A Power Point From a Lecture on Wittgenstein and Rule-Following
In 2012 I gave a seminar
on Wittgenstein’s later philosophy at Tel Aviv University. The following is the
text of one of two powerpoints I used as the basis for lectures in that seminar
(the other powerpoint will be the next post). I am interrupting the flow of this
series of posts, but the second of the two powerpoints - the one I will post next week – talks about
qualia, a notion that will come up in future posts on colors and their “looks”.
RULE-FOLLOWING
IN WITTGENSTEIN, KRIPKE, AND MYSELF
Wittgenstein in Philosophical Investigations §195
(re continuing a series, e.g. 2,4,6,8,…..1000, 1002, 1004,….)
“But I don’t mean that what I do now (in grasping a sense)
determines the future use causally and as a
matter of experience, but that in a queer way, the use itself is in some sense
present.”
I
KNOW I AM NOTORIOUS FOR “CHANGING MY MIND”
Well, upon thinking over what I said about Wittgenstein’s Rule Following argument
last week in this seminar left me feeling dissatisfied, and so once again I
shall revise my view! So here we go.
Wittgenstein
thinks that philosophical problems are only illusions of problems,
but
until we work our way out of the bewitchment, they genuinely do puzzle us. So
it will not beg any questions if I speak of a “rule following puzzle”, rather
than a “rule-following problem”. Whether it is in the end a real problem or a
pseudo-problem, there is a puzzle that Kripke genuinely worries about, and that
Wittgenstein responds to in some way
I proposed a response to the puzzle on
Wittgenstein’s behalf, which I now
think is not Wittgenstein’s,
and at the same time I followed Wittgenstein in dismissing the sort of
puzzlement Kripke exemplifies as misguided. But it should have been clear to me
from my own published criticisms of Wittgenstein’s philosophy of mathematics* that there is a
problem with Wittgenstein’s
dismissive response.
*For example, “On Wittgenstein’s Philosophy of
Mathematics” and “Wittgenstein and the Real
Numbers”
[collected in Philosophy in an Age of Science, Harvard 2012.]
I
shall state the puzzle in my own words, and then present three responses to
it.
THE PUZZZLE:
The puzzle is this: when we grasp a rule
for generating, say, the decimal expansion of pi, the rule determines what the
nth digit of pi is no matter how large n is—for example, it determines whether the trillionth digit is
0,1,2,3,4,5,6,7,8 or 9.
Now, following a rule may not be a
scientific concept, as Wittgenstein stresses but, grasping a rule is a mental
state in an ordinary sense of “mental
state”, and the puzzle is how
creatures like us can have mental states that determine such a large number of
cases.
3 Responses
I shall present three responses to the puzzle:
Kripke’s, Wittgenstein’s, and mine (which I will
not call an “interpretation” of Wittgenstein any
longer).
Kripke’s response to the puzzle: three parts
Part I: Kripke’s Wittgenstein
interpretation (Wittgenstein=Kripgenstein)
Part II: Kripgenstein’s Solution to the Puzzle:
Part III: Kripke’s own view
Part I: Kripke’s Wittgenstein interpretation
(Wittgenstein=Kripgenstein)
According to Kripgenstein, calling a
judgment true is simply endorsing that judgment, i.e., “true” is, as Rorty once put it
a “compliment that we pay” to judgments we agree
with.
Very close to relativism
If people disagree, you can say that one of
them is right and the other is wrong, but if you do so you are simply endorsing
what one of them says and rejecting what the other says.
According to Kripgenstein
There is no fact of the matter about anyone “considered in isolation” as to what they she means
by the formula for, e.g., calculating the decimal expansion of pi. If someone
is able to use the formula well enough (which does not mean she never makes a
mistake)—that is, well enough so
that others say she understands it as they do, then that counts as her
understanding it as they do.
A consequence Kripke does not stress
if the whole community is unable to compute
the trillionth decimal place of pi, then there is no fact about the community
as a whole which is the fact that the meaning of the formula is such that a
certain digit is the one in the trillionth decimal place
Part II: Kripgenstein’s Solution to the Puzzle:
1. The
puzzle assumes that the formula “determines whether the trillionth digit is 0,1,2,3,4,5,6,7,8 or 9.” But if the community cannot use the formula
to determine the trillionth digit, then the way formula is understood does not
determine it either.
