Sunday, April 7, 2013

Letters to a Young Mathematician

(Briefe an einen jungen Mathematiker:  with apologies to Euler.)

This blog has progressively strayed from its original focus on Platonism, owing to its author’s own mathematical incapacity.   Fortunately, great if belated progress is being made in this direction, by a youth of my close acquaintance  -- one who, according to those who know us both, is “much smarter than you are”;  and who, in my own assessment, is just as funny, though (humph!) hardly funnier.   Accordingly, I look forward to bequeathing that aspect of the blog, to abler minds than mine, thereby leaving me the leisure to handle the hamster aspects.

This morning I wrote to him as follows, soliciting contributions. 

*  Real Analysis (or, some topic therein, like Lebesgue) for -- not ‘Dummies’, exactly, but more like:  someone such as you were yourself, mathematically, 8 or 10 years ago:  a diamond in the rough, brilliant but untutored.

* Something re Urysohn Metrization.  This could be e.g.
   (1) A continuation of the “UMT: for real this time” series
   (2) Shedding some light on Uniform Spaces (vide post of that name)
   (3) Looking at metrizability from a different angle entirely, based upon your Real-Analytic work.  After all, most of the foundational topological notions  were originally motivated analytically;  in studying point-set topology before one is analytically mature, the understanding obtained is somewhat abstract and hollow.

* Virtually all of the WDJ posts with “logic” as a Label, concern, not actual mathematical logic, and often not even simple propositional logic, but sheer/mere logical thinking (as opposed, merely, to illogical thinking) as applied to social & legal questions.  Now, you mentioned that your classmates are challenged, not so much (i.e., not yet even) about matters of scope-of-quantifiers or the like, but on the basic matter of logical thought simpliciter.   Essays in this vein might be:

  (1) A psychosocial portrait of the logic class, and the dialectic between teacher and learner.   (This will be a microcosm of our societal intellectual crisis.)
  (2) A cool and even-handed logically-minded analysis of some social phenomenon.
  (3) [something to think about as your experience broadens]  To what extent is Logic fundamental or foundational to mathematics (I anticipate your response:  “very little”;  but after all, the Logicist enterprise has a respectable pedigree, going back to Frege);  or is it more like a separate area within mathematics, coequal with algebra, topology, and number theory;  or is it perhaps almost extra-mathematical, like the History of Mathematics -- something nice to think about, but optional, and unlikely to yield fruitful results ourside itself?


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