[FOM] constructivism and physics

[Neil Tennant]

On Fri, 10 Feb 2006, Bas Spitters wrote:

> Your argument for using classical theorems seems to be that one can translate 
> such statements to a constructive statement. 

This is not how I would put it, so perhaps I need to say more here to
avoid misunderstanding (which I hope the paper itself does not occasion!).

In the paper I make the point that the classical theorem can end up being
*elided* within the overall reasoning involved in testing a scientific
theory. The schema that I used for the hypothetico-deductive method was

	Math axioms
	    :
	Math theorems, Scientific Hypotheses, Boundary Conditions etc.
	\___________________________  _______________________________/
				    \/
				     :
				Prediction  ,   Observation
				\__________  _____________/
					   \/
				  	    :
					Absurdity


where the dots indicate passages of deduction that one's logic needs to
furnish.

The mathematical theorems could, for example, provide solutions for the
differential equations by means of which the scientific hypotheses express
covariances among physical magnitudes.

In such a reductio as the above, the ultimate set of premises from which
absurdity has been derived consists of:

	the mathematical axioms;
	the scientific hypotheses;
	the statements of boundary conditions etc. (to be met by the
		experimental set-up);
	the observation statements concerning the outcome of
		the experiment.

Call this set Delta.

For the classicist, it would not matter if one put double-negations
anywhere within a sentence. Any universal quantifier in the
mathematical axioms and/or the scientific hypotheses may either be
eliminated by means of the classical translation

	forall x Fx  <=>  not some x not Fx.

Metatheorem: If X classically implies absurdity, then X implies absurdity
in intuitionistic relevant logic. 

Apply this result to Delta. Our empirical refutation of the scientific
hypotheses is now seen to go through, modulo the mathematical *axioms*, by
use only of intuitionistic relevant reasoning. 

So, as I remarked in my reply to Harvey Friedman earlier, we just need to
find a powerful enough set of mathematical axioms in a logical form on
which both the classicist and the intuitionist can agree. Then the
Metatheorem shows that strictly *classical* theorems of mathematics are
not in principle needed for *any* empirical refutation of scientific
theories. The result is completely general. (By the way, these
considerations refute a well-known view of Popper, according to which
classical logic is required in order to afford the strongest possible
testing of our theories. Classical logic is *not* needed for this
purpose!)

Even if the traditional intuitionist refuses to accept IZF(C), the result
should be of interest to anyone who thinks it is alright to use the axioms
of set theory as the mathematical axioms in the schema above. What we then
learn is that we shall never need any strictly classical set-theoretic
results in order to apply mathematics in empirical science. Only the
intuitionistically extractable content of those axioms will be needed, in
order to be able to refute a scientific theory by appeal to empirical
evidence.

The Metatheorem is proved as follows. Let Pi be a proof of absurdity from
X. By the G"odel-Glivenko-Gentzen theorem, transform Pi into an
intuitionistic proof Sigma of absurdity from X. Normalize Sigma, to
obtain a normal proof Theta. Then extract what can be called the
"relevant" kernel of Theta. Call this Omega. Omega will be a proof in
intuitionistic relevant logic. Its conclusion will be absurdity, and all
its undischarged assumptions will be in X.

If one had used a strictly classical theorem in the schema above, that
theorem would be elided during the GGG-cum-normalization process. The
reductio Omega would rest only on mathematical *axioms* plus the
contingent premises (scientific hypotheses, boundary condition statements,
observation statements etc.).

Neil Tennant