Abstract Language

Some morals.

• There is no clear generally accepted distinction between abstract and concrete objects. There are various relevant criteria, like causal inaccessibility, conservativeness over prior theories (physical, nominalistic, phenomenal, simpler mathematical theories), lack of spatiotemporal location, being a universal, intrinsicness. Thus, in arguing either for or against accepting abstract objects or language that appears to refer to abstract objects it is critical to specify what class of objects you have in mind. (Equator, number of sheep in the meadow, space-time points, gauge fields, absolute spatial position)
• Lack of causal efficacy or, even stronger, conservativeness do not pose an insuperable epistemological problem: we can, it seems, gain knowledge of many of the "abstract" objects to which people have objected on epistemological grounds on the basis of theories about them or contextual principles that relate them to accessible things.
• All attempts to introduce abstract objects on some clearly specified basis are subject to the Julius Caesar problem.
• However "abstract" objects are introduced, there will not be a proof of consistency or of conservativeness or of eliminability that meets any kind of foundational standards. of the time, that is what those who complain of epistemological difficulties are complaining about. Note that the same is true of "concrete objects" (physical objects over sense data, electrons, …)
• Every method of eliminating objects of any kind is equally a method of introducing them.
• Applicability counts: if there is no relation between some abstract language that appears to make use of referring terms and any of the rest of language, no one will care, no one will defend, …. Even when application is not mentioned, it always plays a role, and role worth spelling out.
• There is no clear generally accepted distinction between objects that exist and ones that don't, or more correctly, between singular terms that "really" refer and others that we can employ in sentences with determinate truth value. There is no realist–nominalist controversy about mathematical objects, there are a host of related debates. A realist (Quine) with a thin notion of realism is nearly indistinguishable from many kinds of nominalists. A nominalist (Field) with a thick criterion of what is nominalistically acceptable is nearly indistinguishable from many kinds of realists. Balaguer.

Putting all of that together, it is not clear that there is any ongoing realist-antirealist debate (which is not to say that there are not debates about more particular issues going on under that label), but there is pretty general agreement about what theories of mathematical objects, at least, are the good ones, and what the grounds are for taking them to be good. I take that to be reason to recast the debates.

What are the real issues? I have no idea, but

• Intrinsicality
• Applicability
• Semantics (Benacerraf, "Mathematical Truth": the problem of giving accounts of the truth conditions for statements about medium-sized dry goods and statements about mathematical objects that make it clear how the truth conditions are related to verification conditions and why they are both accounts of truth conditions.)

As a matter of fact, our practice, at every level, is to accept whatever terms are pragmatically useful and to take them to refer. Every realist-antirealist debate in every area begins by taking some ordinary practice at face value (the face-value practice) and worries about some other practice (the worrisome practice) taking "the" issue to be how the worrisome practice is based on the face-value practice.

Austin noted that 'real' is a trouser word: it's the contrast that wears the trousers.

Real duck vs decoy real decoy vs toy real toy vs store display real store display vs prototype

real cream real hair color

-- ShaughanLavine - 05 Dec 2006