FOM: intuitionistic and classical truth
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Fri Sep 4 17:02:04 EDT 1998
Bill,
your remarks about not understanding what is meant by "saying what truth
*consists in*" do not, as far as I can see, express any sort of philosophical
bafflement, so much as *express a particular (and controversial) philosophical
view about truth*. If you wonder at what truth consists in---thereby
implying that there is nothing for it to consist in---then you are being
a deflationist about truth.
But there is a long tradition (scoffed at, of course, by redundancy theorists,
deflationists and pragmatists such as Rorty) of philosophizing about truth
according to which the truth of a proposition consists in the existence of
an appropriate kind of *truth-maker*. This is a metaphysical intuition driving
the work of philosophers as diverse as, say, David Armstrong (in his
conception of instantiated universals, and highly realist truth-conditions
for causal claims) and, say, Michael Dummett (in his neo-Wittgensteinian
rehabilitation of Brouwer, in which he gives an *anti-realist* account of
the truth-makers---i.e., the proofs---behind the *objective truths* of
mathematics). The anti-realism itself consists, of course, in the view that
one cannot guarantee that every declarative sentence will be determinately
true or false, independently of the means (i.e., proofs) by means of which
one might establish what the truth-value is.
I really do think that Dummett put Brouwer's doctrine into a better light
by explicating Brouwer's "constructions" as intuitionistic proofs, and
re-reading the justificatory burden of Brouwerian intuitions as discharged
by reflecting on the meaning-constituting character of logical rules of
inference (in particular, the introduction rules for logical operators in
a system of natural deduction).
Of course, there is a need to go further than the logical operators, for these
were, for Brouwer, merely part of mathematics generally. One needs to be able
to give a similar "rule-based" or inferential accou of the important
*mathematical* operators, such as "number of", "set of" etc. One way to do
this is to follow Martin-L"of into the intuitionistic theory of types, whereby
one acquires the ontological riches needed for mathematics, but does so by
smooth extrapolation of the Dummettian treatment of (first order) logic.
There are also other ways, tailored to the particular mathematical domain
in question. For example, in the case of arithmetic, it is possible to give
a neo-Fregean but *intuitionistic* derivation of the Peano-Dedekind principles
from some very weak second-order principles. The latter principles are
arguably meaning-constituting (for the notions of zero and successor) in the
same way that, for example, the rule of &-introduction is meaning-consituting
for the operator &.
When one speaks of "classical mathematics" nowadays, one always "up-dates"
the conception under discussion by imagining that one can appeal to the
nicest deductive system (say, a Gentzen-Prawitz system), and a nice model-
theory (or at least truth-definition) couched in the informal language of
set theory. One assumes furthermore that one is entitled to take any of the
branches of mathematis under discussion in its most recent and elegant
characterization. The discussion will then progress against the background
(hopefully) of full knowledge of the G"odel phenomena, the limitations of
formalism, the implicit realism of the classical logical apparatus, the known
"reductions" of different branches of mathematics to set theory, the known
calibrations of existential strength to be had from reverse mathematics, etc.
Why, then, is the intuitionist not allowed a corresponding degree of intellectual freedom in "updating" the doctrine of intuitionism? Why cannot he press into
service, for purposes of foundational debate, all the mathematical *and
philosophical* improvements that have been made since Brouwer's day? Whenever
intuitionism is mentioned among mathematicians, they immediately take its
canonical identity as given by *Brouwerian* doctrines---which, unfortunately,
can be made to appear rather laughable by judicious albeit selective quotations
from that unfortunate crank's work. It is high time for mathematicians and
foundationalists to talk of intuitionism in an up-to-date fashion, in a way
that assumes (among disputants) that one is dealing with the most sophisticated
re-constructive developments in this field since the time of Brouwer. [By the
way, these general remarks are not at all directed at you personally; indeed,
I'm sure you would agree, since I for one would be recommending the Tait
analysis of strict finitism from your J.Phil. article!] We need to have
an updated conception of intuitionism that would be au fait with, say,
Gentzen-Prawtiz proof theory, relative consistency results due to Sieg,
Friedman, Pohlers et al., Martin-L"of's type theory, the meaning-theoretic
foundational work of Dummett, Bishop's constructivism, etc.
But given the dreadful disdain of many a core mathematician for classical
logic---as brought out by Adrian Mathias's compelling papers on Bourbaki---
one could be forgiven for thinking that it will take another century before
the sort of sophisticated appreciation of intuitionism that is needed would
be widespread within the mathematical community.
Neil Tennant
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