[FOM] Empirically relevant undecidability

[Neil Tennant]

Empirically Relevant Undecidability (ERU)

Consider the standard hypothetico-deductive model of scientific
theory-testing, where the descending dots indicate passages of logical
reasoning:

Mathematical Axioms
	 :
	 :				      Initial and	
Mathematical Theorems, Scientific Hypotheses, Boundary Conditions
\________________________________  _____________________________/
                                 \/
				 :		Auxiliary assumptions
				 :		about apparatus
				 :		      :
				 :		      :
		            Predictions   Observations/Measurements
			    \_______________  ____________________/
					    \/
					    :
					    #

Usually, the Mathematical Theorems used are ones that provide, say,
solutions to the differential equations that, according to the
Scientific Hypotheses, govern the time-evolution of certain physical
magnitudes.

The deductive passage from Mathematical Axioms to Mathematical
Theorems is therefore needed in order to serve up the "applied math"
or the "applicable math" that might be required in order to test any
(set H of) Scientific Hypotheses.

Let the axioms form the set T and let a desired "applicable" and
would-be theorem of the mathematical language be S. That is, it would
be nice to have S as a theorem from those axioms T, because S could be
"plugged in" as a Mathematical Theorem in the above schema in order to
test the Scientific Hypotheses H that currently interest us. Let us in
such a situation call S an "H-desirable would-be theorem of T". This
is an intensional notion; for all we know, S is not a theorem of
T. What we *do* know is that, for the purposes of testing H, we would
like S to turn out to be a theorem of T, even if only because T is all
we can currently think of as a basis for proving S.

The foundations of mathematics teaches us that for many powerful
theories T, there are sentences S that are independent of T, provided
that T is consistent.

Question: does anyone know, for the various likely current theories T
(say, ZF; or ZFC; or ZF+V=L; or Z_2; or ...) of a mathematical
sentence S, and any scientific hypothesis H, such that S is an
H-desirable would-be theorem of T, but S is provably independent of T,
on the assumption that T is consistent?

If not, is this because it can be shown that existing theories T are
complete for all those H-desirable would-be theorems, for any
reasonable range of scientific hypotheses H?

Better: could it be shown that *rather weak* theories T are complete
in the foregoing sense? ("weak" to be understood here by reference to
the hierarchy of consistency strengths of mathematical theories)

Put in a nutshell (or two): 
(a) Is our existing mathematics complete enough for all the possible
demands of natural science?
(b) Is a surprisingly weak fragment of existing mathematics complete
enough for all the possible demands of natural science?

I would welcome any help in answering these questions, even if it
consists just in the formulation of further questions aimed at
clarifying the intended use of the informal terms used in posing them.

___________________________________________________________________
Neil W. Tennant
Professor of Philosophy and Adjunct Professor of Cognitive Science

http://people.cohums.ohio-state.edu/tennant9/

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