Symposium: L.E.J. Brouwer, Fifty Years Later
Amsterdam – December 9th, 2016
Free Choice Sequences:
A Temporal Interpretation Compatible with
Acceptance of Classical Mathematics
Saul A. Kripke
CUNY Graduate Center
Saul Kripke Center
Brouwer’s intuitionism
motivations.
had
various
features
and
The most important one was the rejection of nonconstructive arguments, which, according to him,
necessitated a new interpretation of the (syntactic)
logical constants.
Most famously: the interpretations of disjunction and
negation, which implied a rejection of the law of
excluded middle.
Also, of course, the constructive interpretation of
existential quantification: denying that an existential
statement can be made without in principle being able to
provide an instance.
Eventually, though this took time, most
mathematicians simply ignored the Brouwerian
criticisms and developed classical mathematics as
before.
But there was a time when this was a very live
issue.
(As everyone here knows, in spite of his own
position, Brouwer himself did famous work that
was only classically valid.)
And interest in intuitionism is alive for many of us
here, even if we do not reject classical
mathematics.
Another feature of Brouwer’s work came later.
In addition to the determinate or ‘lawlike’
sequences, he proposed that ‘free choice
sequences’ be allowed.
As Heyting says, one does not really need to
suppose that an infinitely proceeding sequence be
determined by a law:
The question how the components of the
sequence are successively determined, whether
by a law, by free choices, by throwing a die, or by
some other means, is entirely irrelevant.
(Intuitionism, an Introduction,1956: 32)
This newer aspect of his work has a more difficult
history.
Errett Bishop, a relatively late admirer of Brouwer’s
constructivism and of his work, categorically rejected
the notion of a free choice sequence. He says:
In Brouwer’s case there seems to have been a
nagging suspicion that unless he personally
intervened to prevent it the continuum would turn
out to be discrete. He therefore introduced the
method
of
free-choice
sequences
for
constructing the continuum, as a consequence of
which the continuum cannot be discrete because
it is not well enough defined. This makes
mathematics so bizarre it becomes unpalatable
to mathematicians, and foredooms the
whole of Brouwer’s program. This is a pity,
because Brouwer had a remarkable insight
into the defects of classical mathematics,
and he made a heroic attempt to set things
right.
(Foundations
of
Constructive
Analysis,1967: 6)
(A similar tendency to sarcasm often appears in
Bishop’s remarks on foundational issues.)
Bishop’s real numbers, or sequences of rationals
defining them, are given by rules.
Martin-Löf’s intuitionistic type theory, and the
homotopy type theory that developed from it, appear
not to use the concept of free choice sequence
either.
These are major applications of intuitionistic ideas,
with connections to theoretical computer science.
And it should also be noted that even some authors
sympathetic to free choice sequences have tried to
show that they can be explained away as a sort of
façon de parler, using axioms of open data, or
analytic (Σ11) data, to reduce discourse about them to
talk of constructive functions.
I am not necessarily opposed to such ideas, but I
do not regard them as important for getting rid of a
fundamental concept.
In any event, the idea of free choice sequences I
will introduce is completely different from that which
motivated these proposals, since it is supposed to
supplement classical mathematics, not intuitionistic
mathematics, conceived as based on any idea of
constructivism.
In this talk I shall outline how a concept of free
choice sequence could be combined with an
acceptance of classical mathematics.
I do not wish to identify classical mathematics with
any particular axiomatization such as ZFC.
Most classical mathematicians other than those
working in set theory itself, probably do not think in
terms of particular sets of axioms, but apply the set
theoretic concepts intuitively.
(Category theory is of course an important part of
contemporary classical mathematics, and we don’t need
to go into the issues about its relation or lack of it to set
theory.
[Intuitionism actually plays some role in category theory,
but this is not important here.])
It would be in consonance with Brouwer’s own attitude
towards intuitionistic mathematics – and also its
exposition in Heyting’s Intuitionism, an Introduction
(1956) and Dummett’s Elements of Intuitionism (1977) –
simply to give the theorems intuitively and not rely on an
axiomatization.
I think this is true of much of classical mathematics also.
Now, let’s consider a classical mathematician
confronted with time.
Here we think of time as represented by discrete
linearly ordered moments.
In other words, with the order type of the positive
integers.
This was Brouwer’s conception as given in his
notion of free choice sequence.
We assume that the instances of time are given only
as a potential totality.
There is no ‘end of time’ from which all the instances
of time can be surveyed.
Of course, there is a considerable idealization involved
here.
The creative subject, who chooses the free choice
sequences, must be assumed to be immortal.
And even when the sequence is determined by a
machine flipping a coin or casting a die, it must be
assumed to be a perpetual motion machine.
On the other hand, there may be some scientific
models in which there is an end of time, even for a
given subject.
And there may be general relativistic models in which
after an end of time, there starts a new sequence of
time.
I do not take such things into account here, nor do I
worry about ‘absolute simultaneity’ and special
relativity.
