I'm doing a visual memory test. On the table in front of me are twelve
green and fourteen red apples, and an empty basket. The lights go out, and
the instructor says to me:
"Put all the green apples into the basket". (1)
I try to do what he says. When the lights go on, you, the instructor's
assistant, are given a form on which you are to tick whether I've
correctly or incorrectly fulfilled the task. You see twelve green and two red apples in the basket. What do you tick?
Today I've been reading Hilbert. I must admit that I don't really
understand his view on the foundations of mathematics. It seems to me that
he always confuses truth with consistency. For example, he writes in his
"New Grounding":
If we can produce [a consistency proof of formalised mathematics], then
we can say that mathematical statements are in fact incontestable and
ultimate truths.
Obviously, Hilbert uses "true" in a very unusual way here: Both ZFC + the
Continuum Hypothesis and ZFC + its negation are consistent. Hence, on
Hilberts account, both CH and its negation are "incontestable and ultimate
truths".
A while ago, I
asked: "Could Frege's ontology be a Henkin model?". I now believe that
this question doesn't make sense: A standard model of second-order logic
is a (standard) Henkin model. I should have asked: "Could Frege's
ontology be a non-standard Henkin model?". Even this question is,
uh, questionable, because the late Frege would have certainly rejected both
a standard and a Henkin semantics, as both of these employ singular terms
to denote the semantic values of function expressions. So I should rather
have asked: "Are Frege's logical and semantical theses satisfiable in a
non-standard Henkin model?" But now, I guess, the answer is trivially Yes,
because nothing you can say in higher-order logic rules out a non-standard
Henkin interpretation. However, my question was not meant to be trivial.
I wanted to know whether Frege is comitted to there being more concepts
(values of second-order quantifiers) than objects (values of first-order
quantifiers), a claim that is true in standard models, but not in some
non-standard models of any (really?)* second-order theory. Unfortunately,
this question can't even be asked without violating Frege's semantical
theses. As he himself notes in a letter to Russell:
This is a problem that cropped up several times in my thesis on Lewis,
but which I never seriously discussed.
Lewis argues, or rather, stipulates, that all fundamental ("perfectly
natural") properties are intrinsic. I agree that fundamental extrinsic
properties would be strange. For if a thing x's being F depends on the
existence and the properties of other things, it seems that F-hood should
be reducible to intrinsic properties (and relations) of all the things
involved. Moreover, fundamental properties are supposed to be the basis for
intrinsic similarity between things, and they could hardly be if they were themselves extrinsic.
A problem from Kit Fine, "The Non-Identity of a Material Thing and Its Matter", Mind 112 (2003):
Suppose a certain piece of well made alloy coincides with a certain
badly made statue. Al makes an inventory of well made things. The only entry on his list is "that piece of alloy". Question: Does the entry on Al's list refer to
a badly made thing?
Kit Fine intuits that the answer is definitely "no", irrespective of the
context in which that question is asked. From which it seems to follow
that the piece of alloy and the statue are not identical. At least I think this is what he thinks would follow. Anyway,
here is an extension of the above story where "the entry in Al's list
refers to a badly made thing" appears to be true.
Strolling through the library, I just came across George Tourlakis' Lectures in Logic and Set
Theory. I wouldn't recommend it as a textbook for logic courses in
philosophy, unless you want to torture your students with a full
proof of Gödel's Second Incompleteness Theorem. But it's nice to have
that proof available somewhere. The second volume on set
theory (unfortunately only on ZFC) also looks useful, if only because there
are so few thoroughgoing introductions to set theory.
Now I'm in a friend's flat, where a lot of books and a TV set have
consipred to distract me. Yesterday I've read Szpilman's "The Pianist" and
watched "The Matrix". I found the latter rather silly and unoriginal, but
maybe I've entirely missed the point. I'll try to find out what's supposed
to be the point as soon as I get a chance to access the net without
increasing other people's telephone bill.
The Frege paper is
finished, by the way. And yes, it's in German.
I've just moved out of my flat. Next I have to find a flat to move into.
This time I've even managed to throw away most of the notes and copies of
papers I used to carry with me each time I moved. I've also thrown away
some other stuff, like two of my three pairs of trousers, so I feel like
quickly approaching my dream lifestyle.
In the meantime, the new semester has begun. I'll probably visit just one
seminar, which is supposed to be about classical logic, though I'm not sure if
e.g. Dov Gabbay's fibring
logics should really be called "classical". Well, maybe I've missed the new era while reading Frege.
If nothing goes terribly wrong, I will finish the Frege paper tomorrow. Though I'm not sure if it's really the same Frege paper I mentioned previously.
Initially I just wanted to put together all the comments on
Fregean thoughts and Rieger's paradox that I had already posted to this
weblog. That looked like a cheap way to get a termpaper. For some reason
however the paper has now evolved into a discussion about the prospects and
dangers of developing a semantics that can be applied to its own
metalanguage.
I'm still working on the Frege paper. Obviously, so far this weblog hasn't cured my perfectionism, which I hoped it would.
Apropos perfectionism: That new (and not very informative) entry on Turing Machines in the Stanford Encyclopedia reminds me of a Turing Machine simlulation I've worked on back in 2000. Here is the rather unusable latest version of that attempt. I've stopped working on it mainly because I wanted the page to automatically draw a flow chart for any machine table that is specified. But I couldn't find an algorithm that prevents overlapping arrows wherever it is possible. (Here is an example of the latest, somewhat funny looking version.)
Magdalena told me that Jay Wallace will offer a seminar next semester here in Berlin. In this respect I fully support the "love it or leave it" messages from the American right, as long as they bring good philosophers to Old Europe...