**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

[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!
Pi is thought likely to be a normal number, though there is at present no known proof of this. However, should such a proof be discovered, the consequence would be that Pi must contain any arbitrary finite sequence of digits (regardless of the numeric base used). Hence using Pi in such arguments is necessarily problematic, hostage to possible future mathematical discoveries.

ReplyDeleteUsing the value of the Nth digit of Pi is probably safer, but... There is a formula, discovered in 1995, which allows the Nth digit of Pi to be computed without computing preceding digits. Admittedly the time required is still a growing (though sub-linear) function of N. And admittedly, even some hypothetical future formula allowing the same in a finite number of steps for any N would be in practice slowed down simply by manipulating larger and larger numbers. But can we guarantee there is no way around that?

Of course, the argument can be modified to use not Pi but some other suitable number. However, in principle, at least, the same difficulty applies regardless of the number chosen .

To my mind this exemplifies a problem facing philosophers when dealing with mathematics -- mathematicians are just too damn clever. :-)