Everything can be assumed to be the ‘proper time’ of a
single subject.
I believe that Brouwer once used the term ‘the
classical continuum’ for the continuum before free
choice sequences, and held that the continuum as
given by free choice sequences is an extension of
it.
Here the term ‘classical continuum’ would mean
the continuum as given by lawlike sequences of
rationals satisfying the usual convergence
conditions (i.e. a constructive continuum).
(It might mean the continuum as a whole taken as
primitive.)
In the conception I have here, the classical continuum
simply consists of arbitrary real numbers, defined in one
of the usual classical ways (Cauchy sequences or
Dedekind cuts).
For the mathematics here is simply the usual classical
mathematics.
The reals determined by free choice sequences are an
extension of the classical continuum.
However, we are considering sequences of natural
numbers, and arbitrary such sequences.
(Or, as I will explain later, arbitrary sequences with an
upper bound condition.)
So, for example, one can consider a coin
flipped independently an arbitrary number of
times.
This might correspond to a lawless sequence.
The total number of possibilities is most
naturally represented as a tree with the topology
of the Cantor set.
I am using the term ‘tree’ rather the intuitionistic
‘spread’ to emphasize that the mathematics is
classical.
No doubt this is a simple example of a ‘fan’ in the
intuitionistic sense.
However, other finitary trees (spreads) could
easily be given, all representing possibilities for
what the sequence might be, but never stating
which branch corresponds to the ‘real’ one.
Brouwer’s proof of the bar theorem, and hence of
the fan theorem, has given rise to a fair amount of
comment and has some difficulty in intuitionistic
mathematics.
(The most hostile comment comes, once again, from
Bishop, who says, “… to accept Brouwer’s argument
as a proof would destroy the character of
mathematics.” (Foundations of Constructive Analysis,
1967: 70)
But he acknowledges that, say, the fan theorem, has
never been counterexampled and thinks the situation
tantalizing since we will never have a proof, nor will
we have a counterexample.)
More sympathetic authors have also regarded
Brouwer’s argument for these theorems as
problematic, and may simply take them as axioms.
In the present case, there is no difficulty about the
fan theorem, which is crucial to the result that a
continuous function on a closed interval is
uniformly continuous.
Since the mathematics of trees with at most finite
branching and where every path has finite length
(a fan) is classical, König’s infinity lemma holds,
and hence the fan theorem holds.
Similarly, there is no difficulty about the bar
theorem.
Now, let me give an example of why, even though one
accepts classical mathematics, the law of excluded
middle will fail for infinitely proceeding sequences
(choice sequences, in the present conception).
For example, if the terms of the sequence are bounded
and are identified with digits of a real number in the unit
interval expanded to an appropriate base, we cannot
assert at any time that the number is algebraic or
transcendental.
Remember that there is no end of time, and that
probabilistic considerations cannot be used even if the
individual terms of the sequence are given by random
throws of a coin or a die.
In such a case, no doubt there is a probability
of 0 that the number will be algebraic, but we
cannot use this to assert that the number will
be transcendental.
(Of course, in the continuum extended by free
choice sequences, real numbers need not
have a determinate expansion to any base, but
this is another matter.)
We are really considering here a family of models
depending on what restrictions are being placed on
the free choice sequence, assumed to be a sequence
of natural numbers.
If there is a bound on the numbers allowed, this will
be reflected in the branching of the tree.
It is however assumed that any classical sequence
subject to the upper bound restriction, if there is one,
is an allowable one in the temporal path.
The creating subject is allowed to choose according
to any temporally possible sequence.
One point needs to be clarified.
Ideally, I would like to include all classically
admissible sequences in those chosen by the
creating subject.
But these will be non-denumerable in number.
And one might think that this is too much of an
idealization.
A creating subject can choose any denumerable
number of sequences simultaneously simply by
alternating his activities from one to another
according to some plan.
But an uncountable totality of sequences cannot be
done in this manner.
So one might more weakly postulate that all definable
sequences are to be included in those chosen by the
creating subject, and these of course could be
denumerable in cardinality.
Which sequences are definable may depend on the
exact language used, a point I have left deliberately
vague.
But it will include all those sequences that are
intuitively thought to be classically definable.
It is important to allow all definable sequences
since we want all classically definable reals to
be included in the model.
Note also that on this conception there is no
analogue of Troelstra’s paradox, other than the
known fact that only countably members of the
continuum are included.
And in an appropriate metalanguage, one
could define a diagonal element, given enough
extra resources.
Also, lawless sequences with any given initial
segment are included, as well as sequences with
restrictions added by one or more free decisions
(which need not be effective).
So that α x continuity holds, or at least α !x
continuity, but not α β continuity, given Kripke’s
schema.
The definable sequences replace the ‘lawlike’
sequences in the usual conception of free choice
sequences.
But here there is no particular effectiveness idea,
just definability.
