FOM: Reply to Tennant re cuts as partitions

[Vaughan Pratt]

From: Neil Tennant <neilt at mercutio.cohums.ohio-state.edu>
>If I were translating [Dedekind] into English, I would have used "partition"
>for "Einteilung", rather than "separation".  It seems pretty clear to
>this reader at least that Dedekind intended his conception of the cut
>to be taken the way Kanovei suggested. Dedekind goes on to say that
>each rational number produces two cuts, depending on whether it is a
>member of the left set or a member of the right set. But nowhere (so
>it seems) can we impose the construal that he *identified* the cut
>with the set A1 rather than with the *partition* involving both A1 and
>A2 symmetrically.

I agree.  This points up a hazard in translating mathematics into logic,
especially pre-20th-century mathematics where these legal niceties
are unknown.  In "there exist x,y," are x and y to be understood as
forming an ordered pair or an unordered pair?

My understanding was that the former is standardly used, in part because
of the clumsiness of the definition of unordered pair when x=y is possible
(not the case here, allowing unordered pairs to be implemented as
doubletons as Friedman did).  Hence a translator needs to judge whether
there is an ambiguity, and if so whether the default is less appropriate
than the alternative.

Vaughan Pratt