[FOM] Counterfactuals in relative computability theory

[Timothy Y. Chow]

On Fri, 19 Aug 2016, Matthias Jenny wrote:
> The reason why I was hesitant to use the expression 'mathematically 
> precise' is because, as I've tried to explain, the way you use this 
> expression seems to me to pick out an epistemological property and not 
> an ontological one.

Thanks for your clarifying comments.

I'm going to take another stab at sketching your viewpoint, by means of a 
fictional story.  You can tell me if I'm on the right track or not.

The Heine-Borel theorem states that a subset of Euclidean space is compact 
if and only if it is closed and bounded.  "Compact" means that every open 
cover has a finite subcover.

On my last trip to Mars I discovered that the Martians are very similar to 
us, except that when they talk about "compact" sets of Euclidean space, 
they take themselves to be using the word "compact" in an informal sense. 
They offer no precise mathematical definition, but they seem to know a 
compact set when they see one.

I also learned that, historically, there was a time when they thought that 
maybe the right way to formally define the predicate "is compact" was "is 
a finite union of closed intervals of finite length."  Then one day 
some clever Martian came up with the set {0} U {1/n : n >= 1} and they all 
agreed that it was compact.

Nowadays, the accepted wisdom among Martians is that a subset of Euclidean 
space is compact if and only if it is closed and bounded.  By a remarkable 
coincidence, they call this the "Heine-Borel thesis."  They don't call 
this a theorem because it identifies a precise mathematical concept 
"closed and bounded" with an informal concept "compact."

My resident Earthling expert on Martian culture, Jenny Matthews, was of 
the opinion that the Martian "Heine-Borel thesis" was really the same as 
the Earthling "Heine-Borel Theorem" and was therefore true in all possible 
worlds.  The Martians were missing a piece of knowledge, namely that their 
"compact sets" were *precisely* the same as "sets with the property that 
every open cover has a finite subcover."  In the absence of this piece of 
knowledge, they felt that the word "compact" was informal and not precise, 
but unbeknownest to them, the true content of their Heine-Borel thesis was 
an equation between two precise concepts, and therefore was true in all 
possible worlds.

Returning to Earth and reflecting on the Church-Turing thesis, I wondered 
if perhaps the predicate "is an algorithm" had the same status as the 
Martian predicate "is compact," in the sense that it referred to some 
precisely delimited property, even though the consensus among Earthlings 
was that "is an algorithm" is informal and imprecise.  Perhaps the 
Church-Turing thesis is just an epistemologically challenged version of an 
unknown Church-Turing theorem that is true in all possible worlds, and is 
therefore also true in all possible worlds?

Is this an accurate reconstruction of your view?

Tim