[FOM] vagueness in mathematics?

[Harvey Friedman]

Tim Chow wrote 2/16/17:

Tim: A rule?  I thought that I knew what you meant by a rule, but now
I'm not so sure.  It seemed quite clear to me that the rule
"1,2,3,1,2,3 repeated indefinitely" meant that the next number was
obviously 4.  But you're telling me that that "rule" dictates that the
next number is 1? Weird.  What do you mean by a "rule" then?

The argument is not just that any finite sequence can be continued in
infinitely many ways.  The argument is that even after you say, with
as much precision and explicitness as the entire mathematical
community can muster, *exactly what the rule* for the sequence is, the
sequence is *still* not determined.  Any finite amount of natural
language conversation, gesticulation, drawing of pictures, training in
logic, building of computers and demonstration of their operation,
holding of international conferences, browbeating, and foaming at the
mouth will still fail to nail down even the simplest possible
so-called "rule." Everyone could agree on every instance of the
alleged rule that has ever come up in the history of mankind but it
could all just be a huge lucky coincidence.

***********************

I take this as W (Wittgenstein) inspired skepticism, or WIS.

The above formulation of WIS, which is about as good very brief
formulation that I am familiar with, is itself a (dogmatic) claim that
if we follow a given rule then we get a given result. The rule in this
case consists of engaging in "natural language conversation,
gesticulation, drawing of pictures, training in logic, building of
computers and demonstration of their operation, holding of
international conferences, browbeating, and foaming at the mouth", and
the result is failure "to nail down even the simplest possible
so-called "rule"".

So WIS seems to me to be compelling us to think in a way that is
essentially the same as the way that WIS is criticizing.

This is a key reason why I think that WIS, prima facie, is not a
serious contribution to the history of ideas, but instead an amusing
clever devious distraction.

I think we are better off trying to reformulate WIS in productive
terms. I think that such a reformulation can be done.

One way of looking at the challenge is to come up with a foundation
for mathematics. Of course, we have a foundation for mathematics that
is much more convincing and much more powerful than we have for other
subjects. But we can try to "perfect" our foundations. We can try to
make it "more primitive" in certain senses. Still, it need to be
noted, that the present foundations is so convincing that it fully
supports spectacular findings along the very familiar Frege, Cantor,
Russell, Zermelo, Goedel line.

Conventional wisdom is that perhaps foundations can always be made
"more primitive". But maybe it can be made "maximally primitive"?
Perhaps in incomparable ways?

There is a kind of related problem in computation. Is there a most
primitive kernel for computation which cannot be improved, and where
there are incompatible maximally primitive kernels?

A real challenge is to make sense of all this, and prove some results.

Harvey Friedman