FOM: intuitionistic mathematics and building bridges

[Neil Tennant]

There's an implicit assumption that, it seems, everyone has been making in 
the discussion about building bridges, and which deserves to be challenged.

The assumption is that, when judging of the adequacy of a particular kind
of `logically parsimonious' mathematics (say, intuitionism), we should be
insisting that it should provide the *theorems* supposedly needed for 
application within our physical theorizing.

But do we really *need* these theorems? This question concerns a need
*in principle*, rather than in practice.

The schematic picture of the logical relations involved in physical theorizing
(and the testing of physical theories) is roughly as follows:

	Mathematical
	axioms
	:
	:
	Mathematical				Initial and
	theorems		   Scientific	boundary     Auxiliary
	(e.g. solns. to d.e.'s) ,  hypotheses , conditions , assumptions
	\_______________________________  _______________________________/
					\/
					 :
					 :
				     Prediction   ,    Observations
				    \___________  ________________/
						\/
						 :
						 :
					      Absurdity

Here the dots indicate deductive progress downwards. The ultimate premisses
of a reductio proof for a scientific theory include the mathematical axioms,
the scientific hypotheses, the initial and boundary conditions characterizing
the experimental situation, the auxiliary assumptions we make about the
measuring apparatus, etc., and the observation statements.

The schema above shows the overall logical form
of a reductio ad absurdum of a set of scientific hypotheses. NOTE THAT THE
MATHEMATICAL THEOREMS ARE NOT AMONG THE OVERALL ASSUMPTIONS FOR THE
SUBPROOF OF THE PREDICTION, NOR ARE THEY AMONG THE OVERALL ASSUMPTIONS FOR
THE REDUCTIO PROOF.  IT IS THE MATHEMATICAL **AXIOMS**, NOT THE THEOREMS,
THAT FEATURE AMONG THE ULTIMATE ASSUMPTIONS ON WHICH THE PREDICTION, AND THE
RESULTING ABSURDITY, LOGICALLY REST.

When the theory makes successful predictions, then the last bit of proof
indicated, namely the one precipitating absurdity after feeding in the 
observation statements, will be missing.

The well-known `Quine-Duhem' problem (which ought to have been called the
`Duhem-PoinCare-Carnap-Quine' problem) is: in the presence of such a
reductio proof, which assumptions in the reductio should be rejected?--
the scientific hypotheses? or the statement of initial and boundary conditions?
or the auxiliary assumptions? or the observation statements?  We might not have
controlled the experimental conditions properly; or the measuring equipment
might have malfunctioned, or been incorrectly calibrated; or we might have
hallucinated, or committed an error of parallax, in taking readings reported
by the observation statements. When these sources of potential error are 
eliminated by repeating the experiment, possibly with different equipment,
and having more observers corroborate the results, the insecurity arising
from a persistent conflict between prediction and observation then focuses
on the scientific hypotheses themselves.

Note, however, that we hardly ever call the mathematical *axioms* into
question. We *might* call into question our earlier *derivations* of the applied
mathematical theorems, wondering whether perhaps there
had been a fallacy committed in the course of `proving' them from the
axioms; but, once we have reassured ourselves on that score by going
through the proofs once more, we refrain from calling the *axioms* into
the circle of Quine-Duhem doubt.

The mathematical theorems stand as `cut' formulae in the proof schema above,
mediating between the pure and the applied parts.
There is the deductive work involved in proving them from the mathematical
axioms. Subsequently, we help ourselves to these theorems `off the shelf',
as it were, when we *apply* them in the context of the scientific hypotheses,
in order to generate predictions about the load-bearing capacity of our 
planned bridge, etc. 

Take any such theorem S so applied.
IN A CUT-FREE PROOF of a prediction from the set 

	Math.axioms + Scientific hypotheses + I & B conditions + Aux Ass

there might not even *be* a sub-proof whose conclusion is S and whose
premisses are all among the mathematical axioms. That is, S might not
be strictly *needed* for the very applications that it facilitates!
It simply stands as a deductive halfway house, mediating between the
pure part and the applied part of our overall reasoning.

Indeed, suppose all universal formulae (x)Fx are replaced, in the obvious
recursive fashion, by ~Ex~Fx. For the classicist, this makes no difference
at all. In the first-order language without the universal quantifier, we
have the following results:

	if X classically implies # then X intuitionistically implies #;
	if X classically implies S then X intuitionistically implies ~~S.

Thus INTUITIONISTIC LOGIC SUFFICES for the `prediction-generating' and
the `theory testing' schemata above. Predictions S are decidable, since they are
expressed as atomic sentences. Thus the inference ~~S |- S is acceptable on
them.

Indeed, INTUITIONISTIC RELEVANT LOGIC SUFFICES in this regard, since in the
foregoing results we can strengthen from "intuitionistically" to "intuitionistically relevantly". 

All that needs to be assured is that the mathematical axioms that result
from translating away all the universal quantifiers remain acceptable to
the intuitionist. 

The weaker logic suffices for mathematical physics, without necessarily
sufficing for the mathematical theorems usually applied within the 
mathematical physics!

Perhaps one explanation for the strong preference for classical logic, and the
classical mathematics that goes with it, is that in the classical case
we *can* separate off the work of proving pure but applicable mathematical 
theorems from the work of applying those theorems when making scientific 
predictions and testing scientific theories. Since the use of cut formulae 
greatly reduces deductive load, they become an intellectual bridge to easier 
conclusions about real bridges.

Neil Tennant