[FOM] Role of Polemics

[Patrick Caldon]

On Thu, Jan 26, 2006 at 10:39:02AM -0000, Jos? F?lix Costa wrote:
> Triggered by Professor Harvey Friedman, I would like fomers to know about a
> very interesting book that introduces the MATHEMATICAL concepts of
> ?scientist?, ?scientific interaction?, etc.:
> 
> Elements of Scientific Inquiry
> Eric Martin and Daniel N. Osherson
> MIT Press

While I'm not sure it's directly relevant to Harvey Friedman's polemic,
if you're interested in this, you should get in touch with Eric Martin
directly - he's done a lot more work in this direction.  There's also
a paper of his with Frank Stephan available at: 

http://math.uni-heidelberg.de/logic/fstephan/topologylogic.ps 

which should give you some idea of the recent emphasis.  

The thrust is to develop a system of reasoning for situations where we
have some background knowledge (e.g. the laws of arithmetic) and some
collection of possible or potential data (e.g. possible physical
observations).  One typical restriction made is to limit the data
to being positive only, and to have the data trickled into the
"scientist" piecemeal.  So for instance, if we have an astronomical
instrument we might observe the position of the sun, and there
will be a predicate "sun_position(Azimuth,Elevation,Time)", of which we
will only see a finite set of positive examples - these are our data.
We would like to infer some sentence describing the position of the
sun.  There are some difficulties - we often have to abandon
non-standard models of our theory and weaken compactness; also we can
find the sun shows up in an unexpected place one morning and may have
to revise our conclusions. 

In this manner the framework allows us to lose "absolute truth"
(which some would consider impossible in the physical sciences) -
often we will conclude a sentence from our theory up-to some number of
potential mind-changes - but we get a notion of how subject to
future mind changes our (classically too strong) inferences become.