FOM: Clarification of LEM in science

[Neil Tennant]

In the light of Harvey's promptings, I should try to clarify the
challenge I posed to Steve. Perhaps it was over-stated by being a
little tongue-in-cheek.

The suggestion is on the table, from a character I shall call the
Classical Realist (CR), that there is an independent reality, and that
*that* is what justifies the use of the strictly classical Law of
Excluded Middle in our thought about that reality. If pressed for
further justification for this claim, CR often offers something along
the following lines:

[CR's argument for the use of LEM:]
We *need* LEM in order to do science. We can't get enough mathematics
for the purposes of application in science unless we allow ourselves
the use of LEM. Moreover (so Popper once claimed in a famous essay) we
*ought* to use classical logic, because only by doing so can we
subject our scientific theories to the most rigorous criticism
possible.
[End of CR's argument for the use of LEM.]

My original posting on this topic was intended to pose a challenge to
this particular line of justification for the use of LEM. But instead
of making matters as clear as I should have, I posed a rhetorical
challenge to LEM itself, to see whether Steve would offer a different
justification than the one just spelled out above on behalf of
CR. That rhetorical challenge has engendered some understandable
confusion.

In order to eliminate any possible confusion over the criticism of
CR's argument above, let me re-pose it more carefully.

1) Choose your favorite classical mathematical theory (e.g. ZFC) that
will provide you with all the mathematics you need for application in
empirical science. 

2) Re-cast this theory in a language without the universal quantifier.
Classically, no principled objection can be raised to this, since
(x)Fx can be defined as ~Ex~Fx.

3) For each axiom, find a classical equivalent, in the reduced
language, that will be intuitionistically acceptable.

[I'm assuming this can be done; if not, someone like Harvey will tell
me. If it proves to be impossible to find intuitionistically
acceptable versions of the axioms, this is not fatal to the
development of my objection to CR; for the objector will then be
challenging CR on his claim that he (CR) is obliged, if he wants to do
science, to use LEM to reason *from* his own axioms.]

4) Now note that every falsification of a scientific theory is a proof
of absurdity from ultimate premisses among which are (i) mathematical
axioms, (ii) scientific hypotheses, (iii) statements of observation,
and sundry other claims. Put all these sentences into one set, and
call it X.

5) Note that all theorems of mathematics that get "applied" play the
role of lemmas, or cut-formulae, within the overall proofs of absurdity
referred to in (4). The applied theorems are not, themselves, in the
set X of ultimate premisses for any such reductio proof.

6) CR is claiming that for at least one such X one of whose contained
scientific hypotheses is false (in the light of the contained
evidential statements), there is a classical refutation of X but no
intuitionistic refutation of X. This is why, according to CR, we
*need*, in principle, to have recourse to LEM when doing (the
mathematics for) science.

7) This claim of CR, however, contradicts the metatheorem that for
every X in the language specified, any classical refutation of X can
be transformed into an intuitionistic refutation of (some subset of)
X.

So CR's argument is infirmed. LEM is simply NOT indispensable, in
principle, for the purposes of hypothetico-deductive empirical
science.

Now, the question arises: can the foregoing considerations be turned
into either (A) an *explanation* of why we use LEM for the mathematics
applied in science, or (B) a *justification* for that use?

On the prospects for an explanation (option (A)): Perhaps there are
considerations of feasibility involved. Perhaps not.  Are there any
formal results out there on blow-up of classical proofs and
refutations when they are turned into intuitionistic ones?

On the prospects for a justification (option (B)): Classical methods
are shown, by the metatheorem cited, to be *conservative* with respect
to intuitionistic methods when only refutations are involved. So, in
the context of a hypothetico-deductive methodology for science, we
shan't make any *mistakes* by using LEM, and we might well enjoy some
proof-theoretic *gains*.  So let's just use LEM. And note that even if
the classical *mathematics* turns out to be non-conservative (over
some significant class of sentences) with respect to intuitionistic
mathematics, the metatheorem shows us, once again, that *no harm can
result from "applying" those strictly classical theorems of
mathematics when doing science*.

But, remember also: there is no principled *need* for the use of LEM
when doing science. My argument is intended to undermine one standard
kind of indispensability argument for the use of LEM. Ironically, the
considerations that I have advanced deliver, instead, a kind of
conservative extension argument *for* the legitimacy of LEM. Perhaps
this should be called a win-win situation!

I'm still puzzled, though, at how (so it would appear) we don't
strictly need those theorems of classical mathematics for the
refutation of false hypotheses about the world. We only need the
mathematical axioms. Here, it would have to be considerations of
feasibility that come into play. We prove the mathematical theorems
and then use them 'off the shelf' in our scientific theorizing because
of the immense economy effected by having them as cut-formulae.

Neil Tennant