Feferman - Does Reductive Proof Theory Have a Viable Rationale?

Feferman - Does Reductive Proof Theory Have a Viable Rationale?
He starts out saying that reductions in science tend to be explanatory, but in mathematics their purpose is foundational.

He debates against Karl-Georg Niebergall (who I met recently) about the merits of implementing reductions as proof-theoretic reduction vs. as relative theory-interpretations. The latter seems to favor the latter. But I don't see why the two "implementations" have to be incompatible.

I like Feferman's coinage of "set-theoretic imperalism", a dogmatic view.


KG Niebergall - On the Logic of Reducibility: Axioms and Examples (doesn't seem to be available unless you are using it from a subscribing institution: I have asked the author for a copy)

Ed Zalta - Neo-Logicism? An Ontological Reduction of Mathematics to Metaphysics
explains mathematics using his theory of abstract objects

Albert Visser - An Overview of Interpretability Logic

Exactly 2 years ago, I had a mini-course with Visser (in Utrecht) about categories of interpretation. Unfortunately, he was too busy to help me through the steep learning curve.

I was interested in applying the ideas of theory interpretations to formalizations of science, and it seems Niebergall has done exactly this.

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Btw, what is the possible-world semantics of a provability logic?
If such a semantics is viable, then it holds that:
a formula phi is provable in world w (written w |= [] phi) if and only if phi is true in all worlds that w points to.

What are these possible worlds and what is the relation between them?

German "der": articles and subordinators

German "der" means:

"the" (masc.) 1st case (nominative),
"the" (fem.) 2nd case (genitive)
and "the" (fem.) 3rd case (dative).
(the 4th case, accusative is "den", "die", "das")

This multiplicity of meanings produces the interesting example: "der Mann, [1] der der Frau [3] das Buch [4] gibt", which means "the man who gives the book to the woman." (interestingly, Google comes up with the same example of "giving a book", 4th hit)

but "der" also means the corresponding subordinating "that"s (which come in all 4 cases).

So "der Mann [1], der das Buch [4] verkauft" means "the man who sells the book" (note that subordination requires a comma).
Here's a more complex example, of a fem.2 subordinating "that": "Das Kind [1] der Frau, [2] der das Buch [4] gekauft hat.", which means "the child of the woman who[that] bought the book"):

What about...
"Die Welt des Mannes ist groß, verglichen mit der der Frau"? ("the world of the man is big, compared to that of the woman")
This was a bit of a puzzle for me: wouldn't it be more natural to say "die der Frau", since "Welt" is feminine?

But I finally figured it out, and the cases tell the whole story:
"Die Welt [1] des Mannes [2] ist groß, verglichen mit der [3] der Frau [2]".

'Objects' of "mit" are always in 3rd case. I mean preposition "mit", of course, not the particle "mit", as in the case of "kommst Du mit?" ("Are you coming along?")

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I think that in German sentences, there can be only one noun in 1st case, i.e. the subject. But since subordination is recursive, this results in a very imperfect system.

For example, ("der Frau, der")* is a regular expression that only generates grammatical sentence fragments, e.g.

"die Frau, die das Buch laß" ("the woman who read(imp.) the book") can form:
"die Frau, die (der Frau, der das Buch laß) hielft." ("the woman who helped the woman who read the book")
"die Frau, die (der Frau, der (der Frau, der das Buch laß) hielft) geschrieben hatte." ("the woman who had written to the woman who helped the woman who read the book") (note the 4 verbs in a row at the end)

When we have multiple subordinations, "subject" and "object" is relative. But the grammar only marks one subject as such, and as a consequence, "subordinated subjects" may not be distinguished from their objects.

The grammar seems to be designed for the simple cases, as if there were no recursion of subordination (is the latter a recent thing, evolutionarily?). I guess it would be cumbersome to have the syntax reflect the semantics: each subject/object would need to be marked twice: once to mark the relationship to its parent, and once to mark the relationship to its child in the parse tree.

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from the German version of "Sesame Street":

Der, die, das.
Wer? Wie? Was?
Wieso? Weshalb? Warum?
Wer nich fragt bleibt dumm!