FOM: Reply to Friedman on scientific method

[Neil Tennant]

Harvey Friedman wrote on Fri Oct 16 09:44 EDT 1998:

> ... it is commonly believed that it is easiest to use
> classical analysis for natural science than intuitionistic analysis,
> including such things as the intermediate value theorem, attainment of
> maxima, etcetera, and to be able to use such things as "if x is not <= 0
> then x > 0," etcetera. These are not intuitionistically provable
> under normal setups.

Yes, I'm aware of that; which is why I made the point that it was
interesting how, upon further reflection, those strictly classical
theorems do not survive the transformation of the scientific
theory-reductio into intuitionistic relevant logic. (The reductio in
question is any successful falsification of a scientific theory by
observational evidence.)

> Any kind of absolute necessity of using classical logic is another matter
> entirely. Claiming that it is far easier to use classical logic is much
> weaker than it being absolutely necessary.

Agreed. The major issue is *justification* v. *convenience*. The
intuitionist's position is that the strictly classical mathematics
involved does not possess a certain kind of *justification*. The
classicist's response (e.g., your own) is that it nevertheless affords
an indispensable *convenience*. There are deep philosophical issues
involved here: can convenience for existing practice ever provide the
kind of justification that is demanded by the philosopher, or by the
epistemologist, or by the meaning theorist, for alleged knowledge that
we would like to call a priori?

In response to my "Informal Claim: All that Logic is needed for in the
course of doing natural science is to produce such *refutations* as
there may be of falsifiable theories about the external world",
Friedman writes

> The status and meaning of your "Informal Claim" is not clear when
> mathematical analysis (rigorous analysis) is involved.

I disagree. Its *meaning* seems pretty clear to me; and as for its
*status*, let us simply clarify that by asking the reader to assume
its truth for the time being. What follows from such an assumption is
still interesting enough to prompt further reflection.

In response to my "Metatheorem: Any classical proof of absurdity from
assumptions X in L can be matched by (indeed: effectively transformed
into) a proof in *intuitionistic relevant logic* of absurdity from
(some subset of) X", Friedman writes

> It is not clear what matched by means here if mathematical analysis
> (rigorous analysis) is involved. Also, in contexts such as arithmetic where
> it is clearer what this means, what does "can be" mean? For example, I
> happen to know that your version of relevant logic is a fragment of cut
> free logic. And mathematics cannot be done *by people* in a cut free
> system.

Let me therefore attempt to clarify the metatheorem further. It says
that any classical proof Pi of absurdity from a set X of assumptions
can, in principle (even if not: feasibly), be effectively transformed
into an intuitionistic relevant proof Pi* of absurdity from (some
subset of X). Two comments are in order:

1) the sense of "can be" involved here is the same as that involved in
Gentzen's Hauptsatz;

2) the need for the proof Pi* to be cut-free arises only when we want
to insist on the adjectival constituent "relevant" in the adjectival
phrase "intuitionistic relevant". If you take the weaker result that
any classical proof Pi of absurdity from a set X of assumptions can be
effectively transformed into an *intuitionistic* proof Pi# of
absurdity from (some subset of X), then there is no longer any need to
insist that Pi# be in cut-free form. Hence the reasoner who wants to
prove the overall result established by Pi# would be free to use
various lemmas (cuts) in the course of doing so.

We are left with the interesting original point: the classicist has no
case to the effect that we *have* to (i.e.: are, in principle, obliged
to) resort to the use classical logic (and mathematics) in order to be
able to refute false scientific theories. The metatheorem shows that
whatever scientific refutations are produced by use of the full
classical apparatus can (even if only in principle) be reproduced by
use only of the intuitionist's apparatus.  Thus there is no
"indispensability in principle" argument for the use of classical
logic and mathematics in scientific method. At best there is only an
"indispensability in practice" argument.

So when Friedman retorts that

> There's no compelling reason not to use classical logic to develop natural
> science, no matter how much of an 'intuitionist" you happen to be in your
> philosophical outlook

he is relying, ultimately, on the *consistency* of classical
methods. And the reason why *it's OK* to use these merely consistent
methods is precisely that one can, in principle, eschew them in favor
of intuitionistic methods. That's the ultimate *justification* we have
for using classical methods. That one might not be able, in practice,
to get by using only the intuitionistic methods is the *explanation of
why we resort to classical methods*, but *not* an ultimate
*justification* of those methods. What I've been trying to do is give
a Hilbert-style justification for the use of classical (cf. "ideal")
methods in refuting scientific theories, in terms of the safer
intuitionistic (cf. "finitary", "real") methods for doing so.

So to Friedman's riposte:

> Given recent work on brain transplants, ... we can or will be able to
> communicate with the outside world without legs, arms, or even a
> voice. So legs, arms, voices are all crutches we can throw away

I ask that he consider this reply: What this would show [if it showed
anything at all] is simply that it is our brains that suffice for
thinking, and that the extraneous things (legs, arms etc.) turn out to
have no role to play in the justification of our methods of
thinking. Compare: it is the intuitionistic methods that suffice for
the discovery of empirical falsity of theories, and extraneous methods
(law of excluded middle, etc.) turn out to have no role to play in the
justification of scientific method.

But of course, there are Wittgensteinian reasons (if Harvey can bear
to see the W-word used!) why brains in vats (or, if not vats, then
whatever life-support systems are involved once the arms, legs
etc. have been removed) might not be capable of world-directed
thoughts. A thinking creature in pursuit of empirical truth (or
falsity) needs, in order to acquire a grasp of observation sentences,
to be able to point to at least some medium-sized physical objects and
perceive them. Thus if you get rid of arms, you'd better provide your
stripped-down former human being a substitute method for
ostending. And if you get rid of eyes, ears, etc., you'd better
provide it with substitute channels for sensory awareness. The brain
needs sensory input in order to do empirical science, and it needs
methods for forming representations of empirical reality that can be
communicated to other brains---whether or not they still have arms,
legs etc.

So your attempted rebuttal by analogy raises perhaps hairier problems
than you anticipated. You are now trespassing on some of the
transcendental preconditions for the very possibility of (even
resolutely intuitionistic) thought about an external world. Such
thinkers might have very principled reasons not to allow you your
thought-experimental premiss that would "strip them down": for it
would deprive them of more than you bargained for.

Neil Tennant