[FOM] Planned Essay

[Karlis.Podnieks@mii.lu.lv]

Reply to Harvey Friedman <friedman at math.ohio-state.edu>:

I would like to call your attention to the following concept by Sergei Yu. 
Maslov (the author of the inverse method) on two kinds of modeling activities:

a) "left-hemispherical" activities - working in a fixed formal theory (on a 
fixed mathematical structure),

b) "right-hemispherical" activities - changing a theory/structure (or, 
inventing a new one).

See
Maslov, S. Yu. Theory of deductive systems and its applications. With a 
foreword by Nina B. Maslova. Translated from the Russian by Michael Gelfond 
and Vladimir Lifschitz. MIT Press Series in the Foundations of Computing. MIT 
Press, Cambridge, MA-London, 1987. xii+151 pp.

See also "G.Mints on S.Maslov" at 
http://www.mathsoc.spb.ru/pers/maslov/mints.html

Thus, in a sense, we should think of mathematics as a "two-dimensional" 
activity: most of our working time is spent along the first dimension (working 
on fixed mathematical structures), but, sometimes, we need also moving along 
the second dimension (changing our structures or, inventing new ones).

Couldn't this also shed some light on the dispute [FOM]Formalism/Platonism? 
(as you put it in 
http://www.cs.nyu.edu/pipermail/fom/2003-October/007557.html).

Aren't ultraformalists under-estimating the second dimension, and thus - 
representing mathematics as an unordered HEAP of formal theories?

Aren't ultraplatonists over-simplifying the second dimension by representing 
mathematics as a series of formal theories approximating a SINGLE, fixed, but 
non-formalizable structure?

Best wishes,
Karlis.Podnieks at mii.lu.lv