have a brilliant and learned friend who is an intellectual historian of note. He and I were walking to lunch in Iowa City one day and I said offhandedly, assuming he would of course know this, that mathematics was one of the great achievements of Western culture. He was so astonished by the claim that he stopped short and argued with me there on the sidewalk by the Old Capitol Mall: “Surely math is like plumbing: useful, but hardly in touch with deeper things; hardly a cultural achievement!” I tried to persuade him that he felt this way only because he had no acquaintance with mathematics, but I don’t think I succeeded.