Apart from the Rortyan relativism about truth
This, I think, is what Wittgenstein
thought, and it agrees with the remark in Remarks on the Foundations of
Mathematics that “even
God’s omniscience” cannot determine anything
about the decimal expansion of pi that human cannot determine as well as with
the emphasis in PI on the claim that the criterion for understanding is how the
speaker applies the formula
Part III: Kripke’s own view
“I can only report that in spite of Wittgenstein’s assurances, the ‘primitive’ interpretation [‘that looks for something
in my present mental state to differentiate between my meaning addition or
quaddition’] often sounds rather good
to me.” [Kripke, Wittgenstein on
Rules and Private Language, p. 67.”
-----------------------------------------------------------------------------------------------------------------------------------------------------
Wittgenstein’s response to the puzzle: two
parts
Part I: Wittgenstein’s “deflationary” attitude to the
rule-following puzzle
Part
II: Wittgenstein’s
verificationist attitude to mathematical truth
Part I: Wittgenstein’s “deflationary” attitude to
the rule-following puzzle
Following a rule has a normative element.
Equally important for our purposes, it implies that a regularity is
present;
The normative element
if one says that someone followed a rule,
one generally means that they followed it correctly; that’s one of the reasons that “he followed the rule for computing
the decimal expansion of pi” is
not a description of the presence of a mechanism.
The need for regularities
Saying that someone followed a rule implies
that a regularity is present. Moreover, the notion “regularity” is not mysterious, I can
teach it to someone by means of examples and corrections. [Philosophical
Investigations §207]
Even if which sequences of events are “uniform” or exhibit “regularities” is an “anthropocentric” matter, that doesn’t mean that the difference
between a regularity (“All
emeralds are green”) and
a sequence that is not a regularity (“All emeralds are grue”) isn’t a real difference.
[Added for this post: I explained to the
seminar that I called this a "deflationary" - as opposed to
metaphysically inflated - remark about rule-following, because there is no
implication that 'regularities' can extend beyond our ability to recognize them
as such]
Part II: Wittgenstein’s verificationist attitude
to mathematical truth
Suppose that people go on and on
calculating the expansion of p. So God, who knows everything, knows whether
they will have reached '777' by the end of the world. But can his omniscience
decide whether they would have reached it after the end of the world? It
cannot. I want to say: Even God can determine something mathematical only by
mathematics. Even for him the rule of expansion cannot decide anything that it
does not decide for us."
[Remarks on the Foundations of
Mathematics,V, §34]
In Kripke’s terminology
—In Kripke’s terminology, this says that if
the whole community is unable to compute far enough to find 777 in the decimal
expansion of pi or to find a proof that
this sequence does not occur in the decimal expansion of pi, then there is no
fact about the community as a whole—not just about any one person “considered
in isolation” which is the fact that the meaning of the formula is such that
777 does (respectively, does not) occur in that decimal expansion.
Wittgenstein’s response (concluded)
In short, the answer to the “puzzle” is that we don’t have the power to
determine the answer in such a large number of cases.
My own view: three parts
Part I: My former interpretation of
Wittgenstein
Part II: why I now reject my former
interpretation
Part III: My response to the puzzle
Part I: My former interpretation of Wittgenstein
(in
last week’s meeting of the seminar)
consisted of the “deflationary
reading” of the rule-following
discussion plus a “quietist” response to our puzzle,
that is, one which denies that there is an intelligible question as whether
our grasp of the formula “determines
the answer in an infinite [or, alternatively, an enormously large finite] class
of cases”.
plus royalistes que le roi
. I now think this response, while
Wittgensteinian in spirit, was actually more “Wittgensteinian” that Wittgenstein himself. (“More royalist than the king”, as the French saying goes.)
I distinguished two senses of “determine”
a mathematical sense of “determine” in
which it is simply a theorem of mathematics that the algorithm for continuing such
series “determines” all the infinitely many members, and an ordinary sense of
“determine”, in which it is plainly false that our understanding of the rule “determines” the series so that we can
continue it to, say, a trillion places.
