[FOM] Wittgenstein Inspired Skepticism

[Harvey Friedman]

http://www.cs.nyu.edu/pipermail/fom/2017-February/020335.html
Tim Chow wrote:
...

If specifying a rule for an operation like addition is entirely
unproblematic, then it's hard to see why Kripke would be so deeply
impressed with the originality and power of this skeptical argument,
and write such a huge long essay about it.

MY RESPONSE

As I indicated before on FOM,
http://www.cs.nyu.edu/pipermail/fom/2017-February/020312.html
I view WIS = Witt inspired Skepticism, is a clever shock meant to
challenge us to make f.o.m. rely on "less commitments". Also to
explore the idea of their being a minimal level of commitments needed
to f.o.m. Or, alternatively, that there is a kind of nonending series
of ever smaller levels of commitments sufficient for f.o.m..

In this way, I do not view WIS as any kind of serious contribution to
f.o.m. Only as a cute tease to get us to think about minimizing
commitments.

Nevertheless, I do realize that many well respected Philosophers - but
I certainly know exceptions - think that WIS is a profound major
philosophical revelation.

Thus WIS and similar missives might be a good way to contrast the aims
of Philosophy with the aims of Foundations. Every really good
predominantly-a-mathematician I know of reacts with a mixture of
amusement, puzzlement, and contempt when confronted with WIS - the
only exception that comes to mind being Tim Chow.

Now perhaps when I get into WIS a little more beyond making fun of it,
I can somehow get the skepticism into my blood. I just don't see the
light at the moment. But I can imagine that somebody might see the
light, even if I don't see it.

Now we all know that in an appropriate sense, f.o.m., at least for
elementary mathematics, relies on an extremely low level of
commitments - more than low enough for just about any usual purpose -
even if it is not enough to literally shut down WIS.

A very well accepted way of introducing an incredible variety of
functions of several variables from N into N is Kleene's primitive
recursion, PR. This seems like a reasonable place to set up a
rule-oriented f.o.m. that should get WIS issues joined. Now just where
and how are we going to apply WIS to the ingredients of PR?

Actually, below we will stop at doubling and try to assess how we want
to attack this with WIS.

1. We take as primitive, the notion of nonnegative integer.

2. We use lower case letters for variables ranging over nonnegative integers.

3. We take as primitive S(x). We can view this as a rule that returns
a nonnegative integer from any nonnegative integer.

4. We take as primitive 0, the "beginning" nonnegative integer.

5. We write D(0) = 0, D(S(x)) = S(S(D(x))).

Now where and how do we want to start complaining?

If we only want to complain about 5, then we need some discussion.
There are some different ways of interpreting 5.

5a. We introduce the *rule* D(0) = 0, D(S(x)) = S(S(D(x))).

5b. We instead introduce a way of specifying and implementing rules
that is more detailed than simply what is done in 5a.

5c. We take as primitive an abstract idea of "rule" which really
amounts to extensional or maybe intensional functions. We state that
"obviously" there is a unique function D satisfying the two equations
in 5a.

For 5a, 5b, we are moving to a fundamental theory of computation, or
rule implementation, and would subject this to WIS, and see where we
get.

For 5c, we are moving to the more set theoretic or mostly extensional
function theoretic approach, and away from directly using rules. We
can then treat rules in terms of a series of states, where the series
of states is itself treated as a function from N into states.

Of course these choices simply do not matter much in the usual f.o.m.,
which is not normally being subjected to WIS. In fact, fairly crude
navigation along 5a-5c is enough even to get into subtle matters as
finitism versus infinitism.

Harvey Friedman