Michael Dummett 
The
following piece is essentially about the antirealist's problems with
mathematical Platonism; as well as about how  and if 
intuitionist mathematics ties in with antirealism.
The
focus will be on Michael Dummett's specific take on intuitionism; as well as
his position on mathematical realism (i.e. Platonism).
Understandably, at least within this limited context, the relation
between truth and proof will also be discussed.
Dummett
on the Platonic Realm
Michael
Dummett asks a (fictional) Platonist the following question:
“'What
makes a mathematical statement true, when it is true?'”
The
Platonist's answer is:
“'The
constitution of mathematical reality.'”
[1982]
Of
course if Dummett hadn't told us that he was talking to a (fictional)
Platonist, the words “mathematical reality” could mean anything.
After all, mathematical inscriptions on a page could be deemed to be,
by some, mathematical reality. As it is, the Platonist's mathematical
reality exists regardless of minds, before minds existed and will
exist after the extinction of minds.
The
antirealist, along with many other philosophers of mathematics and
mathematicians, will now ask:
How
do we gain access to this mathematical reality?
More
relevantly to the antirealist's verificationism, the antirealist will
also ask:
How
do we determine that we've accessed mathematical reality and how
would we know that our mathematical statements match up with that
reality?
Indeed
wouldn't it need to be the case that even the mathematician's
decisionprocures and resultant proofs would also be required to
match parts of this Platonic mathematical reality? No, not according
to (most?) Platonists. The decisionprocedures and resultant proofs
are for us here in the
nonplatonic world. They're the way that mathematicians determine the
truths of their mathematical statements. However, the results
of these decisionprocedures and proofs are indeed in the/a platonic
realm; even though the procedures and proofs aren't.
Having
said all that, and in the hope of capturing the Platonist position, I
find it hard to know what all that means. That is, it's hard to understand
or accept this Platonist separation of truth (or mathematical
results) from decisionprocedures and proofs. The intuitionist, of
course, believes that there are no truths without decisionprocedures
(or “constructions”) and proofs because they quite literally
bring about the truths and even numbers themselves. Mathematical
truths wouldn't exist without them. (Despite that, one may also have
problems with the Platonist construal
of a mathematical reality which aren't particularly antirealist or
intuitionist is nature: e.g., what about our causal links to the
platonic realm?)
What
about other antirealist problems with Platonism (i.e., realism)?
It
can be said that a verifiable statement is also decidable
statement. Thus it may/will be the case that unobservable (in
principle) and observable states will be, respectively, unverifiable
and verifiable. Does this apply to mathematics? Platonist
mathematicians and philosophers say that mathematics doesn't concern
the observable. We can, of course, observe mathematical equations on
paper or on the blackboard. We can even introspect certain
mathematical symbols. Despite these qualifications, in the Platonist
view we're actually observing the symbolic representations of numbers
and mathematics generally – not a number or mathematics itself.
(In addition, a Platonist will happily accept – and even emphasise
– the fact that mathematics and numbers can be applied to the world
or used as the basis of structural descriptions of the world.)
Dummett
on Truth and Proof
Michael
Dummett goes into the technicalities of antirealist truth without
actually mentioning mathematics. In other words, what he says is the case about
all/most domains of truth.
In
terms of the lack of proof of a statement (although proofs aren't
really applicable to verifiable nonmathematical statements), Dummett
says that
“we
cannot assert, in advance of a proof or disproof of a statement, or
an effective method of finding one, that it is either true or false”. [1982]
There is a slight problem with that. The realist  against whom Dummett
is arguing  doesn't say that he knows
that a statement is true or false regardless of proof (or of “an
effective method of finding one”). His position, in this respect,
is effectively the same as the antirealist's. However, the statement
under consideration still has a determinate truthvalue: regardless
of proofs. The realist, sure, can't say that the statement is true.
And he can't say that the statement is false. Though he can say that
the statement is either
determinately true or determinately false regardless of proofs. So,
yes, even the realist, in Dummett's words, “cannot assert” (at this juncture) that “it
is either true or false”. Instead he can simply say: It
is either true or false.