It
is a theorem of mathematics that the digit in the trillionth decimal place =1,
or =2. or =3, or =4, or =5, or =6, or =7, or =8, or =9,
and so this is something we can say. But to
ask: but is a particular member of this disjunction the correct one is to
attempt to step outside of mathematics, and this “stepping outside” of what we ordinarily say (in this case, say when
doing mathematics, as oppose to philosophizing about it) leads to nonsense.
Thus the “puzzle” depends on an
unintelligible use of “determine”.
Part II: why I now reject my former interpretation
The “quietist” part of my former
interpretation amounted to the claim that Wittgenstein would have rejected the
question: “Is there is a fact of the matter as to whether the trillionth digit
of pi =1, or =2. or =3, or =4, or =5, or =6, or =7, or =8, or =9 [assuming we
are unable to find a mathematical answer to the question, even if we try “until
the end of the world]. I gave two arguments for this neither of which seems
good to me after my rethinking last weekend.
The first argument was that the question
assumes that every true judgment must have an “object” which makes it true, e.g. in the case of the judgment “There is nothing red in
this room” the “negative” fact” of the absence of red
things in the room, and we know from lectures Wittgenstein gave that he
rejected this picture—the
picture on which this judgment “corresponds” to something.
But in Remarks on the Foundations of
Mathematics, Part V, §34,
Wittgenstein does manage to make sense of
the question whether there is a fact of the matter as to whether 777 occurs in
the decimal expansion of pi* even if human beings cannot find a mathematical
proof one way or the other, by asking whether an omniscient being could know
the answer.
* [by the way, it does!]
Saying that even a being who knows
everything could not know this clearly implies that there is nothing of this
sort to know. The question as to whether every mathematical question actually
has a determinate answer was not one that Wittgenstein dismissed as nonsense.
My second argument
My second argument was that it was
illegitimate of Kripke to bring in
Remarks on the Foundations of Mathematics in interpreting PI, and that
PI is much less extreme on the decimal-expansion-of-pi question than RFM.
But if we don’t bring in Wittgenstein’s philosophy
of mathematics,
no reason has been given for rejecting the
puzzle as nonsense except the question-begging reason that “if it can’t be asked in ordinary language,
it doesn’t make sense.” Evidently Wittgenstein
didn’t think that asking
whether an omniscient being could know the answer to the question about pi
violated ordinary language. (And that violations of ordinary language are what
is at stake plunges us into the morass of Baker&Hacker vs. “New Wittgenstein”.)
let us look at the passage in question from PI
itself (§516):
It seems clear that we
understand the meaning of the question: 'Does the sequence 7777 [note that the example is slightly different] occur in the
development of pi?' It is an English sentence; it can be shown what it means
for 415 to occur in the development of pi; and similar things. Well. our
understanding of that question reaches just so far, one may say, as such
explanations reach.
It is true that
this does not say that there is no fact to
be known at to whether 7777 does or does not occur in the decimal expansion of
pi; but it is not far away.
No real change in Wittgenstein’s position
The fact that “we know what it means for
415 to occur in the decimal expansion of pi” [=3..14159…]
only shows that we can recognize a calculation showing that a particular
sequence of digits does occur. But if our “understanding of the question” does not reach farther than our ability to recognize mathematical
proofs does, then we have the same picture as in RFM V, §34.
PUTNAM
Part I: My former interpretation of
Wittgenstein
Part II: why I now reject my former
interpretation
Part III: My response to the puzzle
Part III: My own response to the puzzle
My own response to the puzzle depends on my
scientific realism, and goes beyond this seminar to defend it here. [I will
post an optional reading about it; part of the position is the criticism of
Wittgenstein’s mathematical “verificationism” as presupposing
verificationism with respect to physical science (which is widely recognized to
be untenable)]
[Added for this post: The additional reading is ch. 25 of my Philosophy in an Age of Science]
Assuming scientific realism,
Just as we understand what a “regularity” is, even though we can’t
explain how to distinguish between regularities and non-regularities except by
teaching someone the practice of doing so, we also grasp the fact that a rule
determines cases beyond what we can practically calculate, by seeing how that
fact ties in with our practice in doing cosmology, atomic physics, etc.
Am I not saying that I don’t know “how creatures like us can have mental states that determine such large
numbers of cases”? Well, I don’t know how young children
can grasp the idea that there are infinitely many natural numbers, but pace Wittgenstein, many of them do!