FOM: 'constructivism' as 'minimalistic platonism'
Jeffrey Ketland
ketland at ketland.fsnet.co.uk
Sun Jun 4 16:31:20 EDT 2000
This is a reply to Peter Schuster (4 June 2000: 15:09)
Schuster:
>Realism and tertium non datur are indeed one body forever [Dummett].
>By the way, the common use of `either ... or ...' does not seem to
>obeye any restriction like `even if we do not know which of the two
>is the case'; this appears to obtain also for classical mathematics.
I disagree. The connective "either .. or ..." seems to obey precisely that
principle (in my idiolect of English, at least). E.g., when I'm teaching
logic, I say to
my students, "Either my wife is presently eating a currant bun, or she
isn't, but I have no idea which is true" or I say "Either Aristotle was born
on a Tuesday or he wasn't, but I have no idea which is true (and maybe we
will never have any reason to accept or the other)". I see no reason why
this shouldn't work in mathematics either (unless one thinks that
mathematical facts are constructed).
[Simpson's suggestion that constructivists tend to read "P" as "we can know
that P" is along the right lines. It is that reading (or misreading, I'd
say) that largely motivates the rejection of LEM, Bivalence, Tertium non
datur, etc.].
Schuster:
>However, any realism powerful enough to `apply across the board' from
>the finite to the infinite has to be a full-blown one, which not only
>assumes all integers to exist but also that `we must know, we will
>know' [Hilbert] everything about them.
Not so. Realists (and sceptics) advocate the correspondence theory of truth
(or a Tarskian version of it or maybe some slimmed down minimalist or
deflationist theory of truth). On this view, there is no *conceptual link*
between a proposition's truth value and our *beliefs* about its truth value.
Our beliefs are fallible. Indeed, the omniscience claim is usually
associated with *idealism*. I do not know any realist who believes that the
human mind can know everything. In fact, every realist I know would reject
omniscience (or even give technical reasons for thinking that omniscience is
false). E.g., we can use Tarski's definition to get a precise definition of
the set of arithmetic truths. We can show, using Goedel's 1st Incompleteness
Theorem, that this set is not axiomatizable. Whether we (i.e., our minds)
can "know" all of these arithmetic truths is another matter entirely,
occasionally discussed by Goedel. Tarski's definition of truth has nothing
to do with "knowability" or "proof".
(Of course, one can provide an alternative semantics, e.g., based on Kripke
models, for knowable truth: this is how you get the completeness theorems
for intuitionistic logic).
I would suggest that, e.g., Gregory Chaitin's "The Unknowable" (1999), is
based on the *realist* principle that what is true (or exists) and what is
known (or even knowable) needn't be the same. (This is also the sceptic's
view as well: the realist and the sceptic both accept the correspondence
theory of truth, and the mind-independence of reality. The sceptic just says
that we can't *know* reality. The realist says "Don't be so pessimistic! We
can know *some* things about reality, perhaps quite a lot". The
idealist/constructivist *rejects* the correspondence theory and tries to
give an *epistemic* criterion for truth, and this is closely connected with
(if not motivated by) the idealist/subjectivist notion that reality is
"constructed" by us. If reality is constructed by us, then presumably we
*should* be able to know everything about it).
Schuster:
>As soon as one takes serious epistemic and/or temporal matters, one
>indeed risks that tertium non datur goes out of the window,
>an observation which could be made also by most realists
>(except the omniscient ones).
ONLY if you think that the truth value of proposition depends upon our
beliefs or knowledge about it. This attempt to conceptually connect truth
value with some *epistemic property* is a principle that realists (and
sceptics) rightly reject. Why should the truth value of a proposition depend
upon whether we can know it? Just because someone *calls* a sentence S true,
it doesn't mean that S *is* true. People can make mistakes. What does time
have to do with truth values? The truth value of a scientific theory like
General Relativity, e.g., doesn't depend on time. Its truth value doesn't
magically jump from being true to being false as people gather evidence for
or against it. I would say the same about mathematical theories. (I see no
basic difference between mathematical and scientific theories).
Schuster:
> The question, however, is
>whether mathematicians are willing to let their logic depend on such
>issues; my impression is that most of them are not.
Perhaps because working mathematicians know (in their bones) that the
intuitionist is conflating truth with what is knowable (with some odd and
unintuitive consequences concerning what counts as a proof). The working
mathematician, like normal people, in using "A or ~A" actually means "either
A is true or ~A is true, but I don't know which". The intuitionist rejects
the introduction (during a proof) of "A or B" until there is already either
a
proof of A or a proof of B. But I do not see why we need to make this
*extremely restrictive* assumption. More importantly, it seems that most
working mathematicians do not see why they need to make such an assumption
either.
Schuster:
>Classical logic ought rather to be viewed as the logic of what really
>exists AND will entirely be known, sometimes.
I would say that classical logic has precisely nothing to do with "what will
be known" (it makes no claims either way). Sceptics for example are
completely happy with classical logic (even though they think we'll never
know anything).
> Because nobody can know
>what some yet unknown knowledge will be like and whether it will
>eventually be known (unless it IS already known), one must invoke
>some strong omniscience principle to rescue classical logic, since
>otherwise it would be restricted to the knowledge of the day.
This is completely the opposite of the truth. Omniscience is closely related
to *idealism* and is rejected by all the realists and sceptics (that I'm
familiar with). A simple way to express omniscience might be an omniscience
scheme:
(OS) p --> it is possible to know that p
I think there are some constructivistically-inclined (Dummett-influenced)
philosophers who seem to toy with such ideas (Crispin Wright, perhaps).
Realists usually reject such claims about the relation between a fact and
its knowability.
Finally, I would say that the theory of truth is closely connected to logic.
The theory of what we can *know* is connected to human psychology. What we
can know depends upon our brains, our evolutionary past, the structure of
cognition, etc. It depends upon a detailed theory of justification,
evidence, perception, etc. We can build (mathematical) models of the
idealized human knower, and try and figure out how much truth this knower
can get hold of. However, in principle, a proposition can be *true* even if
we shall never know it.
That's realism.
Regards - Jeff
Dr Jeffrey Ketland
Department of Philosophy C15 Trent Building
University of Nottingham NG7 2RD
Tel: 0115 951 5843
E-mail: Jeffrey.Ketland at nottingham.ac.uk
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