Having
said all that, I see very little point in saying that a mathematical
statement is determinedly true or false regardless of
decisionprocedures and their resultant proofs. What does this claim
amount to? Where does it get us? It's effectively equivalent to
Bertrand
Russell's teapot
flying around somewhere in a distant galaxy. Yes, there could be such
a thing. However, we can never establish that there is such a thing.
Therefore what, exactly, is the point of saying that “there's a
flying teapot somewhere in a distant galaxy”?
Dummett
(in his own way) puts the gist of the last paragraph in more
circumspect and, indeed, Dummettian prose. On the proofindependent
mathematical statement, Dummett says that
“we
shall be unable to conceive of a statement as being true although we
shall never know it to be true, although we can suppose a true
statement as yet unproved”. [1982]
Yes, what's the point of “suppos[ing] a true statement as
yet unproved”? As it is, however, there are indeed unprovable
truths in mathematics. Dummett himself puts that in this way:
“A
Platonist will admit that, for a given statement, there may be
neither a proof nor a disproof of it to be found.”
[1982]
Despite
that, what Dummett has
just said
is most certainly applicable to nonmathematical statements; whether
about distant galaxies, the past, other minds and other problematic
areas. This is even more the case when it comes to statements about
the exact number of people in a given aeroplane at a given time; or,
more mundanely, about whether or not Jesus
H. Corbett is dead at
time t.
Truthconditions,
Proof and Truth
Clearly
it appears to be problematic to think in terms of truthconditions
when it comes to mathematical statements. (A Platonist, perhaps,
could think in terms of such truthconditions being in an abstract
realm; which are accessed through “intuition”, “direct insight”
or  metaphorical  “seeing”.) More concretely, it seems odd to
demand truthconditions for the statement 5
times 56 = 280. Strictly speaking, it has no
truthconditions. So what does it have? According to antirealists or
intuitionists, there is a procedure which can result in a determinate
result. Is that decisionprocedure also the proof of the mathematical
statement? Is the way of determining the truth of a mathematical
statement also a proof of the statement? Yes, but truth is not proof.
The intuitionist believes that a proof leads to the truth of a
mathematical statement. Without proof there is no truth. Nonetheless,
a proof isn't the same thing as a truth.
Mathematical
statements, when true, are decidable, not verifiable (because
unobservable). Despite that, the decidability of mathematical
statements (in a way) does the job that verifiability does when it comes
to statements which can be observed (or only observed “in
principle”). That is, there's a decisionprocedure for deciding the
truth of mathematical statements.
In
reference to all the above, an antirealist philosopher of
mathematics would say that mathematical truth depends on mathematical
proof. And mathematical proof is itself a question of decidability.
That is, from proof comes truth. And proof, in mathematics, is very
much like verification when it comes to the observable realm. That
means that proof, in intuitionist mathematics, satisfies the
antirealist position.
Nonetheless,
the realist would ask about the situation in which there is no proof
of a mathematical statement or equation. He would follow that by
saying that such a statement would still be determinately true
regardless of whether or not it had been proved. (This is/was the case
with Goldbach's Conjecture and Fermat's Theorem. The latter was proved by Andrew Wiles.)
Intuitionism
Objectivity
in mathematics isn't a question of objects. Instead, objectivity can
be said to be about the objectivity of the procedures and proofs
which lead to truth. On the other hand, if this were a question of
truthconditions, then it may have been the case that objects do
enter the equation. As it is, according to the intuitionist, this
isn't the case for mathematics.
Can't
we be realists about numbers, functions,sets, etc.? Aren't they
abstract objects? And if they are abstract objects, then don't
we have truthconditions (of some kind) because we have objects (of
some kind)?
On
the other hand, what if numbers are simply “free creations”, as
Richard Dedekind believed?
