FOM: Re: fom-digest V1 #245

[Vaughan Pratt]

Hi, Jan.  We haven't talked in quite a while, 27 years I think.

What you're arguing has its counterpart in physics.  We could with
Hamilton postulate an independent notion of momentum, as half the basis
of Hamiltonian mechanics, position constituting the other half.  But we
could just as well deny the independent existence of momentum and instead
obtain it as the product of mass with velocity, and further obtain the
latter as the derivative of position with respect to time, the basis for
Lagrangian mechanics.  Both points of view are not only valid but useful.
Furthermore we know in great detail how to pass between them.

By the same token we can, according to a popular thesis, reduce all
thought to the mechanics of the thinker.  But it is sometimes useful
to postulate a "domain of thought" and make a connection between the
process of thinking and its referent.

The model theory of first-order reasoning can be understood as
characterizing the connection between first-order formulas and relational
structures in remarkable detail.  Equations and algebras enjoy a similar
connection.  Each side of model theory, the formal and the platonic,
can be understood as having an independent existence, yet with each
perfectly mirroring the other.  Which side of the mirror is real is not
as cut-and-dried with the mirror that reflects logic and mathematics in
each other as with the mirror on the wall.

>The main meaning of "abstract" which I know is "admits many interpretations".

That's the meaning of "ambiguous."  Identifying the abstract with the
ambiguous does both a great injustice.  Any actor whose depiction of
the death of Macbeth is open to interpretation invites chuckles from
the audience.  When done properly there is only one interpretation,
that Macbeth is now dead (and the actor presumably still alive).

My sense is that you are reluctant to pass from "the actor depicts the
death of Macbeth" to "Macbeth dies."  But like the passage from Lagrangian
to Hamiltonian mechanics, this too can be a useful passage.

Best regards
Vaughan