Now, let us return to issue of the law of excluded
middle and its failure on the present conception.
We are assuming a model with branching time, but
no one knows which branch is actual.
When does the failure set in?
According to one reading of Aristotle’s De
Interpretatione (and my impression is that for a long
time it was the received reading), the failure sets in at
the very outset.
In this reading no one can say that either there will be a
sea battle tomorrow or there will not, since the future is
indeterminate.
One would have to change the example to correspond
with free choice sequences.
Or make the value of the sequence dependent on
whether there will be a sea battle tomorrow.
But the idea is clear enough.
In this model the idea of the branching time model
consists essentially in treating it with the semantics of
Kripke models: a disjunction is true at a point iff one
disjunct is.
But the motivation is quite different from the one I
originally gave for such models.
The original idea was that the points on a branch
represent ‘evidential situations’ and later points
represent situations in which one gets more evidence.
One may remain stuck at a single point for an
arbitrarily long time, but only if one gets more evidence
can one assert a disjunction without being able to
assert a given disjunct.
This might correspond reasonably well to the
intuitionistic idea that proving a disjunction amounts to
having a method to prove at least one of the disjuncts.
Here, on the Aristotelian model, the idea is different,
since the nodes of the tree represent points in time,
where later ones represent later points.
Then the law of excluded middle fails simply because
the future is not determined, though if one waits until
the next day, one will know which disjunct is true.
Now, this might be a view, but I think it does not so
well accord with our intuitions about free choice
sequences.
If we can wait and see, in some finite time, which of
two disjuncts is true – here especially for the law of
excluded middle – then one can assert it in advance.
In Aristotle’s example, there either will be a sea battle
tomorrow or they will not, though we have to wait till
tomorrow to see which is true.
Putting things a bit more formally:
If there is a bar on the branching time tree, so that any
path along it will decide between the two disjuncts
eventually, then the disjunction can be asserted in
advance, though we have to wait a bit to see which
one holds.
This means that the semantics corresponds to
Beth models.
Remembering that Konig’s lemma and the fan
theorem hold for finitary spreads, there will
actually be in the case of finite branching an
upper bound for how long one has to wait.
So here, following Brouwerian intuitionism, we
assume the semantics of the connectives for free
choice sequences to be given in terms of Beth
models.
We otherwise continue to take the underlying
mathematics to be classical.
Thus the idea is that the ordinary classical
continuum is to be supplemented by one given in
terms of Brouwerian free choice sequences.
For these, the usual intuitionistic mathematics is
assumed, just as in textbooks such as those of
Heyting (1956) and Dummett (1977).
I do wish, however, to make some comment on
the
role
of
Kripke’s
schema
in
this
supplementation.
I originally proposed it in the weak form:
α[(¬A ↔ xαx = 0)
( xαx ≠ 0 → A)]
But there is a stronger and much simpler form:
α[A ↔ xαx ≠ 0]
I still prefer the weak form. (And I would hold this
preference in ordinary intuitionistic mathematics too,
as well as on the present conception.)
Moreover models have been found in which the
strong form holds. Also, as van Dalen (1977) remarks,
the strong schema is trivially true in classical
mathematics.
What is the reason?
Intuitionistically, to prove a conditional A→B, one
must have a technique so that from any proof of A
one can get a proof of B.
Now, the idea of the schema, based on Brouwer’s
own arguments about the creative subject, is that one
imagines a sequence in time that is 0 as long as A
has not been proved but is 1 as soon as it has been
proved.
Then one claims that a proof that the sequence is
always 0 amounts to a proof that A can never be
proved, i.e. that it is absurd.
If the sequence gets the value 1, then A has been
proved.
What would justify the strong form?
Well, as Myhill says, if A is ever proved, it is
proved at some definite time, and then one gets a
value of α that is not equal to 0 simply by looking
at your watch (or calendar, as the case may be).
However:
Is the idea of the intuitionistic conditional
(remember that from a proof of A, one can get a
proof of B) such that intuitionistic proofs, which
indeed must take place at some definite time, have
the time of their occurrence as part of the proof, so
that the same proof would be different if it
happened to occur at a different time?
I think not.
In that case, the strong form does not appear to be
justified, but the weak form is.
Remark:
Heyting allows his character representing
intuitionist to object even to the weak form:
the
… if a law is passed throughout the world
prohibiting the making of any mathematical
deduction whatever, then the proof of α ≠ 0
fails. (1956: 116. In effect, (¬A ↔ xαx = 0)
fails.)
But the idea of such a law simply gets rid of the entire
intuitionistic interpretation of the conditional, since
A→B is supposed to mean that from any proof of A,
one can get a proof of B, and this becomes a vacuous
notion because of the supposed law.)
So, what have I argued here?
That a Brouwerian theory of free choice
sequences could be added to classical
mathematics, without any constructive doubts
as to its validity.
I have tried to be deliberately informal, in the
spirit that Brouwer himself would have
preferred.
Thank you!