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*To*: fom@math.psu.edu*Subject*: Re: FOM: Re: fom-digest V1 #245*From*: Vaughan Pratt <pratt@CS.Stanford.EDU>*Date*: Fri, 10 Dec 1999 21:03:13 -0800*In-reply-to*: Your message of "Fri, 10 Dec 1999 19:45:16 MST." <Pine.GSO.4.05.9912101556530.16195-100000@euclid.Colorado.EDU>*Sender*: owner-fom@math.psu.edu

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

**References**:**Jan Mycielski**- FOM: Re: fom-digest V1 #245

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