In that case, numbers are created or constructed by the
mathematician. The free creation of a number would still be the
creation of a determinate
something; just as when a person makes (or “constructs”) a
toy dog whose nature becomes determinate.
In
terms of an intuitionist/antirealist position on free creations.
There must still be decisionprocedures which can come up with
definite results even if numbers are “constructed”. The toy dog
just mentioned was indeed constructed. Still, we have various ways of
deciding its nature. That is, like the constructed number or
equation, we have ways of determining the nature of the toy dog. The
realist, on the other hand, would say that any truths about the toy
dog would hold even though no one could ever gain access to it (say,
after the creator died, etc.).
Brouwer/Heyting
on Maths as a Human Activity
L.E.J.
Brouwer, according
to Dirk van Dalen and Mark van Atten [2007, 513],
thought of mathematics as
an “activity rather than a theory” . In that simple sense,
truthconditions or a Platonic reality don't give maths a realist
foundation. More importantly, perhaps, “[m]athematical truth
doesn't consist in correspondence to an independent reality”. In a
certain sense, this means that such a construal of mathematics is
beyond the antirealism/realism debate in that verification or
observation isn't even possible in principle.
The
basic intuitionist position on mathematical truth was also put by
Arend Heyting. Michael
Dummett puts Heyting's position this way:
“...
the only admissible notion of truth is one directly connected with
our capacity for recognising a statement as true: the supposition
that a statement is true is the supposition that there is a
mathematical construction constituting a proof of that statement.”
[1982]
In
terms of mathematics, that “capacity for recognising a statement
as true” would depend on a decisionprocedure (or construction) for
determining a proof of that statement. As can be seen, everything in
the quote above seems to refer to human actions – even if
human cognitive actions. A mathematical construction is a
cognitive activity. A proof is also a result of a cognitive activity.
Indeed recognising a statement to be true is a cognitive activity.
What we don't have (in the quote above) is any reference to anything
outside these cognitive actions (such as truthconditions, states of
affairs, facts, etc.). Indeed there isn't even a mention of abstract
objects as such. However, that doesn't automatically mean that
abstract objects aren't illicitly or tacitly referred to.
For
example, the word “truth”may refer to the end result of a
mathematical construction which works as a proof that a statement is
true. But what does the word “true” refer to or mean? To the
proof itself? To the construction of the proof itself? In that case,
perhaps we have:
truth
= proof.
Or
alternatively:
a
construction (or decisionprocedure) = (a) truth
Dummett's
Problem With Intuitionism
It
may seem odd that the intuitionists' rebellion against “mystical
Platonism” should rely, instead, on the happenings which go on in
the privacy of a mathematician's head. At least that's how Dummett
saw it. Dummett's position is, of course, Wittgensteinian [Dummett,
1978, 215247]. Yet, in a sense, that's exactly what intuitionists or
mathematicians do. That is, even if we supply a retrospective
externalist (or “broad content”) account of the meaning/s, etc.
of those mental constructions, and also argue that they have
externalist/broad features (even during the private acts of mental
construction), at the initial stage this mathematical mental activity
is still individualistic (or private). The functions, numbers and
symbols are of course public or communal. Nonetheless, the
mental/cognitive acts of construction are, in an obvious sense,
private. So it remains the case that, on the one hand, mental
objects and actions aren't really private in that their meanings,
senses, extensions or whatever are externally determined.
Nonetheless, it's still the case that the mental constructions of
numbers or mathematical statements are private. And surely no
intuitionist would have denied any of that. (Perhaps we can say said
that Dummett was trying too hard to be a Wittgensteinian.)
It
is still the case that private mental constructions can't be the
subject of a decisionprocedure by other mathematicians (even
if the mathematical symbols, functions, etc. are bone
fide externalist items). This of course ceases to be the
case once the mathematical statements are written down or notated in
some other way.
References
Dalen,
Dirk van and Mark van Atten (2007) 'Intuitionism'.
Dummett,
Michael. (1982) 'Realism'.

(1978) Truth
and Other Enigmas.
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